mann-type viscosity approximation methods for multivalued variational inclusions with finitely many...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 328740 18 pageshttpdxdoiorg1011552013328740

Research ArticleMann-Type Viscosity Approximation Methods for MultivaluedVariational Inclusions with Finitely Many Variational InequalityConstraints in Banach Spaces

Lu-Chuan Ceng1 Abdul Latif2 and Abdullah E Al-Mazrooei3

1 Department of Mathematics Shanghai Normal University Scientific Computing Key Laboratory of Shanghai UniversitiesShanghai 200234 China

2Department of Mathematics Faculty of Science King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia3 Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Abdul Latif alatifkauedusa

Received 13 September 2013 Accepted 23 October 2013

Academic Editor Chi-Ming Chen

Copyright copy 2013 Lu-Chuan Ceng et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI)which are also common ones of finitely many variational inequality problems and common fixed points of a countable familyof nonexpansive mappings in real smooth Banach spaces Here the Mann-type viscosity approximation methods are based on theMann iterationmethod and viscosity approximationmethodWe consider and analyzeMann-type viscosity iterative algorithms notonly in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space havinga uniformly Gateaux differentiable norm Under suitable assumptions we derive some strong convergence theorems In additionwe also give some applications of these theorems for instance we prove strong convergence theorems for finding a common fixedpoint of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformlyconvex and 2-uniformly smooth Banach spaces The results presented in this paper improve extend supplement and develop thecorresponding results announced in the earlier and very recent literature

1 Introduction

Let 119883 be a real Banach space whose dual space is denoted by119883lowastThe normalized duality mapping 119869 119883 rarr 2

119883lowast

is definedby

119869 (119909) = 119909lowast

isin 119883lowast

⟨119909 119909lowast

⟩ = 1199092

=1003817100381710038171003817119909lowast1003817100381710038171003817

2

forall119909 isin 119883

(1)where ⟨sdot sdot⟩ denotes the generalized duality pairing It is animmediate consequence of the Hahn-Banach theorem that119869(119909) is nonempty for each 119909 isin 119883 Let 119880 = 119909 isin 119883 119909 = 1

denote the unite sphere of 119883 A Banach space 119883 is said to beuniformly convex if for each 120598 isin (0 2] there exists 120575 gt 0

such that for all 119909 119910 isin 119880

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 ge 120598 997904rArr

1003817100381710038171003817119909 + 1199101003817100381710038171003817

2le 1 minus 120575 (2)

It is known that a uniformly convex Banach space is reflexiveand strict convex A Banach space 119883 is said to be smooth ifthe limit

lim119905rarr0

1003817100381710038171003817119909 + 1199051199101003817100381710038171003817 minus 119909

119905

(3)

exists for all 119909 119910 isin 119880 in this case 119883 is also said tohave a Gateaux differentiable norm 119883 is said to have auniformly Gateaux differentiable norm if for each 119910 isin

119880 the limit is attained uniformly for 119909 isin 119880 Moreoverit is said to be uniformly smooth if this limit is attaineduniformly for 119909 119910 isin 119880 The norm of 119883 is said to be theFrechet differential if for each 119909 isin 119880 this limit is attaineduniformly for 119910 isin 119880 In addition we define a function 120588

2 Abstract and Applied Analysis

[0infin) rarr [0infin) called the modulus of smoothness of 119883 asfollows

120588 (120591) = sup 12(1003817100381710038171003817119909 + 119910

1003817100381710038171003817 +1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)

minus1 119909 119910 isin 119883 119909 = 110038171003817100381710038171199101003817100381710038171003817 = 120591

(4)

It is known that 119883 is uniformly smooth if and only iflim120591rarr0

120588(120591)120591 = 0 Let 119902 be a fixed real number with 1 lt 119902 le

2 Then a Banach space119883 is said to be 119902-uniformly smooth ifthere exists a constant 119888 gt 0 such that 120588(120591) le 119888120591

119902 for all 120591 gt 0It is well-known that no Banach space is 119902-uniformly smoothfor 119902 gt 2 In addition it is also known that 119869 is single-valuedif and only if119883 is smooth whereas if119883 is uniformly smooththen the mapping 119869 is norm-to-norm uniformly continuouson bounded subsets of 119883 If 119883 has a uniformly Gateauxdifferentiable norm then the duality mapping 119869 is norm-to-weaklowast uniformly continuous on bounded subsets of119883

Let119862 be a nonempty closed convex subset of a real Banachspace119883 A mapping 119879 119862 rarr 119862 is called nonexpansive if

1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119862 (5)

The set of fixed points of 119879 is denoted by Fix(119879) We use thenotation to indicate the weak convergence and the one rarrto indicate the strong convergence

Definition 1 Let119860 119862 rarr 119883 be a mapping of 119862 into119883 Then119860 is said to be

(i) accretive if for each 119909 119910 isin 119862 there exists 119895(119909 minus 119910) isin

119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ ge 0 (6)

where 119869 is the normalized duality mapping

(ii) 120572-strongly accretive if for each 119909 119910 isin 119862 there exists119895(119909 minus 119910) isin 119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ ge 1205721003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

(7)

for some 120572 isin (0 1)

(iii) 120573-inverse strongly accretive if for each 119909 119910 isin 119862 thereexists 119895(119909 minus 119910) isin 119869(119909 minus 119910) such that


1003817100381710038171003817

2

(8)

for some 120573 gt 0

(iv) 120582-strictly pseudocontractive if for each 119909 119910 isin 119862 thereexists 119895(119909 minus 119910) isin 119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

minus 1205821003817100381710038171003817119909 minus 119910 minus (119860119909 minus 119860119910)

1003817100381710038171003817

2

(9)


Let119883 be a real smoothBanach space Let119862 be a nonemptyclosed convex subset of119883 and let 119860 119862 rarr 119883 be a nonlinearmapping The so-called variational inequality problem (VIP)is the problem of finding 119909lowast isin 119862 such that

⟨119860119909lowast

119869 (119909 minus 119909lowast

)⟩ ge 0 forall119909 isin 119862 (10)

which was considered by Aoyama et al [1] Note that VIP(10) is connected with the fixed point problem for nonlinearmapping (see eg [2]) the problem of finding a zero point ofa nonlinear operator (see eg [3]) and so on In particularwhenever119883 = 119867 a Hilbert space the VIP (10) reduces to theclassical VIP of finding 119909lowast isin 119862 such that

⟨119860119909lowast

119909 minus 119909lowast

⟩ ge 0 forall119909 isin 119862 (11)

whose solution set is denoted by VI(119862 119860) Recently in orderto find a solution of VIP (10) Aoyama et al [1] introducedMann-type iterative scheme for an accretive operator 119860 asfollows

119909119899+1

= 120572119899119909119899+ (1 minus 120572

119899)Π119862(119909119899minus 120582119899119860119909119899) forall119899 ge 1 (12)

whereΠ119862is a sunny nonexpansive retraction from119883 onto119862

Then they proved a weak convergence theorem

Definition 2 Let 119862 be a nonempty convex subset of a realBanach space119883 Let 119879

119894119873

119894=1be a finite family of nonexpansive

mappings of 119862 into itself and let 1205821 120582

119873be real numbers

such that 0 le 120582119894le 1 for every 119894 = 1 119873 Define a mapping

119870 119862 rarr 119862 as follows

1198801= 12058211198791+ (1 minus 120582

1) 119868

1198802= 120582211987921198801+ (1 minus 120582

2) 1198801

1198803= 120582311987931198802+ (1 minus 120582

3) 1198802

119880119873minus1

= 120582119873minus1

119879119873minus1

119880119873minus2

+ (1 minus 120582119873minus1

) 119880119873minus2

119870 = 119880119873= 120582119873119879119873119880119873minus1

+ (1 minus 120582119873) 119880119873minus1

(13)

Such a mapping 119870 is called the 119870-mapping generated by1198791 119879

119873and 120582

1 120582

119873

Lemma 3 (see [4]) Let 119862 be a nonempty closed convex subsetof a strictly convex Banach space Let 119879

119894119873

119894=1be a finite family

of nonexpansive mappings of 119862 into itself with cap119873119894=1

Fix(119879119894) = 0

and let 1205821 120582

119873be real numbers such that 0 lt 120582

119894lt 1

for every 119894 = 1 119873 minus 1 and 0 lt 120582119873

le 1 Let 119870 bethe 119870-mapping generated by 119879

1 119879

119873and 120582

1 120582

119873 Then

Fix(K) = cap119873

119894=1Fix(119879119894)

From Lemma 3 it is easy to see that the 119870-mapping is anonexpansive mapping

On the other hand let 119862119861(119883) be the family of allnonempty closed and bounded subsets of a real smooth

Abstract and Applied Analysis 3

Banach space 119883 Also we denote by 119867(sdot sdot) the Hausdorffmetric on 119862119861(119883) defined by

119867(119860 119861) = maxsup119909isin119861

inf119910isin119860

119889 (119909 119910) sup119909isin119860

inf119910isin119861

119889 (119909 119910)

forall119860 119861 isin 119862119861 (119883)

(14)

Let 119879 119865 119883 rarr 119862119861(119883) be two multivalued mappings let 119860

119863(119860) sub 119883 rarr 2119883 be an 119898-accretive mapping let 119892 119883 rarr

119863(119860)be a single-valuedmapping and let119873(sdot sdot) 119883times119883 rarr 119883

be a nonlinear mapping Then for any given V isin 119883 120582 gt 0Chidume et al [5] introduced and studied the multivaluedvariational inclusion (MVVI) of finding 119909 isin 119863(119860) such that(119909 119908 119896) is a solution of the following

V isin 119873 (119908 119896) + 120582119860 (119892 (119909)) forall119908 isin 119879119909 119896 isin 119865119909 (15)

If V = 0 and 120582 = 1 then the MVVI (15) reduces to theproblem of finding 119909 isin 119863(119860) such that (119909 119908 119896) is a solutionof the following

0 isin 119873 (119908 119896) + 119860 (119892 (119909)) forall119908 isin T119909 119896 isin 119865119909 (16)

We denote by Γ the set of such solutions 119909 for MVVI (16)The authors [5] established an existence theorem for

MVVI (15) in a smooth Banach space119883 and then proved thatthe sequence generated by their iterative algorithm convergesstrongly to a solution of MVVI (16)

Theorem 4 (see [5 Theorem 32]) Let 119883 be a real smoothBanach space Let 119879 119865 119883 rarr 119862119861(119883) and 119860 119863(119860) sub

119883 rarr 2119883 be three multivalued mappings let 119892 119883 rarr 119863(119860)

be a single-valued mapping and let 119873(sdot sdot) 119883 times 119883 rarr 119883

be a single-valued continuous mapping satisfying the followingconditions

(C1) 119860 ∘ 119892 119883 rarr 2119883 is 119898-accretive and 119867-uniformly

continuous(C2) 119879 119883 rarr 119862119861(119883) is119867-uniformly continuous(C3) 119865 119883 rarr 119862119861(119883) is119867-uniformly continuous(C4) the mapping 119909 997891rarr 119873(119909 119910) is 120601-strongly accretive and

120583-119867-Lipschitz with respect to the mapping 119879 where 120601

[0infin) rarr [0infin) is a strictly increasing function with120601(0) = 0

(C5) the mapping 119910 997891rarr 119873(119909 119910) is accretive and 120585-119867-Lipschitz with respect to the mapping 119865

For arbitrary 1199090isin 119863(119860) define the sequence 119909

119899 iteratively

by

119909119899+1

= 119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899) 119906119899isin 119860 (119892 (119909

119899)) (17)

where 119906119899 is defined by

1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817

le (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899))) forall119899 ge 0

(18)

for any 119908119899isin 119879119909119899 119896119899isin 119865119909119899 and some 120576 gt 0 where 120590

119899 is a

positive real sequence such that lim119899rarrinfin

120590119899= 0sum

infin

119899=0120590119899= infin

Then there exists 119889 gt 0 such that for 0 lt 120590119899le 119889 and for all

119899 ge 0 119909119899 converges strongly to 119909 isin Γ and for any 119908 isin 119879119909

and 119896 isin 119865119909 (119909 119908 119896) is a solution of the MVVI (16)

Let 119862 be a nonempty closed convex subset of a realsmooth Banach space119883 and letΠ

119862be a sunny nonexpansive

retraction from 119883 onto 119862 Let 119891 119862 rarr 119862 be a contractionwith coefficient 120588 isin (0 1) Motivated and inspired by theresearch going on this area we introduceMann-type viscosityapproximation methods for finding solutions of the MVVI(16) which are also common ones of finitely many variationalinequality problems and common fixed points of a countablefamily of nonexpansive mappings Here the Mann-typeviscosity approximationmethods are based on theMann iter-ation method and viscosity approximation method We con-sider and analyze Mann-type viscosity iterative algorithmsnot only in the setting of uniformly convex and 2-uniformlysmooth Banach space but also in a uniformly convex Banachspace having a uniformlyGateaux differentiable normUndersuitable assumptions we derive some strong convergencetheorems In addition we also give some applications ofthese theorems for instance we prove strong convergencetheorems for finding a common fixed point of a finite familyof 120578119894-strictly pseudocontractive mappings (119894 = 1 119873) and

a countable family of nonexpansive mappings in uniformlyconvex and 2-uniformly smooth Banach spaces The resultspresented in this paper improve extend supplement anddevelop the corresponding results announced in the earlierand very recent literature see for example [6ndash11]

2 Preliminaries

Let 119883 be a real Banach space with dual 119883lowast We denote by 119869the normalized duality mapping from119883 to 2119883

lowast

defined by

119869 (119909) = 119909lowast

isin 119883lowast

⟨119909 119909lowast

⟩ = 1199092

=1003817100381710038171003817119909lowast1003817100381710038171003817

2

(19)

where ⟨sdot sdot⟩ denotes the generalized duality pairingThrough-out this paper the single-valued normalized duality map isstill denoted by 119869 Unless otherwise stated we assume that119883is a smooth Banach space with dual119883lowast

A multivalued mapping 119860 119863(119860) sube 119883 rarr 2119883 is said to

be(i) accretive if

⟨119906 minus V 119869 (119909 minus 119910)⟩ ge 0 forall119906 isin 119860119909 V isin 119860119910 (20)

(ii) 119898-accretive if119860 is accretive and (119868 + 119903119860)(119863(119860)) = 119883for all 119903 gt 0 where 119868 is the identity mapping

(iii) 120577-inverse strongly accretive if there exists a constant120577 gt 0 such that

⟨119906 minus V 119869 (119909 minus 119910)⟩ ge 120577119906 minus V2 forall119906 isin 119860119909 V isin 119860119910 (21)

(iv) 120601-strongly accretive if there exists a strictly increasingcontinuous function 120601 [0infin) rarr [0infin) with120601(0) = 0 such that

⟨119906 minus V 119869 (119909 minus 119910)⟩

ge 120601 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119906 isin 119860119909 V isin 119860119910(22)


(v) 120601-expansive if

119906 minus V ge 120601 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817) forall119906 isin 119860119909 V isin 119860119910 (23)

It is easy to see that if 119860 is 120601-strongly accretive then 119860 is120601-expansive

A mapping 119879 119883 rarr 119862119861(119883) is said to be 119867-uniformlycontinuous if for any given 120576 gt 0 there exists a 120575 gt 0 suchthat whenever 119909 minus 119910 lt 120575 then119867(119879119909 119879119910) lt 120576

A mapping119873 119883 times119883 rarr 119883 is 120601-strongly accretive withrespect to 119879 119883 rarr 119862119861(119883) in the first argument if

⟨119873 (119906 119911) minus 119873 (V 119911) 119869 (119909 minus 119910)⟩ ge 120601 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

forall119906 isin 119879119909 V isin 119879119910(24)

A mapping 119878 119883 rarr 2119883 is called lower semicontinuous

if 119878minus1(119874) = 119909 isin 119883 119878119909 cap 119874 = 0 is open in 119883 whenever119874 sub 119884 is open

We list some propositions and lemmas that will be usedin the sequel

Proposition 5 (see [12]) Let 120582119899 and 119887

119899 be sequences of

nonnegative numbers and 120572119899 sub (0 1) a sequence satisfying

the conditions that 120582119899 is bounded suminfin

119899=0120572119899= infin and 119887

119899rarr

0 as 119899 rarr infin Let the recursive inequality

1205822

119899+1le 1205822

119899minus 2120572119899120595 (120582119899+1

) + 2120572119899119887119899120582119899+1

forall119899 ge 0 (25)

be given where 120595 [0infin) rarr [0infin) is a strictly increasingfunction such that it is positive on (0infin) and 120595(0) = 0 Then120582119899rarr 0 as 119899 rarr infin

Proposition6 (see [13]) Let119883 be a real smooth Banach spaceLet 119879 and 119865 119883 rarr 2

119883 be two multivalued mappings and let119873(sdot sdot) 119883 times 119883 rarr 119883 be a nonlinear mapping satisfying thefollowing conditions

(i) the mapping 119909 997891rarr 119873(119909 119910) is 120601-strongly accretive withrespect to the mapping 119879

(ii) the mapping 119910 997891rarr 119873(119909 119910) is accretive with respect tothe mapping 119865

Then the mapping 119878 119883 rarr 2119883 defined by 119878119909 = 119873(119879119909 119865119909)

is 120601-strongly accretive

Proposition 7 (see [14]) Let119883 be a real Banach space and let119878 119883 rarr 2

119883

0 be a lower semicontinuous and 120601-stronglyaccretive mapping then for any 119909 isin 119883 119878119909 is a one-point setthat is 119878 is a single-valued mapping

Lemma 8 can be found in [15] Lemma 9 is an immediateconsequence of the subdifferential inequality of the function(12) sdot

2

Lemma 8 Let 119904119899 be a sequence of nonnegative real numbers

satisfying

119904119899+1

le (1 minus 120572119899) 119904119899+ 120572119899120573119899+ 120574119899 forall119899 ge 0 (26)

where 120572119899 120573119899 and 120574

119899 satisfy the following conditions

(i) 120572119899 sub [0 1] and suminfin

119899=0120572119899= infin

(ii) lim sup119899rarrinfin

120573119899le 0

(iii) 120574119899ge 0 for all 119899 ge 0 and suminfin

119899=0120574119899lt infin

Then lim sup119899rarrinfin

119904119899= 0

Lemma 9 In a smooth Banach space 119883 there holds theinequality

1003817100381710038171003817119909 + 1199101003817100381710038171003817

2

le 1199092

+ 2 ⟨119910 119869 (119909 + 119910)⟩ forall119909 119910 isin 119883 (27)

Lemma 10 (see [1]) Let119862 be a nonempty closed convex subsetof a smooth Banach space 119883 Let Π


retraction from 119883 onto 119862 and let 119860 be an accretive operator of119862 into119883 Then for all 120582 gt 0

VI (119862 119860) = Fix (Π119862(119868 minus 120582119860)) (28)

Let 119863 be a subset of 119862 and let Π be a mapping of 119862 into119863 Then Π is said to be sunny if

Π [Π (119909) + 119905 (119909 minus Π (119909))] = Π (119909) (29)

whenever Π(119909) + 119905(119909 minus Π(119909)) isin 119862 for 119909 isin 119862 and 119905 ge 0 Amapping Π of 119862 into itself is called a retraction if Π2 = Π Ifa mappingΠ of 119862 into itself is a retraction thenΠ(119911) = 119911 forevery 119911 isin 119877(Π) where 119877(Π) is the range of Π A subset 119863 of119862 is called a sunny nonexpansive retract of 119862 if there exists asunny nonexpansive retraction from119862 onto119863The followinglemma concerns the sunny nonexpansive retraction

Lemma 11 (see [16]) Let119862 be a nonempty closed convex subsetof a real smooth Banach space 119883 Let 119863 be a nonempty subsetof 119862 Let Π be a retraction of 119862 onto 119863 Then the following areequivalent

(i) Π is sunny and nonexpansive

(ii) Π(119909) minus Π(119910)2 le ⟨119909 minus 119910 119869(Π(119909) minus Π(119910))⟩ for all119909 119910 isin 119862

(iii) ⟨119909 minus Π(119909) 119869(119910 minus Π(119909))⟩ le 0 for all 119909 isin 119862 119910 isin 119863

It is well known that if 119883 = 119867 a Hilbert space thena sunny nonexpansive retraction Π

119862is coincident with the

metric projection from 119883 onto 119862 that is Π119862

= 119875119862 If 119862

is a nonempty closed convex subset of a strictly convex anduniformly smooth Banach space 119883 and if 119879 119862 rarr 119862 isa nonexpansive mapping with the fixed point set Fix(119879) = 0then the set Fix(119879) is a sunny nonexpansive retract of 119862

Lemma 12 (see [17]) Let 119883 be a uniformly convex Banachspace and 119861

119903(0) = 119909 isin 119883 119909 le 119903 119903 gt 0 Then there

exists a continuous strictly increasing and convex function120593 [0infin] rarr [0infin] 120593(0) = 0 such that

1003817100381710038171003817120572119909 + 120573119910 + 1205741199111003817100381710038171003817

2

le 1205721199092

+ 12057310038171003817100381710038171199101003817100381710038171003817

2

+ 1205741199112

minus 120572120573120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)

(30)

for all 119909 119910 119911 isin 119861119903(0) and all 120572 120573 120574 isin [0 1]with 120572+120573+120574 = 1


Lemma 13 (see [18]) Let 119862 be a nonempty closed convexsubset of a Banach space 119883 Let 119878

0 1198781 be a sequence of

mappings of119862 into itself Suppose thatsuminfin119899=1

sup119878119899119909minus119878119899minus1

119909

119909 isin 119862 lt infin Then for each 119910 isin 119862 119878119899119910 converges strongly

to some point of 119862 Moreover let 119878 be a mapping of 119862 intoitself defined by 119878119910 = lim

119899rarrinfin119878119899119910 for all 119910 isin 119862 Then

lim119899rarrinfin

sup119878119909 minus 119878119899119909 119909 isin 119862 = 0

Let 119862 be a nonempty closed convex subset of a Banachspace119883 and let 119879 119862 rarr 119862 be a nonexpansive mapping withFix(119879) = 0 As previous letΞ

119862be the set of all contractions on

119862 For 119905 isin (0 1) and 119891 isin Ξ119862 let 119909

119905isin 119862 be the unique fixed

point of the contraction 119909 997891rarr 119905119891(119909) + (1 minus 119905)119879119909 on 119862 that is

119909119905= 119905119891 (119909

119905) + (1 minus 119905) 119879119909

119905 (31)

Lemma 14 (see [19]) Let 119883 be a uniformly smooth Banachspace or a reflexive and strictly convex Banach space with auniformly Gateaux differentiable norm Let 119862 be a nonemptyclosed convex subset of 119883 let 119879 119862 rarr 119862 be a nonexpansivemapping with Fix(119879) = 0 and let 119891 isin Ξ

119862 Then the net 119909

119905

defined by 119909119905= 119905119891(119909

119905) + (1 minus 119905)119879119909

119905converges strongly to a

point in Fix(119879) If one defines a mapping119876 Ξ119862rarr Fix(119879) by

119876(119891) = 119904minus lim119905rarr0

119909119905 for all 119891 isin Ξ

119862 then119876(119891) solves the VIP

as follows

⟨(119868 minus 119891)119876 (119891) 119869 (119876 (119891) minus 119901)⟩ le 0

forall119891 isin Ξ119862 119901 isin Fix (119879)

(32)

Lemma 15 (see [20]) Let 119862 be a nonempty closed convexsubset of a strictly convex Banach space 119883 Let 119879

119899infin

119899=0

be a sequence of nonexpansive mappings on 119862 Suppose⋂infin

119899=0Fix(119879119899) is nonempty Let 120582

119899 be a sequence of positive

numbers with suminfin119899=0

120582119899= 1 Then a mapping 119878 on 119862 defined by

119878119909 = suminfin

119899=0120582119899119879119899119909 for 119909 isin 119862 is defined well and nonexpansive

and Fix(119878) = ⋂infin

119899=0Fix(119879119899) holds

Lemma 16 (see [21]) Given a number 119903 gt 0 A real Banachspace 119883 is uniformly convex if and only if there exists acontinuous strictly increasing function 120593 [0infin) rarr [0infin)120593(0) = 0 such that

1003817100381710038171003817120582119909 + (1 minus 120582) 1199101003817100381710038171003817

2

le 1205821199092

+ (1 minus 120582)10038171003817100381710038171199101003817100381710038171003817

2

minus 120582 (1 minus 120582) 120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)

(33)

for all 120582 isin [0 1] and 119909 119910 isin 119883 such that 119909 le 119903 and 119910 le 119903

3 Mann-Type Viscosity Algorithms inUniformly Convex and 2-UniformlySmooth Banach Spaces

In this section we introduce Mann-type viscosity iterativealgorithms in uniformly convex and 2-uniformly smoothBanach spaces and show strong convergence theorems Wewill use the following useful lemma

Lemma 17 Let119862 be a nonempty closed convex subset of a real2-uniformly smooth Banach space 119883 Let 119860 119862 rarr 119883 be an120572-inverse strongly accretive mapping Then one has1003817100381710038171003817(119868 minus 120582119860) 119909 minus (119868 minus 120582119860) 119910

1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 2120582 (1205821205812

minus 120572)1003817100381710038171003817119860119909 minus 119860119910

1003817100381710038171003817

2

forall119909 119910 isin 119862

(34)

where 120582 gt 0 In particular if 0 lt 120582 le 1205721205812 then 119868 minus 120582119860 is

nonexpansive

Theorem 18 Let 119883 be a uniformly convex and 2-uniformlysmooth Banach space and let 119862 be a nonempty closed convexsubset of 119883 such that 119862 plusmn 119862 sub 119862 Let Π

119862be a sunny

nonexpansive retraction from 119883 onto 119862 Let 119879 119865 119883 rarr

119862119861(119883) and 119860 119862 rarr 2119862 be three multivalued mappings

let 119892 119883 rarr 119862 be a single-valued mapping and let 119873(sdot sdot)

119883 times119883 rarr 119862 be a single-valued continuous mapping satisfyingconditions (C1)ndash(C5) in Theorem 4 Consider that

(C6) 119873(119879119909 119865119909) + 119860(119892(119909)) 119883 rarr 2119862

0 is 120577-inversestrongly accretive with 120577 ge 120581

2

Let 119860119894 119862 rarr 119883 be an 120572

119894-inverse strongly accretive

mapping for each 119894 = 1 119873 Define the mapping 119866119894

119862 rarr 119862 by 119866119894= Π119862(119868 minus 120582

119894119860119894) for 119894 = 1 119873 where

120582119894isin (0 120572

1198941205812

) and 120581 is the 2-uniformly smooth constant of119883Let 119861 119862 rarr 119862 be the 119870-mapping generated by 119866

1 119866

119873

and 1205881 120588

119873 where 120588

119894isin (0 1) for all 119894 = 1 119873 minus 1 and

120588119873isin (0 1] Let 119891 119862 rarr 119862 be a contraction with coefficient

120588 isin (0 1) Let 119878119894infin

119894=0be a countable family of nonexpansive

mappings of 119862 into itself such that Δ = ⋂infin

119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899

and 120598119899 are the sequences in [0 1] 120572

119899+ 120573119899+ 120574119899+ 120575119899= 1

and satisfy the following conditions

(i) suminfin119899=1

(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim

119899rarrinfin120572119899= 0 and suminfin

119899=0120572119899= infin

(iii) 120574119899 120575119899 sub [119888 119889] for some 119888 119889 isin (0 1)

(iv) 0 lt lim inf119899rarrinfin

120573119899le lim sup

119899rarrinfin120573119899lt 1

(v) 0 lt lim inf119899rarrinfin

120590119899le lim sup

119899rarrinfin120590119899lt 1

(vi) 0 lt lim inf119899rarrinfin

120598119899le lim sup

119899rarrinfin120598119899lt 1

For arbitrary 1199090isin 119862 define the sequence 119909

119899 iteratively by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(35)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(36)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909

119899 and some 120576 gt 0 Assume

that suminfin119899=0

sup119909isin119863

119878119899+1

119909 minus 119878119899119909 lt infin for any bounded subset


119863 of 119862 and let 119878 be a mapping of 119862 into itself defined by119878119909 = lim

119899rarrinfin119878119899119909 for all 119909 isin 119862 and suppose that Fix(119878) =

⋂infin

119894=0Fix(119878119894) Then 119909

119899 converges strongly to 119902 isin Δ which

solves the following VIP

⟨119902 minus 119891 (119902) 119869 (119902 minus 119901)⟩ le 0 forall119901 isin Δ (37)

and for any 119908 isin 119879119902 and 119896 isin 119865119902 (119902 119908 119896) is a solution of theMVVI (16)

Proof First of all by Lemma 17 we know that 119868 minus 120582119894119860119894is

a nonexpansive mapping where 120582119894isin (0 120572

1198941205812

) for each119894 = 1 119873 Hence from the nonexpansivity of Π

119862 it

follows that 119866119894is a nonexpansive mapping for each 119894 =

1 119873 Since 119861 119862 rarr 119862 is the 119870-mapping generatedby 1198661 119866

119873and 120588

1 120588

119873 by Lemma 3 we deduce that

Fix(119861) = cap119873

119894=1Fix(119866

119894) Utilizing Lemma 10 and the definition

of 119866119894 we get Fix(119866

119894) = VI(119862 119860

119894) for each 119894 = 1 119873 Thus

we have

Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (38)

Now let us show that for any V isin 119862 120582 gt 0 there exists apoint 119909 isin 119862 such that (119909 119908 119896) is a solution of the MVVI (15)for any 119908 isin 119879119909 and 119896 isin 119865119909 Indeed following the argumentidea in the proof of Chidume et al [5 Theorem 31] we put119881119909 = 119873(119879119909 119865119909) for all 119909 isin 119883 Then by Proposition 6119881 is 120601-strongly accretive Since119879 and119865 are119867-uniformly continuousand 119873(sdot sdot) is continuous 119881119909 is continuous and hence lowersemicontinuous Thus by Proposition 7 119881119909 is single-valuedMoreover since 119881 is 120601-strongly accretive and by assumption119860 ∘ 119892 119883 rarr 2

119862 is119898-accretive we have that 119881 + 120582119860 ∘ 119892 is an119898-accretive and 120601-strongly accretive mapping and hence byCioranescu [22 page 184] for any 119909 isin 119883 we have that (119881 +

120582119860∘119892)(119909) is closed and boundedTherefore byMorales [23]119881+120582119860∘119892 is surjective Hence for any V isin 119883 and 120582 gt 0 thereexists 119909 isin 119863(119860) = 119862 such that V isin 119881119909+120582119860(119892(119909)) = 119873(119908 119896)+

120582119860(119892(119909)) where 119908 isin 119879119909 and 119896 isin 119865119909 In addition in termsof Proposition 7 we know that 119881 + 120582119860 ∘ 119892 is a single-valuedmapping Assume that119873(119879119909 119865119909) + 120582119860(119892(119909)) 119883 rarr 119862 is 120577-inverse strongly accretive with 120577 ge 120581

2 Then by Lemma 17 weconclude that the mapping 119909 997891rarr 119909 minus (119873(119879119909 119865119909) + 120582119860(119892(119909)))

is nonexpansiveWithout loss of generality we may assume that V = 0 and

120582 = 1 Let 119901 isin Δ and let 119903(ge 119891(119901)minus119901(1minus120588)) be sufficientlylarge such that 119909

0isin 119861119903(119901) = 119861 Then 119901 isin 119863(119860) = 119862 such

that 0 isin 119873(119908 119896) + 119860 ∘ 119892(119901) for any 119908 isin 119879119901 and 119896 isin 119865119901Let119872 = sup119906 119906 isin 119873(119908 119896) + 119860(119892(119909)) 119909 isin 119861 119908 isin 119879119909119896 isin 119865119909 Then as 119860 ∘ 119892 119879 and 119865 are119867-uniformly continuouson 119883 for 120576

1= 120601(119903)8(1 + 120576) 120576

2= 120601(119903)8120583(1 + 120576) and 120576

3=

120601(119903)8120585(1 + 120576) there exist 1205751 1205752 1205753gt 0 such that for any

119909 119910 isin 119883 119909 minus 119910 lt 1205751 119909 minus 119910 lt 120575

2and 119909 minus 119910 lt 120575

3imply

119867(119860 ∘ 119892(119909) 119860 ∘ 119892(119910)) lt 1205761 119867(119879119909 119879119910) lt 120576

2and119867(119865119909 119865119910) lt

1205763 respectivelyLet us show that 119909

119899isin 119861 for all 119899 ge 0 We show this by

induction First 1199090isin 119861 by construction Assume that 119909

119899isin 119861

We show that 119909119899+1

isin 119861 If possible we assume that 119909119899+1

notin 119861then 119909

119899+1minus 119901 gt 119903 Further from (35) it follows that

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817120572119899 (119891 (119909119899) minus 119901) + 120573119899 (119909119899 minus 119901)

+120574119899(119861119909119899minus 119901) + 120575

119899(119878119899119909119899minus 119901)

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120572119899(1 minus 120588)

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(39)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

= ⟨120598119899[119909119899minus 119901 minus 120590

119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

= ⟨120598119899(119909119899minus 119901) + (1 minus 120598

119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le1003817100381710038171003817120572119899 (119909119899 minus 119901) + (1 minus 120572119899) (119910119899 minus 119901)

1003817100381710038171003817

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le max1003817100381710038171003817119909119899 minus p1003817100381710038171003817 1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817

1 minus 1205881003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


le1

2(max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

+1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

)

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

(40)

which immediately yields1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899+1 119896119899+1

) + 119906119899+1

119869 (119909119899+1

minus 119901)⟩

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899

minus (119873 (119908119899+1

119896119899+1

) + 119906119899+1

) 119869 (119909119899+1

minus 119901)⟩

(41)

Since 119873(sdot sdot) is 120601-strongly accretive with respect to 119879 and119860(119892(sdot)) is accretive we deduce from (41) that

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899+1

119896119899)1003817100381710038171003817

+1003817100381710038171003817119873 (119908

119899+1 119896119899) minus 119873 (119908

119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

(42)

Again from (35) we have that1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119909119899 minus 119901 minus 120590119899 (119873 (119908119899 119896119899) + 119906119899)1003817100381710038171003817

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205901198991003817100381710038171003817119873 (119908

119899 119896119899) + 119906119899

1003817100381710038171003817]

+ (1 minus 120598119899)max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le 120598119899[119903 + 120590

119899119872] + (1 minus 120598

119899) 119903

le 2119903

(43)

Also from Proposition 7 119881119909 = 119873(119879119909 119865119909) is a single-valuedmapping that is for any 119896 1198961015840 isin 119865119909 and 1199081199081015840 isin 119879119909 we have119873(119908 119896) = 119873(119908 119896

1015840

) and 119873(119908 119896) = 119873(1199081015840

119896) On the otherhand it follows from Nadler [24] that for 119896

119899+1isin 119865119909119899+1

and119908119899+1

isin 119879119909119899+1

there exist 1198961015840119899isin 119865119909119899and 1199081015840

119899isin 119879119909119899such that

10038171003817100381710038171003817119896119899+1

minus 1198961015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119865119909

119899+1 119865119909119899) (44)

10038171003817100381710038171003817119908119899+1

minus 1199081015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119879119909

119899+1 119879119909119899) (45)

respectively Therefore from (42) and (36) we have

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[10038171003817100381710038171003817119873 (119908119899+1

119896119899+1

) minus 119873 (119908119899+1

1198961015840

119899)10038171003817100381710038171003817

+10038171003817100381710038171003817119873 (119908119899+1

119896119899) minus 119873 (119908

1015840

119899 119896119899)10038171003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817 ] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120585 (1 + 120576)119867 (119865119909

119899+1 119865119909119899)

+ 120583 (1 + 120576)119867 (119879119909119899+1

119879119909119899)

+ (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899)))] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120601 (119903)

8+120601 (119903)

8+120601 (119903)

8] 2119903

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903 + 120572

119899120590119899

3

2120601 (119903) 119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

(46)


So we get 119909119899+1

minus 119901 le 119903 a contradiction Therefore 119909119899 is

boundedLet us show that lim

119899rarrinfin119909119899minus 119909119899+1

= 0 andlim119899rarrinfin

119909119899minus 119910119899 = 0

Indeed we define 119866 119862 rarr 119862 by 119866119909 = 119909 minus (119873(119879119909 119865119909) +

119860(119892(119909))) for all 119909 isin 119862 Then 119866 is a nonexpansive mappingand the iterative scheme (35) can be rewritten as follows

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899]

+ (1 minus 120598119899) 119910119899 forall119899 ge 0

(47)

Taking into account condition (iv) we may assume that120573119899 sub [119886 119887] for some 119886 119887 isin (0 1) From (47) we can rewrite

119910119899by

119910119899= 120573119899119909119899+ (1 minus 120573

119899) 119911119899 (48)

where 119911119899= (120572119899119891(119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)(1 minus 120573

119899) Now we

have1003817100381710038171003817119911119899+1 minus 119911119899

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

120572119899+1

119891 (119909119899+1

) + 120574119899+1

119861119909119899+1

+ 120575119899+1

119878119899+1

119909119899+1

1 minus 120573119899+1

minus120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899

1 minus 120573119899

100381710038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

+119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=1

1 minus 120573119899+1

1003817100381710038171003817119910119899+1 minus 120573119899+1119909119899+1 minus (119910119899 minus 120573119899119909119899)1003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816

1

1 minus 120573119899+1

minus1

1 minus 120573119899

10038161003816100381610038161003816100381610038161003816

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1


+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1

1003817100381710038171003817120572119899+1119891 (119909119899+1) + 120574119899+1119861119909119899+1 + 120575119899+1119878119899+1119909119899+1

minus (120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)1003817100381710038171003817

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

(120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817

+ 120574119899+1

1003817100381710038171003817119861119909119899+1 minus 1198611199091198991003817100381710038171003817

+ 120575119899+1

1003817100381710038171003817119878119899+1119909119899+1 minus 1198781198991199091198991003817100381710038171003817

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817)

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817 + 120574119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119878119899+1119909119899+1 minus 119878119899+1119909119899

1003817100381710038171003817

+1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1205881003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 + 120574119899+11003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817

=1 minus 120573119899+1

minus 120572119899+1

(1 minus 120588)

1 minus 120573119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+120575119899+1

1 minus 120573119899+1

1003817100381710038171003817119878119899+1119909119899 minus 1198781198991199091198991003817100381710038171003817

+1

1 minus 120573119899+1

[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817


le (1 minus120572119899+1

(1 minus 120588)

1 minus 120573119899+1

)1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816 +1003816100381610038161003816120573119899+1 minus 120573119899

1003816100381610038161003816

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816]

(49)

where 1(1 minus 119887)2sup119899ge0

119891(119909119899) + 119861119909

119899 + 119878

119899119909119899 le 119872

0for

some1198720gt 0 By simple calculation we have

119910119899minus 119910119899minus1

= 120573119899(119909119899minus 119909119899minus1

) + (120573119899minus 120573119899minus1

)

times (119909119899minus1

minus 119911119899minus1

) + (1 minus 120573119899) (119911119899minus 119911119899minus1

)

(50)

So from (49) we get

1003817100381710038171003817119910119899 minus 119910119899minus11003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816

1003817100381710038171003817119909119899minus1 minus 119911119899minus11003817100381710038171003817

+ (1 minus 120573119899)1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


+ (1 minus 120573119899) (1 minus

120572119899(1 minus 120588)

1 minus 120573119899

)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119909119899minus1 minus 119878119899minus1119909119899minus1

1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

(51)

Also for convenience we write

119909119899+1

= 120598119899119899+ (1 minus 120598

119899) 119910119899

119899= 120590119899119866119909119899+ (1 minus 120590

119899) 119909119899

(52)

By simple calculation we get

119909119899+1

minus 119909119899= 120598119899(119899minus 119899minus1

) + (120598119899minus 120598119899minus1

)

times (119899minus1



)

119899minus 119899minus1

= 120590119899(119866119909119899minus 119866119909119899minus1

) + (120590119899minus 120590119899minus1

)

times (119866119909119899minus1



)

(53)

From (51) and (53) we deduce that

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817

le 120590119899

1003817100381710038171003817119866119909119899 minus 119866119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816

times1003817100381710038171003817119866119909119899minus1 minus 119909119899minus1

1003817100381710038171003817 + (1 minus 120590119899)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


(54)

and hence1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817 +

1003816100381610038161003816120598119899 minus 120598119899minus11003816100381610038161003816



1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816

1003817100381710038171003817119866119909119899minus1 minus 119909119899minus11003817100381710038171003817]

+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


+ (1 minus 120598119899) (1 minus 120572

119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le [1 minus (1 minus 120598119899) 120572119899(1 minus 120588)]

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816

1003817100381710038171003817119899minus1 minus 119910119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878119899minus1119909119899minus1

1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817


1003817100381710038171003817

+ 1198721[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816 +1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816]

(55)

where sup119899ge1

119866119909119899minus1

minus119909119899minus1

+ 119899minus1


+ 119909119899minus1


+

1198720 le 119872

1for some 119872

1gt 0 Utilizing Lemma 17 we

conclude from (55) conditions (i) (ii) and (vi) and theassumption on 119878

119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 (56)


Furthermore utilizing Lemma 16 we obtain from (39) and(47) that

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120598119899 [(1 minus 120590119899) (119909119899 minus 119901) + 120590119899 (119866119909119899 minus 119901)]

+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119901) + 120590119899 (119866119909119899 minus 119901)1003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)

times 120593 (1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899) + 120590119899 (119866119909119899 minus 119910119899)

1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)

which immediately yields

120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)

So from (56) and conditions (ii) (v) and (vi) we get

lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin

120593 (1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899) + 120590119899 (119866119909119899 minus 119910119899)

1003817100381710038171003817) = 0

(59)

which together with the properties of 120593 and 1205931implies that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899) + 120590119899 (119866119909119899 minus 119910119899)1003817100381710038171003817 = 0

(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817

=1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899) + 120590119899 (119866119909119899 minus 119910119899) + 120590119899 (119909119899 minus 119866119909119899)

1003817100381710038171003817

le1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899) + 120590119899 (119866119909119899 minus 119910119899)

1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)

Hence from (60) it follows that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)

Let us show that lim119899rarrinfin

119909119899minus119861119909119899 = 0 and lim

119899rarrinfin119909119899minus

119878119909119899 = 0Indeed from the definition of 119910

119899 we can rewrite 119910

119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)

Utilizing Lemma 12 from (63) we have

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)

From (62) and conditions (ii) (iii) and (iv) we have

lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)

From the properties of 1205932 we have

lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)

By Lemma 16 we deduce from the definition of 1199111015840

119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)

From (67) and condition (iii) we have

lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)

From the definition of 119910119899 we can rewrite 119910

119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)

Utilizing Lemma 12 from (72) and the convexity of sdot 2we have

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)

From (62) (74) and conditions (ii) (iii) and (iv) we have

lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)

By the properties of 1205934 we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)

Utilizing Lemma 13 we conclude from (78) that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)

Define a mapping119882119909 = (1 minus 1205791minus 1205792)119861119909 + 120579

1119878119909 + 120579

2119866119909

where 1205791 1205792isin (0 1) are two constants with 120579

1+ 1205792lt 1 Then

by Lemma 15 we have Fix(119882) = Fix(119861)capFix(119878)capFix(119866) = ΔWe observe that

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817

=1003817100381710038171003817(1 minus 1205791 minus 1205792) (119909119899 minus 119861119909119899)

+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 1205791minus 1205792)1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)

From (60) (76) and (80) we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that

lim sup119899rarrinfin

⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)

where 119902 = 119904 minus lim119905rarr0

119909119905with 119909

119905being the fixed point of the

contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)

Then 119909119905solves the fixed point equation 119909

119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)

By Lemma 9 we conclude that

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin

(87)

It follows from (86) that

⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)

Letting 119899 rarr infin in (88) and noticing (87) we derive


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)

where1198722gt 0 is a constant such that 119909

119905minus 1199091198992

le 1198722for all

119905 isin (0 1) and 119899 ge 0 Taking 119905 rarr 0 in (89) we have

lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)

On the other hand we have

⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩

= ⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ minus ⟨119891 (119902) minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩

minus ⟨119891 (119902) minus 119909119905 119869 (119909119899minus 119909119905)⟩ + ⟨119891 (119902) minus 119909

119905 119869 (119909119899minus 119909119905)⟩

minus ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

= ⟨119891 (119902) minus 119902 119869 (119909119899minus 119902) minus 119869 (119909

119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩

le lim sup119899rarrinfin

⟨119891 (119902) minus 119902 119869 (119909119899minus 119902) minus 119869 (119909

119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905

1003817100381710038171003817 lim sup119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ lim sup119899rarrinfin

⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)

Taking into account that 119909119905rarr 119902 as 119905 rarr 0 we have


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩

= lim sup119905rarr0


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩

le lim sup119905rarr0



119899minus 119909119905)⟩

(93)

Since 119883 has a uniformly Frechet differentiable norm theduality mapping 119869 is norm-to-norm uniformly continuouson bounded subsets of 119883 Consequently the two limits areinterchangeable and hence (83) holds Noticing that 119869 isnorm-to-norm uniformly continuous on bounded subsets of119883 we deduce from (62) that


⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

= lim sup119899rarrinfin

(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)

+ ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902) minus 119869 (119909

119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)

Finally let us show that 119909119899rarr 119902 as 119899 rarr infin Indeed

utilizing Lemma 9 we obtain from (47) that

1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2

=1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902) + 120574119899 (119861119909119899 minus 119902)

+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)

times [(1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]

= [1 minus (1 minus 120598119899) 120572119899(1 minus 120588)]

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)

Applying Lemma 8 to (96) we conclude from conditions (ii)and (vi) and (94) that 119909

119899rarr 119902 as 119899 rarr infin This completes

the proof

Corollary 19 Let 119883 be a uniformly convex and 2-uniformlysmooth Banach space and let 119862 be a nonempty closed convexsubset of 119883 such that 119862 plusmn 119862 sub 119862 Let Π

119862be a sunny




119883 times119883 rarr 119862 be a single-valued continuous mapping satisfyingconditions (C1)ndash(C5) in Theorem 4 and (C6) 119873(119879119909 119865119909) +

119860(119892(119909)) 119883 rarr 2119862

0 is 120577-inverse strongly accretive with120577 ge 1205812

Let 119879119894 119862 rarr 119883 be a 120578

119894-strictly pseudocontractive

mapping for each 119894 = 1 119873 Define the mapping119866119894 119862 rarr

119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812

) and 120581 is the 2-uniformly smooth constant of119883 Let119861 119862 rarr 119862 be the 119870-mapping generated by 119866

1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and

120598119899 are the sequences in [0 1] 120572

119899+120573119899+120574119899+120575119899= 1 and satisfy

the following conditions


(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin

(ii) lim119899rarrinfin

120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909

119899and some 120576 gt 0 Assume


sup119909isin119863

119878119899+1


119863 of 119862 and let 119878 be a mapping of 119862 into itself definedby 119878119909 = lim


⋂infin

119894=0Fix(119878119894) Then 119909





Proof Since 119879119894is a 120578119894-strictly pseudocontractive mapping for

each 119894 = 1 119873 it is known that 119860119894= 119868 minus 119879

119894is 120578119894-inverse

strongly accretive for each 119894 = 1 119873 In Theorem 18 weput 119866

119894= Π119862(119868 minus 120582i119860 119894) for 119894 = 1 119873 where 120582

119894isin (0 120578

1198941205812

)


It is not hard to see that Fix(119879119894) = VI(119862 119860

119894) As a matter of

fact we have for 120582119894gt 0

119906 isin VI (119862 119860119894)

lArrrArr ⟨119860119894119906 119869 (119910 minus 119906)⟩ ge 0 forall119910 isin 119862

lArrrArr ⟨119906 minus 120582119894119860119894119906 minus 119906 119869 (119906 minus 119910)⟩ ge 0 forall119910 isin 119862

lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)

lArrrArr ⟨119906 minus 120582119894119906 + 120582119894119879119894119906 minus 119906 119869 (119906 minus 119910)⟩ ge 0 forall119910 isin 119862

lArrrArr ⟨119906 minus 119879119894119906 119869 (119906 minus 119910)⟩ le 0 forall119910 isin 119862

lArrrArr 119906 = 119879119894119906

lArrrArr 119906 isin Fix (119879119894)

(100)

Accordingly we conclude that Δ = ⋂infin

119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore

the desired result follows fromTheorem 18

Remark 20 Theorem 18 improves extends supplements anddevelops [5 Theorem 32] and [25 Theorem 31] in thefollowing aspects

(i) Kangtunyakarnrsquos problem of finding a point of Fix(119878)capFix(119881) cap (cap

119873

119894=1VI(119862 119860

119894)) (see [25 Theorem 11]) is extended

to develop our problem of finding a point of⋂infin119894=0

Fix(119878119894)capΓcap

(cap119873

119894=1VI(119862 119860

119894)) inTheorem 18 because119861

119860= 119878((1minus120572)119868+120572119881)

is nonexpansive with 120572 isin (0 1205781205812

) and Fix(119861119860) = Fix(119878) cap

Fix(119881) (see [25 Lemma 212]) It is clear that the problemof finding a point of ⋂infin

119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in

Theorem 18 ismore general andmore subtle than the problemof finding a point of Γ in [5 Theorem 32]

(ii) The iterative scheme in [25 Theorem 31] is extendedto develop the iterative scheme (35) of Theorem 18 by virtueof the iterative schemes of [5 Theorem 32] The iterativescheme (35) of Theorem 18 is more advantageous and moreflexible than the iterative scheme of [10Theorem 32] becauseit can be applied to solving three problems (ie MVVI (16) afinite family of VIPs and the fixed point problem of 119878

119899) and

involves several parameter sequences 120572119899 120573119899 120574119899 120575119899

120590119899 and 120598

119899

(iii) Theorem 18 extends and generalizes [5 Theorems32] to the setting of a countable family of nonexpansivemappings and a finite family of VIPs In the meantimeTheorem 18 extends and generalizes Kangtunyakarn [25Theorem 31] to the setting of the MVVI (16)

(iv) The iterative scheme (35) in Theorem 18 is verydifferent fromevery one in [5Theorem32] and [25Theorem31] because every iterative scheme in [25 Theorem 31] and[5Theorem 32] is one-step iterative scheme and the iterativescheme (35) inTheorem 18 is the combination of two iterativeschemes in [25 Theorem 31] and [5 Theorem 32]

(v) No boundedness condition on the ranges 119877(119868 minus

119873(119879(sdot) 119865(sdot))) and 119877(119860(119892(sdot))) is imposed inTheorems 18

4 Mann-Type Viscosity Algorithms ina Uniformly Convex Banach Space Havinga Uniformly Gaacuteteaux Differentiable Norm

In this section we introduce Mann-type viscosity iterativealgorithms in a uniformly convex Banach space having auniformly Gateaux differentiable norm and show strongconvergence theorems First we give the following usefullemma

Lemma 21 Let 119862 be a nonempty closed convex subset of asmooth Banach space 119883 and let 119860 119862 rarr 119883 be a 120585-strictlypseudocontractive and ]-strongly accretive mapping with 120585 +

] ge 1 Then for 120582 isin (0 1] one has

1003817100381710038171003817(119868 minus 120582119860) 119909 minus (119868 minus 120582119860) 1199101003817100381710038171003817

le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119862

(101)

In particular if 1 minus (120585(1 + 120585))(1 minus radic(1 minus ])120585) le 120582 le 1 then119868 minus 120582119860 is nonexpansive

Theorem 22 Let 119883 be a nonempty closed convex subset of auniformly convex Banach spacewhich has a uniformlyGateauxdifferentiable norm and let 119862 be a nonempty closed convexsubset of 119883 such that 119862 plusmn 119862 sub 119862 Let Π

119862be a sunny





(H6) 119873(119879119909 119865119909) + 119860(119892(119909)) 119883 rarr 119862 is 1205850-strictly

pseudocontractive and ]0-strongly accretive with 120585

0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585

119894-strictly pseudocontractive and

]119894-strongly accretive with 120585

119894+ ]119894ge 1 for each 119894 = 1 119873

Define the mapping 119866119894 119862 rarr 119862 by 119866

119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585

119894))(1 minus radic(1 minus ]

119894)120585119894) le 120582

119894le 1 for

each 119894 = 1 119873 Let 119861 119862 rarr 119862 be the 119870-mappinggenerated by 119866

1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862

be a contraction with coefficient 120588 isin (0 1) Let 119878119894infin

119894=0be a

countable family of nonexpansive mappings of 119862 into itselfsuch that Δ = ⋂

infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598

119899 are the sequences in

[0 1] 120572119899+120573119899+120574119899+120575119899= 1 and satisfy the following conditions


(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909





Proof First of all by Lemma 21 we know that 119868 minus 120582119894119860119894

is a nonexpansive mapping where 1 minus (120585119894(1 + 120585

119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Hence from

the nonexpansivity ofΠ119862 it follows that119866

119894is a nonexpansive

mapping for each 119894 = 1 119873 Since 119861 119862 rarr 119862 isthe 119870-mapping generated by 119866

1 119866

119873and 120588

1 120588

119873 by

Lemma 3 we deduce that Fix(119861) = cap119873

119894=1Fix(119866

119894) Utilizing

Lemma 10 and the definition of119866119894 we get Fix(119866

119894) = VI(119862 119860

119894)

for each 119894 = 1 119873 Thus we have

Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)

Repeating the same arguments as those in the proof ofTheorem 18 we can prove that for any V isin 119862 120582 gt 0 thereexists a point 119909 isin 119862 such that (119909 119908 119896) is a solution of theMVVI (15) for any119908 isin 119879119909 and 119896 isin 119865119909 In addition in termsof Proposition 7 we know that 119881 + 120582119860 ∘ 119892 is a single-valuedmapping due to the fact that119881+120582119860∘119892 is 120601-strongly accretiveAssume that 119873(119879119909 119865119909) + 119860(119892(119909)) 119883 rarr 119862 is 120585

0-strictly


0+ ]0ge 1

Then by Lemma 21 we conclude that the mapping 119909 997891rarr 119909 minus

(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive

Without loss of generality we may assume that V = 0 and120582 = 1 Let 119901 isin Δ and let 119903(ge 119891(119901)minus119901(1minus120588)) be sufficientlylarge such that 119909

0isin 119861119903(119901) = 119861 Observe that

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)

Utilizing (106) and repeating the same arguments as those inthe proof of Theorem 18 we can derive 119909

119899isin 119861 for all 119899 ge 0

Hence 119909119899 is bounded


119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)

Repeating the same arguments as those of (56) (60) (62)(76) and (80) in the proof of Theorem 18 we can obtain that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1

Then by Lemma 15 we have that Fix(119882) = Fix(119861) cap Fix(119878) capFix(119866) = Δ We observe that

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817

=1003817100381710038171003817(1 minus 1205791 minus 1205792) (119909119899 minus B119909

119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)

From (109) we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)


Now we claim thatlim sup119899rarrinfin

⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Repeating the same arguments as those of (93) in the

proof of Theorem 18 we can deduce that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)

Since 119883 has a uniformly Gateaux differentiable norm theduality mapping 119869 is norm-to-weak lowast uniformly continuouson bounded subsets of 119883 Consequently the two limits areinterchangeable and hence (112) holds Noticing that 119869 isnorm-to-weak lowast uniformly continuous on bounded subsetsof119883 we conclude from (108) that


⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)


repeating the same arguments as those (96) in the proof ofTheorem 18 we can deduce from (107) that1003817100381710038171003817119909119899+1 minus 119902

1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)

Applying Lemma 8 to (116) we infer from conditions (ii) and(vi) and (115) that 119909

119899rarr 119902 as 119899 rarr infin This completes the

proof

Corollary 23 Let 119883 be a uniformly convex Banach spacewhich has a uniformly Gateaux differentiable norm and let119862 be a nonempty closed convex subset of 119883 such that 119862 plusmn

119862 sub 119862 Let Π119862be a sunny nonexpansive retraction from 119883

onto 119862 Let 119879 119865 119883 rarr 119862119861(119883) and 119860 119862 rarr 2119862 be

three multivalued mappings let 119892 119883 rarr 119862 be a single-valued mapping and let 119873(sdot sdot) 119883 times 119883 rarr 119862 be a single-valued continuous mapping satisfying conditions (C1)ndash(C5) inTheorem 4 and (H6) 119873(119879119909 119865119909) + 119860(119892(119909)) 119883 rarr 119862 is 120585

0-

strictly pseudocontractive and ]0-strongly accretive with 120585

0+

]0ge 1

For each 119894 = 1 119873 let 119879119894 119862 rarr 119862 be a self-mapping

such that 119868 minus 119879119894 119862 rarr 119883 is 120585


and ]119894-strongly accretive with 120585

119894+ ]119894ge 1 Define the mapping

119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let

119861 119862 rarr 119862 be the 119870-mapping generated by 1198661 119866

119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909





Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR for the technicaland financial support This research was partially supported


(to Lu-Chuan Ceng) by the National Science Foundationof China (11071169) the Innovation Program of ShanghaiMunicipal Education Commission (09ZZ133) and the PhDProgram Foundation of Ministry of Education of China(20123127110002) Finally the authors thank the referees fortheir valuable comments and appreciation

References

[1] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006

[2] L C Ceng and A Petrusel ldquoKrasnoselski-Mann iterations forhierarchical fixed point problems for a finite family of nonselfmappings in Banach spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 3 pp 617ndash639 2010

[3] L-C Ceng H-K Xu and J-C Yao ldquoStrong convergence ofan iterative method with perturbedmappings for nonexpansiveand accretive operatorsrdquo Numerical Functional Analysis andOptimization vol 29 no 3-4 pp 324ndash345 2008

[4] A Kangtunyakarn and S Suantai ldquoA new mapping for findingcommon solutions of equilibrium problems and fixed pointproblems of finite family of nonexpansive mappingsrdquoNonlinearAnalysis Theory Methods amp Applications vol 71 no 10 pp4448ndash4460 2009

[5] C E Chidume H Zegeye and K R Kazmi ldquoExistence andconvergence theorems for a class of multi-valued variationalinclusions in Banach spacesrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 59 no 5 pp 649ndash656 2004

[6] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011

[7] L-C Ceng Q H Ansari M M Wong and J-C Yao ldquoManntype hybrid extragradient method for variational inequalitiesvariational inclusions and fixed point problemsrdquo Fixed PointTheory vol 13 no 2 pp 403ndash422 2012

[8] L-C Ceng Q H Ansari N-C Wong and J-C YaoldquoAn extragradient-like approximation method for variationalinequalities and fixed point problemsrdquo Fixed Point Theory andApplications vol 2011 article 22 2011

[9] L-C Ceng N Hadjisavvas and N-C Wong ldquoStrong conver-gence theorem by a hybrid extragradient-like approximationmethod for variational inequalities and fixed point problemsrdquoJournal of Global Optimization vol 46 no 4 pp 635ndash646 2010

[10] L-C Ceng S-M Guu and J-C Yao ldquoFinding commonsolutions of a variational inequality a general system of vari-ational inequalities and a fixed-point problem via a hybridextragradientmethodrdquoFixed PointTheory andApplications vol2011 Article ID 626159 22 pages 2011

[11] L-C Zeng and J-C Yao ldquoStrong convergence theorem by anextragradient method for fixed point problems and variationalinequality problemsrdquo Taiwanese Journal of Mathematics vol 10no 5 pp 1293ndash1303 2006

[12] C E Chidume and H Zegeye ldquoApproximation methods fornonlinear operator equationsrdquo Proceedings of the AmericanMathematical Society vol 131 no 8 pp 2467ndash2478 2003

[13] S S Chang J K Kim and K H Kim ldquoOn the existenceand iterative approximation problems of solutions for set-valued variational inclusions in Banach spacesrdquo Journal ofMathematical Analysis and Applications vol 268 no 1 pp 89ndash108 2002

[14] X He ldquoOn 120593-strongly accretive mappings and some set-valuedvariational problemsrdquo Journal of Mathematical Analysis andApplications vol 277 no 2 pp 504ndash511 2003

[15] H-K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 66 no 1 pp 240ndash2562002

[16] S Reich ldquoWeak convergence theorems for nonexpansive map-pings in Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 67 no 2 pp 274ndash276 1979

[17] Y J Cho H Zhou and G Guo ldquoWeak and strong convergencetheorems for three-step iterations with errors for asymptoti-cally nonexpansive mappingsrdquo Computers amp Mathematics withApplications vol 47 no 4-5 pp 707ndash717 2004

[18] K Aoyama Y Kimura W Takahashi andM Toyoda ldquoApprox-imation of common fixed points of a countable family ofnonexpansive mappings in a Banach spacerdquoNonlinear AnalysisTheory Methods amp Applications vol 67 no 8 pp 2350ndash23602007

[19] J S Jung ldquoIterative approaches to common fixed points of non-expansive mappings in Banach spacesrdquo Journal of MathematicalAnalysis and Applications vol 302 no 2 pp 509ndash520 2005

[20] R E Bruck Jr ldquoProperties of fixed-point sets of nonexpansivemappings in Banach spacesrdquo Transactions of the AmericanMathematical Society vol 179 pp 251ndash262 1973

[21] H K Xu ldquoInequalities in Banach spaces with applicationsrdquoNonlinear Analysis Theory Methods amp Applications vol 16 no12 pp 1127ndash1138 1991

[22] I Cioranescu Geometry of Banach Spaces Duality Mappingsand Nonlinear Problems vol 62 Kluwer Academic PublishersGroup Amsterdam The Netherland 1990

[23] CHMorales ldquoSurjectivity theorems formultivaluedmappingsof accretive typerdquo Commentationes Mathematicae UniversitatisCarolinae vol 26 no 2 pp 397ndash413 1985

[24] S B Nadler Jr ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 pp 475ndash488 1969

[25] A Kangtunyakarn ldquoIterative scheme for a nonexpansive map-ping an 120578-strictly pseudo-contractive mapping and variationalinequality problems in a uniformly convex and 2-uniformlysmooth Banach spacerdquo Fixed PointTheory andApplications vol2013 article 23 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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[0infin) rarr [0infin) called the modulus of smoothness of 119883 asfollows

120588 (120591) = sup 12(1003817100381710038171003817119909 + 119910

1003817100381710038171003817 +1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)

minus1 119909 119910 isin 119883 119909 = 110038171003817100381710038171199101003817100381710038171003817 = 120591

(4)

It is known that 119883 is uniformly smooth if and only iflim120591rarr0

120588(120591)120591 = 0 Let 119902 be a fixed real number with 1 lt 119902 le

2 Then a Banach space119883 is said to be 119902-uniformly smooth ifthere exists a constant 119888 gt 0 such that 120588(120591) le 119888120591

119902 for all 120591 gt 0It is well-known that no Banach space is 119902-uniformly smoothfor 119902 gt 2 In addition it is also known that 119869 is single-valuedif and only if119883 is smooth whereas if119883 is uniformly smooththen the mapping 119869 is norm-to-norm uniformly continuouson bounded subsets of 119883 If 119883 has a uniformly Gateauxdifferentiable norm then the duality mapping 119869 is norm-to-weaklowast uniformly continuous on bounded subsets of119883

Let119862 be a nonempty closed convex subset of a real Banachspace119883 A mapping 119879 119862 rarr 119862 is called nonexpansive if

1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119862 (5)

The set of fixed points of 119879 is denoted by Fix(119879) We use thenotation to indicate the weak convergence and the one rarrto indicate the strong convergence

Definition 1 Let119860 119862 rarr 119883 be a mapping of 119862 into119883 Then119860 is said to be

(i) accretive if for each 119909 119910 isin 119862 there exists 119895(119909 minus 119910) isin

119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ ge 0 (6)

where 119869 is the normalized duality mapping

(ii) 120572-strongly accretive if for each 119909 119910 isin 119862 there exists119895(119909 minus 119910) isin 119869(119909 minus 119910) such that


1003817100381710038171003817

2

(7)


(iii) 120573-inverse strongly accretive if for each 119909 119910 isin 119862 thereexists 119895(119909 minus 119910) isin 119869(119909 minus 119910) such that


1003817100381710038171003817

2

(8)

for some 120573 gt 0

(iv) 120582-strictly pseudocontractive if for each 119909 119910 isin 119862 thereexists 119895(119909 minus 119910) isin 119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

minus 1205821003817100381710038171003817119909 minus 119910 minus (119860119909 minus 119860119910)

1003817100381710038171003817

2

(9)


Let119883 be a real smoothBanach space Let119862 be a nonemptyclosed convex subset of119883 and let 119860 119862 rarr 119883 be a nonlinearmapping The so-called variational inequality problem (VIP)is the problem of finding 119909lowast isin 119862 such that

⟨119860119909lowast

119869 (119909 minus 119909lowast

)⟩ ge 0 forall119909 isin 119862 (10)

which was considered by Aoyama et al [1] Note that VIP(10) is connected with the fixed point problem for nonlinearmapping (see eg [2]) the problem of finding a zero point ofa nonlinear operator (see eg [3]) and so on In particularwhenever119883 = 119867 a Hilbert space the VIP (10) reduces to theclassical VIP of finding 119909lowast isin 119862 such that

⟨119860119909lowast

119909 minus 119909lowast

⟩ ge 0 forall119909 isin 119862 (11)

whose solution set is denoted by VI(119862 119860) Recently in orderto find a solution of VIP (10) Aoyama et al [1] introducedMann-type iterative scheme for an accretive operator 119860 asfollows

119909119899+1

= 120572119899119909119899+ (1 minus 120572

119899)Π119862(119909119899minus 120582119899119860119909119899) forall119899 ge 1 (12)

whereΠ119862is a sunny nonexpansive retraction from119883 onto119862

Then they proved a weak convergence theorem

Definition 2 Let 119862 be a nonempty convex subset of a realBanach space119883 Let 119879

119894119873

119894=1be a finite family of nonexpansive

mappings of 119862 into itself and let 1205821 120582

119873be real numbers

such that 0 le 120582119894le 1 for every 119894 = 1 119873 Define a mapping

119870 119862 rarr 119862 as follows

1198801= 12058211198791+ (1 minus 120582

1) 119868

1198802= 120582211987921198801+ (1 minus 120582

2) 1198801

1198803= 120582311987931198802+ (1 minus 120582

3) 1198802

119880119873minus1

= 120582119873minus1

119879119873minus1

119880119873minus2

+ (1 minus 120582119873minus1

) 119880119873minus2

119870 = 119880119873= 120582119873119879119873119880119873minus1

+ (1 minus 120582119873) 119880119873minus1

(13)

Such a mapping 119870 is called the 119870-mapping generated by1198791 119879

119873and 120582

1 120582

119873

Lemma 3 (see [4]) Let 119862 be a nonempty closed convex subsetof a strictly convex Banach space Let 119879

119894119873

119894=1be a finite family

of nonexpansive mappings of 119862 into itself with cap119873119894=1

Fix(119879119894) = 0

and let 1205821 120582

119873be real numbers such that 0 lt 120582

119894lt 1

for every 119894 = 1 119873 minus 1 and 0 lt 120582119873

le 1 Let 119870 bethe 119870-mapping generated by 119879

1 119879

119873and 120582

1 120582

119873 Then

Fix(K) = cap119873

119894=1Fix(119879119894)

From Lemma 3 it is easy to see that the 119870-mapping is anonexpansive mapping

On the other hand let 119862119861(119883) be the family of allnonempty closed and bounded subsets of a real smooth



119867(119860 119861) = maxsup119909isin119861

inf119910isin119860

119889 (119909 119910) sup119909isin119860

inf119910isin119861

119889 (119909 119910)

forall119860 119861 isin 119862119861 (119883)

(14)




















119899 iteratively

by

119909119899+1

= 119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899) 119906119899isin 119860 (119892 (119909

119899)) (17)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817

le (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899))) forall119899 ge 0

(18)


119899 is a


120590119899= 0sum

infin

119899=0120590119899= infin








2 Preliminaries


lowast

defined by

119869 (119909) = 119909lowast

isin 119883lowast

⟨119909 119909lowast

⟩ = 1199092

=1003817100381710038171003817119909lowast1003817100381710038171003817

2

(19)



be(i) accretive if






⟨119906 minus V 119869 (119909 minus 119910)⟩

ge 120601 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)1003817100381710038171003817119909 minus 119910




119906 minus V ge 120601 (1003817100381710038171003817119909 minus 119910






1003817100381710038171003817)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817









119899=0120572119899= infin and 119887

119899rarr


1205822

119899+1le 1205822

119899minus 2120572119899120595 (120582119899+1

) + 2120572119899119887119899120582119899+1

forall119899 ge 0 (25)









119883



2


satisfying

119904119899+1


where 120572119899 120573119899 and 120574



119899=0120572119899= infin


120573119899le 0


119899=0120574119899lt infin


119904119899= 0


1003817100381710038171003817119909 + 1199101003817100381710038171003817

2

le 1199092

+ 2 ⟨119910 119869 (119909 + 119910)⟩ forall119909 119910 isin 119883 (27)




VI (119862 119860) = Fix (Π119862(119868 minus 120582119860)) (28)


Π [Π (119909) + 119905 (119909 minus Π (119909))] = Π (119909) (29)









= 119875119862 If 119862





1003817100381710038171003817120572119909 + 120573119910 + 1205741199111003817100381710038171003817

2

le 1205721199092

+ 12057310038171003817100381710038171199101003817100381710038171003817

2

+ 1205741199112

minus 120572120573120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)

(30)






sup119878119899119909minus119878119899minus1

119909




lim119899rarrinfin

sup119878119909 minus 119878119899119909 119909 isin 119862 = 0






119909119905= 119905119891 (119909

119905) + (1 minus 119905) 119879119909

119905 (31)


119862 Then the net 119909

119905

defined by 119909119905= 119905119891(119909

119905) + (1 minus 119905)119879119909



119876(119891) = 119904minus lim119905rarr0

119909119905 for all 119891 isin Ξ


as follows

⟨(119868 minus 119891)119876 (119891) 119869 (119876 (119891) minus 119901)⟩ le 0


(32)


119899infin

119899=0






119878119909 = suminfin



119899=0Fix(119879119899) holds


1003817100381710038171003817120582119909 + (1 minus 120582) 1199101003817100381710038171003817

2

le 1205821199092

+ (1 minus 120582)10038171003817100381710038171199101003817100381710038171003817

2


1003817100381710038171003817)

(33)





1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 2120582 (1205821205812

minus 120572)1003817100381710038171003817119860119909 minus 119860119910

1003817100381710038171003817

2

forall119909 119910 isin 119862

(34)


nonexpansive


119862be a sunny





(C6) 119873(119879119909 119865119909) + 119860(119892(119909)) 119883 rarr 2119862


2

Let 119860119894 119862 rarr 119883 be an 120572



119862 rarr 119862 by 119866119894= Π119862(119868 minus 120582

119894119860119894) for 119894 = 1 119873 where

120582119894isin (0 120572

1198941205812


1 119866

119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899


119899+ 120573119899+ 120574119899+ 120575119899= 1



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(35)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(36)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1





⋂infin

119894=0Fix(119878119894) Then 119909







1198941205812


119862 it



119873and 120588

1 120588


Fix(119861) = cap119873

119894=1Fix(119866


of 119866119894 we get Fix(119866

119894) = VI(119862 119860

119894) for each 119894 = 1 119873 Thus

we have

Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (38)










1= 120601(119903)8(1 + 120576) 120576

2= 120601(119903)8120583(1 + 120576) and 120576

3=



2and 119909 minus 119910 lt 120575

3imply

119867(119860 ∘ 119892(119909) 119860 ∘ 119892(119910)) lt 1205761 119867(119879119909 119879119910) lt 120576

2and119867(119865119909 119865119910) lt




119899isin 119861

We show that 119909119899+1


notin 119861then 119909


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817120572119899 (119891 (119909119899) minus 119901) + 120573119899 (119909119899 minus 119901)

+120574119899(119861119909119899minus 119901) + 120575

119899(119878119899119909119899minus 119901)

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120572119899(1 minus 120588)

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(39)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

= ⟨120598119899[119909119899minus 119901 minus 120590

119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

= ⟨120598119899(119909119899minus 119901) + (1 minus 120598

119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1 minus 1205881003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


le1

2(max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

+1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

)

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

(40)


1003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899+1 119896119899+1

) + 119906119899+1

119869 (119909119899+1

minus 119901)⟩

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899

minus (119873 (119908119899+1

119896119899+1

) + 119906119899+1

) 119869 (119909119899+1

minus 119901)⟩

(41)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899+1

119896119899)1003817100381710038171003817

+1003817100381710038171003817119873 (119908

119899+1 119896119899) minus 119873 (119908

119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

(42)


1003817100381710038171003817

le 120598119899

1003817100381710038171003817119909119899 minus 119901 minus 120590119899 (119873 (119908119899 119896119899) + 119906119899)1003817100381710038171003817

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205901198991003817100381710038171003817119873 (119908

119899 119896119899) + 119906119899

1003817100381710038171003817]

+ (1 minus 120598119899)max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le 120598119899[119903 + 120590

119899119872] + (1 minus 120598

119899) 119903

le 2119903

(43)


1015840

) and 119873(119908 119896) = 119873(1199081015840


119899+1isin 119865119909119899+1

and119908119899+1

isin 119879119909119899+1


119899isin 119879119909119899such that

10038171003817100381710038171003817119896119899+1

minus 1198961015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119865119909

119899+1 119865119909119899) (44)

10038171003817100381710038171003817119908119899+1

minus 1199081015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119879119909

119899+1 119879119909119899) (45)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[10038171003817100381710038171003817119873 (119908119899+1

119896119899+1

) minus 119873 (119908119899+1

1198961015840

119899)10038171003817100381710038171003817

+10038171003817100381710038171003817119873 (119908119899+1

119896119899) minus 119873 (119908

1015840

119899 119896119899)10038171003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817 ] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120585 (1 + 120576)119867 (119865119909

119899+1 119865119909119899)

+ 120583 (1 + 120576)119867 (119879119909119899+1

119879119909119899)

+ (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899)))] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120601 (119903)

8+120601 (119903)

8+120601 (119903)

8] 2119903

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903 + 120572

119899120590119899

3

2120601 (119903) 119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

(46)


So we get 119909119899+1



119899rarrinfin119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899]

+ (1 minus 120598119899) 119910119899 forall119899 ge 0

(47)


119910119899by

119910119899= 120573119899119909119899+ (1 minus 120573

119899) 119911119899 (48)

where 119911119899= (120572119899119891(119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)(1 minus 120573

119899) Now we

have1003817100381710038171003817119911119899+1 minus 119911119899

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

120572119899+1

119891 (119909119899+1

) + 120574119899+1

119861119909119899+1

+ 120575119899+1

119878119899+1

119909119899+1

1 minus 120573119899+1

minus120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899

1 minus 120573119899

100381710038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

+119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=1

1 minus 120573119899+1


+

10038161003816100381610038161003816100381610038161003816

1

1 minus 120573119899+1

minus1

1 minus 120573119899

10038161003816100381610038161003816100381610038161003816

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1


+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1

1003817100381710038171003817120572119899+1119891 (119909119899+1) + 120574119899+1119861119909119899+1 + 120575119899+1119878119899+1119909119899+1

minus (120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)1003817100381710038171003817

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

(120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817

+ 120574119899+1

1003817100381710038171003817119861119909119899+1 minus 1198611199091198991003817100381710038171003817

+ 120575119899+1

1003817100381710038171003817119878119899+1119909119899+1 minus 1198781198991199091198991003817100381710038171003817

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817)

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817 + 120574119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119878119899+1119909119899+1 minus 119878119899+1119909119899

1003817100381710038171003817

+1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1205881003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 + 120574119899+11003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817

=1 minus 120573119899+1

minus 120572119899+1

(1 minus 120588)

1 minus 120573119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+120575119899+1

1 minus 120573119899+1

1003817100381710038171003817119878119899+1119909119899 minus 1198781198991199091198991003817100381710038171003817

+1

1 minus 120573119899+1

[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817


le (1 minus120572119899+1

(1 minus 120588)

1 minus 120573119899+1

)1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816 +1003816100381610038161003816120573119899+1 minus 120573119899

1003816100381610038161003816

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816]

(49)


119891(119909119899) + 119861119909

119899 + 119878

119899119909119899 le 119872

0for


119910119899minus 119910119899minus1

= 120573119899(119909119899minus 119909119899minus1

) + (120573119899minus 120573119899minus1

)




)

(50)

So from (49) we get

1003817100381710038171003817119910119899 minus 119910119899minus11003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


+ (1 minus 120573119899) (1 minus

120572119899(1 minus 120588)

1 minus 120573119899

)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817


1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

(51)


119909119899+1

= 120598119899119899+ (1 minus 120598

119899) 119910119899

119899= 120590119899119866119909119899+ (1 minus 120590

119899) 119909119899

(52)


119909119899+1


) + (120598119899minus 120598119899minus1

)

times (119899minus1



)


= 120590119899(119866119909119899minus 119866119909119899minus1

) + (120590119899minus 120590119899minus1

)

times (119866119909119899minus1



)

(53)


1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817

le 120590119899

1003817100381710038171003817119866119909119899 minus 119866119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


(54)

and hence1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817 +

1003816100381610038161003816120598119899 minus 120598119899minus11003816100381610038161003816



1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


+ (1 minus 120598119899) (1 minus 120572

119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817


1003817100381710038171003817

+ 1198721[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816 +1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816]

(55)

where sup119899ge1

119866119909119899minus1


+ 119899minus1


+ 119909119899minus1


+

1198720 le 119872

1for some 119872



119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 (56)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)


120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)


lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin


1003817100381710038171003817) = 0

(59)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin


(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817


1003817100381710038171003817


1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)


119909119899minus119861119909119899 = 0 and lim




119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119867(119860 119861) = maxsup119909isin119861

inf119910isin119860

119889 (119909 119910) sup119909isin119860

inf119910isin119861

119889 (119909 119910)

forall119860 119861 isin 119862119861 (119883)

(14)




















119899 iteratively

by

119909119899+1

= 119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899) 119906119899isin 119860 (119892 (119909

119899)) (17)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817

le (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899))) forall119899 ge 0

(18)


119899 is a


120590119899= 0sum

infin

119899=0120590119899= infin








2 Preliminaries


lowast

defined by

119869 (119909) = 119909lowast

isin 119883lowast

⟨119909 119909lowast

⟩ = 1199092

=1003817100381710038171003817119909lowast1003817100381710038171003817

2

(19)



be(i) accretive if






⟨119906 minus V 119869 (119909 minus 119910)⟩

ge 120601 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)1003817100381710038171003817119909 minus 119910




119906 minus V ge 120601 (1003817100381710038171003817119909 minus 119910






1003817100381710038171003817)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817









119899=0120572119899= infin and 119887

119899rarr


1205822

119899+1le 1205822

119899minus 2120572119899120595 (120582119899+1

) + 2120572119899119887119899120582119899+1

forall119899 ge 0 (25)









119883



2


satisfying

119904119899+1


where 120572119899 120573119899 and 120574



119899=0120572119899= infin


120573119899le 0


119899=0120574119899lt infin


119904119899= 0


1003817100381710038171003817119909 + 1199101003817100381710038171003817

2

le 1199092

+ 2 ⟨119910 119869 (119909 + 119910)⟩ forall119909 119910 isin 119883 (27)




VI (119862 119860) = Fix (Π119862(119868 minus 120582119860)) (28)


Π [Π (119909) + 119905 (119909 minus Π (119909))] = Π (119909) (29)









= 119875119862 If 119862





1003817100381710038171003817120572119909 + 120573119910 + 1205741199111003817100381710038171003817

2

le 1205721199092

+ 12057310038171003817100381710038171199101003817100381710038171003817

2

+ 1205741199112

minus 120572120573120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)

(30)






sup119878119899119909minus119878119899minus1

119909




lim119899rarrinfin

sup119878119909 minus 119878119899119909 119909 isin 119862 = 0






119909119905= 119905119891 (119909

119905) + (1 minus 119905) 119879119909

119905 (31)


119862 Then the net 119909

119905

defined by 119909119905= 119905119891(119909

119905) + (1 minus 119905)119879119909



119876(119891) = 119904minus lim119905rarr0

119909119905 for all 119891 isin Ξ


as follows

⟨(119868 minus 119891)119876 (119891) 119869 (119876 (119891) minus 119901)⟩ le 0


(32)


119899infin

119899=0






119878119909 = suminfin



119899=0Fix(119879119899) holds


1003817100381710038171003817120582119909 + (1 minus 120582) 1199101003817100381710038171003817

2

le 1205821199092

+ (1 minus 120582)10038171003817100381710038171199101003817100381710038171003817

2


1003817100381710038171003817)

(33)





1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 2120582 (1205821205812

minus 120572)1003817100381710038171003817119860119909 minus 119860119910

1003817100381710038171003817

2

forall119909 119910 isin 119862

(34)


nonexpansive


119862be a sunny





(C6) 119873(119879119909 119865119909) + 119860(119892(119909)) 119883 rarr 2119862


2

Let 119860119894 119862 rarr 119883 be an 120572



119862 rarr 119862 by 119866119894= Π119862(119868 minus 120582

119894119860119894) for 119894 = 1 119873 where

120582119894isin (0 120572

1198941205812


1 119866

119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899


119899+ 120573119899+ 120574119899+ 120575119899= 1



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(35)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(36)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1





⋂infin

119894=0Fix(119878119894) Then 119909







1198941205812


119862 it



119873and 120588

1 120588


Fix(119861) = cap119873

119894=1Fix(119866


of 119866119894 we get Fix(119866

119894) = VI(119862 119860

119894) for each 119894 = 1 119873 Thus

we have

Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (38)










1= 120601(119903)8(1 + 120576) 120576

2= 120601(119903)8120583(1 + 120576) and 120576

3=



2and 119909 minus 119910 lt 120575

3imply

119867(119860 ∘ 119892(119909) 119860 ∘ 119892(119910)) lt 1205761 119867(119879119909 119879119910) lt 120576

2and119867(119865119909 119865119910) lt




119899isin 119861

We show that 119909119899+1


notin 119861then 119909


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817120572119899 (119891 (119909119899) minus 119901) + 120573119899 (119909119899 minus 119901)

+120574119899(119861119909119899minus 119901) + 120575

119899(119878119899119909119899minus 119901)

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120572119899(1 minus 120588)

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(39)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

= ⟨120598119899[119909119899minus 119901 minus 120590

119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

= ⟨120598119899(119909119899minus 119901) + (1 minus 120598

119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1 minus 1205881003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


le1

2(max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

+1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

)

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

(40)


1003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899+1 119896119899+1

) + 119906119899+1

119869 (119909119899+1

minus 119901)⟩

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899

minus (119873 (119908119899+1

119896119899+1

) + 119906119899+1

) 119869 (119909119899+1

minus 119901)⟩

(41)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899+1

119896119899)1003817100381710038171003817

+1003817100381710038171003817119873 (119908

119899+1 119896119899) minus 119873 (119908

119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

(42)


1003817100381710038171003817

le 120598119899

1003817100381710038171003817119909119899 minus 119901 minus 120590119899 (119873 (119908119899 119896119899) + 119906119899)1003817100381710038171003817

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205901198991003817100381710038171003817119873 (119908

119899 119896119899) + 119906119899

1003817100381710038171003817]

+ (1 minus 120598119899)max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le 120598119899[119903 + 120590

119899119872] + (1 minus 120598

119899) 119903

le 2119903

(43)


1015840

) and 119873(119908 119896) = 119873(1199081015840


119899+1isin 119865119909119899+1

and119908119899+1

isin 119879119909119899+1


119899isin 119879119909119899such that

10038171003817100381710038171003817119896119899+1

minus 1198961015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119865119909

119899+1 119865119909119899) (44)

10038171003817100381710038171003817119908119899+1

minus 1199081015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119879119909

119899+1 119879119909119899) (45)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[10038171003817100381710038171003817119873 (119908119899+1

119896119899+1

) minus 119873 (119908119899+1

1198961015840

119899)10038171003817100381710038171003817

+10038171003817100381710038171003817119873 (119908119899+1

119896119899) minus 119873 (119908

1015840

119899 119896119899)10038171003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817 ] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120585 (1 + 120576)119867 (119865119909

119899+1 119865119909119899)

+ 120583 (1 + 120576)119867 (119879119909119899+1

119879119909119899)

+ (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899)))] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120601 (119903)

8+120601 (119903)

8+120601 (119903)

8] 2119903

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903 + 120572

119899120590119899

3

2120601 (119903) 119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

(46)


So we get 119909119899+1



119899rarrinfin119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899]

+ (1 minus 120598119899) 119910119899 forall119899 ge 0

(47)


119910119899by

119910119899= 120573119899119909119899+ (1 minus 120573

119899) 119911119899 (48)

where 119911119899= (120572119899119891(119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)(1 minus 120573

119899) Now we

have1003817100381710038171003817119911119899+1 minus 119911119899

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

120572119899+1

119891 (119909119899+1

) + 120574119899+1

119861119909119899+1

+ 120575119899+1

119878119899+1

119909119899+1

1 minus 120573119899+1

minus120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899

1 minus 120573119899

100381710038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

+119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=1

1 minus 120573119899+1


+

10038161003816100381610038161003816100381610038161003816

1

1 minus 120573119899+1

minus1

1 minus 120573119899

10038161003816100381610038161003816100381610038161003816

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1


+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1

1003817100381710038171003817120572119899+1119891 (119909119899+1) + 120574119899+1119861119909119899+1 + 120575119899+1119878119899+1119909119899+1

minus (120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)1003817100381710038171003817

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

(120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817

+ 120574119899+1

1003817100381710038171003817119861119909119899+1 minus 1198611199091198991003817100381710038171003817

+ 120575119899+1

1003817100381710038171003817119878119899+1119909119899+1 minus 1198781198991199091198991003817100381710038171003817

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817)

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817 + 120574119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119878119899+1119909119899+1 minus 119878119899+1119909119899

1003817100381710038171003817

+1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1205881003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 + 120574119899+11003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817

=1 minus 120573119899+1

minus 120572119899+1

(1 minus 120588)

1 minus 120573119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+120575119899+1

1 minus 120573119899+1

1003817100381710038171003817119878119899+1119909119899 minus 1198781198991199091198991003817100381710038171003817

+1

1 minus 120573119899+1

[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817


le (1 minus120572119899+1

(1 minus 120588)

1 minus 120573119899+1

)1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816 +1003816100381610038161003816120573119899+1 minus 120573119899

1003816100381610038161003816

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816]

(49)


119891(119909119899) + 119861119909

119899 + 119878

119899119909119899 le 119872

0for


119910119899minus 119910119899minus1

= 120573119899(119909119899minus 119909119899minus1

) + (120573119899minus 120573119899minus1

)




)

(50)

So from (49) we get

1003817100381710038171003817119910119899 minus 119910119899minus11003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


+ (1 minus 120573119899) (1 minus

120572119899(1 minus 120588)

1 minus 120573119899

)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817


1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

(51)


119909119899+1

= 120598119899119899+ (1 minus 120598

119899) 119910119899

119899= 120590119899119866119909119899+ (1 minus 120590

119899) 119909119899

(52)


119909119899+1


) + (120598119899minus 120598119899minus1

)

times (119899minus1



)


= 120590119899(119866119909119899minus 119866119909119899minus1

) + (120590119899minus 120590119899minus1

)

times (119866119909119899minus1



)

(53)


1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817

le 120590119899

1003817100381710038171003817119866119909119899 minus 119866119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


(54)

and hence1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817 +

1003816100381610038161003816120598119899 minus 120598119899minus11003816100381610038161003816



1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


+ (1 minus 120598119899) (1 minus 120572

119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817


1003817100381710038171003817

+ 1198721[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816 +1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816]

(55)

where sup119899ge1

119866119909119899minus1


+ 119899minus1


+ 119909119899minus1


+

1198720 le 119872

1for some 119872



119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 (56)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)


120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)


lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin


1003817100381710038171003817) = 0

(59)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin


(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817


1003817100381710038171003817


1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)


119909119899minus119861119909119899 = 0 and lim




119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 minus V ge 120601 (1003817100381710038171003817119909 minus 119910






1003817100381710038171003817)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817









119899=0120572119899= infin and 119887

119899rarr


1205822

119899+1le 1205822

119899minus 2120572119899120595 (120582119899+1

) + 2120572119899119887119899120582119899+1

forall119899 ge 0 (25)









119883



2


satisfying

119904119899+1


where 120572119899 120573119899 and 120574



119899=0120572119899= infin


120573119899le 0


119899=0120574119899lt infin


119904119899= 0


1003817100381710038171003817119909 + 1199101003817100381710038171003817

2

le 1199092

+ 2 ⟨119910 119869 (119909 + 119910)⟩ forall119909 119910 isin 119883 (27)




VI (119862 119860) = Fix (Π119862(119868 minus 120582119860)) (28)


Π [Π (119909) + 119905 (119909 minus Π (119909))] = Π (119909) (29)









= 119875119862 If 119862





1003817100381710038171003817120572119909 + 120573119910 + 1205741199111003817100381710038171003817

2

le 1205721199092

+ 12057310038171003817100381710038171199101003817100381710038171003817

2

+ 1205741199112

minus 120572120573120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817)

(30)






sup119878119899119909minus119878119899minus1

119909




lim119899rarrinfin

sup119878119909 minus 119878119899119909 119909 isin 119862 = 0






119909119905= 119905119891 (119909

119905) + (1 minus 119905) 119879119909

119905 (31)


119862 Then the net 119909

119905

defined by 119909119905= 119905119891(119909

119905) + (1 minus 119905)119879119909



119876(119891) = 119904minus lim119905rarr0

119909119905 for all 119891 isin Ξ


as follows

⟨(119868 minus 119891)119876 (119891) 119869 (119876 (119891) minus 119901)⟩ le 0


(32)


119899infin

119899=0






119878119909 = suminfin



119899=0Fix(119879119899) holds


1003817100381710038171003817120582119909 + (1 minus 120582) 1199101003817100381710038171003817

2

le 1205821199092

+ (1 minus 120582)10038171003817100381710038171199101003817100381710038171003817

2


1003817100381710038171003817)

(33)





1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 2120582 (1205821205812

minus 120572)1003817100381710038171003817119860119909 minus 119860119910

1003817100381710038171003817

2

forall119909 119910 isin 119862

(34)


nonexpansive


119862be a sunny





(C6) 119873(119879119909 119865119909) + 119860(119892(119909)) 119883 rarr 2119862


2

Let 119860119894 119862 rarr 119883 be an 120572



119862 rarr 119862 by 119866119894= Π119862(119868 minus 120582

119894119860119894) for 119894 = 1 119873 where

120582119894isin (0 120572

1198941205812


1 119866

119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899


119899+ 120573119899+ 120574119899+ 120575119899= 1



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(35)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(36)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1





⋂infin

119894=0Fix(119878119894) Then 119909







1198941205812


119862 it



119873and 120588

1 120588


Fix(119861) = cap119873

119894=1Fix(119866


of 119866119894 we get Fix(119866

119894) = VI(119862 119860

119894) for each 119894 = 1 119873 Thus

we have

Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (38)










1= 120601(119903)8(1 + 120576) 120576

2= 120601(119903)8120583(1 + 120576) and 120576

3=



2and 119909 minus 119910 lt 120575

3imply

119867(119860 ∘ 119892(119909) 119860 ∘ 119892(119910)) lt 1205761 119867(119879119909 119879119910) lt 120576

2and119867(119865119909 119865119910) lt




119899isin 119861

We show that 119909119899+1


notin 119861then 119909


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817120572119899 (119891 (119909119899) minus 119901) + 120573119899 (119909119899 minus 119901)

+120574119899(119861119909119899minus 119901) + 120575

119899(119878119899119909119899minus 119901)

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120572119899(1 minus 120588)

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(39)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

= ⟨120598119899[119909119899minus 119901 minus 120590

119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

= ⟨120598119899(119909119899minus 119901) + (1 minus 120598

119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1 minus 1205881003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


le1

2(max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

+1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

)

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

(40)


1003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899+1 119896119899+1

) + 119906119899+1

119869 (119909119899+1

minus 119901)⟩

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899

minus (119873 (119908119899+1

119896119899+1

) + 119906119899+1

) 119869 (119909119899+1

minus 119901)⟩

(41)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899+1

119896119899)1003817100381710038171003817

+1003817100381710038171003817119873 (119908

119899+1 119896119899) minus 119873 (119908

119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

(42)


1003817100381710038171003817

le 120598119899

1003817100381710038171003817119909119899 minus 119901 minus 120590119899 (119873 (119908119899 119896119899) + 119906119899)1003817100381710038171003817

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205901198991003817100381710038171003817119873 (119908

119899 119896119899) + 119906119899

1003817100381710038171003817]

+ (1 minus 120598119899)max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le 120598119899[119903 + 120590

119899119872] + (1 minus 120598

119899) 119903

le 2119903

(43)


1015840

) and 119873(119908 119896) = 119873(1199081015840


119899+1isin 119865119909119899+1

and119908119899+1

isin 119879119909119899+1


119899isin 119879119909119899such that

10038171003817100381710038171003817119896119899+1

minus 1198961015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119865119909

119899+1 119865119909119899) (44)

10038171003817100381710038171003817119908119899+1

minus 1199081015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119879119909

119899+1 119879119909119899) (45)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[10038171003817100381710038171003817119873 (119908119899+1

119896119899+1

) minus 119873 (119908119899+1

1198961015840

119899)10038171003817100381710038171003817

+10038171003817100381710038171003817119873 (119908119899+1

119896119899) minus 119873 (119908

1015840

119899 119896119899)10038171003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817 ] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120585 (1 + 120576)119867 (119865119909

119899+1 119865119909119899)

+ 120583 (1 + 120576)119867 (119879119909119899+1

119879119909119899)

+ (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899)))] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120601 (119903)

8+120601 (119903)

8+120601 (119903)

8] 2119903

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903 + 120572

119899120590119899

3

2120601 (119903) 119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

(46)


So we get 119909119899+1



119899rarrinfin119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899]

+ (1 minus 120598119899) 119910119899 forall119899 ge 0

(47)


119910119899by

119910119899= 120573119899119909119899+ (1 minus 120573

119899) 119911119899 (48)

where 119911119899= (120572119899119891(119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)(1 minus 120573

119899) Now we

have1003817100381710038171003817119911119899+1 minus 119911119899

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

120572119899+1

119891 (119909119899+1

) + 120574119899+1

119861119909119899+1

+ 120575119899+1

119878119899+1

119909119899+1

1 minus 120573119899+1

minus120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899

1 minus 120573119899

100381710038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

+119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=1

1 minus 120573119899+1


+

10038161003816100381610038161003816100381610038161003816

1

1 minus 120573119899+1

minus1

1 minus 120573119899

10038161003816100381610038161003816100381610038161003816

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1


+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1

1003817100381710038171003817120572119899+1119891 (119909119899+1) + 120574119899+1119861119909119899+1 + 120575119899+1119878119899+1119909119899+1

minus (120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)1003817100381710038171003817

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

(120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817

+ 120574119899+1

1003817100381710038171003817119861119909119899+1 minus 1198611199091198991003817100381710038171003817

+ 120575119899+1

1003817100381710038171003817119878119899+1119909119899+1 minus 1198781198991199091198991003817100381710038171003817

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817)

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817 + 120574119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119878119899+1119909119899+1 minus 119878119899+1119909119899

1003817100381710038171003817

+1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1205881003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 + 120574119899+11003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817

=1 minus 120573119899+1

minus 120572119899+1

(1 minus 120588)

1 minus 120573119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+120575119899+1

1 minus 120573119899+1

1003817100381710038171003817119878119899+1119909119899 minus 1198781198991199091198991003817100381710038171003817

+1

1 minus 120573119899+1

[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817


le (1 minus120572119899+1

(1 minus 120588)

1 minus 120573119899+1

)1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816 +1003816100381610038161003816120573119899+1 minus 120573119899

1003816100381610038161003816

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816]

(49)


119891(119909119899) + 119861119909

119899 + 119878

119899119909119899 le 119872

0for


119910119899minus 119910119899minus1

= 120573119899(119909119899minus 119909119899minus1

) + (120573119899minus 120573119899minus1

)




)

(50)

So from (49) we get

1003817100381710038171003817119910119899 minus 119910119899minus11003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


+ (1 minus 120573119899) (1 minus

120572119899(1 minus 120588)

1 minus 120573119899

)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817


1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

(51)


119909119899+1

= 120598119899119899+ (1 minus 120598

119899) 119910119899

119899= 120590119899119866119909119899+ (1 minus 120590

119899) 119909119899

(52)


119909119899+1


) + (120598119899minus 120598119899minus1

)

times (119899minus1



)


= 120590119899(119866119909119899minus 119866119909119899minus1

) + (120590119899minus 120590119899minus1

)

times (119866119909119899minus1



)

(53)


1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817

le 120590119899

1003817100381710038171003817119866119909119899 minus 119866119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


(54)

and hence1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817 +

1003816100381610038161003816120598119899 minus 120598119899minus11003816100381610038161003816



1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


+ (1 minus 120598119899) (1 minus 120572

119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817


1003817100381710038171003817

+ 1198721[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816 +1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816]

(55)

where sup119899ge1

119866119909119899minus1


+ 119899minus1


+ 119909119899minus1


+

1198720 le 119872

1for some 119872



119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 (56)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)


120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)


lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin


1003817100381710038171003817) = 0

(59)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin


(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817


1003817100381710038171003817


1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)


119909119899minus119861119909119899 = 0 and lim




119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













sup119878119899119909minus119878119899minus1

119909




lim119899rarrinfin

sup119878119909 minus 119878119899119909 119909 isin 119862 = 0






119909119905= 119905119891 (119909

119905) + (1 minus 119905) 119879119909

119905 (31)


119862 Then the net 119909

119905

defined by 119909119905= 119905119891(119909

119905) + (1 minus 119905)119879119909



119876(119891) = 119904minus lim119905rarr0

119909119905 for all 119891 isin Ξ


as follows

⟨(119868 minus 119891)119876 (119891) 119869 (119876 (119891) minus 119901)⟩ le 0


(32)


119899infin

119899=0






119878119909 = suminfin



119899=0Fix(119879119899) holds


1003817100381710038171003817120582119909 + (1 minus 120582) 1199101003817100381710038171003817

2

le 1205821199092

+ (1 minus 120582)10038171003817100381710038171199101003817100381710038171003817

2


1003817100381710038171003817)

(33)





1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 2120582 (1205821205812

minus 120572)1003817100381710038171003817119860119909 minus 119860119910

1003817100381710038171003817

2

forall119909 119910 isin 119862

(34)


nonexpansive


119862be a sunny





(C6) 119873(119879119909 119865119909) + 119860(119892(119909)) 119883 rarr 2119862


2

Let 119860119894 119862 rarr 119883 be an 120572



119862 rarr 119862 by 119866119894= Π119862(119868 minus 120582

119894119860119894) for 119894 = 1 119873 where

120582119894isin (0 120572

1198941205812


1 119866

119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899


119899+ 120573119899+ 120574119899+ 120575119899= 1



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(35)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(36)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1





⋂infin

119894=0Fix(119878119894) Then 119909







1198941205812


119862 it



119873and 120588

1 120588


Fix(119861) = cap119873

119894=1Fix(119866


of 119866119894 we get Fix(119866

119894) = VI(119862 119860

119894) for each 119894 = 1 119873 Thus

we have

Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (38)










1= 120601(119903)8(1 + 120576) 120576

2= 120601(119903)8120583(1 + 120576) and 120576

3=



2and 119909 minus 119910 lt 120575

3imply

119867(119860 ∘ 119892(119909) 119860 ∘ 119892(119910)) lt 1205761 119867(119879119909 119879119910) lt 120576

2and119867(119865119909 119865119910) lt




119899isin 119861

We show that 119909119899+1


notin 119861then 119909


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817120572119899 (119891 (119909119899) minus 119901) + 120573119899 (119909119899 minus 119901)

+120574119899(119861119909119899minus 119901) + 120575

119899(119878119899119909119899minus 119901)

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120572119899(1 minus 120588)

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(39)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

= ⟨120598119899[119909119899minus 119901 minus 120590

119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

= ⟨120598119899(119909119899minus 119901) + (1 minus 120598

119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1 minus 1205881003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


le1

2(max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

+1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

)

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

(40)


1003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899+1 119896119899+1

) + 119906119899+1

119869 (119909119899+1

minus 119901)⟩

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899

minus (119873 (119908119899+1

119896119899+1

) + 119906119899+1

) 119869 (119909119899+1

minus 119901)⟩

(41)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899+1

119896119899)1003817100381710038171003817

+1003817100381710038171003817119873 (119908

119899+1 119896119899) minus 119873 (119908

119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

(42)


1003817100381710038171003817

le 120598119899

1003817100381710038171003817119909119899 minus 119901 minus 120590119899 (119873 (119908119899 119896119899) + 119906119899)1003817100381710038171003817

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205901198991003817100381710038171003817119873 (119908

119899 119896119899) + 119906119899

1003817100381710038171003817]

+ (1 minus 120598119899)max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le 120598119899[119903 + 120590

119899119872] + (1 minus 120598

119899) 119903

le 2119903

(43)


1015840

) and 119873(119908 119896) = 119873(1199081015840


119899+1isin 119865119909119899+1

and119908119899+1

isin 119879119909119899+1


119899isin 119879119909119899such that

10038171003817100381710038171003817119896119899+1

minus 1198961015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119865119909

119899+1 119865119909119899) (44)

10038171003817100381710038171003817119908119899+1

minus 1199081015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119879119909

119899+1 119879119909119899) (45)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[10038171003817100381710038171003817119873 (119908119899+1

119896119899+1

) minus 119873 (119908119899+1

1198961015840

119899)10038171003817100381710038171003817

+10038171003817100381710038171003817119873 (119908119899+1

119896119899) minus 119873 (119908

1015840

119899 119896119899)10038171003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817 ] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120585 (1 + 120576)119867 (119865119909

119899+1 119865119909119899)

+ 120583 (1 + 120576)119867 (119879119909119899+1

119879119909119899)

+ (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899)))] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120601 (119903)

8+120601 (119903)

8+120601 (119903)

8] 2119903

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903 + 120572

119899120590119899

3

2120601 (119903) 119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

(46)


So we get 119909119899+1



119899rarrinfin119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899]

+ (1 minus 120598119899) 119910119899 forall119899 ge 0

(47)


119910119899by

119910119899= 120573119899119909119899+ (1 minus 120573

119899) 119911119899 (48)

where 119911119899= (120572119899119891(119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)(1 minus 120573

119899) Now we

have1003817100381710038171003817119911119899+1 minus 119911119899

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

120572119899+1

119891 (119909119899+1

) + 120574119899+1

119861119909119899+1

+ 120575119899+1

119878119899+1

119909119899+1

1 minus 120573119899+1

minus120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899

1 minus 120573119899

100381710038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

+119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=1

1 minus 120573119899+1


+

10038161003816100381610038161003816100381610038161003816

1

1 minus 120573119899+1

minus1

1 minus 120573119899

10038161003816100381610038161003816100381610038161003816

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1


+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1

1003817100381710038171003817120572119899+1119891 (119909119899+1) + 120574119899+1119861119909119899+1 + 120575119899+1119878119899+1119909119899+1

minus (120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)1003817100381710038171003817

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

(120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817

+ 120574119899+1

1003817100381710038171003817119861119909119899+1 minus 1198611199091198991003817100381710038171003817

+ 120575119899+1

1003817100381710038171003817119878119899+1119909119899+1 minus 1198781198991199091198991003817100381710038171003817

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817)

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817 + 120574119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119878119899+1119909119899+1 minus 119878119899+1119909119899

1003817100381710038171003817

+1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1205881003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 + 120574119899+11003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817

=1 minus 120573119899+1

minus 120572119899+1

(1 minus 120588)

1 minus 120573119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+120575119899+1

1 minus 120573119899+1

1003817100381710038171003817119878119899+1119909119899 minus 1198781198991199091198991003817100381710038171003817

+1

1 minus 120573119899+1

[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817


le (1 minus120572119899+1

(1 minus 120588)

1 minus 120573119899+1

)1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816 +1003816100381610038161003816120573119899+1 minus 120573119899

1003816100381610038161003816

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816]

(49)


119891(119909119899) + 119861119909

119899 + 119878

119899119909119899 le 119872

0for


119910119899minus 119910119899minus1

= 120573119899(119909119899minus 119909119899minus1

) + (120573119899minus 120573119899minus1

)




)

(50)

So from (49) we get

1003817100381710038171003817119910119899 minus 119910119899minus11003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


+ (1 minus 120573119899) (1 minus

120572119899(1 minus 120588)

1 minus 120573119899

)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817


1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

(51)


119909119899+1

= 120598119899119899+ (1 minus 120598

119899) 119910119899

119899= 120590119899119866119909119899+ (1 minus 120590

119899) 119909119899

(52)


119909119899+1


) + (120598119899minus 120598119899minus1

)

times (119899minus1



)


= 120590119899(119866119909119899minus 119866119909119899minus1

) + (120590119899minus 120590119899minus1

)

times (119866119909119899minus1



)

(53)


1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817

le 120590119899

1003817100381710038171003817119866119909119899 minus 119866119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


(54)

and hence1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817 +

1003816100381610038161003816120598119899 minus 120598119899minus11003816100381610038161003816



1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


+ (1 minus 120598119899) (1 minus 120572

119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817


1003817100381710038171003817

+ 1198721[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816 +1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816]

(55)

where sup119899ge1

119866119909119899minus1


+ 119899minus1


+ 119909119899minus1


+

1198720 le 119872

1for some 119872



119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 (56)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)


120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)


lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin


1003817100381710038171003817) = 0

(59)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin


(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817


1003817100381710038171003817


1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)


119909119899minus119861119909119899 = 0 and lim




119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












⋂infin

119894=0Fix(119878119894) Then 119909







1198941205812


119862 it



119873and 120588

1 120588


Fix(119861) = cap119873

119894=1Fix(119866


of 119866119894 we get Fix(119866

119894) = VI(119862 119860

119894) for each 119894 = 1 119873 Thus

we have

Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (38)










1= 120601(119903)8(1 + 120576) 120576

2= 120601(119903)8120583(1 + 120576) and 120576

3=



2and 119909 minus 119910 lt 120575

3imply

119867(119860 ∘ 119892(119909) 119860 ∘ 119892(119910)) lt 1205761 119867(119879119909 119879119910) lt 120576

2and119867(119865119909 119865119910) lt




119899isin 119861

We show that 119909119899+1


notin 119861then 119909


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817120572119899 (119891 (119909119899) minus 119901) + 120573119899 (119909119899 minus 119901)

+120574119899(119861119909119899minus 119901) + 120575

119899(119878119899119909119899minus 119901)

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120572119899(1 minus 120588)

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(39)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

= ⟨120598119899[119909119899minus 119901 minus 120590

119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

= ⟨120598119899(119909119899minus 119901) + (1 minus 120598

119899) (119910119899minus 119901) 119869 (119909

119899+1minus 119901)⟩

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120598119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

le (120598119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120598119899)

max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


1003817100381710038171003817

1 minus 1205881003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩


le1

2(max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

+1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

)

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

(40)


1003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899+1 119896119899+1

) + 119906119899+1

119869 (119909119899+1

minus 119901)⟩

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899

minus (119873 (119908119899+1

119896119899+1

) + 119906119899+1

) 119869 (119909119899+1

minus 119901)⟩

(41)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899+1

119896119899)1003817100381710038171003817

+1003817100381710038171003817119873 (119908

119899+1 119896119899) minus 119873 (119908

119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

(42)


1003817100381710038171003817

le 120598119899

1003817100381710038171003817119909119899 minus 119901 minus 120590119899 (119873 (119908119899 119896119899) + 119906119899)1003817100381710038171003817

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205901198991003817100381710038171003817119873 (119908

119899 119896119899) + 119906119899

1003817100381710038171003817]

+ (1 minus 120598119899)max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le 120598119899[119903 + 120590

119899119872] + (1 minus 120598

119899) 119903

le 2119903

(43)


1015840

) and 119873(119908 119896) = 119873(1199081015840


119899+1isin 119865119909119899+1

and119908119899+1

isin 119879119909119899+1


119899isin 119879119909119899such that

10038171003817100381710038171003817119896119899+1

minus 1198961015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119865119909

119899+1 119865119909119899) (44)

10038171003817100381710038171003817119908119899+1

minus 1199081015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119879119909

119899+1 119879119909119899) (45)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[10038171003817100381710038171003817119873 (119908119899+1

119896119899+1

) minus 119873 (119908119899+1

1198961015840

119899)10038171003817100381710038171003817

+10038171003817100381710038171003817119873 (119908119899+1

119896119899) minus 119873 (119908

1015840

119899 119896119899)10038171003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817 ] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120585 (1 + 120576)119867 (119865119909

119899+1 119865119909119899)

+ 120583 (1 + 120576)119867 (119879119909119899+1

119879119909119899)

+ (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899)))] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120601 (119903)

8+120601 (119903)

8+120601 (119903)

8] 2119903

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903 + 120572

119899120590119899

3

2120601 (119903) 119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

(46)


So we get 119909119899+1



119899rarrinfin119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899]

+ (1 minus 120598119899) 119910119899 forall119899 ge 0

(47)


119910119899by

119910119899= 120573119899119909119899+ (1 minus 120573

119899) 119911119899 (48)

where 119911119899= (120572119899119891(119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)(1 minus 120573

119899) Now we

have1003817100381710038171003817119911119899+1 minus 119911119899

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

120572119899+1

119891 (119909119899+1

) + 120574119899+1

119861119909119899+1

+ 120575119899+1

119878119899+1

119909119899+1

1 minus 120573119899+1

minus120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899

1 minus 120573119899

100381710038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

+119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=1

1 minus 120573119899+1


+

10038161003816100381610038161003816100381610038161003816

1

1 minus 120573119899+1

minus1

1 minus 120573119899

10038161003816100381610038161003816100381610038161003816

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1


+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1

1003817100381710038171003817120572119899+1119891 (119909119899+1) + 120574119899+1119861119909119899+1 + 120575119899+1119878119899+1119909119899+1

minus (120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)1003817100381710038171003817

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

(120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817

+ 120574119899+1

1003817100381710038171003817119861119909119899+1 minus 1198611199091198991003817100381710038171003817

+ 120575119899+1

1003817100381710038171003817119878119899+1119909119899+1 minus 1198781198991199091198991003817100381710038171003817

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817)

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817 + 120574119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119878119899+1119909119899+1 minus 119878119899+1119909119899

1003817100381710038171003817

+1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1205881003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 + 120574119899+11003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817

=1 minus 120573119899+1

minus 120572119899+1

(1 minus 120588)

1 minus 120573119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+120575119899+1

1 minus 120573119899+1

1003817100381710038171003817119878119899+1119909119899 minus 1198781198991199091198991003817100381710038171003817

+1

1 minus 120573119899+1

[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817


le (1 minus120572119899+1

(1 minus 120588)

1 minus 120573119899+1

)1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816 +1003816100381610038161003816120573119899+1 minus 120573119899

1003816100381610038161003816

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816]

(49)


119891(119909119899) + 119861119909

119899 + 119878

119899119909119899 le 119872

0for


119910119899minus 119910119899minus1

= 120573119899(119909119899minus 119909119899minus1

) + (120573119899minus 120573119899minus1

)




)

(50)

So from (49) we get

1003817100381710038171003817119910119899 minus 119910119899minus11003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


+ (1 minus 120573119899) (1 minus

120572119899(1 minus 120588)

1 minus 120573119899

)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817


1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

(51)


119909119899+1

= 120598119899119899+ (1 minus 120598

119899) 119910119899

119899= 120590119899119866119909119899+ (1 minus 120590

119899) 119909119899

(52)


119909119899+1


) + (120598119899minus 120598119899minus1

)

times (119899minus1



)


= 120590119899(119866119909119899minus 119866119909119899minus1

) + (120590119899minus 120590119899minus1

)

times (119866119909119899minus1



)

(53)


1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817

le 120590119899

1003817100381710038171003817119866119909119899 minus 119866119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


(54)

and hence1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817 +

1003816100381610038161003816120598119899 minus 120598119899minus11003816100381610038161003816



1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


+ (1 minus 120598119899) (1 minus 120572

119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817


1003817100381710038171003817

+ 1198721[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816 +1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816]

(55)

where sup119899ge1

119866119909119899minus1


+ 119899minus1


+ 119909119899minus1


+

1198720 le 119872

1for some 119872



119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 (56)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)


120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)


lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin


1003817100381710038171003817) = 0

(59)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin


(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817


1003817100381710038171003817


1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)


119909119899minus119861119909119899 = 0 and lim




119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le1

2(max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

+1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

)

minus 120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

(40)


1003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899 119869 (119909119899+1

minus 119901)⟩

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899⟨119873 (119908

119899+1 119896119899+1

) + 119906119899+1

119869 (119909119899+1

minus 119901)⟩

minus 2120572119899120590119899⟨119873 (119908

119899 119896119899) + 119906119899

minus (119873 (119908119899+1

119896119899+1

) + 119906119899+1

) 119869 (119909119899+1

minus 119901)⟩

(41)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817)1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

+ 2120572119899120590119899[1003817100381710038171003817119873 (119908

119899+1 119896119899+1

) minus 119873 (119908119899+1

119896119899)1003817100381710038171003817

+1003817100381710038171003817119873 (119908

119899+1 119896119899) minus 119873 (119908

119899 119896119899)1003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817]1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

(42)


1003817100381710038171003817

le 120598119899

1003817100381710038171003817119909119899 minus 119901 minus 120590119899 (119873 (119908119899 119896119899) + 119906119899)1003817100381710038171003817

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205901198991003817100381710038171003817119873 (119908

119899 119896119899) + 119906119899

1003817100381710038171003817]

+ (1 minus 120598119899)max1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

le 120598119899[119903 + 120590

119899119872] + (1 minus 120598

119899) 119903

le 2119903

(43)


1015840

) and 119873(119908 119896) = 119873(1199081015840


119899+1isin 119865119909119899+1

and119908119899+1

isin 119879119909119899+1


119899isin 119879119909119899such that

10038171003817100381710038171003817119896119899+1

minus 1198961015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119865119909

119899+1 119865119909119899) (44)

10038171003817100381710038171003817119908119899+1

minus 1199081015840

119899

10038171003817100381710038171003817le (1 + 120576)119867 (119879119909

119899+1 119879119909119899) (45)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[10038171003817100381710038171003817119873 (119908119899+1

119896119899+1

) minus 119873 (119908119899+1

1198961015840

119899)10038171003817100381710038171003817

+10038171003817100381710038171003817119873 (119908119899+1

119896119899) minus 119873 (119908

1015840

119899 119896119899)10038171003817100381710038171003817

+1003817100381710038171003817119906119899+1 minus 119906119899

1003817100381710038171003817 ] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120585 (1 + 120576)119867 (119865119909

119899+1 119865119909119899)

+ 120583 (1 + 120576)119867 (119879119909119899+1

119879119909119899)

+ (1 + 120576)119867 (119860 (119892 (119909119899+1

)) 119860 (119892 (119909119899)))] 2119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903

+ 2120572119899120590119899[120601 (119903)

8+120601 (119903)

8+120601 (119903)

8] 2119903

= max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

minus 2120572119899120590119899120601 (119903) 119903 + 120572

119899120590119899

3

2120601 (119903) 119903

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

100381710038171003817100381710038171003817100381710038171003817

119891 (119901) minus 119901

1 minus 120588

100381710038171003817100381710038171003817100381710038171003817

2

(46)


So we get 119909119899+1



119899rarrinfin119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899]

+ (1 minus 120598119899) 119910119899 forall119899 ge 0

(47)


119910119899by

119910119899= 120573119899119909119899+ (1 minus 120573

119899) 119911119899 (48)

where 119911119899= (120572119899119891(119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)(1 minus 120573

119899) Now we

have1003817100381710038171003817119911119899+1 minus 119911119899

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

120572119899+1

119891 (119909119899+1

) + 120574119899+1

119861119909119899+1

+ 120575119899+1

119878119899+1

119909119899+1

1 minus 120573119899+1

minus120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899

1 minus 120573119899

100381710038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

+119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=1

1 minus 120573119899+1


+

10038161003816100381610038161003816100381610038161003816

1

1 minus 120573119899+1

minus1

1 minus 120573119899

10038161003816100381610038161003816100381610038161003816

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1


+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1

1003817100381710038171003817120572119899+1119891 (119909119899+1) + 120574119899+1119861119909119899+1 + 120575119899+1119878119899+1119909119899+1

minus (120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)1003817100381710038171003817

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

(120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817

+ 120574119899+1

1003817100381710038171003817119861119909119899+1 minus 1198611199091198991003817100381710038171003817

+ 120575119899+1

1003817100381710038171003817119878119899+1119909119899+1 minus 1198781198991199091198991003817100381710038171003817

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817)

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817 + 120574119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119878119899+1119909119899+1 minus 119878119899+1119909119899

1003817100381710038171003817

+1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1205881003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 + 120574119899+11003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817

=1 minus 120573119899+1

minus 120572119899+1

(1 minus 120588)

1 minus 120573119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+120575119899+1

1 minus 120573119899+1

1003817100381710038171003817119878119899+1119909119899 minus 1198781198991199091198991003817100381710038171003817

+1

1 minus 120573119899+1

[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817


le (1 minus120572119899+1

(1 minus 120588)

1 minus 120573119899+1

)1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816 +1003816100381610038161003816120573119899+1 minus 120573119899

1003816100381610038161003816

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816]

(49)


119891(119909119899) + 119861119909

119899 + 119878

119899119909119899 le 119872

0for


119910119899minus 119910119899minus1

= 120573119899(119909119899minus 119909119899minus1

) + (120573119899minus 120573119899minus1

)




)

(50)

So from (49) we get

1003817100381710038171003817119910119899 minus 119910119899minus11003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


+ (1 minus 120573119899) (1 minus

120572119899(1 minus 120588)

1 minus 120573119899

)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817


1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

(51)


119909119899+1

= 120598119899119899+ (1 minus 120598

119899) 119910119899

119899= 120590119899119866119909119899+ (1 minus 120590

119899) 119909119899

(52)


119909119899+1


) + (120598119899minus 120598119899minus1

)

times (119899minus1



)


= 120590119899(119866119909119899minus 119866119909119899minus1

) + (120590119899minus 120590119899minus1

)

times (119866119909119899minus1



)

(53)


1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817

le 120590119899

1003817100381710038171003817119866119909119899 minus 119866119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


(54)

and hence1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817 +

1003816100381610038161003816120598119899 minus 120598119899minus11003816100381610038161003816



1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


+ (1 minus 120598119899) (1 minus 120572

119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817


1003817100381710038171003817

+ 1198721[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816 +1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816]

(55)

where sup119899ge1

119866119909119899minus1


+ 119899minus1


+ 119909119899minus1


+

1198720 le 119872

1for some 119872



119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 (56)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)


120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)


lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin


1003817100381710038171003817) = 0

(59)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin


(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817


1003817100381710038171003817


1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)


119909119899minus119861119909119899 = 0 and lim




119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










So we get 119909119899+1



119899rarrinfin119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899]

+ (1 minus 120598119899) 119910119899 forall119899 ge 0

(47)


119910119899by

119910119899= 120573119899119909119899+ (1 minus 120573

119899) 119911119899 (48)

where 119911119899= (120572119899119891(119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)(1 minus 120573

119899) Now we

have1003817100381710038171003817119911119899+1 minus 119911119899

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

120572119899+1

119891 (119909119899+1

) + 120574119899+1

119861119909119899+1

+ 120575119899+1

119878119899+1

119909119899+1

1 minus 120573119899+1

minus120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899

1 minus 120573119899

100381710038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

+119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119910119899+1

minus 120573119899+1

119909119899+1

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899+1

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

119910119899minus 120573119899119909119899

1 minus 120573119899+1

minus119910119899minus 120573119899119909119899

1 minus 120573119899

10038171003817100381710038171003817100381710038171003817

=1

1 minus 120573119899+1


+

10038161003816100381610038161003816100381610038161003816

1

1 minus 120573119899+1

minus1

1 minus 120573119899

10038161003816100381610038161003816100381610038161003816

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1


+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

=1

1 minus 120573119899+1

1003817100381710038171003817120572119899+1119891 (119909119899+1) + 120574119899+1119861119909119899+1 + 120575119899+1119878119899+1119909119899+1

minus (120572119899119891 (119909119899) + 120574119899119861119909119899+ 120575119899119878119899119909119899)1003817100381710038171003817

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

(120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817

+ 120574119899+1

1003817100381710038171003817119861119909119899+1 minus 1198611199091198991003817100381710038171003817

+ 120575119899+1

1003817100381710038171003817119878119899+1119909119899+1 minus 1198781198991199091198991003817100381710038171003817

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817)

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1003817100381710038171003817119891 (119909119899+1) minus 119891 (119909119899)1003817100381710038171003817 + 120574119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119878119899+1119909119899+1 minus 119878119899+1119909119899

1003817100381710038171003817

+1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817119910119899 minus 1205731198991199091198991003817100381710038171003817

le1

1 minus 120573119899+1

[120572119899+1

1205881003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 + 120574119899+11003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

+ 120575119899+1

(1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817)

+1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

10038171003817100381710038171198611199091198991003817100381710038171003817 +

1003816100381610038161003816120575119899+1 minus 1205751198991003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817

=1 minus 120573119899+1

minus 120572119899+1

(1 minus 120588)

1 minus 120573119899+1

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+120575119899+1

1 minus 120573119899+1

1003817100381710038171003817119878119899+1119909119899 minus 1198781198991199091198991003817100381710038171003817

+1

1 minus 120573119899+1

[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899)1003817100381710038171003817 +

1003816100381610038161003816120574119899+1 minus 1205741198991003816100381610038161003816

times1003817100381710038171003817119861119909119899

1003817100381710038171003817 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816

10038171003817100381710038171198781198991199091198991003817100381710038171003817]

+

1003816100381610038161003816120573119899+1 minus 1205731198991003816100381610038161003816

(1 minus 120573119899) (1 minus 120573

119899+1)

1003817100381710038171003817120572119899119891 (119909119899) + 120574119899119861119909119899 + 1205751198991198781198991199091198991003817100381710038171003817


le (1 minus120572119899+1

(1 minus 120588)

1 minus 120573119899+1

)1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816 +1003816100381610038161003816120573119899+1 minus 120573119899

1003816100381610038161003816

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816]

(49)


119891(119909119899) + 119861119909

119899 + 119878

119899119909119899 le 119872

0for


119910119899minus 119910119899minus1

= 120573119899(119909119899minus 119909119899minus1

) + (120573119899minus 120573119899minus1

)




)

(50)

So from (49) we get

1003817100381710038171003817119910119899 minus 119910119899minus11003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


+ (1 minus 120573119899) (1 minus

120572119899(1 minus 120588)

1 minus 120573119899

)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817


1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

(51)


119909119899+1

= 120598119899119899+ (1 minus 120598

119899) 119910119899

119899= 120590119899119866119909119899+ (1 minus 120590

119899) 119909119899

(52)


119909119899+1


) + (120598119899minus 120598119899minus1

)

times (119899minus1



)


= 120590119899(119866119909119899minus 119866119909119899minus1

) + (120590119899minus 120590119899minus1

)

times (119866119909119899minus1



)

(53)


1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817

le 120590119899

1003817100381710038171003817119866119909119899 minus 119866119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


(54)

and hence1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817 +

1003816100381610038161003816120598119899 minus 120598119899minus11003816100381610038161003816



1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


+ (1 minus 120598119899) (1 minus 120572

119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817


1003817100381710038171003817

+ 1198721[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816 +1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816]

(55)

where sup119899ge1

119866119909119899minus1


+ 119899minus1


+ 119909119899minus1


+

1198720 le 119872

1for some 119872



119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 (56)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)


120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)


lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin


1003817100381710038171003817) = 0

(59)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin


(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817


1003817100381710038171003817


1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)


119909119899minus119861119909119899 = 0 and lim




119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le (1 minus120572119899+1

(1 minus 120588)

1 minus 120573119899+1

)1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899+1119909119899 minus 119878119899119909119899

1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899+1 minus 120572119899

1003816100381610038161003816 +1003816100381610038161003816120573119899+1 minus 120573119899

1003816100381610038161003816

+1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816 +1003816100381610038161003816120575119899+1 minus 120575119899

1003816100381610038161003816]

(49)


119891(119909119899) + 119861119909

119899 + 119878

119899119909119899 le 119872

0for


119910119899minus 119910119899minus1

= 120573119899(119909119899minus 119909119899minus1

) + (120573119899minus 120573119899minus1

)




)

(50)

So from (49) we get

1003817100381710038171003817119910119899 minus 119910119899minus11003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816


+ (1 minus 120573119899) (1 minus

120572119899(1 minus 120588)

1 minus 120573119899

)1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817


1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

le (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]

(51)


119909119899+1

= 120598119899119899+ (1 minus 120598

119899) 119910119899

119899= 120590119899119866119909119899+ (1 minus 120590

119899) 119909119899

(52)


119909119899+1


) + (120598119899minus 120598119899minus1

)

times (119899minus1



)


= 120590119899(119866119909119899minus 119866119909119899minus1

) + (120590119899minus 120590119899minus1

)

times (119866119909119899minus1



)

(53)


1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817

le 120590119899

1003817100381710038171003817119866119909119899 minus 119866119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817 +

1003816100381610038161003816120590119899 minus 120590119899minus11003816100381610038161003816



1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


(54)

and hence1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le 120598119899

1003817100381710038171003817119899 minus 119899minus11003817100381710038171003817 +

1003816100381610038161003816120598119899 minus 120598119899minus11003816100381610038161003816



1003817100381710038171003817

le 120598119899[1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


+ (1 minus 120598119899) (1 minus 120572

119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816


+1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816


1003816100381610038161003816120573119899 minus 120573119899minus11003816100381610038161003816



1003817100381710038171003817

+ 1198720[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816]


1003817100381710038171003817119909119899 minus 119909119899minus11003817100381710038171003817


1003817100381710038171003817

+ 1198721[1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573119899minus1

1003816100381610038161003816

+1003816100381610038161003816120574119899 minus 120574119899minus1

1003816100381610038161003816 +1003816100381610038161003816120575119899 minus 120575119899minus1

1003816100381610038161003816

+1003816100381610038161003816120590119899 minus 120590119899minus1

1003816100381610038161003816 +1003816100381610038161003816120598119899 minus 120598119899minus1

1003816100381610038161003816]

(55)

where sup119899ge1

119866119909119899minus1


+ 119899minus1


+ 119909119899minus1


+

1198720 le 119872

1for some 119872



119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 (56)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)


120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)


lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin


1003817100381710038171003817) = 0

(59)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin


(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817


1003817100381710038171003817


1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)


119909119899minus119861119909119899 = 0 and lim




119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119901)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119866119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

minus 120598119899(1 minus 120598

119899)


1003817100381710038171003817)

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) ]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817]2

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

= 120598119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)]

+ (1 minus 120598119899) [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) ]

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

minus 120598119899(1 minus 120598

119899) 120593 (

1003817100381710038171003817(1 minus 120590119899) (119909119899 minus 119910119899)

+120590119899(119866119909119899minus 119910119899)1003817100381710038171003817)

(57)


120598119899120590119899(1 minus 120590

119899) 1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) + 120598119899 (1 minus 120598119899)


1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817)

(58)


lim119899rarrinfin

1205931(1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817) = 0

lim119899rarrinfin


1003817100381710038171003817) = 0

(59)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0

lim119899rarrinfin


(60)

Note that

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817


1003817100381710038171003817


1003817100381710038171003817 + 1205901198991003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817


1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817

(61)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (62)


119909119899minus119861119909119899 = 0 and lim




119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ (120574119899+ 120575119899)120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

= 120572119899119891 (119909119899) + 120573119899119909119899+ 1198901198991199111015840

119899

(63)

where 119890119899= 120574119899+ 120575119899and 1199111015840119899= (120574119899119861119909119899+ 120575119899119878119899119909119899)(120574119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120572119899(119891 (119909119899) minus 119901) + 120573

119899(119909119899minus 119901) + 119890

119899(1199111015840

119899minus 119901)

10038171003817100381710038171003817

2


le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 119890119899

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

+ 119890119899

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120574119899+ 120575119899

) (119861119909119899minus 119901) +

120575119899

120574119899+ 120575119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899((1 minus

120575119899

120574119899+ 120575119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

+120575119899

120574119899+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817)

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) + 119890119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus 1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

(64)

which implies that

1205731198991198901198991205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817)

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(65)


lim119899rarrinfin

1205932(100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817) = 0 (66)


lim119899rarrinfin

100381710038171003817100381710038171199111015840

119899minus 119909119899

10038171003817100381710038171003817= 0 (67)


119899the

following

100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120574119899119861119909119899+ 120575119899119878119899119909119899

120574119899+ 120575119899

minus 119901

10038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

(1 minus120575119899

120575119899+ 120574119899

) (119861119909119899minus 119901) +

120575119899

120575119899+ 120574119899

(119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

le (1 minus120575119899

120575119899+ 120574119899

)1003817100381710038171003817119861119909119899 minus 119901

1003817100381710038171003817

2

+120575119899

120575119899+ 120574119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817B119909119899 minus 119878119899119909119899

1003817100381710038171003817)

(68)

which implies that

120575119899

120575119899+ 120574119899

(1 minus120575119899

120575119899+ 120574119899

)1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +100381710038171003817100381710038171199111015840

119899minus 119901

10038171003817100381710038171003817)10038171003817100381710038171003817119909119899minus 1199111015840

119899

10038171003817100381710038171003817

(69)


lim119899rarrinfin

1205933(1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817) = 0 (70)


lim119899rarrinfin

1003817100381710038171003817119861119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (71)


119899by

119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

= 120573119899119909119899+ 120574119899119861119909119899+ (120572119899+ 120575119899)120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

= 120573119899119909119899+ 120574119899119861119909119899+ 11988911989911991110158401015840

119899

(72)

where 119889119899= 120572119899+ 120575119899and 11991110158401015840119899= (120572119899119891(119909119899) + 120575119899119878119899119909119899)(120572119899+ 120575119899)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817120573119899(119909119899minus 119901) + 120574

119899(119861119909119899minus 119901) + 119889

119899(11991110158401015840

119899minus 119901)

10038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2


+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 119889119899

1003817100381710038171003817100381711991110158401015840

119899minus 119901

10038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

100381710038171003817100381710038171003817100381710038171003817

120572119899119891 (119909119899) + 120575119899119878119899119909119899

120572119899+ 120575119899

minus 119901

100381710038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899

10038171003817100381710038171003817100381710038171003817

120572119899

120572119899+ 120575119899

(119891 (119909119899) minus 119901)

+(1 minus120572119899

120572119899+ 120575119899

) (119878119899119909119899minus 119901)

10038171003817100381710038171003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 119889119899[

120572119899

120572119899+ 120575119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+(1 minus120572119899

120572119899+ 120575119899

)1003817100381710038171003817119878119899119909119899 minus 119901

1003817100381710038171003817

2

]

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

minus 1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

(73)

which implies that

1205731198991205741198991205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

+ 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 + 1205721198991003817100381710038171003817119891 (119909119899) minus 119901

1003817100381710038171003817

2

(74)


lim119899rarrinfin

1205934(1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817) = 0 (75)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0 (76)

From (71) (76) and1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119861119909119899

1003817100381710038171003817 +1003817100381710038171003817119861119909119899 minus 119878119899119909119899

1003817100381710038171003817 (77)

we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 = 0 (78)

Observe that

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 1198781198991199091198991003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 1198781199091198991003817100381710038171003817 (79)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0 (80)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1 Then


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


+1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(81)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (82)

Now we claim that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (83)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (84)


119905= 119905119891(119909

119905) + (1 minus

119905)119882119909119905 Thus we have

119909119905minus 119909119899= (1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899) (85)


1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

=1003817100381710038171003817(1 minus 119905) (119882119909

119905minus 119909119899) + 119905 (119891 (119909

119905) minus 119909119899)1003817100381710038171003817

2

le (1 minus 119905)21003817100381710038171003817119882119909119905minus 119909119899

1003817100381710038171003817

2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩


le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le (1 minus 119905)2

(1003817100381710038171003817119882119909119905minus119882119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

le (1 minus 119905)2

(1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817)2

+ 2119905 ⟨119891 (119909119905) minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 119905)2

[1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

times1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119882119909119899minus 119909119899

1003817100381710038171003817

2

]

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩

+ 2119905 ⟨119909119905minus 119909119899 119869 (119909119905minus 119909119899)⟩

= (1 minus 2119905 + 1199052

)1003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817

2

+ 119891119899(119905)

+ 2119905 ⟨119891 (119909119905) minus 119909119905 119869 (119909119905minus 119909119899)⟩ + 2119905

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

(86)

where

119891119899(119905) = (1 minus 119905)

2

(21003817100381710038171003817119909119905 minus 119909119899

1003817100381710038171003817 +1003817100381710038171003817119909119899 minus119882119909

119899

1003817100381710038171003817)

times1003817100381710038171003817119909119899 minus119882119909

119899


(87)


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

2

1003817100381710038171003817119909119905 minus 1199091198991003817100381710038171003817

2

+1

2119905119891119899(119905) (88)



⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le

119905

21198722 (89)


119905minus 1199091198992

le 1198722for all


lim sup119905rarr0


⟨119909119905minus 119891 (119909

119905) 119869 (119909

119905minus 119909119899)⟩ le 0 (90)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119902 119869 (119909119899minus 119909119905)⟩


119905 119869 (119909119899minus 119909119905)⟩


+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩


119899minus 119909119905)⟩

+ ⟨119909119905minus 119902 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119902) minus 119891 (119909119905) 119869 (119909

119899minus 119909119905)⟩

+ ⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(91)

It follows that


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



119899minus 119909119905)⟩

+1003817100381710038171003817119909119905 minus q1003817100381710038171003817 lim sup

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817

+ 1205881003817100381710038171003817119902 minus 119909119905


1003817100381710038171003817119909119899 minus 1199091199051003817100381710038171003817


⟨119891 (119909119905) minus 119909119905 119869 (119909119899minus 119909119905)⟩

(92)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(93)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(94)



1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817

2


+120575119899(119878119899119909119899minus 119902) + 120572

119899(119891 (119902) minus 119902)

1003817100381710038171003817

2

le1003817100381710038171003817120572119899 (119891 (119909119899) minus 119891 (119902)) + 120573119899 (119909119899 minus 119902)

+120574119899(119861119909119899minus 119902) + 120575

119899(119878119899119909119899minus 119902)

1003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119902)1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199021003817100381710038171003817

2

+ 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 1205721198991205881003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120573119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 120574119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

+ 120575119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩

(95)

and hence

1003817100381710038171003817119909119899+11199021003817100381710038171003817

2


+ (1 minus 120598119899) (119910119899minus 119902)

1003817100381710038171003817

2

le 120598119899


2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+120590119899

1003817100381710038171003817119866119909119899 minus 1199021003817100381710038171003817

2

] + (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899[(1 minus 120590

119899)1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2

+ 120590119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

]

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

= 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)1003817100381710038171003817119910119899 minus 119902

1003817100381710038171003817

2

le 120598119899

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899)


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+2120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩]


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ 2 (1 minus 120598119899) 120572119899⟨119891 (119902) minus 119902 119869 (119910

119899minus 119902)⟩


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(96)



the proof


119862be a sunny





119860(119892(119909)) 119883 rarr 2119862


Let 119879119894 119862 rarr 119883 be a 120578



119862 by 119866119894= Π119862(119868 minus 120582

119894(119868 minus 119879

119894)) for 119894 = 1 119873 where 120582

119894isin

(0 1205781198941205812


1 119866

119873and

1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(97)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(98)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894is 120578119894-inverse



119894isin (0 120578

1198941205812

)





119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119906 isin VI (119862 119860119894)



lArrrArr 119906 = Π119862(119906 minus 120582

119894119860119894119906)

lArrrArr 119906 = Π119862(119906 minus 120582

119894119906 + 120582119894119879119894119906)



lArrrArr 119906 = 119879119894119906


(100)


119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1VI(119862 119860

119894)) = ⋂

infin

119894=0Fix(119878119894) cap Γcap (cap

119873

119894=1Fix(119879119894)) Therefore




119873

119894=1VI(119862 119860




(cap119873

119894=1VI(119862 119860


119860= 119878((1minus120572)119868+120572119881)


) and Fix(119861119860) = Fix(119878) cap


119894=0Fix(119878119894) cap Γ cap (cap

119873

119894=1VI(119862 119860

119894)) in



119899) and


120590119899 and 120598

119899










le radic1 minus ]

120585+ (1 minus 120582) (1 +

1

120585)


(101)



119862be a sunny







0+ ]0ge 1

Let 119860119894 119862 rarr 119883 be 120585



119894+ ]119894ge 1 for each 119894 = 1 119873


119894= Π119862(119868 minus 120582

119894119860119894)

where 1 minus (120585119894(1 + 120585


119894)120585119894) le 120582

119894le 1 for


1 119866

119873and 120588

1 120588

119873 where 120588

119894isin (0 1)

for all 119894 = 1 119873 minus 1 and 120588119873

isin (0 1] Let 119891 119862 rarr 119862


119894=0be a


infin

119894=0Fix(119878119894)capΓcap(cap

119873

119894=1VI(119862 119860

119894)) = 0 Suppose

that 120572119899 120573119899 120574119899 120575119899 120590119899 and 120598




(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin


120572119899= 0 and suminfin

119899=0120572119899= infin




120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(102)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(103)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







119894))(1 minus

radic(1 minus ]119894)120585119894) le 120582





1 119866

119873and 120588

1 120588

119873 by


119894=1Fix(119866

119894) Utilizing


119894) = VI(119862 119860

119894)


Fix (119861) =119873

⋂

119894=1

Fix (119866119894) =

119873

⋂

119894=1

VI (119862 119860119894) (105)


0-strictly


0+ ]0ge 1


(119873(119879119909 119865119909) + 120582119860(119892(119909))) is nonexpansive



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

le 120572119899

1003817100381710038171003817119891 (119909119899) minus 1199011003817100381710038171003817 + 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

+ 120574119899

1003817100381710038171003817119861119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119878119899119909119899 minus 1199011003817100381710038171003817

le 120572119899(1205881003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119891 (119901) minus 119901

1003817100381710038171003817) + 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ 120574119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120575119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

= (1 minus 120572119899(1 minus 120588))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120572119899

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

le max1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

1003817100381710038171003817119891 (119901) minus 1199011003817100381710038171003817

1 minus 120588

(106)


119899isin 119861 for all 119899 ge 0



119909119899minus 119909119899+1


119909119899minus 119910119899 = 0



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[(1 minus 120590

119899) 119909119899+ 120590119899119866119909119899] + (1 minus 120598

119899) 119910119899

forall119899 ge 0

(107)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (108)

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1198611199091198991003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 = 0

(109)


1119878119909 + 120579

2119866119909


1+ 1205792lt 1


1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817


119899)

+ 1205791(119909119899minus 119878119909119899) + 1205792(119909119899minus 119866119909119899)1003817100381710038171003817


1003817100381710038171003817

+ 1205791

1003817100381710038171003817119909119899 minus 1198781199091198991003817100381710038171003817 + 1205792

1003817100381710038171003817119909119899 minus 1198661199091198991003817100381710038171003817

(110)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus119882119909119899

1003817100381710038171003817 = 0 (111)



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0 (112)


119909119905with 119909


contraction

119909 997891997888rarr 119905119891 (119909) + (1 minus 119905)119882119909 (113)


119905= 119905119891(119909

119905) + (1 minus




⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩



⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩




119899minus 119909119905)⟩

(114)



⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩


(⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩


119899minus 119902)⟩)


⟨119891 (119902) minus 119902 119869 (119909119899minus 119902)⟩ le 0

(115)



1003817100381710038171003817

2


1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2

+ (1 minus 120598119899) 120572119899(1 minus 120588)

2 ⟨119891 (119902) minus 119902 119869 (119910119899minus 119902)⟩

1 minus 120588

(116)



proof





0-


0+

]0ge 1






119866119894 119862 rarr 119862 by 119866

119894= (1 minus 120582

119894)119868 + 120582

119894119879119894where 1 minus (120585

119894(1 +


119894)120585119894) le 120582

119894le 1 for each 119894 = 1 119873 Let


119873

and 1205881 120588

119873 where 120588



120588 isin (0 1) Let 119878119894infin



119894=0Fix(119878119894) cap Γ cap

(cap119873

119894=1Fix(119879119894)) = 0 Suppose that 120572

119899 120573119899 120574119899 120575119899 120590119899 and


119899+120573119899+120574119899+120575119899= 1 and satisfy



(|120572119899minus120572119899minus1

|+ |120573119899minus120573119899minus1

|+ |120574119899minus120574119899minus1

|+ |120575119899minus120575119899minus1

|+

|120590119899minus 120590119899minus1

| + |120598119899minus 120598119899minus1

|) lt infin(ii) lim


119899=0120572119899= infin



120573119899le lim sup

119899rarrinfin120573119899lt 1


120590119899le lim sup

119899rarrinfin120590119899lt 1


120598119899le lim sup

119899rarrinfin120598119899lt 1



119910119899= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119861119909119899+ 120575119899119878119899119909119899

119909119899+1

= 120598119899[119909119899minus 120590119899(119873 (119908

119899 119896119899) + 119906119899)]

+ (1 minus 120598119899) 119910119899 119906119899isin 119860 (119892 (119909

119899)) forall119899 ge 0

(117)


1003817100381710038171003817119906119899 minus 119906119899+11003817100381710038171003817 le (1 + 120576)119867 (119860 (119892 (119909

119899+1)) 119860 (119892 (119909

119899)))

forall119899 ge 0

(118)

for any 119908119899isin 119879119909

119899 119896119899isin 119865119909



sup119909isin119863

119878119899+1




⋂infin

119894=0Fix(119878119894) Then 119909







Acknowledgments




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









mann-type viscosity approximation methods for multivalued variational inclusions with finitely many...

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