Research ArticleViscosity Projection Algorithms for Pseudocontractive Mappingsin Hilbert Spaces

Xiujuan Pan12 Shin Min Kang3 and Young Chel Kwun4

1 School of Science Tianjin Polytechnic University Tianjin 300387 China2 College of Management and Economics Tianjin University Tianjin 300072 China3Department of Mathematics and the RINS Gyeongsang National University Jinju 660-701 Republic of Korea4Department of Mathematics Dong-A University Busan 614-714 Republic of Korea

Correspondence should be addressed to Young Chel Kwun yckwundauackr

Received 6 December 2013 Accepted 28 December 2013 Published 9 February 2014

Academic Editor Chong Li

Copyright copy 2014 Xiujuan Pan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractivemapping in Hilbert spaces Strong convergence theorem is demonstrated Consequently as an application we can approximateto the minimum-norm fixed point of the pseudocontractive mapping

1 Introduction

Let H be a real Hilbert space with inner product ⟨sdot sdot⟩ andnorm sdot respectively Let C be a nonempty closed convexsubset of H

Recall that a mapping T C rarr C is said to be

(i) 119871-Lipschitz hArr there exists a constant 119871 gt 0 such thatT119906minusTV le 119871119906minusV for all 119906 V isin C if 119871 isin (0 1) thenT is said to be contractive if 119871 = 1 then T is said tobe nonexpansive

(ii) pseudocontractive

hArr ⟨T119906 minus TV 119906 minus V⟩ le 119906 minus V2hArr T119906 minus TV2 le 119906 minus V2+(119868 minus T)119906 minus (119868 minus T)V)2hArr ⟨119906 minus V (119868 minus T)119906 minus (119868 minus T)V⟩ ge 0

for all 119906 V isin C

Interest in pseudocontractive mappings stems mainlyfrom their firm connection with the class of nonlinearaccretive operators It is a classical result see Deimling [1]that if T is an accretive operator then the solutions of theequations T119909 = 0 correspond to the equilibrium pointsof some evolution systems This explains the importancefrom this point of view of the improvement brought by the

Ishikawa iteration which was introduced by Ishikawa [2] in1974The original result of Ishikawa is stated in the following

Theorem 1 Let C be a convex compact subset of a Hilbertspace H and let T C rarr C be an 119871-Lipschitzianpseudocontractive mapping with Fix(T) = 0 For any 119909

0isin C

define the sequence 119909119899 iteratively by

119910119899= (1 minus 120578

119899) 119909119899+ 120578119899T119909119899

119909119899+1

= (1 minus 120585119899) 119909119899+ 120585119899T119910119899

(1)

for all 119899 isin N where 120585119899 sub [0 1] and 120578

119899 sub [0 1] satisfy

the conditions lim119899rarrinfin

120578119899

= 0 and suminfin

119899=1120585119899120578119899

= infin Thenthe sequence 119909

119899 generated by (1) converges strongly to a fixed

point of T

The iteration (1) is now referred to as the Ishikawaiterative sequence However we note that 119862 is compactsubset Now we know that strong convergence has not beenachieved without compactness assumption on the involvedoperation or the underlying spaces A counter example canbe found in Chidume and Mutangadura [3]

In order to obtain a strong convergence result for pseu-docontractive mappings without the compactness assump-tion in [4] Zhou coupled the Ishikawa algorithm with the

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 791209 8 pageshttpdxdoiorg1011552014791209

2 Abstract and Applied Analysis

hybrid technique and presented the following algorithm forLipschitz pseudocontractive mappings

119910119899= (1 minus 120585

119899) 119909119899+ 120585119899T119909119899

119911119899= (1 minus 120578

119899) 119909119899+ 120578119899T119910119899

C119899= 119911 isin C

1003817100381710038171003817119911119899 minus 11991110038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119911

10038171003817100381710038172

minus 120585119899120578119899(1 minus 2120585

119899minus 1205852

1198991198712)1003817100381710038171003817119909119899 minus T119909

119899

10038171003817100381710038172

Q119899= 119911 isin C ⟨119909

119899minus 119911 119909

0minus 119909119899⟩ ge 0

119909119899+1

= projC119899capQ119899

1199090 119899 isin N

(2)

Zhou proved that the sequence 119909119899 generated by (2) con-

verges strongly to the fixed point of T Further in [5] Yaoet al introduced the hybrid Mann algorithm as follows

LetC be a nonempty closed convex subset of a realHilbertspaceH Let 120585

119899 be a sequence in (0 1) Let 119909

0isin H ForC

1=

C and 1199091= projC

1

1199090 define a sequence 119909

119899 of C as follows

119910119899= (1 minus 120585

119899) 119909119899+ 120585119899T119909119899

C119899+1

= 119911 isin C1198991003817100381710038171003817120585119899 (119868 minus T) 119910

119899

10038171003817100381710038172

le 2120585119899⟨119909119899minus 119911 (119868 minus T) 119910

119899⟩

119909119899+1

= projC119899+1

1199090 119899 isin N

(3)

Note that in iterations (2) and (3) we need to computethe half-spaces C

119899(andor Q

119899) Very recently Zegeye et al

[6] further studied the convergence analysis of the Ishikawaiteration (1)They proved ingeniously the strong convergenceof the Ishikawa iteration (1) However we have to assumethat the interior of Fix(T) is nonempty This appears veryrestrictive since even in R with the usual norm Lipschitzpseudocontractivemapswith finite number of fixed points donot enjoy this condition that intFix(T) = 0 For some relatedworks please refer to [7ndash19]

On the other hand we notice that it is quite often toseek a particular solution of a given nonlinear problem inparticular the minimum-norm solution For instance givena closed convex subsetC of a Hilbert spaceH

1and a bounded

linear operator B H1

rarr H2 where H

2is another Hilbert

space The C-constrained pseudoinverse of B BdaggerC is thendefined as the minimum-norm solution of the constrainedminimization problem

Bdagger

C (119887) = argmin119909isinC

B119909 minus 119887 (4)

which is equivalent to the fixed point problem

119906 = projC (119906 minus 120583Blowast(B119906 minus 119887)) (5)

where Blowast is the adjoint of B 120583 gt 0 is a constant and 119887 isin H2

is such that projB(C)

(119887) isin B(C)It is therefore an interesting problem to invent iterative

algorithms that can generate sequences which convergestrongly to the minimum-norm solution of a given fixed

point problem The purpose of this paper is to solve sucha problem for pseudocontractions More precisely we willintroduce an explicit projection algorithm with viscositytechnique for finding the fixed points of a Lipschitzianpseudocontractive mapping Strong convergence theorem isdemonstrated Consequently as an application we can findthe minimum-norm fixed point of the pseudocontractivemapping

2 Preliminaries

Recall that themetric projection projC H rarr C satisfies 119906minusprojC119906 = inf119906 minus V V isin C The metric projection projCis a typical firmly nonexpansive mapping The characteristicinequality of the projection is ⟨119906minusprojC119906 VminusprojC119906⟩ le 0 forall 119906 isin H V isin C

In the sequel we will use the following expressions

(i) Fix(T) denotes the set of fixed points of T (ii) 119909119899 119909dagger denotes the weak convergence of 119909

119899to 119909dagger

(iii) 119909119899rarr 119909dagger denotes the strong convergence of 119909

119899to 119909dagger

The following lemmas will be useful for the next section

Lemma2 (see [20]) LetC be a nonempty closed convex subsetof a real Hilbert space H Let T C rarr C be a nonexpansivemapping with Fix(T) = 0 Then

C sup 119906

119899 119906Dagger

(I minus T) 119906119899997888rarr ] 997904rArr (I minus T) 119906

Dagger= ] (6)

Lemma 3 (see [21]) LetC be a nonempty closed convex subsetof a realHilbert spaceH Assume that amappingA C rarr H ismonotone and weakly continuous along segments (ie A(119909 +

119905119910) rarr A(119909) weakly as 119905 rarr 0 whenever 119909 + 119905119910 isin C for119909 119910 isin C) Then the variational inequality

119909Daggerisin 119862 ⟨A119909

Dagger 119909 minus 119909

Dagger⟩ ge 0 forall119909 isin C (7)

is equivalent to the dual variational inequality

119909Daggerisin C ⟨A119909 119909 minus 119909

Dagger⟩ ge 0 forall119909 isin C (8)

Lemma 4 (see [22]) Assume that the sequence 119886119899 satisfies

119886119899ge 0 and 119886

119899+1le (1 minus ]

119899)119886119899+ 120589119899]119899where ]

119899 is a sequence

in (0 1) and 120589119899 is a sequence such that suminfin

119899=1]119899

= infin andlim sup

119899rarrinfin120589119899le 0 (or suminfin

119899=1|120589119899]119899| lt infin) Then lim

119899rarrinfin119886119899=

0

3 Main Results

In order to prove our main result we need the followingproposition

Proposition 5 Let C be a nonempty closed convex subset ofa real Hilbert space H Let W C rarr C be a nonexpansivemapping with Fix(W) = 0 Let 984858 C rarr H be a 120581-contraction

For each 119905 isin (0 1) let the net 119906119905 be defined by

119906119905= W projC [119905984858 (119906

119905) + (1 minus 119905) 119906

119905] (9)

Abstract and Applied Analysis 3

Then as 119905 rarr 0+ the net 119906

119905 converges strongly to a point

119909Daggerisin Fix(W) which solves the following variational inequality

119909Daggerisin Fix (W) ⟨(119868 minus 984858) 119909

Dagger 119911 minus 119909

Dagger⟩ ge 0 119911 isin Fix (W)

(10)Proof For 119905 isin (0 1) define a mappingW

119905 C rarr C by

W119905119906 = W projC [119905984858 (119906) + (1 minus 119905) 119906] 119906 isin C (11)

For any 119906 V isin C we have1003817100381710038171003817W119905119906 minusW

119905V1003817100381710038171003817 =

1003817100381710038171003817W projC [119905984858 (119906) + (1 minus 119905) 119906]

minusW projC [119905984858 (V) + (1 minus 119905) V]1003817100381710038171003817

le 1199051003817100381710038171003817984858 (119906) minus 984858 (V)1003817100381710038171003817 + (1 minus 119905) 119906 minus V

le [1 minus (1 minus 120581) 119905] 119906 minus V

(12)

Hence W119905is a 1 minus (1 minus 120581)119905-contraction on C with 119906

119905isin C as

its unique fixed point So 119906119905 is well defined

Let 119906 isin Fix(W) From (9) we have1003817100381710038171003817119906119905 minus 119906

1003817100381710038171003817 =1003817100381710038171003817W projC [119905984858 (119906

119905) + (1 minus 119905) 119906

119905] minusW projC119906

1003817100381710038171003817

le 1199051003817100381710038171003817984858 (119906119905) minus 984858 (119906)

1003817100381710038171003817 + 1199051003817100381710038171003817984858 (119906) minus 119906

1003817100381710038171003817 + (1 minus 119905)1003817100381710038171003817119906119905 minus 119906

1003817100381710038171003817

le [1 minus (1 minus 120581) 119905]1003817100381710038171003817119906119905 minus 119906

1003817100381710038171003817 + 1199051003817100381710038171003817984858 (119906) minus 119906

1003817100381710038171003817

(13)It follows that

1003817100381710038171003817119906119905 minus 1199061003817100381710038171003817 le

1003817100381710038171003817984858 (119906) minus 1199061003817100381710038171003817

1 minus 120581 (14)

Thus 119906119905 and 984858(119906

119905) are bounded

Again from (9) we get1003817100381710038171003817119906119905 minusW119906

119905

1003817100381710038171003817 =1003817100381710038171003817W projC [119905984858 (119906

119905) + (1 minus 119905) 119906

119905] minusW projC119906119905

1003817100381710038171003817

le 1199051003817100381710038171003817984858 (119906119905) minus 119906

119905

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0+

(15)

Let 119905119899 sub (0 1) be a sequence such that 119905

119899rarr 0+ as 119899 rarr infin

Put 119906119899= 119906119905119899

From (15) we have1003817100381710038171003817119906119899 minusW119906

119899

1003817100381710038171003817 997888rarr 0 (16)From (9) we obtain1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

=1003817100381710038171003817W projC[119905984858(119906119905) + (1 minus 119905)119906

119905] minusW projC119906

10038171003817100381710038172

le1003817100381710038171003817119906119905 minus 119906 + 119905 (984858 (119906

119905) minus 119906119905)10038171003817100381710038172

=1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

+ 2119905 ⟨984858 (119906119905) minus 119906119905 119906119905minus 119906⟩

+ 11990521003817100381710038171003817984858(119906119905) minus 119906

119905

10038171003817100381710038172

=1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

+ 2119905 ⟨984858 (119906119905) minus 984858 (119906) 119906

119905minus 119906⟩

+ 2119905 ⟨984858 (119906) minus 119906 119906119905minus 119906⟩

+ 2119905⟨119906 minus 119906119905 119906119905minus 119906⟩ + 119905

21003817100381710038171003817984858(119906119905) minus 119906119905

10038171003817100381710038172

le [1 minus 2 (1 minus 120581) 119905]1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

+ 2119905 ⟨984858 (119906) minus 119906 119906119905minus 119906⟩ + 119905

21003817100381710038171003817984858(119906119905) minus 119906119905

10038171003817100381710038172

(17)

It follows that

1003817100381710038171003817119906119905 minus 11990610038171003817100381710038172

le1

1 minus 120581⟨984858 (119906) minus 119906 119906

119905minus 119906⟩ + 119905119872 (18)

where119872 gt 0 is a constant such that

119872 gt1

2 (1 minus 120581)sup

1003817100381710038171003817984858(119906119905) minus 119906119905

10038171003817100381710038172

119905 isin (0 1) (19)

In particular we have

1003817100381710038171003817119906119899 minus 11990610038171003817100381710038172

le1

1 minus 120581⟨984858 (119906) minus 119906 119906

119899minus 119906⟩ + 119905

119899119872 119906 isin Fix (W)

(20)

Noting that 119906119899 is bounded without loss of generality we

assume that 119906119899 119909Dagger It is obvious that 119909Dagger isin C From (16)

and Lemma 2 we deduce 119909Dagger isin Fix(W) Substitute 119909Dagger for 119906 in(20) to get

10038171003817100381710038171003817119906119899minus 119909Dagger10038171003817100381710038171003817

2

le1

1 minus 120581⟨984858 (119909Dagger) minus 119909Dagger 119909119899minus 119909Dagger⟩ + 119905119899119872 (21)

Since 119906119899 119909Dagger we deduce from (21) that 119906

119899rarr 119909Dagger The net

119906119905 is therefore relatively compact as 119905 rarr 0

+ in the normtopology

In (20) letting 119899 rarr infin we get

10038171003817100381710038171003817119909Daggerminus 119906

10038171003817100381710038171003817

2

le1

1 minus 120581⟨984858 (119906) minus 119906 119909

Daggerminus 119906⟩ 119906 isin Fix (W)

(22)

Therefore 119909Dagger solves the variational inequality

119909Daggerisin Fix (W) ⟨(119868 minus 984858) 119906 119906 minus 119909

Dagger⟩ ge 0 119906 isin Fix (W)

(23)

By Lemma 3 (23) equals its dual variational inequality

119909Daggerisin Fix (W) ⟨(I minus 984858) 119909

Dagger 119906 minus 119909

Dagger⟩ ge 0 119906 isin Fix (W)

(24)

This indicates that 119909Dagger = (projFix(W)984858)119909Dagger That is 119909Dagger is the

unique fixed point in Fix(W) of the contraction projFix(W)984858So the entire net 119906

119905 converges in norm to 119909

Dagger as 119905 rarr 0+

This completes the proof

Corollary 6 Let C be a nonempty closed convex subset of areal Hilbert space H Let W C rarr C be a nonexpansivemapping with Fix(W) = 0

For each 119905 isin (0 1) let the net 119906119905 be defined by

119906119905= W projC [(1 minus 119905) 119906

119905] (25)

Then as 119905 rarr 0+ the net 119906

119905 converges strongly to the

minimum-norm fixed point ofW

Proof As a matter of fact in (9) we choose 984858 = 0 and then(9) reduces to (25) From Proposition 5 (24) is reduced to

0 le ⟨119909Dagger 119906 minus 119909

Dagger⟩ 119906 isin Fix (W) (26)

4 Abstract and Applied Analysis

Equivalently

10038171003817100381710038171003817119909Dagger10038171003817100381710038171003817

2

le ⟨119909Dagger 119906⟩ 119906 isin Fix (W) (27)

This implies that10038171003817100381710038171003817119909Dagger10038171003817100381710038171003817

le 119906 119906 isin Fix (W) (28)

Therefore 119909Dagger is the minimum-norm fixed point of W Thiscompletes the proof

Algorithm 7 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontractionand let 984858 C rarr H be a 120581-contraction Let 120585

119899 sub [0 1] and

120578119899 sub [0 1] be two real number sequences For 119909

0isin C we

define a sequence 119909119899 iteratively by

119909119899+1

= projC [120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0

(29)

Theorem 8 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 and 984858 C rarr H a 120581-contraction Suppose the following conditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0

Then the sequence 119909119899 generated by the algorithm (29)

converges strongly to 119909Dagger= (projFix(T)984858)119909

Dagger

Proof First we prove that the sequence 119909119899 is bounded We

will show this fact by induction According to conditions (C1)and (C2) there exists a sufficiently large positive integer 119898

such that

1 minus2

1 minus 120581(119871 + 1) (119871 + 3) (120585

119899+ 2120578119899+

1205782

119899

120585119899

) gt 0 119899 ge 119898

(30)

Fix a 119901 isin Fix(T) and take a constant1198721gt 0 such that

max 10038171003817100381710038171199090 minus 1199011003817100381710038171003817

10038171003817100381710038171199091 minus 1199011003817100381710038171003817

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817 2

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

le 1198721

(31)

Next we show that 119909119898+1

minus 119901 le 1198721

Set

119910119898

= 120585119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898

thus 119909119898+1

= projC [119910119898]

(32)

Then we have

⟨119909119898+1

minus 119910119898 119909119898+1

minus 119901⟩ le 0 (33)

Since T is pseudocontractive I minus T is monotone So we have

⟨(I minus T) 119909119898+1

minus (I minus T) 119901 119909119898+1

minus 119901⟩ ge 0 (34)

From (29) (33) and (34) we obtain

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

= ⟨119909119898+1

minus 119901 119909119898+1

minus 119901⟩

= ⟨119909119898+1

minus 119910119898 119909119898+1

minus 119901⟩ + ⟨119910119898minus 119901 119909

119898+1minus 119901⟩

le ⟨119910119898minus 119901 119909

119898+1minus 119901⟩

= ⟨119909119898minus 119901 119909

119898+1minus 119901⟩

minus 120585119898⟨119909119898minus 984858 (119909

119898) 119909119898+1

minus 119901⟩

+ 120578119898⟨T119909119898minus 119909119898 119909119898+1

minus 119901⟩

= ⟨119909119898minus 119901 119909

119898+1minus 119901⟩

+ 120585119898⟨119909119898+1

minus 119909119898 119909119898+1

minus 119901⟩

+ 120585119898⟨984858 (119909119898) minus 984858 (119901) 119909

119898+1minus 119901⟩

+ 120585119898⟨119891 (119901) minus 119901 119909

119898+1minus 119901⟩

minus 120585119898⟨119909119898+1

minus 119901 119909119898+1

minus 119901⟩

+ 120578119898⟨T119909119898minus T119909119898+1

119909119898+1

minus 119901⟩

+ 120578119898⟨119909119898+1

minus 119909119898 119909119898+1

minus 119901⟩

minus 120578119898⟨119909119898+1

minus T119909119898+1

119909119898+1

minus 119901⟩

le1003817100381710038171003817119909119898 minus 119901

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817119909119898+1 minus 119909119898

1003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 1205851198981205811003817100381710038171003817119909119898 minus 119901

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

+ 120585119898

1003817100381710038171003817984858 (119901) minus 11990110038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 minus 120585119898

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

+ 120578119898(1003817100381710038171003817T119909119898 minus T119909

119898+1

1003817100381710038171003817 +1003817100381710038171003817119909119898+1 minus 119909

119898

1003817100381710038171003817)

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 minus 120585119898

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

+ (119871 + 1) (120585119898+ 120578119898)1003817100381710038171003817119909119898+1 minus 119909

119898

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

(35)

It follows that

(1 + 120585119898)1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

+120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

+ (119871 + 1) (120585119898+ 120578119898)1003817100381710038171003817119909119898+1 minus 119909

119898

1003817100381710038171003817

(36)

Abstract and Applied Analysis 5

By (29) we have

1003817100381710038171003817119909119898+1 minus 119909119898

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898]

minusprojC [119909119898]1003817100381710038171003817

le1003817100381710038171003817120585119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898minus 119909119898

1003817100381710038171003817

le 120585119898(1003817100381710038171003817984858 (119909119898) minus 984858 (119901)

1003817100381710038171003817 +1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ 120578119898(1003817100381710038171003817T119909119898 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

le 120585119898(1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 + (1 + 120581)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ (119871 + 1) 120578119898

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

le (119871 + 1 + 120581) (120585119898+ 120578119898)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

le (119871 + 3) (120585119898+ 120578119898)1198721

(37)

Substitute (37) into (36) to obtain

(1 + 120585119898)1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

+ (119871 + 1) (119871 + 3) (120585119898+ 120578119898)2

1198721

le (1 +1 + 120581

2120585119898)1198721+ (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1198721

(38)

that is1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le [1 minus((1 minus 120581) 120585

1198982) minus (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1 + 120585119898

]1198721

= 1 minus ((((1 minus 120581) 120585

119898

2)[1 minus

2

1 minus 120581(119871 + 1) (119871 + 3)

times (120585119898+ 2120578119898+ (

1205782

119898

120585119898

))])

times (1 + 120585119898)minus1

)1198721

le 1198721

(39)

By induction we get1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 le 1198721 forall119899 ge 0 (40)

which implies that 119909119899 is bounded and so is T119909

119899 Now we

take a constant1198722gt 0 such that

1198722= sup119899

1003817100381710038171003817119909119899

1003817100381710038171003817 +1003817100381710038171003817T119909119899 minus 119909

119899

1003817100381710038171003817 (41)

Set S = (2I minus T)minus1 (ie S is a resolvent of the monotone

operator I minus T) We then have that S is a nonexpansive selfmapping of C and Fix(S) = Fix(T)

By Proposition 5 we know that whenever 120574119899 sub (0 1)

and 120574119899rarr 0+ the sequence 119911

119899 defined by

119911119899= S projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] (42)

converges strongly to the fixed point 119909Dagger of S (and of T asFix(S) = Fix(T)) Without loss of generality we may assumethat 119911

119899 le 119872

2for all 119899

It suffices to prove that 119909119899+1

minus 119911119899 rarr 0 as 119899 rarr infin (for

some 120574119899rarr 0+) To this end we rewrite (42) as

(2I minus T) 119911119899= projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] 119899 ge 0 (43)

By using the property of the metric projection we have

⟨120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus (2119911119899minus T119911119899) 119909119899+1

minus (2119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩

+ ⟨T119911119899minus 119911119899 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

+1003817100381710038171003817119911119899 minus T119911

119899

10038171003817100381710038172

le ⟨120574119899(119911119899minus 984858 (119911

119899)) T119911

119899minus 119911119899⟩

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

le 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

997904rArr ⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩

le1003817100381710038171003817119911119899 minus 984858 (119911

119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

(44)

Note that

1003817100381710038171003817119911119899 minus T119911119899

1003817100381710038171003817 =1003817100381710038171003817projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] minus 119911119899

1003817100381710038171003817

le1003817100381710038171003817120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus 119911119899

1003817100381710038171003817

= 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817

(45)

Hence we get

⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩ le 120574

119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(46)

6 Abstract and Applied Analysis

From (42) we have1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 =1003817100381710038171003817S projC [120574

119899+1984858 (119911119899+1

) + (1 minus 120574119899+1

) 119911119899+1

]

minusS projC [120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899]1003817100381710038171003817

le1003817100381710038171003817120574119899+1984858 (119911119899+1) + (1 minus 120574

119899+1) 119911119899+1

minus120574119899984858 (119911119899) minus (1 minus 120574

119899) 119911119899

1003817100381710038171003817

=1003817100381710038171003817(1 minus 120574

119899+1) (119911119899+1

minus 119911119899) + (120574119899minus 120574119899+1

)

times (119911119899minus 984858 (119911

119899)) + 120574

119899+1(984858 (119911119899+1

) minus 984858 (119911119899))1003817100381710038171003817

le [1 minus (1 minus 120581) 120574119899+1

]1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574

119899

10038161003816100381610038161003817100381710038171003817119911119899 minus 984858 (119911

119899)1003817100381710038171003817

(47)

It follows that1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 le

1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

(1 minus 120581) 120574119899+1

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817 (48)

Set

120574119899=

120585119899

120578119899

(49)

By condition (C2) 120574119899

rarr 0+ and 120574

119899isin (0 1) for 119899 large

enough Hence by (46) and (48) we have

⟨minus(119911119899minus 984858 (119911

119899)) +

120578119899(T119911119899minus 119911119899)

120585119899

119909119899+1

minus 119911119899⟩

le120585119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

le120585119899

120578119899

1198722

2

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817 le

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

1198722

(50)

By (29) we have1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minusprojC1199091198991003817100381710038171003817

le 120585119899

1003817100381710038171003817119909119899 minus 984858 (119909119899)1003817100381710038171003817 + 120578119899

1003817100381710038171003817T119909119899 minus 119909119899

1003817100381710038171003817

le (120585119899+ 120578119899)1198722

(51)

Next we estimate 119909119899+1

minus 119911119899+1

Since 119909119899+1

= projC[119910119899]⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩ le 0 Then we have

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

= ⟨119909119899+1

minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩

+ ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

le ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨[120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minus119911119899 119909119899+1

minus 119911119899⟩

= (1 minus 120585119899minus 120578119899) ⟨119909119899minus 119911119899 119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899minus T119909119899+1

119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899+1

minus T119911119899 119909119899+1

minus 119911119899⟩

+ ⟨120585119899(984858 (119909119899) minus 119911119899)

+ 120578119899(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

(52)

It follows that

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus 120585119899minus 120578119899)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 1205781198991198711003817100381710038171003817119909119899 minus 119909

119899+1

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+ 120585119899⟨984858 (119911

119899) minus 119911119899+

120578119899

120585119899

(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

le1 minus 120585119899minus 120578119899

2(1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

)

+1205782

119899

2

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1198712

2

1003817100381710038171003817119909119899 minus 119909119899+1

10038171003817100381710038172

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1205852

119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(53)

Thus

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le1 minus 120585119899minus 120578119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899 minus 119911119899

10038171003817100381710038172

+1198712

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119909119899

10038171003817100381710038172

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

+1205782

119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+(120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

11987121198722

2+

21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1198722

2

+1205782

119899

1 + 120585119899minus 120578119899

41198722

2

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)

times (1003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)2

Abstract and Applied Analysis 7

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

119872

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

(54)

where the finite constant119872 gt 0 is given by

119872 = max 119871211987222 41198722

2

1198722sup119899

(21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

(55)

Set

120575119899=

2120585119899

1 + 120585119899minus 120578119899

asymp 2120585119899

(as 119899 997888rarr infin)

120579119899=

120585119899120578119899minus1

minus 120585119899minus1

120578119899

21205852119899120578119899minus1

+1

2(120585119899+ 2120578119899+

1205782

119899

120585119899

) +120585119899

120578119899

+1205782

119899

2120585119899

119872

(56)

Then (54) can be rewritten as1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

le (1 minus 120575119899)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+ 120575119899120579119899 (57)

By the conditions (C1)ndash(C3) we deuce that

lim119899rarrinfin

120575119899= 0

infin

sum119899=1

120575119899= infin lim

119899rarrinfin120579119899= 0 (58)

From Lemma 4 and (57) we get 119909119899+1

minus 1199111198992

rarr 0 as 119899 rarr

infin This completes the proof

Algorithm 9 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontraction Let120585119899 sub [0 1] and 120578

119899 sub [0 1] be two real number sequences

For 1199090isin C we define a sequence 119909

119899 iteratively by

119909119899+1

= projC [(1 minus 120585119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0 (59)

Corollary 10 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 Suppose the followingconditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0Then the sequence 119909

119899 generated by the algorithm (59)

converges strongly to the minimum-norm fixed point of T

Conflict of Interests

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

Acknowledgment

The first author was supported by the National NaturalScience Foundation of China Grant no 11226125

References

[1] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 no 4 pp 365ndash374 1974

[2] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974

[3] C E Chidume and S A Mutangadura ldquoAn example onthe mann iteration method for lipschitz pseudocontractionsrdquoProceedings of the American Mathematical Society vol 129 no8 pp 2359ndash2363 2001

[4] H Zhou ldquoConvergence theorems of fixed points for Lipschitzpseudo-contractions inHilbert spacesrdquo Journal ofMathematicalAnalysis and Applications vol 343 no 1 pp 546ndash556 2008

[5] Y Yao Y-C Liou and G Marino ldquoA hybrid algorithmfor pseudo-contractive mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 10 pp 4997ndash5002 2009

[6] H Zegeye N Shahzad and M A Alghamdi ldquoConvergence ofIshikawarsquos iteration method for pseudocontractive mappingsrdquoNonlinear Analysis Theory Methods and Applications vol 74no 18 pp 7304ndash7311 2011

[7] L-C Ceng A Petrusel and J-C Yao ldquoStrong convergence ofmodified implicit iterative algorithmswith perturbedmappingsfor continuous pseudocontractive mappingsrdquo Applied Mathe-matics and Computation vol 209 no 2 pp 162ndash176 2009

[8] L-C Ceng A Petrusel and J-C Yao ldquoIterative approximationof fixed points for asymptotically strict pseudocontractive typemappings in the intermediate senserdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 587ndash606 2011

8 Abstract and Applied Analysis

[9] L-C Ceng and J-C Yao ldquoStrong convergence theorems forvariational inequalities and fixed point problems of asymptot-ically strict pseudocontractive mappings in the intermediatesenserdquo Acta Applicandae Mathematicae vol 115 no 2 pp 167ndash191 2011

[10] J C Yao and L C Zeng ldquoStrong convergence of averagedapproximants for asymptotically pseudocontractive mappingsin Banach spacesrdquo Journal of Nonlinear and Convex Analysisvol 8 pp 451ndash462 2007

[11] L C Ceng Q H Ansari and C F Wen ldquoImplicit relaxed andhybrid methods with regularization for minimization problemsand asymptotically strict pseudocontractive mappings in theintermediate senserdquo Abstract and Applied Analysis vol 2013Article ID 854297 14 pages 2013

[12] C E ChidumeMAbbas andBAli ldquoConvergence of theManniteration algorithm for a class of pseudocontractive mappingsrdquoApplied Mathematics and Computation vol 194 no 1 pp 1ndash62007

[13] C E Chidume and H Zegeye ldquoIterative solution of nonlinearequations of accretive and pseudocontractive typesrdquo Journal ofMathematical Analysis and Applications vol 282 no 2 pp 756ndash765 2003

[14] L Ciric A Rafiq N Cakic and J S Ume ldquoImplicit Mannfixedpoint iterations for pseudo-contractivemappingsrdquoAppliedMathematics Letters vol 22 no 4 pp 581ndash584 2009

[15] C Moore and B V C Nnoli ldquoStrong convergence of averagedapproximants for Lipschitz pseudocontractive mapsrdquo Journal ofMathematical Analysis and Applications vol 260 no 1 pp 269ndash278 2001

[16] A Udomene ldquoPath convergence approximation of fixed pointsand variational solutions of Lipschitz pseudocontractions inBanach spacesrdquoNonlinear Analysis Theory Methods and Appli-cations vol 67 no 8 pp 2403ndash2414 2007

[17] H Zegeye N Shahzad and T Mekonen ldquoViscosity approx-imation methods for pseudocontractive mappings in Banachspacesrdquo Applied Mathematics and Computation vol 185 no 1pp 538ndash546 2007

[18] Q-B Zhang andC-Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008

[19] H Zhou ldquoStrong convergence of an explicit iterative algorithmfor continuous pseudo-contractions in Banach spacesrdquo Nonlin-ear Analysis Theory Methods and Applications vol 70 no 11pp 4039ndash4046 2009

[20] K Geobel and W A Kirk Topics in Metric Fixed PointTheory vol 28 Cambridge Studies in Advanced MathematicsCambridge University Press 1990

[21] X Lu H-K Xu and X Yin ldquoHybrid methods for a class ofmonotone variational inequalitiesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 3-4 pp 1032ndash1041 2009

[22] H K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 2 pp 1ndash17 2002

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

hybrid technique and presented the following algorithm forLipschitz pseudocontractive mappings

119910119899= (1 minus 120585

119899) 119909119899+ 120585119899T119909119899

119911119899= (1 minus 120578

119899) 119909119899+ 120578119899T119910119899

C119899= 119911 isin C

1003817100381710038171003817119911119899 minus 11991110038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119911

10038171003817100381710038172

minus 120585119899120578119899(1 minus 2120585

119899minus 1205852

1198991198712)1003817100381710038171003817119909119899 minus T119909

119899

10038171003817100381710038172

Q119899= 119911 isin C ⟨119909

119899minus 119911 119909

0minus 119909119899⟩ ge 0

119909119899+1

= projC119899capQ119899

1199090 119899 isin N

(2)

Zhou proved that the sequence 119909119899 generated by (2) con-

verges strongly to the fixed point of T Further in [5] Yaoet al introduced the hybrid Mann algorithm as follows

LetC be a nonempty closed convex subset of a realHilbertspaceH Let 120585

119899 be a sequence in (0 1) Let 119909

0isin H ForC

1=

C and 1199091= projC

1

1199090 define a sequence 119909

119899 of C as follows

119910119899= (1 minus 120585

119899) 119909119899+ 120585119899T119909119899

C119899+1

= 119911 isin C1198991003817100381710038171003817120585119899 (119868 minus T) 119910

119899

10038171003817100381710038172

le 2120585119899⟨119909119899minus 119911 (119868 minus T) 119910

119899⟩

119909119899+1

= projC119899+1

1199090 119899 isin N

(3)

Note that in iterations (2) and (3) we need to computethe half-spaces C

119899(andor Q

119899) Very recently Zegeye et al

[6] further studied the convergence analysis of the Ishikawaiteration (1)They proved ingeniously the strong convergenceof the Ishikawa iteration (1) However we have to assumethat the interior of Fix(T) is nonempty This appears veryrestrictive since even in R with the usual norm Lipschitzpseudocontractivemapswith finite number of fixed points donot enjoy this condition that intFix(T) = 0 For some relatedworks please refer to [7ndash19]

On the other hand we notice that it is quite often toseek a particular solution of a given nonlinear problem inparticular the minimum-norm solution For instance givena closed convex subsetC of a Hilbert spaceH

1and a bounded

linear operator B H1

rarr H2 where H

2is another Hilbert

space The C-constrained pseudoinverse of B BdaggerC is thendefined as the minimum-norm solution of the constrainedminimization problem

Bdagger

C (119887) = argmin119909isinC

B119909 minus 119887 (4)

which is equivalent to the fixed point problem

119906 = projC (119906 minus 120583Blowast(B119906 minus 119887)) (5)

where Blowast is the adjoint of B 120583 gt 0 is a constant and 119887 isin H2

is such that projB(C)

(119887) isin B(C)It is therefore an interesting problem to invent iterative

algorithms that can generate sequences which convergestrongly to the minimum-norm solution of a given fixed

point problem The purpose of this paper is to solve sucha problem for pseudocontractions More precisely we willintroduce an explicit projection algorithm with viscositytechnique for finding the fixed points of a Lipschitzianpseudocontractive mapping Strong convergence theorem isdemonstrated Consequently as an application we can findthe minimum-norm fixed point of the pseudocontractivemapping

2 Preliminaries

Recall that themetric projection projC H rarr C satisfies 119906minusprojC119906 = inf119906 minus V V isin C The metric projection projCis a typical firmly nonexpansive mapping The characteristicinequality of the projection is ⟨119906minusprojC119906 VminusprojC119906⟩ le 0 forall 119906 isin H V isin C

In the sequel we will use the following expressions

(i) Fix(T) denotes the set of fixed points of T (ii) 119909119899 119909dagger denotes the weak convergence of 119909

119899to 119909dagger

(iii) 119909119899rarr 119909dagger denotes the strong convergence of 119909

119899to 119909dagger

The following lemmas will be useful for the next section

Lemma2 (see [20]) LetC be a nonempty closed convex subsetof a real Hilbert space H Let T C rarr C be a nonexpansivemapping with Fix(T) = 0 Then

C sup 119906

119899 119906Dagger

(I minus T) 119906119899997888rarr ] 997904rArr (I minus T) 119906

Dagger= ] (6)

Lemma 3 (see [21]) LetC be a nonempty closed convex subsetof a realHilbert spaceH Assume that amappingA C rarr H ismonotone and weakly continuous along segments (ie A(119909 +

119905119910) rarr A(119909) weakly as 119905 rarr 0 whenever 119909 + 119905119910 isin C for119909 119910 isin C) Then the variational inequality

119909Daggerisin 119862 ⟨A119909

Dagger 119909 minus 119909

Dagger⟩ ge 0 forall119909 isin C (7)

is equivalent to the dual variational inequality

119909Daggerisin C ⟨A119909 119909 minus 119909

Dagger⟩ ge 0 forall119909 isin C (8)

Lemma 4 (see [22]) Assume that the sequence 119886119899 satisfies

119886119899ge 0 and 119886

119899+1le (1 minus ]

119899)119886119899+ 120589119899]119899where ]

119899 is a sequence

in (0 1) and 120589119899 is a sequence such that suminfin

119899=1]119899

= infin andlim sup

119899rarrinfin120589119899le 0 (or suminfin

119899=1|120589119899]119899| lt infin) Then lim

119899rarrinfin119886119899=

0

3 Main Results

In order to prove our main result we need the followingproposition

Proposition 5 Let C be a nonempty closed convex subset ofa real Hilbert space H Let W C rarr C be a nonexpansivemapping with Fix(W) = 0 Let 984858 C rarr H be a 120581-contraction

For each 119905 isin (0 1) let the net 119906119905 be defined by

119906119905= W projC [119905984858 (119906

119905) + (1 minus 119905) 119906

119905] (9)

Abstract and Applied Analysis 3

Then as 119905 rarr 0+ the net 119906

119905 converges strongly to a point

119909Daggerisin Fix(W) which solves the following variational inequality

119909Daggerisin Fix (W) ⟨(119868 minus 984858) 119909

Dagger 119911 minus 119909

Dagger⟩ ge 0 119911 isin Fix (W)

(10)Proof For 119905 isin (0 1) define a mappingW

119905 C rarr C by

W119905119906 = W projC [119905984858 (119906) + (1 minus 119905) 119906] 119906 isin C (11)

For any 119906 V isin C we have1003817100381710038171003817W119905119906 minusW

119905V1003817100381710038171003817 =

1003817100381710038171003817W projC [119905984858 (119906) + (1 minus 119905) 119906]

minusW projC [119905984858 (V) + (1 minus 119905) V]1003817100381710038171003817

le 1199051003817100381710038171003817984858 (119906) minus 984858 (V)1003817100381710038171003817 + (1 minus 119905) 119906 minus V

le [1 minus (1 minus 120581) 119905] 119906 minus V

(12)

Hence W119905is a 1 minus (1 minus 120581)119905-contraction on C with 119906

119905isin C as

its unique fixed point So 119906119905 is well defined

Let 119906 isin Fix(W) From (9) we have1003817100381710038171003817119906119905 minus 119906

1003817100381710038171003817 =1003817100381710038171003817W projC [119905984858 (119906

119905) + (1 minus 119905) 119906

119905] minusW projC119906

1003817100381710038171003817

le 1199051003817100381710038171003817984858 (119906119905) minus 984858 (119906)

1003817100381710038171003817 + 1199051003817100381710038171003817984858 (119906) minus 119906

1003817100381710038171003817 + (1 minus 119905)1003817100381710038171003817119906119905 minus 119906

1003817100381710038171003817

le [1 minus (1 minus 120581) 119905]1003817100381710038171003817119906119905 minus 119906

1003817100381710038171003817 + 1199051003817100381710038171003817984858 (119906) minus 119906

1003817100381710038171003817

(13)It follows that

1003817100381710038171003817119906119905 minus 1199061003817100381710038171003817 le

1003817100381710038171003817984858 (119906) minus 1199061003817100381710038171003817

1 minus 120581 (14)

Thus 119906119905 and 984858(119906

119905) are bounded

Again from (9) we get1003817100381710038171003817119906119905 minusW119906

119905

1003817100381710038171003817 =1003817100381710038171003817W projC [119905984858 (119906

119905) + (1 minus 119905) 119906

119905] minusW projC119906119905

1003817100381710038171003817

le 1199051003817100381710038171003817984858 (119906119905) minus 119906

119905

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0+

(15)

Let 119905119899 sub (0 1) be a sequence such that 119905

119899rarr 0+ as 119899 rarr infin

Put 119906119899= 119906119905119899

From (15) we have1003817100381710038171003817119906119899 minusW119906

119899

1003817100381710038171003817 997888rarr 0 (16)From (9) we obtain1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

=1003817100381710038171003817W projC[119905984858(119906119905) + (1 minus 119905)119906

119905] minusW projC119906

10038171003817100381710038172

le1003817100381710038171003817119906119905 minus 119906 + 119905 (984858 (119906

119905) minus 119906119905)10038171003817100381710038172

=1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

+ 2119905 ⟨984858 (119906119905) minus 119906119905 119906119905minus 119906⟩

+ 11990521003817100381710038171003817984858(119906119905) minus 119906

119905

10038171003817100381710038172

=1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

+ 2119905 ⟨984858 (119906119905) minus 984858 (119906) 119906

119905minus 119906⟩

+ 2119905 ⟨984858 (119906) minus 119906 119906119905minus 119906⟩

+ 2119905⟨119906 minus 119906119905 119906119905minus 119906⟩ + 119905

21003817100381710038171003817984858(119906119905) minus 119906119905

10038171003817100381710038172

le [1 minus 2 (1 minus 120581) 119905]1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

+ 2119905 ⟨984858 (119906) minus 119906 119906119905minus 119906⟩ + 119905

21003817100381710038171003817984858(119906119905) minus 119906119905

10038171003817100381710038172

(17)

It follows that

1003817100381710038171003817119906119905 minus 11990610038171003817100381710038172

le1

1 minus 120581⟨984858 (119906) minus 119906 119906

119905minus 119906⟩ + 119905119872 (18)

where119872 gt 0 is a constant such that

119872 gt1

2 (1 minus 120581)sup

1003817100381710038171003817984858(119906119905) minus 119906119905

10038171003817100381710038172

119905 isin (0 1) (19)

In particular we have

1003817100381710038171003817119906119899 minus 11990610038171003817100381710038172

le1

1 minus 120581⟨984858 (119906) minus 119906 119906

119899minus 119906⟩ + 119905

119899119872 119906 isin Fix (W)

(20)

Noting that 119906119899 is bounded without loss of generality we

assume that 119906119899 119909Dagger It is obvious that 119909Dagger isin C From (16)

and Lemma 2 we deduce 119909Dagger isin Fix(W) Substitute 119909Dagger for 119906 in(20) to get

10038171003817100381710038171003817119906119899minus 119909Dagger10038171003817100381710038171003817

2

le1

1 minus 120581⟨984858 (119909Dagger) minus 119909Dagger 119909119899minus 119909Dagger⟩ + 119905119899119872 (21)

Since 119906119899 119909Dagger we deduce from (21) that 119906

119899rarr 119909Dagger The net

119906119905 is therefore relatively compact as 119905 rarr 0

+ in the normtopology

In (20) letting 119899 rarr infin we get

10038171003817100381710038171003817119909Daggerminus 119906

10038171003817100381710038171003817

2

le1

1 minus 120581⟨984858 (119906) minus 119906 119909

Daggerminus 119906⟩ 119906 isin Fix (W)

(22)

Therefore 119909Dagger solves the variational inequality

119909Daggerisin Fix (W) ⟨(119868 minus 984858) 119906 119906 minus 119909

Dagger⟩ ge 0 119906 isin Fix (W)

(23)

By Lemma 3 (23) equals its dual variational inequality

119909Daggerisin Fix (W) ⟨(I minus 984858) 119909

Dagger 119906 minus 119909

Dagger⟩ ge 0 119906 isin Fix (W)

(24)

This indicates that 119909Dagger = (projFix(W)984858)119909Dagger That is 119909Dagger is the

unique fixed point in Fix(W) of the contraction projFix(W)984858So the entire net 119906

119905 converges in norm to 119909

Dagger as 119905 rarr 0+

This completes the proof

Corollary 6 Let C be a nonempty closed convex subset of areal Hilbert space H Let W C rarr C be a nonexpansivemapping with Fix(W) = 0

For each 119905 isin (0 1) let the net 119906119905 be defined by

119906119905= W projC [(1 minus 119905) 119906

119905] (25)

Then as 119905 rarr 0+ the net 119906

119905 converges strongly to the

minimum-norm fixed point ofW

Proof As a matter of fact in (9) we choose 984858 = 0 and then(9) reduces to (25) From Proposition 5 (24) is reduced to

0 le ⟨119909Dagger 119906 minus 119909

Dagger⟩ 119906 isin Fix (W) (26)

4 Abstract and Applied Analysis

Equivalently

10038171003817100381710038171003817119909Dagger10038171003817100381710038171003817

2

le ⟨119909Dagger 119906⟩ 119906 isin Fix (W) (27)

This implies that10038171003817100381710038171003817119909Dagger10038171003817100381710038171003817

le 119906 119906 isin Fix (W) (28)

Therefore 119909Dagger is the minimum-norm fixed point of W Thiscompletes the proof

Algorithm 7 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontractionand let 984858 C rarr H be a 120581-contraction Let 120585

119899 sub [0 1] and

120578119899 sub [0 1] be two real number sequences For 119909

0isin C we

define a sequence 119909119899 iteratively by

119909119899+1

= projC [120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0

(29)

Theorem 8 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 and 984858 C rarr H a 120581-contraction Suppose the following conditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0

Then the sequence 119909119899 generated by the algorithm (29)

converges strongly to 119909Dagger= (projFix(T)984858)119909

Dagger

Proof First we prove that the sequence 119909119899 is bounded We

will show this fact by induction According to conditions (C1)and (C2) there exists a sufficiently large positive integer 119898

such that

1 minus2

1 minus 120581(119871 + 1) (119871 + 3) (120585

119899+ 2120578119899+

1205782

119899

120585119899

) gt 0 119899 ge 119898

(30)

Fix a 119901 isin Fix(T) and take a constant1198721gt 0 such that

max 10038171003817100381710038171199090 minus 1199011003817100381710038171003817

10038171003817100381710038171199091 minus 1199011003817100381710038171003817

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817 2

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

le 1198721

(31)

Next we show that 119909119898+1

minus 119901 le 1198721

Set

119910119898

= 120585119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898

thus 119909119898+1

= projC [119910119898]

(32)

Then we have

⟨119909119898+1

minus 119910119898 119909119898+1

minus 119901⟩ le 0 (33)

Since T is pseudocontractive I minus T is monotone So we have

⟨(I minus T) 119909119898+1

minus (I minus T) 119901 119909119898+1

minus 119901⟩ ge 0 (34)

From (29) (33) and (34) we obtain

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

= ⟨119909119898+1

minus 119901 119909119898+1

minus 119901⟩

= ⟨119909119898+1

minus 119910119898 119909119898+1

minus 119901⟩ + ⟨119910119898minus 119901 119909

119898+1minus 119901⟩

le ⟨119910119898minus 119901 119909

119898+1minus 119901⟩

= ⟨119909119898minus 119901 119909

119898+1minus 119901⟩

minus 120585119898⟨119909119898minus 984858 (119909

119898) 119909119898+1

minus 119901⟩

+ 120578119898⟨T119909119898minus 119909119898 119909119898+1

minus 119901⟩

= ⟨119909119898minus 119901 119909

119898+1minus 119901⟩

+ 120585119898⟨119909119898+1

minus 119909119898 119909119898+1

minus 119901⟩

+ 120585119898⟨984858 (119909119898) minus 984858 (119901) 119909

119898+1minus 119901⟩

+ 120585119898⟨119891 (119901) minus 119901 119909

119898+1minus 119901⟩

minus 120585119898⟨119909119898+1

minus 119901 119909119898+1

minus 119901⟩

+ 120578119898⟨T119909119898minus T119909119898+1

119909119898+1

minus 119901⟩

+ 120578119898⟨119909119898+1

minus 119909119898 119909119898+1

minus 119901⟩

minus 120578119898⟨119909119898+1

minus T119909119898+1

119909119898+1

minus 119901⟩

le1003817100381710038171003817119909119898 minus 119901

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817119909119898+1 minus 119909119898

1003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 1205851198981205811003817100381710038171003817119909119898 minus 119901

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

+ 120585119898

1003817100381710038171003817984858 (119901) minus 11990110038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 minus 120585119898

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

+ 120578119898(1003817100381710038171003817T119909119898 minus T119909

119898+1

1003817100381710038171003817 +1003817100381710038171003817119909119898+1 minus 119909

119898

1003817100381710038171003817)

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 minus 120585119898

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

+ (119871 + 1) (120585119898+ 120578119898)1003817100381710038171003817119909119898+1 minus 119909

119898

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

(35)

It follows that

(1 + 120585119898)1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

+120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

+ (119871 + 1) (120585119898+ 120578119898)1003817100381710038171003817119909119898+1 minus 119909

119898

1003817100381710038171003817

(36)

Abstract and Applied Analysis 5

By (29) we have

1003817100381710038171003817119909119898+1 minus 119909119898

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898]

minusprojC [119909119898]1003817100381710038171003817

le1003817100381710038171003817120585119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898minus 119909119898

1003817100381710038171003817

le 120585119898(1003817100381710038171003817984858 (119909119898) minus 984858 (119901)

1003817100381710038171003817 +1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ 120578119898(1003817100381710038171003817T119909119898 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

le 120585119898(1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 + (1 + 120581)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ (119871 + 1) 120578119898

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

le (119871 + 1 + 120581) (120585119898+ 120578119898)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

le (119871 + 3) (120585119898+ 120578119898)1198721

(37)

Substitute (37) into (36) to obtain

(1 + 120585119898)1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

+ (119871 + 1) (119871 + 3) (120585119898+ 120578119898)2

1198721

le (1 +1 + 120581

2120585119898)1198721+ (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1198721

(38)

that is1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le [1 minus((1 minus 120581) 120585

1198982) minus (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1 + 120585119898

]1198721

= 1 minus ((((1 minus 120581) 120585

119898

2)[1 minus

2

1 minus 120581(119871 + 1) (119871 + 3)

times (120585119898+ 2120578119898+ (

1205782

119898

120585119898

))])

times (1 + 120585119898)minus1

)1198721

le 1198721

(39)

By induction we get1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 le 1198721 forall119899 ge 0 (40)

which implies that 119909119899 is bounded and so is T119909

119899 Now we

take a constant1198722gt 0 such that

1198722= sup119899

1003817100381710038171003817119909119899

1003817100381710038171003817 +1003817100381710038171003817T119909119899 minus 119909

119899

1003817100381710038171003817 (41)

Set S = (2I minus T)minus1 (ie S is a resolvent of the monotone

operator I minus T) We then have that S is a nonexpansive selfmapping of C and Fix(S) = Fix(T)

By Proposition 5 we know that whenever 120574119899 sub (0 1)

and 120574119899rarr 0+ the sequence 119911

119899 defined by

119911119899= S projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] (42)

converges strongly to the fixed point 119909Dagger of S (and of T asFix(S) = Fix(T)) Without loss of generality we may assumethat 119911

119899 le 119872

2for all 119899

It suffices to prove that 119909119899+1

minus 119911119899 rarr 0 as 119899 rarr infin (for

some 120574119899rarr 0+) To this end we rewrite (42) as

(2I minus T) 119911119899= projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] 119899 ge 0 (43)

By using the property of the metric projection we have

⟨120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus (2119911119899minus T119911119899) 119909119899+1

minus (2119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩

+ ⟨T119911119899minus 119911119899 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

+1003817100381710038171003817119911119899 minus T119911

119899

10038171003817100381710038172

le ⟨120574119899(119911119899minus 984858 (119911

119899)) T119911

119899minus 119911119899⟩

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

le 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

997904rArr ⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩

le1003817100381710038171003817119911119899 minus 984858 (119911

119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

(44)

Note that

1003817100381710038171003817119911119899 minus T119911119899

1003817100381710038171003817 =1003817100381710038171003817projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] minus 119911119899

1003817100381710038171003817

le1003817100381710038171003817120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus 119911119899

1003817100381710038171003817

= 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817

(45)

Hence we get

⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩ le 120574

119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(46)

6 Abstract and Applied Analysis

From (42) we have1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 =1003817100381710038171003817S projC [120574

119899+1984858 (119911119899+1

) + (1 minus 120574119899+1

) 119911119899+1

]

minusS projC [120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899]1003817100381710038171003817

le1003817100381710038171003817120574119899+1984858 (119911119899+1) + (1 minus 120574

119899+1) 119911119899+1

minus120574119899984858 (119911119899) minus (1 minus 120574

119899) 119911119899

1003817100381710038171003817

=1003817100381710038171003817(1 minus 120574

119899+1) (119911119899+1

minus 119911119899) + (120574119899minus 120574119899+1

)

times (119911119899minus 984858 (119911

119899)) + 120574

119899+1(984858 (119911119899+1

) minus 984858 (119911119899))1003817100381710038171003817

le [1 minus (1 minus 120581) 120574119899+1

]1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574

119899

10038161003816100381610038161003817100381710038171003817119911119899 minus 984858 (119911

119899)1003817100381710038171003817

(47)

It follows that1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 le

1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

(1 minus 120581) 120574119899+1

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817 (48)

Set

120574119899=

120585119899

120578119899

(49)

By condition (C2) 120574119899

rarr 0+ and 120574

119899isin (0 1) for 119899 large

enough Hence by (46) and (48) we have

⟨minus(119911119899minus 984858 (119911

119899)) +

120578119899(T119911119899minus 119911119899)

120585119899

119909119899+1

minus 119911119899⟩

le120585119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

le120585119899

120578119899

1198722

2

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817 le

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

1198722

(50)

By (29) we have1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minusprojC1199091198991003817100381710038171003817

le 120585119899

1003817100381710038171003817119909119899 minus 984858 (119909119899)1003817100381710038171003817 + 120578119899

1003817100381710038171003817T119909119899 minus 119909119899

1003817100381710038171003817

le (120585119899+ 120578119899)1198722

(51)

Next we estimate 119909119899+1

minus 119911119899+1

Since 119909119899+1

= projC[119910119899]⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩ le 0 Then we have

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

= ⟨119909119899+1

minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩

+ ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

le ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨[120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minus119911119899 119909119899+1

minus 119911119899⟩

= (1 minus 120585119899minus 120578119899) ⟨119909119899minus 119911119899 119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899minus T119909119899+1

119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899+1

minus T119911119899 119909119899+1

minus 119911119899⟩

+ ⟨120585119899(984858 (119909119899) minus 119911119899)

+ 120578119899(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

(52)

It follows that

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus 120585119899minus 120578119899)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 1205781198991198711003817100381710038171003817119909119899 minus 119909

119899+1

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+ 120585119899⟨984858 (119911

119899) minus 119911119899+

120578119899

120585119899

(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

le1 minus 120585119899minus 120578119899

2(1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

)

+1205782

119899

2

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1198712

2

1003817100381710038171003817119909119899 minus 119909119899+1

10038171003817100381710038172

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1205852

119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(53)

Thus

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le1 minus 120585119899minus 120578119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899 minus 119911119899

10038171003817100381710038172

+1198712

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119909119899

10038171003817100381710038172

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

+1205782

119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+(120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

11987121198722

2+

21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1198722

2

+1205782

119899

1 + 120585119899minus 120578119899

41198722

2

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)

times (1003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)2

Abstract and Applied Analysis 7

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

119872

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

(54)

where the finite constant119872 gt 0 is given by

119872 = max 119871211987222 41198722

2

1198722sup119899

(21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

(55)

Set

120575119899=

2120585119899

1 + 120585119899minus 120578119899

asymp 2120585119899

(as 119899 997888rarr infin)

120579119899=

120585119899120578119899minus1

minus 120585119899minus1

120578119899

21205852119899120578119899minus1

+1

2(120585119899+ 2120578119899+

1205782

119899

120585119899

) +120585119899

120578119899

+1205782

119899

2120585119899

119872

(56)

Then (54) can be rewritten as1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

le (1 minus 120575119899)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+ 120575119899120579119899 (57)

By the conditions (C1)ndash(C3) we deuce that

lim119899rarrinfin

120575119899= 0

infin

sum119899=1

120575119899= infin lim

119899rarrinfin120579119899= 0 (58)

From Lemma 4 and (57) we get 119909119899+1

minus 1199111198992

rarr 0 as 119899 rarr

infin This completes the proof

Algorithm 9 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontraction Let120585119899 sub [0 1] and 120578

119899 sub [0 1] be two real number sequences

For 1199090isin C we define a sequence 119909

119899 iteratively by

119909119899+1

= projC [(1 minus 120585119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0 (59)

Corollary 10 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 Suppose the followingconditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0Then the sequence 119909

119899 generated by the algorithm (59)

converges strongly to the minimum-norm fixed point of T

Conflict of Interests

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

Acknowledgment

The first author was supported by the National NaturalScience Foundation of China Grant no 11226125

References

[1] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 no 4 pp 365ndash374 1974

[2] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974

[3] C E Chidume and S A Mutangadura ldquoAn example onthe mann iteration method for lipschitz pseudocontractionsrdquoProceedings of the American Mathematical Society vol 129 no8 pp 2359ndash2363 2001

[4] H Zhou ldquoConvergence theorems of fixed points for Lipschitzpseudo-contractions inHilbert spacesrdquo Journal ofMathematicalAnalysis and Applications vol 343 no 1 pp 546ndash556 2008

[5] Y Yao Y-C Liou and G Marino ldquoA hybrid algorithmfor pseudo-contractive mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 10 pp 4997ndash5002 2009

[6] H Zegeye N Shahzad and M A Alghamdi ldquoConvergence ofIshikawarsquos iteration method for pseudocontractive mappingsrdquoNonlinear Analysis Theory Methods and Applications vol 74no 18 pp 7304ndash7311 2011

[7] L-C Ceng A Petrusel and J-C Yao ldquoStrong convergence ofmodified implicit iterative algorithmswith perturbedmappingsfor continuous pseudocontractive mappingsrdquo Applied Mathe-matics and Computation vol 209 no 2 pp 162ndash176 2009

[8] L-C Ceng A Petrusel and J-C Yao ldquoIterative approximationof fixed points for asymptotically strict pseudocontractive typemappings in the intermediate senserdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 587ndash606 2011

8 Abstract and Applied Analysis

[9] L-C Ceng and J-C Yao ldquoStrong convergence theorems forvariational inequalities and fixed point problems of asymptot-ically strict pseudocontractive mappings in the intermediatesenserdquo Acta Applicandae Mathematicae vol 115 no 2 pp 167ndash191 2011

[10] J C Yao and L C Zeng ldquoStrong convergence of averagedapproximants for asymptotically pseudocontractive mappingsin Banach spacesrdquo Journal of Nonlinear and Convex Analysisvol 8 pp 451ndash462 2007

[11] L C Ceng Q H Ansari and C F Wen ldquoImplicit relaxed andhybrid methods with regularization for minimization problemsand asymptotically strict pseudocontractive mappings in theintermediate senserdquo Abstract and Applied Analysis vol 2013Article ID 854297 14 pages 2013

[12] C E ChidumeMAbbas andBAli ldquoConvergence of theManniteration algorithm for a class of pseudocontractive mappingsrdquoApplied Mathematics and Computation vol 194 no 1 pp 1ndash62007

[13] C E Chidume and H Zegeye ldquoIterative solution of nonlinearequations of accretive and pseudocontractive typesrdquo Journal ofMathematical Analysis and Applications vol 282 no 2 pp 756ndash765 2003

[14] L Ciric A Rafiq N Cakic and J S Ume ldquoImplicit Mannfixedpoint iterations for pseudo-contractivemappingsrdquoAppliedMathematics Letters vol 22 no 4 pp 581ndash584 2009

[15] C Moore and B V C Nnoli ldquoStrong convergence of averagedapproximants for Lipschitz pseudocontractive mapsrdquo Journal ofMathematical Analysis and Applications vol 260 no 1 pp 269ndash278 2001

[16] A Udomene ldquoPath convergence approximation of fixed pointsand variational solutions of Lipschitz pseudocontractions inBanach spacesrdquoNonlinear Analysis Theory Methods and Appli-cations vol 67 no 8 pp 2403ndash2414 2007

[17] H Zegeye N Shahzad and T Mekonen ldquoViscosity approx-imation methods for pseudocontractive mappings in Banachspacesrdquo Applied Mathematics and Computation vol 185 no 1pp 538ndash546 2007

[18] Q-B Zhang andC-Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008

[19] H Zhou ldquoStrong convergence of an explicit iterative algorithmfor continuous pseudo-contractions in Banach spacesrdquo Nonlin-ear Analysis Theory Methods and Applications vol 70 no 11pp 4039ndash4046 2009

[20] K Geobel and W A Kirk Topics in Metric Fixed PointTheory vol 28 Cambridge Studies in Advanced MathematicsCambridge University Press 1990

[21] X Lu H-K Xu and X Yin ldquoHybrid methods for a class ofmonotone variational inequalitiesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 3-4 pp 1032ndash1041 2009

[22] H K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 2 pp 1ndash17 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Then as 119905 rarr 0+ the net 119906

119905 converges strongly to a point

119909Daggerisin Fix(W) which solves the following variational inequality

119909Daggerisin Fix (W) ⟨(119868 minus 984858) 119909

Dagger 119911 minus 119909

Dagger⟩ ge 0 119911 isin Fix (W)

(10)Proof For 119905 isin (0 1) define a mappingW

119905 C rarr C by

W119905119906 = W projC [119905984858 (119906) + (1 minus 119905) 119906] 119906 isin C (11)

For any 119906 V isin C we have1003817100381710038171003817W119905119906 minusW

119905V1003817100381710038171003817 =

1003817100381710038171003817W projC [119905984858 (119906) + (1 minus 119905) 119906]

minusW projC [119905984858 (V) + (1 minus 119905) V]1003817100381710038171003817

le 1199051003817100381710038171003817984858 (119906) minus 984858 (V)1003817100381710038171003817 + (1 minus 119905) 119906 minus V

le [1 minus (1 minus 120581) 119905] 119906 minus V

(12)

Hence W119905is a 1 minus (1 minus 120581)119905-contraction on C with 119906

119905isin C as

its unique fixed point So 119906119905 is well defined

Let 119906 isin Fix(W) From (9) we have1003817100381710038171003817119906119905 minus 119906

1003817100381710038171003817 =1003817100381710038171003817W projC [119905984858 (119906

119905) + (1 minus 119905) 119906

119905] minusW projC119906

1003817100381710038171003817

le 1199051003817100381710038171003817984858 (119906119905) minus 984858 (119906)

1003817100381710038171003817 + 1199051003817100381710038171003817984858 (119906) minus 119906

1003817100381710038171003817 + (1 minus 119905)1003817100381710038171003817119906119905 minus 119906

1003817100381710038171003817

le [1 minus (1 minus 120581) 119905]1003817100381710038171003817119906119905 minus 119906

1003817100381710038171003817 + 1199051003817100381710038171003817984858 (119906) minus 119906

1003817100381710038171003817

(13)It follows that

1003817100381710038171003817119906119905 minus 1199061003817100381710038171003817 le

1003817100381710038171003817984858 (119906) minus 1199061003817100381710038171003817

1 minus 120581 (14)

Thus 119906119905 and 984858(119906

119905) are bounded

Again from (9) we get1003817100381710038171003817119906119905 minusW119906

119905

1003817100381710038171003817 =1003817100381710038171003817W projC [119905984858 (119906

119905) + (1 minus 119905) 119906

119905] minusW projC119906119905

1003817100381710038171003817

le 1199051003817100381710038171003817984858 (119906119905) minus 119906

119905

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0+

(15)

Let 119905119899 sub (0 1) be a sequence such that 119905

119899rarr 0+ as 119899 rarr infin

Put 119906119899= 119906119905119899

From (15) we have1003817100381710038171003817119906119899 minusW119906

119899

1003817100381710038171003817 997888rarr 0 (16)From (9) we obtain1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

=1003817100381710038171003817W projC[119905984858(119906119905) + (1 minus 119905)119906

119905] minusW projC119906

10038171003817100381710038172

le1003817100381710038171003817119906119905 minus 119906 + 119905 (984858 (119906

119905) minus 119906119905)10038171003817100381710038172

=1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

+ 2119905 ⟨984858 (119906119905) minus 119906119905 119906119905minus 119906⟩

+ 11990521003817100381710038171003817984858(119906119905) minus 119906

119905

10038171003817100381710038172

=1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

+ 2119905 ⟨984858 (119906119905) minus 984858 (119906) 119906

119905minus 119906⟩

+ 2119905 ⟨984858 (119906) minus 119906 119906119905minus 119906⟩

+ 2119905⟨119906 minus 119906119905 119906119905minus 119906⟩ + 119905

21003817100381710038171003817984858(119906119905) minus 119906119905

10038171003817100381710038172

le [1 minus 2 (1 minus 120581) 119905]1003817100381710038171003817119906119905 minus 119906

10038171003817100381710038172

+ 2119905 ⟨984858 (119906) minus 119906 119906119905minus 119906⟩ + 119905

21003817100381710038171003817984858(119906119905) minus 119906119905

10038171003817100381710038172

(17)

It follows that

1003817100381710038171003817119906119905 minus 11990610038171003817100381710038172

le1

1 minus 120581⟨984858 (119906) minus 119906 119906

119905minus 119906⟩ + 119905119872 (18)

where119872 gt 0 is a constant such that

119872 gt1

2 (1 minus 120581)sup

1003817100381710038171003817984858(119906119905) minus 119906119905

10038171003817100381710038172

119905 isin (0 1) (19)

In particular we have

1003817100381710038171003817119906119899 minus 11990610038171003817100381710038172

le1

1 minus 120581⟨984858 (119906) minus 119906 119906

119899minus 119906⟩ + 119905

119899119872 119906 isin Fix (W)

(20)

Noting that 119906119899 is bounded without loss of generality we

assume that 119906119899 119909Dagger It is obvious that 119909Dagger isin C From (16)

and Lemma 2 we deduce 119909Dagger isin Fix(W) Substitute 119909Dagger for 119906 in(20) to get

10038171003817100381710038171003817119906119899minus 119909Dagger10038171003817100381710038171003817

2

le1

1 minus 120581⟨984858 (119909Dagger) minus 119909Dagger 119909119899minus 119909Dagger⟩ + 119905119899119872 (21)

Since 119906119899 119909Dagger we deduce from (21) that 119906

119899rarr 119909Dagger The net

119906119905 is therefore relatively compact as 119905 rarr 0

+ in the normtopology

In (20) letting 119899 rarr infin we get

10038171003817100381710038171003817119909Daggerminus 119906

10038171003817100381710038171003817

2

le1

1 minus 120581⟨984858 (119906) minus 119906 119909

Daggerminus 119906⟩ 119906 isin Fix (W)

(22)

Therefore 119909Dagger solves the variational inequality

119909Daggerisin Fix (W) ⟨(119868 minus 984858) 119906 119906 minus 119909

Dagger⟩ ge 0 119906 isin Fix (W)

(23)

By Lemma 3 (23) equals its dual variational inequality

119909Daggerisin Fix (W) ⟨(I minus 984858) 119909

Dagger 119906 minus 119909

Dagger⟩ ge 0 119906 isin Fix (W)

(24)

This indicates that 119909Dagger = (projFix(W)984858)119909Dagger That is 119909Dagger is the

unique fixed point in Fix(W) of the contraction projFix(W)984858So the entire net 119906

119905 converges in norm to 119909

Dagger as 119905 rarr 0+

This completes the proof

Corollary 6 Let C be a nonempty closed convex subset of areal Hilbert space H Let W C rarr C be a nonexpansivemapping with Fix(W) = 0

For each 119905 isin (0 1) let the net 119906119905 be defined by

119906119905= W projC [(1 minus 119905) 119906

119905] (25)

Then as 119905 rarr 0+ the net 119906

119905 converges strongly to the

minimum-norm fixed point ofW

Proof As a matter of fact in (9) we choose 984858 = 0 and then(9) reduces to (25) From Proposition 5 (24) is reduced to

0 le ⟨119909Dagger 119906 minus 119909

Dagger⟩ 119906 isin Fix (W) (26)

4 Abstract and Applied Analysis

Equivalently

10038171003817100381710038171003817119909Dagger10038171003817100381710038171003817

2

le ⟨119909Dagger 119906⟩ 119906 isin Fix (W) (27)

This implies that10038171003817100381710038171003817119909Dagger10038171003817100381710038171003817

le 119906 119906 isin Fix (W) (28)

Therefore 119909Dagger is the minimum-norm fixed point of W Thiscompletes the proof

Algorithm 7 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontractionand let 984858 C rarr H be a 120581-contraction Let 120585

119899 sub [0 1] and

120578119899 sub [0 1] be two real number sequences For 119909

0isin C we

define a sequence 119909119899 iteratively by

119909119899+1

= projC [120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0

(29)

Theorem 8 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 and 984858 C rarr H a 120581-contraction Suppose the following conditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0

Then the sequence 119909119899 generated by the algorithm (29)

converges strongly to 119909Dagger= (projFix(T)984858)119909

Dagger

Proof First we prove that the sequence 119909119899 is bounded We

will show this fact by induction According to conditions (C1)and (C2) there exists a sufficiently large positive integer 119898

such that

1 minus2

1 minus 120581(119871 + 1) (119871 + 3) (120585

119899+ 2120578119899+

1205782

119899

120585119899

) gt 0 119899 ge 119898

(30)

Fix a 119901 isin Fix(T) and take a constant1198721gt 0 such that

max 10038171003817100381710038171199090 minus 1199011003817100381710038171003817

10038171003817100381710038171199091 minus 1199011003817100381710038171003817

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817 2

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

le 1198721

(31)

Next we show that 119909119898+1

minus 119901 le 1198721

Set

119910119898

= 120585119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898

thus 119909119898+1

= projC [119910119898]

(32)

Then we have

⟨119909119898+1

minus 119910119898 119909119898+1

minus 119901⟩ le 0 (33)

Since T is pseudocontractive I minus T is monotone So we have

⟨(I minus T) 119909119898+1

minus (I minus T) 119901 119909119898+1

minus 119901⟩ ge 0 (34)

From (29) (33) and (34) we obtain

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

= ⟨119909119898+1

minus 119901 119909119898+1

minus 119901⟩

= ⟨119909119898+1

minus 119910119898 119909119898+1

minus 119901⟩ + ⟨119910119898minus 119901 119909

119898+1minus 119901⟩

le ⟨119910119898minus 119901 119909

119898+1minus 119901⟩

= ⟨119909119898minus 119901 119909

119898+1minus 119901⟩

minus 120585119898⟨119909119898minus 984858 (119909

119898) 119909119898+1

minus 119901⟩

+ 120578119898⟨T119909119898minus 119909119898 119909119898+1

minus 119901⟩

= ⟨119909119898minus 119901 119909

119898+1minus 119901⟩

+ 120585119898⟨119909119898+1

minus 119909119898 119909119898+1

minus 119901⟩

+ 120585119898⟨984858 (119909119898) minus 984858 (119901) 119909

119898+1minus 119901⟩

+ 120585119898⟨119891 (119901) minus 119901 119909

119898+1minus 119901⟩

minus 120585119898⟨119909119898+1

minus 119901 119909119898+1

minus 119901⟩

+ 120578119898⟨T119909119898minus T119909119898+1

119909119898+1

minus 119901⟩

+ 120578119898⟨119909119898+1

minus 119909119898 119909119898+1

minus 119901⟩

minus 120578119898⟨119909119898+1

minus T119909119898+1

119909119898+1

minus 119901⟩

le1003817100381710038171003817119909119898 minus 119901

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817119909119898+1 minus 119909119898

1003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 1205851198981205811003817100381710038171003817119909119898 minus 119901

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

+ 120585119898

1003817100381710038171003817984858 (119901) minus 11990110038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 minus 120585119898

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

+ 120578119898(1003817100381710038171003817T119909119898 minus T119909

119898+1

1003817100381710038171003817 +1003817100381710038171003817119909119898+1 minus 119909

119898

1003817100381710038171003817)

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 minus 120585119898

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

+ (119871 + 1) (120585119898+ 120578119898)1003817100381710038171003817119909119898+1 minus 119909

119898

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

(35)

It follows that

(1 + 120585119898)1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

+120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

+ (119871 + 1) (120585119898+ 120578119898)1003817100381710038171003817119909119898+1 minus 119909

119898

1003817100381710038171003817

(36)

Abstract and Applied Analysis 5

By (29) we have

1003817100381710038171003817119909119898+1 minus 119909119898

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898]

minusprojC [119909119898]1003817100381710038171003817

le1003817100381710038171003817120585119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898minus 119909119898

1003817100381710038171003817

le 120585119898(1003817100381710038171003817984858 (119909119898) minus 984858 (119901)

1003817100381710038171003817 +1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ 120578119898(1003817100381710038171003817T119909119898 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

le 120585119898(1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 + (1 + 120581)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ (119871 + 1) 120578119898

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

le (119871 + 1 + 120581) (120585119898+ 120578119898)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

le (119871 + 3) (120585119898+ 120578119898)1198721

(37)

Substitute (37) into (36) to obtain

(1 + 120585119898)1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

+ (119871 + 1) (119871 + 3) (120585119898+ 120578119898)2

1198721

le (1 +1 + 120581

2120585119898)1198721+ (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1198721

(38)

that is1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le [1 minus((1 minus 120581) 120585

1198982) minus (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1 + 120585119898

]1198721

= 1 minus ((((1 minus 120581) 120585

119898

2)[1 minus

2

1 minus 120581(119871 + 1) (119871 + 3)

times (120585119898+ 2120578119898+ (

1205782

119898

120585119898

))])

times (1 + 120585119898)minus1

)1198721

le 1198721

(39)

By induction we get1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 le 1198721 forall119899 ge 0 (40)

which implies that 119909119899 is bounded and so is T119909

119899 Now we

take a constant1198722gt 0 such that

1198722= sup119899

1003817100381710038171003817119909119899

1003817100381710038171003817 +1003817100381710038171003817T119909119899 minus 119909

119899

1003817100381710038171003817 (41)

Set S = (2I minus T)minus1 (ie S is a resolvent of the monotone

operator I minus T) We then have that S is a nonexpansive selfmapping of C and Fix(S) = Fix(T)

By Proposition 5 we know that whenever 120574119899 sub (0 1)

and 120574119899rarr 0+ the sequence 119911

119899 defined by

119911119899= S projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] (42)

converges strongly to the fixed point 119909Dagger of S (and of T asFix(S) = Fix(T)) Without loss of generality we may assumethat 119911

119899 le 119872

2for all 119899

It suffices to prove that 119909119899+1

minus 119911119899 rarr 0 as 119899 rarr infin (for

some 120574119899rarr 0+) To this end we rewrite (42) as

(2I minus T) 119911119899= projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] 119899 ge 0 (43)

By using the property of the metric projection we have

⟨120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus (2119911119899minus T119911119899) 119909119899+1

minus (2119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩

+ ⟨T119911119899minus 119911119899 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

+1003817100381710038171003817119911119899 minus T119911

119899

10038171003817100381710038172

le ⟨120574119899(119911119899minus 984858 (119911

119899)) T119911

119899minus 119911119899⟩

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

le 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

997904rArr ⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩

le1003817100381710038171003817119911119899 minus 984858 (119911

119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

(44)

Note that

1003817100381710038171003817119911119899 minus T119911119899

1003817100381710038171003817 =1003817100381710038171003817projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] minus 119911119899

1003817100381710038171003817

le1003817100381710038171003817120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus 119911119899

1003817100381710038171003817

= 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817

(45)

Hence we get

⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩ le 120574

119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(46)

6 Abstract and Applied Analysis

From (42) we have1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 =1003817100381710038171003817S projC [120574

119899+1984858 (119911119899+1

) + (1 minus 120574119899+1

) 119911119899+1

]

minusS projC [120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899]1003817100381710038171003817

le1003817100381710038171003817120574119899+1984858 (119911119899+1) + (1 minus 120574

119899+1) 119911119899+1

minus120574119899984858 (119911119899) minus (1 minus 120574

119899) 119911119899

1003817100381710038171003817

=1003817100381710038171003817(1 minus 120574

119899+1) (119911119899+1

minus 119911119899) + (120574119899minus 120574119899+1

)

times (119911119899minus 984858 (119911

119899)) + 120574

119899+1(984858 (119911119899+1

) minus 984858 (119911119899))1003817100381710038171003817

le [1 minus (1 minus 120581) 120574119899+1

]1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574

119899

10038161003816100381610038161003817100381710038171003817119911119899 minus 984858 (119911

119899)1003817100381710038171003817

(47)

It follows that1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 le

1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

(1 minus 120581) 120574119899+1

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817 (48)

Set

120574119899=

120585119899

120578119899

(49)

By condition (C2) 120574119899

rarr 0+ and 120574

119899isin (0 1) for 119899 large

enough Hence by (46) and (48) we have

⟨minus(119911119899minus 984858 (119911

119899)) +

120578119899(T119911119899minus 119911119899)

120585119899

119909119899+1

minus 119911119899⟩

le120585119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

le120585119899

120578119899

1198722

2

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817 le

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

1198722

(50)

By (29) we have1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minusprojC1199091198991003817100381710038171003817

le 120585119899

1003817100381710038171003817119909119899 minus 984858 (119909119899)1003817100381710038171003817 + 120578119899

1003817100381710038171003817T119909119899 minus 119909119899

1003817100381710038171003817

le (120585119899+ 120578119899)1198722

(51)

Next we estimate 119909119899+1

minus 119911119899+1

Since 119909119899+1

= projC[119910119899]⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩ le 0 Then we have

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

= ⟨119909119899+1

minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩

+ ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

le ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨[120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minus119911119899 119909119899+1

minus 119911119899⟩

= (1 minus 120585119899minus 120578119899) ⟨119909119899minus 119911119899 119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899minus T119909119899+1

119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899+1

minus T119911119899 119909119899+1

minus 119911119899⟩

+ ⟨120585119899(984858 (119909119899) minus 119911119899)

+ 120578119899(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

(52)

It follows that

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus 120585119899minus 120578119899)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 1205781198991198711003817100381710038171003817119909119899 minus 119909

119899+1

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+ 120585119899⟨984858 (119911

119899) minus 119911119899+

120578119899

120585119899

(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

le1 minus 120585119899minus 120578119899

2(1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

)

+1205782

119899

2

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1198712

2

1003817100381710038171003817119909119899 minus 119909119899+1

10038171003817100381710038172

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1205852

119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(53)

Thus

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le1 minus 120585119899minus 120578119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899 minus 119911119899

10038171003817100381710038172

+1198712

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119909119899

10038171003817100381710038172

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

+1205782

119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+(120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

11987121198722

2+

21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1198722

2

+1205782

119899

1 + 120585119899minus 120578119899

41198722

2

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)

times (1003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)2

Abstract and Applied Analysis 7

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

119872

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

(54)

where the finite constant119872 gt 0 is given by

119872 = max 119871211987222 41198722

2

1198722sup119899

(21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

(55)

Set

120575119899=

2120585119899

1 + 120585119899minus 120578119899

asymp 2120585119899

(as 119899 997888rarr infin)

120579119899=

120585119899120578119899minus1

minus 120585119899minus1

120578119899

21205852119899120578119899minus1

+1

2(120585119899+ 2120578119899+

1205782

119899

120585119899

) +120585119899

120578119899

+1205782

119899

2120585119899

119872

(56)

Then (54) can be rewritten as1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

le (1 minus 120575119899)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+ 120575119899120579119899 (57)

By the conditions (C1)ndash(C3) we deuce that

lim119899rarrinfin

120575119899= 0

infin

sum119899=1

120575119899= infin lim

119899rarrinfin120579119899= 0 (58)

From Lemma 4 and (57) we get 119909119899+1

minus 1199111198992

rarr 0 as 119899 rarr

infin This completes the proof

Algorithm 9 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontraction Let120585119899 sub [0 1] and 120578

119899 sub [0 1] be two real number sequences

For 1199090isin C we define a sequence 119909

119899 iteratively by

119909119899+1

= projC [(1 minus 120585119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0 (59)

Corollary 10 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 Suppose the followingconditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0Then the sequence 119909

119899 generated by the algorithm (59)

converges strongly to the minimum-norm fixed point of T

Conflict of Interests

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

Acknowledgment

The first author was supported by the National NaturalScience Foundation of China Grant no 11226125

References

[1] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 no 4 pp 365ndash374 1974

[2] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974

[3] C E Chidume and S A Mutangadura ldquoAn example onthe mann iteration method for lipschitz pseudocontractionsrdquoProceedings of the American Mathematical Society vol 129 no8 pp 2359ndash2363 2001

[4] H Zhou ldquoConvergence theorems of fixed points for Lipschitzpseudo-contractions inHilbert spacesrdquo Journal ofMathematicalAnalysis and Applications vol 343 no 1 pp 546ndash556 2008

[5] Y Yao Y-C Liou and G Marino ldquoA hybrid algorithmfor pseudo-contractive mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 10 pp 4997ndash5002 2009

[6] H Zegeye N Shahzad and M A Alghamdi ldquoConvergence ofIshikawarsquos iteration method for pseudocontractive mappingsrdquoNonlinear Analysis Theory Methods and Applications vol 74no 18 pp 7304ndash7311 2011

[7] L-C Ceng A Petrusel and J-C Yao ldquoStrong convergence ofmodified implicit iterative algorithmswith perturbedmappingsfor continuous pseudocontractive mappingsrdquo Applied Mathe-matics and Computation vol 209 no 2 pp 162ndash176 2009

[8] L-C Ceng A Petrusel and J-C Yao ldquoIterative approximationof fixed points for asymptotically strict pseudocontractive typemappings in the intermediate senserdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 587ndash606 2011

8 Abstract and Applied Analysis

[9] L-C Ceng and J-C Yao ldquoStrong convergence theorems forvariational inequalities and fixed point problems of asymptot-ically strict pseudocontractive mappings in the intermediatesenserdquo Acta Applicandae Mathematicae vol 115 no 2 pp 167ndash191 2011

[10] J C Yao and L C Zeng ldquoStrong convergence of averagedapproximants for asymptotically pseudocontractive mappingsin Banach spacesrdquo Journal of Nonlinear and Convex Analysisvol 8 pp 451ndash462 2007

[11] L C Ceng Q H Ansari and C F Wen ldquoImplicit relaxed andhybrid methods with regularization for minimization problemsand asymptotically strict pseudocontractive mappings in theintermediate senserdquo Abstract and Applied Analysis vol 2013Article ID 854297 14 pages 2013

[12] C E ChidumeMAbbas andBAli ldquoConvergence of theManniteration algorithm for a class of pseudocontractive mappingsrdquoApplied Mathematics and Computation vol 194 no 1 pp 1ndash62007

[13] C E Chidume and H Zegeye ldquoIterative solution of nonlinearequations of accretive and pseudocontractive typesrdquo Journal ofMathematical Analysis and Applications vol 282 no 2 pp 756ndash765 2003

[14] L Ciric A Rafiq N Cakic and J S Ume ldquoImplicit Mannfixedpoint iterations for pseudo-contractivemappingsrdquoAppliedMathematics Letters vol 22 no 4 pp 581ndash584 2009

[15] C Moore and B V C Nnoli ldquoStrong convergence of averagedapproximants for Lipschitz pseudocontractive mapsrdquo Journal ofMathematical Analysis and Applications vol 260 no 1 pp 269ndash278 2001

[16] A Udomene ldquoPath convergence approximation of fixed pointsand variational solutions of Lipschitz pseudocontractions inBanach spacesrdquoNonlinear Analysis Theory Methods and Appli-cations vol 67 no 8 pp 2403ndash2414 2007

[17] H Zegeye N Shahzad and T Mekonen ldquoViscosity approx-imation methods for pseudocontractive mappings in Banachspacesrdquo Applied Mathematics and Computation vol 185 no 1pp 538ndash546 2007

[18] Q-B Zhang andC-Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008

[19] H Zhou ldquoStrong convergence of an explicit iterative algorithmfor continuous pseudo-contractions in Banach spacesrdquo Nonlin-ear Analysis Theory Methods and Applications vol 70 no 11pp 4039ndash4046 2009

[20] K Geobel and W A Kirk Topics in Metric Fixed PointTheory vol 28 Cambridge Studies in Advanced MathematicsCambridge University Press 1990

[21] X Lu H-K Xu and X Yin ldquoHybrid methods for a class ofmonotone variational inequalitiesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 3-4 pp 1032ndash1041 2009

[22] H K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 2 pp 1ndash17 2002

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Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Equivalently

10038171003817100381710038171003817119909Dagger10038171003817100381710038171003817

2

le ⟨119909Dagger 119906⟩ 119906 isin Fix (W) (27)

This implies that10038171003817100381710038171003817119909Dagger10038171003817100381710038171003817

le 119906 119906 isin Fix (W) (28)

Therefore 119909Dagger is the minimum-norm fixed point of W Thiscompletes the proof

Algorithm 7 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontractionand let 984858 C rarr H be a 120581-contraction Let 120585

119899 sub [0 1] and

120578119899 sub [0 1] be two real number sequences For 119909

0isin C we

define a sequence 119909119899 iteratively by

119909119899+1

= projC [120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0

(29)

Theorem 8 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 and 984858 C rarr H a 120581-contraction Suppose the following conditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0

Then the sequence 119909119899 generated by the algorithm (29)

converges strongly to 119909Dagger= (projFix(T)984858)119909

Dagger

Proof First we prove that the sequence 119909119899 is bounded We

will show this fact by induction According to conditions (C1)and (C2) there exists a sufficiently large positive integer 119898

such that

1 minus2

1 minus 120581(119871 + 1) (119871 + 3) (120585

119899+ 2120578119899+

1205782

119899

120585119899

) gt 0 119899 ge 119898

(30)

Fix a 119901 isin Fix(T) and take a constant1198721gt 0 such that

max 10038171003817100381710038171199090 minus 1199011003817100381710038171003817

10038171003817100381710038171199091 minus 1199011003817100381710038171003817

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817 2

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

le 1198721

(31)

Next we show that 119909119898+1

minus 119901 le 1198721

Set

119910119898

= 120585119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898

thus 119909119898+1

= projC [119910119898]

(32)

Then we have

⟨119909119898+1

minus 119910119898 119909119898+1

minus 119901⟩ le 0 (33)

Since T is pseudocontractive I minus T is monotone So we have

⟨(I minus T) 119909119898+1

minus (I minus T) 119901 119909119898+1

minus 119901⟩ ge 0 (34)

From (29) (33) and (34) we obtain

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

= ⟨119909119898+1

minus 119901 119909119898+1

minus 119901⟩

= ⟨119909119898+1

minus 119910119898 119909119898+1

minus 119901⟩ + ⟨119910119898minus 119901 119909

119898+1minus 119901⟩

le ⟨119910119898minus 119901 119909

119898+1minus 119901⟩

= ⟨119909119898minus 119901 119909

119898+1minus 119901⟩

minus 120585119898⟨119909119898minus 984858 (119909

119898) 119909119898+1

minus 119901⟩

+ 120578119898⟨T119909119898minus 119909119898 119909119898+1

minus 119901⟩

= ⟨119909119898minus 119901 119909

119898+1minus 119901⟩

+ 120585119898⟨119909119898+1

minus 119909119898 119909119898+1

minus 119901⟩

+ 120585119898⟨984858 (119909119898) minus 984858 (119901) 119909

119898+1minus 119901⟩

+ 120585119898⟨119891 (119901) minus 119901 119909

119898+1minus 119901⟩

minus 120585119898⟨119909119898+1

minus 119901 119909119898+1

minus 119901⟩

+ 120578119898⟨T119909119898minus T119909119898+1

119909119898+1

minus 119901⟩

+ 120578119898⟨119909119898+1

minus 119909119898 119909119898+1

minus 119901⟩

minus 120578119898⟨119909119898+1

minus T119909119898+1

119909119898+1

minus 119901⟩

le1003817100381710038171003817119909119898 minus 119901

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817119909119898+1 minus 119909119898

1003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 1205851198981205811003817100381710038171003817119909119898 minus 119901

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

+ 120585119898

1003817100381710038171003817984858 (119901) minus 11990110038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 minus 120585119898

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

+ 120578119898(1003817100381710038171003817T119909119898 minus T119909

119898+1

1003817100381710038171003817 +1003817100381710038171003817119909119898+1 minus 119909

119898

1003817100381710038171003817)

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

times1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 minus 120585119898

1003817100381710038171003817119909119898+1 minus 11990110038171003817100381710038172

+ (119871 + 1) (120585119898+ 120578119898)1003817100381710038171003817119909119898+1 minus 119909

119898

10038171003817100381710038171003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

(35)

It follows that

(1 + 120585119898)1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817 le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

+120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

+ (119871 + 1) (120585119898+ 120578119898)1003817100381710038171003817119909119898+1 minus 119909

119898

1003817100381710038171003817

(36)

Abstract and Applied Analysis 5

By (29) we have

1003817100381710038171003817119909119898+1 minus 119909119898

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898]

minusprojC [119909119898]1003817100381710038171003817

le1003817100381710038171003817120585119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898minus 119909119898

1003817100381710038171003817

le 120585119898(1003817100381710038171003817984858 (119909119898) minus 984858 (119901)

1003817100381710038171003817 +1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ 120578119898(1003817100381710038171003817T119909119898 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

le 120585119898(1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 + (1 + 120581)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ (119871 + 1) 120578119898

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

le (119871 + 1 + 120581) (120585119898+ 120578119898)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

le (119871 + 3) (120585119898+ 120578119898)1198721

(37)

Substitute (37) into (36) to obtain

(1 + 120585119898)1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

+ (119871 + 1) (119871 + 3) (120585119898+ 120578119898)2

1198721

le (1 +1 + 120581

2120585119898)1198721+ (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1198721

(38)

that is1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le [1 minus((1 minus 120581) 120585

1198982) minus (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1 + 120585119898

]1198721

= 1 minus ((((1 minus 120581) 120585

119898

2)[1 minus

2

1 minus 120581(119871 + 1) (119871 + 3)

times (120585119898+ 2120578119898+ (

1205782

119898

120585119898

))])

times (1 + 120585119898)minus1

)1198721

le 1198721

(39)

By induction we get1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 le 1198721 forall119899 ge 0 (40)

which implies that 119909119899 is bounded and so is T119909

119899 Now we

take a constant1198722gt 0 such that

1198722= sup119899

1003817100381710038171003817119909119899

1003817100381710038171003817 +1003817100381710038171003817T119909119899 minus 119909

119899

1003817100381710038171003817 (41)

Set S = (2I minus T)minus1 (ie S is a resolvent of the monotone

operator I minus T) We then have that S is a nonexpansive selfmapping of C and Fix(S) = Fix(T)

By Proposition 5 we know that whenever 120574119899 sub (0 1)

and 120574119899rarr 0+ the sequence 119911

119899 defined by

119911119899= S projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] (42)

converges strongly to the fixed point 119909Dagger of S (and of T asFix(S) = Fix(T)) Without loss of generality we may assumethat 119911

119899 le 119872

2for all 119899

It suffices to prove that 119909119899+1

minus 119911119899 rarr 0 as 119899 rarr infin (for

some 120574119899rarr 0+) To this end we rewrite (42) as

(2I minus T) 119911119899= projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] 119899 ge 0 (43)

By using the property of the metric projection we have

⟨120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus (2119911119899minus T119911119899) 119909119899+1

minus (2119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩

+ ⟨T119911119899minus 119911119899 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

+1003817100381710038171003817119911119899 minus T119911

119899

10038171003817100381710038172

le ⟨120574119899(119911119899minus 984858 (119911

119899)) T119911

119899minus 119911119899⟩

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

le 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

997904rArr ⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩

le1003817100381710038171003817119911119899 minus 984858 (119911

119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

(44)

Note that

1003817100381710038171003817119911119899 minus T119911119899

1003817100381710038171003817 =1003817100381710038171003817projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] minus 119911119899

1003817100381710038171003817

le1003817100381710038171003817120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus 119911119899

1003817100381710038171003817

= 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817

(45)

Hence we get

⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩ le 120574

119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(46)

6 Abstract and Applied Analysis

From (42) we have1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 =1003817100381710038171003817S projC [120574

119899+1984858 (119911119899+1

) + (1 minus 120574119899+1

) 119911119899+1

]

minusS projC [120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899]1003817100381710038171003817

le1003817100381710038171003817120574119899+1984858 (119911119899+1) + (1 minus 120574

119899+1) 119911119899+1

minus120574119899984858 (119911119899) minus (1 minus 120574

119899) 119911119899

1003817100381710038171003817

=1003817100381710038171003817(1 minus 120574

119899+1) (119911119899+1

minus 119911119899) + (120574119899minus 120574119899+1

)

times (119911119899minus 984858 (119911

119899)) + 120574

119899+1(984858 (119911119899+1

) minus 984858 (119911119899))1003817100381710038171003817

le [1 minus (1 minus 120581) 120574119899+1

]1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574

119899

10038161003816100381610038161003817100381710038171003817119911119899 minus 984858 (119911

119899)1003817100381710038171003817

(47)

It follows that1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 le

1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

(1 minus 120581) 120574119899+1

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817 (48)

Set

120574119899=

120585119899

120578119899

(49)

By condition (C2) 120574119899

rarr 0+ and 120574

119899isin (0 1) for 119899 large

enough Hence by (46) and (48) we have

⟨minus(119911119899minus 984858 (119911

119899)) +

120578119899(T119911119899minus 119911119899)

120585119899

119909119899+1

minus 119911119899⟩

le120585119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

le120585119899

120578119899

1198722

2

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817 le

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

1198722

(50)

By (29) we have1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minusprojC1199091198991003817100381710038171003817

le 120585119899

1003817100381710038171003817119909119899 minus 984858 (119909119899)1003817100381710038171003817 + 120578119899

1003817100381710038171003817T119909119899 minus 119909119899

1003817100381710038171003817

le (120585119899+ 120578119899)1198722

(51)

Next we estimate 119909119899+1

minus 119911119899+1

Since 119909119899+1

= projC[119910119899]⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩ le 0 Then we have

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

= ⟨119909119899+1

minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩

+ ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

le ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨[120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minus119911119899 119909119899+1

minus 119911119899⟩

= (1 minus 120585119899minus 120578119899) ⟨119909119899minus 119911119899 119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899minus T119909119899+1

119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899+1

minus T119911119899 119909119899+1

minus 119911119899⟩

+ ⟨120585119899(984858 (119909119899) minus 119911119899)

+ 120578119899(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

(52)

It follows that

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus 120585119899minus 120578119899)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 1205781198991198711003817100381710038171003817119909119899 minus 119909

119899+1

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+ 120585119899⟨984858 (119911

119899) minus 119911119899+

120578119899

120585119899

(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

le1 minus 120585119899minus 120578119899

2(1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

)

+1205782

119899

2

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1198712

2

1003817100381710038171003817119909119899 minus 119909119899+1

10038171003817100381710038172

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1205852

119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(53)

Thus

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le1 minus 120585119899minus 120578119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899 minus 119911119899

10038171003817100381710038172

+1198712

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119909119899

10038171003817100381710038172

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

+1205782

119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+(120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

11987121198722

2+

21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1198722

2

+1205782

119899

1 + 120585119899minus 120578119899

41198722

2

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)

times (1003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)2

Abstract and Applied Analysis 7

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

119872

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

(54)

where the finite constant119872 gt 0 is given by

119872 = max 119871211987222 41198722

2

1198722sup119899

(21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

(55)

Set

120575119899=

2120585119899

1 + 120585119899minus 120578119899

asymp 2120585119899

(as 119899 997888rarr infin)

120579119899=

120585119899120578119899minus1

minus 120585119899minus1

120578119899

21205852119899120578119899minus1

+1

2(120585119899+ 2120578119899+

1205782

119899

120585119899

) +120585119899

120578119899

+1205782

119899

2120585119899

119872

(56)

Then (54) can be rewritten as1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

le (1 minus 120575119899)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+ 120575119899120579119899 (57)

By the conditions (C1)ndash(C3) we deuce that

lim119899rarrinfin

120575119899= 0

infin

sum119899=1

120575119899= infin lim

119899rarrinfin120579119899= 0 (58)

From Lemma 4 and (57) we get 119909119899+1

minus 1199111198992

rarr 0 as 119899 rarr

infin This completes the proof

Algorithm 9 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontraction Let120585119899 sub [0 1] and 120578

119899 sub [0 1] be two real number sequences

For 1199090isin C we define a sequence 119909

119899 iteratively by

119909119899+1

= projC [(1 minus 120585119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0 (59)

Corollary 10 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 Suppose the followingconditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0Then the sequence 119909

119899 generated by the algorithm (59)

converges strongly to the minimum-norm fixed point of T

Conflict of Interests

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

Acknowledgment

The first author was supported by the National NaturalScience Foundation of China Grant no 11226125

References

[1] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 no 4 pp 365ndash374 1974

[2] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974

[3] C E Chidume and S A Mutangadura ldquoAn example onthe mann iteration method for lipschitz pseudocontractionsrdquoProceedings of the American Mathematical Society vol 129 no8 pp 2359ndash2363 2001

[4] H Zhou ldquoConvergence theorems of fixed points for Lipschitzpseudo-contractions inHilbert spacesrdquo Journal ofMathematicalAnalysis and Applications vol 343 no 1 pp 546ndash556 2008

[5] Y Yao Y-C Liou and G Marino ldquoA hybrid algorithmfor pseudo-contractive mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 10 pp 4997ndash5002 2009

[6] H Zegeye N Shahzad and M A Alghamdi ldquoConvergence ofIshikawarsquos iteration method for pseudocontractive mappingsrdquoNonlinear Analysis Theory Methods and Applications vol 74no 18 pp 7304ndash7311 2011

[7] L-C Ceng A Petrusel and J-C Yao ldquoStrong convergence ofmodified implicit iterative algorithmswith perturbedmappingsfor continuous pseudocontractive mappingsrdquo Applied Mathe-matics and Computation vol 209 no 2 pp 162ndash176 2009

[8] L-C Ceng A Petrusel and J-C Yao ldquoIterative approximationof fixed points for asymptotically strict pseudocontractive typemappings in the intermediate senserdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 587ndash606 2011

8 Abstract and Applied Analysis

[9] L-C Ceng and J-C Yao ldquoStrong convergence theorems forvariational inequalities and fixed point problems of asymptot-ically strict pseudocontractive mappings in the intermediatesenserdquo Acta Applicandae Mathematicae vol 115 no 2 pp 167ndash191 2011

[10] J C Yao and L C Zeng ldquoStrong convergence of averagedapproximants for asymptotically pseudocontractive mappingsin Banach spacesrdquo Journal of Nonlinear and Convex Analysisvol 8 pp 451ndash462 2007

[11] L C Ceng Q H Ansari and C F Wen ldquoImplicit relaxed andhybrid methods with regularization for minimization problemsand asymptotically strict pseudocontractive mappings in theintermediate senserdquo Abstract and Applied Analysis vol 2013Article ID 854297 14 pages 2013

[12] C E ChidumeMAbbas andBAli ldquoConvergence of theManniteration algorithm for a class of pseudocontractive mappingsrdquoApplied Mathematics and Computation vol 194 no 1 pp 1ndash62007

[13] C E Chidume and H Zegeye ldquoIterative solution of nonlinearequations of accretive and pseudocontractive typesrdquo Journal ofMathematical Analysis and Applications vol 282 no 2 pp 756ndash765 2003

[14] L Ciric A Rafiq N Cakic and J S Ume ldquoImplicit Mannfixedpoint iterations for pseudo-contractivemappingsrdquoAppliedMathematics Letters vol 22 no 4 pp 581ndash584 2009

[15] C Moore and B V C Nnoli ldquoStrong convergence of averagedapproximants for Lipschitz pseudocontractive mapsrdquo Journal ofMathematical Analysis and Applications vol 260 no 1 pp 269ndash278 2001

[16] A Udomene ldquoPath convergence approximation of fixed pointsand variational solutions of Lipschitz pseudocontractions inBanach spacesrdquoNonlinear Analysis Theory Methods and Appli-cations vol 67 no 8 pp 2403ndash2414 2007

[17] H Zegeye N Shahzad and T Mekonen ldquoViscosity approx-imation methods for pseudocontractive mappings in Banachspacesrdquo Applied Mathematics and Computation vol 185 no 1pp 538ndash546 2007

[18] Q-B Zhang andC-Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008

[19] H Zhou ldquoStrong convergence of an explicit iterative algorithmfor continuous pseudo-contractions in Banach spacesrdquo Nonlin-ear Analysis Theory Methods and Applications vol 70 no 11pp 4039ndash4046 2009

[20] K Geobel and W A Kirk Topics in Metric Fixed PointTheory vol 28 Cambridge Studies in Advanced MathematicsCambridge University Press 1990

[21] X Lu H-K Xu and X Yin ldquoHybrid methods for a class ofmonotone variational inequalitiesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 3-4 pp 1032ndash1041 2009

[22] H K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 2 pp 1ndash17 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

By (29) we have

1003817100381710038171003817119909119898+1 minus 119909119898

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898]

minusprojC [119909119898]1003817100381710038171003817

le1003817100381710038171003817120585119898984858 (119909119898) + (1 minus 120585

119898minus 120578119898) 119909119898+ 120578119898T119909119898minus 119909119898

1003817100381710038171003817

le 120585119898(1003817100381710038171003817984858 (119909119898) minus 984858 (119901)

1003817100381710038171003817 +1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ 120578119898(1003817100381710038171003817T119909119898 minus 119901

1003817100381710038171003817 +1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

le 120585119898(1003817100381710038171003817984858 (119901) minus 119901

1003817100381710038171003817 + (1 + 120581)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817)

+ (119871 + 1) 120578119898

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817

le (119871 + 1 + 120581) (120585119898+ 120578119898)1003817100381710038171003817119909119898 minus 119901

1003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

le (119871 + 3) (120585119898+ 120578119898)1198721

(37)

Substitute (37) into (36) to obtain

(1 + 120585119898)1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le (1 + 120585119898120581)

1003817100381710038171003817119909119898 minus 1199011003817100381710038171003817 + 120585119898

1003817100381710038171003817984858 (119901) minus 1199011003817100381710038171003817

+ (119871 + 1) (119871 + 3) (120585119898+ 120578119898)2

1198721

le (1 +1 + 120581

2120585119898)1198721+ (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1198721

(38)

that is1003817100381710038171003817119909119898+1 minus 119901

1003817100381710038171003817

le [1 minus((1 minus 120581) 120585

1198982) minus (119871 + 1) (119871 + 3) (120585

119898+ 120578119898)2

1 + 120585119898

]1198721

= 1 minus ((((1 minus 120581) 120585

119898

2)[1 minus

2

1 minus 120581(119871 + 1) (119871 + 3)

times (120585119898+ 2120578119898+ (

1205782

119898

120585119898

))])

times (1 + 120585119898)minus1

)1198721

le 1198721

(39)

By induction we get1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 le 1198721 forall119899 ge 0 (40)

which implies that 119909119899 is bounded and so is T119909

119899 Now we

take a constant1198722gt 0 such that

1198722= sup119899

1003817100381710038171003817119909119899

1003817100381710038171003817 +1003817100381710038171003817T119909119899 minus 119909

119899

1003817100381710038171003817 (41)

Set S = (2I minus T)minus1 (ie S is a resolvent of the monotone

operator I minus T) We then have that S is a nonexpansive selfmapping of C and Fix(S) = Fix(T)

By Proposition 5 we know that whenever 120574119899 sub (0 1)

and 120574119899rarr 0+ the sequence 119911

119899 defined by

119911119899= S projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] (42)

converges strongly to the fixed point 119909Dagger of S (and of T asFix(S) = Fix(T)) Without loss of generality we may assumethat 119911

119899 le 119872

2for all 119899

It suffices to prove that 119909119899+1

minus 119911119899 rarr 0 as 119899 rarr infin (for

some 120574119899rarr 0+) To this end we rewrite (42) as

(2I minus T) 119911119899= projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] 119899 ge 0 (43)

By using the property of the metric projection we have

⟨120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus (2119911119899minus T119911119899) 119909119899+1

minus (2119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩

+ ⟨T119911119899minus 119911119899 119909119899+1

minus 119911119899minus (119911119899minus T119911119899)⟩ le 0

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

+1003817100381710038171003817119911119899 minus T119911

119899

10038171003817100381710038172

le ⟨120574119899(119911119899minus 984858 (119911

119899)) T119911

119899minus 119911119899⟩

997904rArr ⟨minus120574119899(119911119899minus 984858 (119911

119899)) + T119911

119899minus 119911119899 119909119899+1

minus 119911119899⟩

le 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

997904rArr ⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩

le1003817100381710038171003817119911119899 minus 984858 (119911

119899)10038171003817100381710038171003817100381710038171003817T119911119899 minus 119911

119899

1003817100381710038171003817

(44)

Note that

1003817100381710038171003817119911119899 minus T119911119899

1003817100381710038171003817 =1003817100381710038171003817projC [120574

119899984858 (119911119899) + (1 minus 120574

119899) 119911119899] minus 119911119899

1003817100381710038171003817

le1003817100381710038171003817120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899minus 119911119899

1003817100381710038171003817

= 120574119899

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817

(45)

Hence we get

⟨minus (119911119899minus 984858 (119911

119899)) +

T119911119899minus 119911119899

120574119899

119909119899+1

minus 119911119899⟩ le 120574

119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(46)

6 Abstract and Applied Analysis

From (42) we have1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 =1003817100381710038171003817S projC [120574

119899+1984858 (119911119899+1

) + (1 minus 120574119899+1

) 119911119899+1

]

minusS projC [120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899]1003817100381710038171003817

le1003817100381710038171003817120574119899+1984858 (119911119899+1) + (1 minus 120574

119899+1) 119911119899+1

minus120574119899984858 (119911119899) minus (1 minus 120574

119899) 119911119899

1003817100381710038171003817

=1003817100381710038171003817(1 minus 120574

119899+1) (119911119899+1

minus 119911119899) + (120574119899minus 120574119899+1

)

times (119911119899minus 984858 (119911

119899)) + 120574

119899+1(984858 (119911119899+1

) minus 984858 (119911119899))1003817100381710038171003817

le [1 minus (1 minus 120581) 120574119899+1

]1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574

119899

10038161003816100381610038161003817100381710038171003817119911119899 minus 984858 (119911

119899)1003817100381710038171003817

(47)

It follows that1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 le

1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

(1 minus 120581) 120574119899+1

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817 (48)

Set

120574119899=

120585119899

120578119899

(49)

By condition (C2) 120574119899

rarr 0+ and 120574

119899isin (0 1) for 119899 large

enough Hence by (46) and (48) we have

⟨minus(119911119899minus 984858 (119911

119899)) +

120578119899(T119911119899minus 119911119899)

120585119899

119909119899+1

minus 119911119899⟩

le120585119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

le120585119899

120578119899

1198722

2

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817 le

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

1198722

(50)

By (29) we have1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minusprojC1199091198991003817100381710038171003817

le 120585119899

1003817100381710038171003817119909119899 minus 984858 (119909119899)1003817100381710038171003817 + 120578119899

1003817100381710038171003817T119909119899 minus 119909119899

1003817100381710038171003817

le (120585119899+ 120578119899)1198722

(51)

Next we estimate 119909119899+1

minus 119911119899+1

Since 119909119899+1

= projC[119910119899]⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩ le 0 Then we have

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

= ⟨119909119899+1

minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩

+ ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

le ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨[120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minus119911119899 119909119899+1

minus 119911119899⟩

= (1 minus 120585119899minus 120578119899) ⟨119909119899minus 119911119899 119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899minus T119909119899+1

119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899+1

minus T119911119899 119909119899+1

minus 119911119899⟩

+ ⟨120585119899(984858 (119909119899) minus 119911119899)

+ 120578119899(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

(52)

It follows that

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus 120585119899minus 120578119899)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 1205781198991198711003817100381710038171003817119909119899 minus 119909

119899+1

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+ 120585119899⟨984858 (119911

119899) minus 119911119899+

120578119899

120585119899

(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

le1 minus 120585119899minus 120578119899

2(1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

)

+1205782

119899

2

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1198712

2

1003817100381710038171003817119909119899 minus 119909119899+1

10038171003817100381710038172

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1205852

119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(53)

Thus

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le1 minus 120585119899minus 120578119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899 minus 119911119899

10038171003817100381710038172

+1198712

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119909119899

10038171003817100381710038172

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

+1205782

119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+(120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

11987121198722

2+

21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1198722

2

+1205782

119899

1 + 120585119899minus 120578119899

41198722

2

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)

times (1003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)2

Abstract and Applied Analysis 7

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

119872

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

(54)

where the finite constant119872 gt 0 is given by

119872 = max 119871211987222 41198722

2

1198722sup119899

(21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

(55)

Set

120575119899=

2120585119899

1 + 120585119899minus 120578119899

asymp 2120585119899

(as 119899 997888rarr infin)

120579119899=

120585119899120578119899minus1

minus 120585119899minus1

120578119899

21205852119899120578119899minus1

+1

2(120585119899+ 2120578119899+

1205782

119899

120585119899

) +120585119899

120578119899

+1205782

119899

2120585119899

119872

(56)

Then (54) can be rewritten as1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

le (1 minus 120575119899)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+ 120575119899120579119899 (57)

By the conditions (C1)ndash(C3) we deuce that

lim119899rarrinfin

120575119899= 0

infin

sum119899=1

120575119899= infin lim

119899rarrinfin120579119899= 0 (58)

From Lemma 4 and (57) we get 119909119899+1

minus 1199111198992

rarr 0 as 119899 rarr

infin This completes the proof

Algorithm 9 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontraction Let120585119899 sub [0 1] and 120578

119899 sub [0 1] be two real number sequences

For 1199090isin C we define a sequence 119909

119899 iteratively by

119909119899+1

= projC [(1 minus 120585119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0 (59)

Corollary 10 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 Suppose the followingconditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0Then the sequence 119909

119899 generated by the algorithm (59)

converges strongly to the minimum-norm fixed point of T

Conflict of Interests

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

Acknowledgment

The first author was supported by the National NaturalScience Foundation of China Grant no 11226125

References

[1] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 no 4 pp 365ndash374 1974

[2] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974

[3] C E Chidume and S A Mutangadura ldquoAn example onthe mann iteration method for lipschitz pseudocontractionsrdquoProceedings of the American Mathematical Society vol 129 no8 pp 2359ndash2363 2001

[4] H Zhou ldquoConvergence theorems of fixed points for Lipschitzpseudo-contractions inHilbert spacesrdquo Journal ofMathematicalAnalysis and Applications vol 343 no 1 pp 546ndash556 2008

[5] Y Yao Y-C Liou and G Marino ldquoA hybrid algorithmfor pseudo-contractive mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 10 pp 4997ndash5002 2009

[6] H Zegeye N Shahzad and M A Alghamdi ldquoConvergence ofIshikawarsquos iteration method for pseudocontractive mappingsrdquoNonlinear Analysis Theory Methods and Applications vol 74no 18 pp 7304ndash7311 2011

[7] L-C Ceng A Petrusel and J-C Yao ldquoStrong convergence ofmodified implicit iterative algorithmswith perturbedmappingsfor continuous pseudocontractive mappingsrdquo Applied Mathe-matics and Computation vol 209 no 2 pp 162ndash176 2009

[8] L-C Ceng A Petrusel and J-C Yao ldquoIterative approximationof fixed points for asymptotically strict pseudocontractive typemappings in the intermediate senserdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 587ndash606 2011

8 Abstract and Applied Analysis

[9] L-C Ceng and J-C Yao ldquoStrong convergence theorems forvariational inequalities and fixed point problems of asymptot-ically strict pseudocontractive mappings in the intermediatesenserdquo Acta Applicandae Mathematicae vol 115 no 2 pp 167ndash191 2011

[10] J C Yao and L C Zeng ldquoStrong convergence of averagedapproximants for asymptotically pseudocontractive mappingsin Banach spacesrdquo Journal of Nonlinear and Convex Analysisvol 8 pp 451ndash462 2007

[11] L C Ceng Q H Ansari and C F Wen ldquoImplicit relaxed andhybrid methods with regularization for minimization problemsand asymptotically strict pseudocontractive mappings in theintermediate senserdquo Abstract and Applied Analysis vol 2013Article ID 854297 14 pages 2013

[12] C E ChidumeMAbbas andBAli ldquoConvergence of theManniteration algorithm for a class of pseudocontractive mappingsrdquoApplied Mathematics and Computation vol 194 no 1 pp 1ndash62007

[13] C E Chidume and H Zegeye ldquoIterative solution of nonlinearequations of accretive and pseudocontractive typesrdquo Journal ofMathematical Analysis and Applications vol 282 no 2 pp 756ndash765 2003

[14] L Ciric A Rafiq N Cakic and J S Ume ldquoImplicit Mannfixedpoint iterations for pseudo-contractivemappingsrdquoAppliedMathematics Letters vol 22 no 4 pp 581ndash584 2009

[15] C Moore and B V C Nnoli ldquoStrong convergence of averagedapproximants for Lipschitz pseudocontractive mapsrdquo Journal ofMathematical Analysis and Applications vol 260 no 1 pp 269ndash278 2001

[16] A Udomene ldquoPath convergence approximation of fixed pointsand variational solutions of Lipschitz pseudocontractions inBanach spacesrdquoNonlinear Analysis Theory Methods and Appli-cations vol 67 no 8 pp 2403ndash2414 2007

[17] H Zegeye N Shahzad and T Mekonen ldquoViscosity approx-imation methods for pseudocontractive mappings in Banachspacesrdquo Applied Mathematics and Computation vol 185 no 1pp 538ndash546 2007

[18] Q-B Zhang andC-Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008

[19] H Zhou ldquoStrong convergence of an explicit iterative algorithmfor continuous pseudo-contractions in Banach spacesrdquo Nonlin-ear Analysis Theory Methods and Applications vol 70 no 11pp 4039ndash4046 2009

[20] K Geobel and W A Kirk Topics in Metric Fixed PointTheory vol 28 Cambridge Studies in Advanced MathematicsCambridge University Press 1990

[21] X Lu H-K Xu and X Yin ldquoHybrid methods for a class ofmonotone variational inequalitiesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 3-4 pp 1032ndash1041 2009

[22] H K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 2 pp 1ndash17 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

From (42) we have1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 =1003817100381710038171003817S projC [120574

119899+1984858 (119911119899+1

) + (1 minus 120574119899+1

) 119911119899+1

]

minusS projC [120574119899984858 (119911119899) + (1 minus 120574

119899) 119911119899]1003817100381710038171003817

le1003817100381710038171003817120574119899+1984858 (119911119899+1) + (1 minus 120574

119899+1) 119911119899+1

minus120574119899984858 (119911119899) minus (1 minus 120574

119899) 119911119899

1003817100381710038171003817

=1003817100381710038171003817(1 minus 120574

119899+1) (119911119899+1

minus 119911119899) + (120574119899minus 120574119899+1

)

times (119911119899minus 984858 (119911

119899)) + 120574

119899+1(984858 (119911119899+1

) minus 984858 (119911119899))1003817100381710038171003817

le [1 minus (1 minus 120581) 120574119899+1

]1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817

+1003816100381610038161003816120574119899+1 minus 120574

119899

10038161003816100381610038161003817100381710038171003817119911119899 minus 984858 (119911

119899)1003817100381710038171003817

(47)

It follows that1003817100381710038171003817119911119899+1 minus 119911

119899

1003817100381710038171003817 le

1003816100381610038161003816120574119899+1 minus 120574119899

1003816100381610038161003816

(1 minus 120581) 120574119899+1

1003817100381710038171003817119911119899 minus 984858 (119911119899)1003817100381710038171003817 (48)

Set

120574119899=

120585119899

120578119899

(49)

By condition (C2) 120574119899

rarr 0+ and 120574

119899isin (0 1) for 119899 large

enough Hence by (46) and (48) we have

⟨minus(119911119899minus 984858 (119911

119899)) +

120578119899(T119911119899minus 119911119899)

120585119899

119909119899+1

minus 119911119899⟩

le120585119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

le120585119899

120578119899

1198722

2

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817 le

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

1198722

(50)

By (29) we have1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817

=1003817100381710038171003817projC [120585

119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minusprojC1199091198991003817100381710038171003817

le 120585119899

1003817100381710038171003817119909119899 minus 984858 (119909119899)1003817100381710038171003817 + 120578119899

1003817100381710038171003817T119909119899 minus 119909119899

1003817100381710038171003817

le (120585119899+ 120578119899)1198722

(51)

Next we estimate 119909119899+1

minus 119911119899+1

Since 119909119899+1

= projC[119910119899]⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩ le 0 Then we have

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

= ⟨119909119899+1

minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨119909119899+1

minus 119910119899 119909119899+1

minus 119911119899⟩

+ ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

le ⟨119910119899minus 119911119899 119909119899+1

minus 119911119899⟩

= ⟨[120585119899984858 (119909119899) + (1 minus 120585

119899minus 120578119899) 119909119899+ 120578119899T119909119899]

minus119911119899 119909119899+1

minus 119911119899⟩

= (1 minus 120585119899minus 120578119899) ⟨119909119899minus 119911119899 119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899minus T119909119899+1

119909119899+1

minus 119911119899⟩

+ 120578119899⟨T119909119899+1

minus T119911119899 119909119899+1

minus 119911119899⟩

+ ⟨120585119899(984858 (119909119899) minus 119911119899)

+ 120578119899(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

(52)

It follows that

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus 120585119899minus 120578119899)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 1205781198991198711003817100381710038171003817119909119899 minus 119909

119899+1

10038171003817100381710038171003817100381710038171003817119909119899+1 minus 119911

119899

1003817100381710038171003817

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+ 120585119899⟨984858 (119911

119899) minus 119911119899+

120578119899

120585119899

(T119911119899minus 119911119899) 119909119899+1

minus 119911119899⟩

le1 minus 120585119899minus 120578119899

2(1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

)

+1205782

119899

2

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1198712

2

1003817100381710038171003817119909119899 minus 119909119899+1

10038171003817100381710038172

+ 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

+1205852

119899

120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

(53)

Thus

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le1 minus 120585119899minus 120578119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899 minus 119911119899

10038171003817100381710038172

+1198712

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119909119899

10038171003817100381710038172

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1003817100381710038171003817119911119899 minus 984858(119911119899)10038171003817100381710038172

+1205782

119899

1 + 120585119899minus 120578119899

1003817100381710038171003817119909119899+1 minus 119911119899

10038171003817100381710038172

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899

10038171003817100381710038172

+(120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

11987121198722

2+

21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

1198722

2

+1205782

119899

1 + 120585119899minus 120578119899

41198722

2

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)

times (1003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)2

Abstract and Applied Analysis 7

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

119872

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

(54)

where the finite constant119872 gt 0 is given by

119872 = max 119871211987222 41198722

2

1198722sup119899

(21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

(55)

Set

120575119899=

2120585119899

1 + 120585119899minus 120578119899

asymp 2120585119899

(as 119899 997888rarr infin)

120579119899=

120585119899120578119899minus1

minus 120585119899minus1

120578119899

21205852119899120578119899minus1

+1

2(120585119899+ 2120578119899+

1205782

119899

120585119899

) +120585119899

120578119899

+1205782

119899

2120585119899

119872

(56)

Then (54) can be rewritten as1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

le (1 minus 120575119899)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+ 120575119899120579119899 (57)

By the conditions (C1)ndash(C3) we deuce that

lim119899rarrinfin

120575119899= 0

infin

sum119899=1

120575119899= infin lim

119899rarrinfin120579119899= 0 (58)

From Lemma 4 and (57) we get 119909119899+1

minus 1199111198992

rarr 0 as 119899 rarr

infin This completes the proof

Algorithm 9 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontraction Let120585119899 sub [0 1] and 120578

119899 sub [0 1] be two real number sequences

For 1199090isin C we define a sequence 119909

119899 iteratively by

119909119899+1

= projC [(1 minus 120585119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0 (59)

Corollary 10 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 Suppose the followingconditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0Then the sequence 119909

119899 generated by the algorithm (59)

converges strongly to the minimum-norm fixed point of T

Conflict of Interests

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

Acknowledgment

The first author was supported by the National NaturalScience Foundation of China Grant no 11226125

References

[1] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 no 4 pp 365ndash374 1974

[2] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974

[3] C E Chidume and S A Mutangadura ldquoAn example onthe mann iteration method for lipschitz pseudocontractionsrdquoProceedings of the American Mathematical Society vol 129 no8 pp 2359ndash2363 2001

[4] H Zhou ldquoConvergence theorems of fixed points for Lipschitzpseudo-contractions inHilbert spacesrdquo Journal ofMathematicalAnalysis and Applications vol 343 no 1 pp 546ndash556 2008

[5] Y Yao Y-C Liou and G Marino ldquoA hybrid algorithmfor pseudo-contractive mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 10 pp 4997ndash5002 2009

[6] H Zegeye N Shahzad and M A Alghamdi ldquoConvergence ofIshikawarsquos iteration method for pseudocontractive mappingsrdquoNonlinear Analysis Theory Methods and Applications vol 74no 18 pp 7304ndash7311 2011

[7] L-C Ceng A Petrusel and J-C Yao ldquoStrong convergence ofmodified implicit iterative algorithmswith perturbedmappingsfor continuous pseudocontractive mappingsrdquo Applied Mathe-matics and Computation vol 209 no 2 pp 162ndash176 2009

[8] L-C Ceng A Petrusel and J-C Yao ldquoIterative approximationof fixed points for asymptotically strict pseudocontractive typemappings in the intermediate senserdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 587ndash606 2011

8 Abstract and Applied Analysis

[9] L-C Ceng and J-C Yao ldquoStrong convergence theorems forvariational inequalities and fixed point problems of asymptot-ically strict pseudocontractive mappings in the intermediatesenserdquo Acta Applicandae Mathematicae vol 115 no 2 pp 167ndash191 2011

[10] J C Yao and L C Zeng ldquoStrong convergence of averagedapproximants for asymptotically pseudocontractive mappingsin Banach spacesrdquo Journal of Nonlinear and Convex Analysisvol 8 pp 451ndash462 2007

[11] L C Ceng Q H Ansari and C F Wen ldquoImplicit relaxed andhybrid methods with regularization for minimization problemsand asymptotically strict pseudocontractive mappings in theintermediate senserdquo Abstract and Applied Analysis vol 2013Article ID 854297 14 pages 2013

[12] C E ChidumeMAbbas andBAli ldquoConvergence of theManniteration algorithm for a class of pseudocontractive mappingsrdquoApplied Mathematics and Computation vol 194 no 1 pp 1ndash62007

[13] C E Chidume and H Zegeye ldquoIterative solution of nonlinearequations of accretive and pseudocontractive typesrdquo Journal ofMathematical Analysis and Applications vol 282 no 2 pp 756ndash765 2003

[14] L Ciric A Rafiq N Cakic and J S Ume ldquoImplicit Mannfixedpoint iterations for pseudo-contractivemappingsrdquoAppliedMathematics Letters vol 22 no 4 pp 581ndash584 2009

[15] C Moore and B V C Nnoli ldquoStrong convergence of averagedapproximants for Lipschitz pseudocontractive mapsrdquo Journal ofMathematical Analysis and Applications vol 260 no 1 pp 269ndash278 2001

[16] A Udomene ldquoPath convergence approximation of fixed pointsand variational solutions of Lipschitz pseudocontractions inBanach spacesrdquoNonlinear Analysis Theory Methods and Appli-cations vol 67 no 8 pp 2403ndash2414 2007

[17] H Zegeye N Shahzad and T Mekonen ldquoViscosity approx-imation methods for pseudocontractive mappings in Banachspacesrdquo Applied Mathematics and Computation vol 185 no 1pp 538ndash546 2007

[18] Q-B Zhang andC-Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008

[19] H Zhou ldquoStrong convergence of an explicit iterative algorithmfor continuous pseudo-contractions in Banach spacesrdquo Nonlin-ear Analysis Theory Methods and Applications vol 70 no 11pp 4039ndash4046 2009

[20] K Geobel and W A Kirk Topics in Metric Fixed PointTheory vol 28 Cambridge Studies in Advanced MathematicsCambridge University Press 1990

[21] X Lu H-K Xu and X Yin ldquoHybrid methods for a class ofmonotone variational inequalitiesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 3-4 pp 1032ndash1041 2009

[22] H K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 2 pp 1ndash17 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003817100381710038171003817119911119899 minus 119911119899minus1

1003817100381710038171003817

times (21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

le (1 minus2120585119899

1 + 120585119899minus 120578119899

)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+1

1 + 120585119899minus 120578119899

1003816100381610038161003816120585119899120578119899minus1 minus 120585119899minus1

120578119899

1003816100381610038161003816

120585119899120578119899minus1

119872

+ (120585119899+ 120578119899)2

1 + 120585119899minus 120578119899

+21205852

119899

(1 + 120585119899minus 120578119899) 120578119899

+1205782

119899

1 + 120585119899minus 120578119899

119872

(54)

where the finite constant119872 gt 0 is given by

119872 = max 119871211987222 41198722

2

1198722sup119899

(21003817100381710038171003817119909119899 minus 119911

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119911119899 minus 119911

119899minus1

1003817100381710038171003817)

(55)

Set

120575119899=

2120585119899

1 + 120585119899minus 120578119899

asymp 2120585119899

(as 119899 997888rarr infin)

120579119899=

120585119899120578119899minus1

minus 120585119899minus1

120578119899

21205852119899120578119899minus1

+1

2(120585119899+ 2120578119899+

1205782

119899

120585119899

) +120585119899

120578119899

+1205782

119899

2120585119899

119872

(56)

Then (54) can be rewritten as1003817100381710038171003817119909119899+1 minus 119911

119899

10038171003817100381710038172

le (1 minus 120575119899)1003817100381710038171003817119909119899 minus 119911

119899minus1

10038171003817100381710038172

+ 120575119899120579119899 (57)

By the conditions (C1)ndash(C3) we deuce that

lim119899rarrinfin

120575119899= 0

infin

sum119899=1

120575119899= infin lim

119899rarrinfin120579119899= 0 (58)

From Lemma 4 and (57) we get 119909119899+1

minus 1199111198992

rarr 0 as 119899 rarr

infin This completes the proof

Algorithm 9 Let C be a nonempty closed subset of a realHilbert space H Let T C rarr C be a pseudocontraction Let120585119899 sub [0 1] and 120578

119899 sub [0 1] be two real number sequences

For 1199090isin C we define a sequence 119909

119899 iteratively by

119909119899+1

= projC [(1 minus 120585119899minus 120578119899) 119909119899+ 120578119899T119909119899] 119899 ge 0 (59)

Corollary 10 Let C be a nonempty closed convex subset of areal Hilbert space H Let T C rarr C be an 119871-Lipschitzianand pseudocontraction with Fix(T) = 0 Suppose the followingconditions are satisfied

(C1) lim119899rarrinfin

120585119899= 0 and sum

infin

119899=0120585119899= infin

(C2) lim119899rarrinfin

(120585119899120578119899) = lim

119899rarrinfin(1205782

119899120585119899) = 0

(C3) lim119899rarrinfin

((120585119899120578119899minus1

minus 120585119899minus1

120578119899)1205852

119899120578119899minus1

) = 0Then the sequence 119909

119899 generated by the algorithm (59)

converges strongly to the minimum-norm fixed point of T

Conflict of Interests

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

Acknowledgment

The first author was supported by the National NaturalScience Foundation of China Grant no 11226125

References

[1] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 no 4 pp 365ndash374 1974

[2] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974

[3] C E Chidume and S A Mutangadura ldquoAn example onthe mann iteration method for lipschitz pseudocontractionsrdquoProceedings of the American Mathematical Society vol 129 no8 pp 2359ndash2363 2001

[4] H Zhou ldquoConvergence theorems of fixed points for Lipschitzpseudo-contractions inHilbert spacesrdquo Journal ofMathematicalAnalysis and Applications vol 343 no 1 pp 546ndash556 2008

[5] Y Yao Y-C Liou and G Marino ldquoA hybrid algorithmfor pseudo-contractive mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 10 pp 4997ndash5002 2009

[6] H Zegeye N Shahzad and M A Alghamdi ldquoConvergence ofIshikawarsquos iteration method for pseudocontractive mappingsrdquoNonlinear Analysis Theory Methods and Applications vol 74no 18 pp 7304ndash7311 2011

[7] L-C Ceng A Petrusel and J-C Yao ldquoStrong convergence ofmodified implicit iterative algorithmswith perturbedmappingsfor continuous pseudocontractive mappingsrdquo Applied Mathe-matics and Computation vol 209 no 2 pp 162ndash176 2009

[8] L-C Ceng A Petrusel and J-C Yao ldquoIterative approximationof fixed points for asymptotically strict pseudocontractive typemappings in the intermediate senserdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 587ndash606 2011

8 Abstract and Applied Analysis

[9] L-C Ceng and J-C Yao ldquoStrong convergence theorems forvariational inequalities and fixed point problems of asymptot-ically strict pseudocontractive mappings in the intermediatesenserdquo Acta Applicandae Mathematicae vol 115 no 2 pp 167ndash191 2011

[10] J C Yao and L C Zeng ldquoStrong convergence of averagedapproximants for asymptotically pseudocontractive mappingsin Banach spacesrdquo Journal of Nonlinear and Convex Analysisvol 8 pp 451ndash462 2007

[11] L C Ceng Q H Ansari and C F Wen ldquoImplicit relaxed andhybrid methods with regularization for minimization problemsand asymptotically strict pseudocontractive mappings in theintermediate senserdquo Abstract and Applied Analysis vol 2013Article ID 854297 14 pages 2013

[12] C E ChidumeMAbbas andBAli ldquoConvergence of theManniteration algorithm for a class of pseudocontractive mappingsrdquoApplied Mathematics and Computation vol 194 no 1 pp 1ndash62007

[13] C E Chidume and H Zegeye ldquoIterative solution of nonlinearequations of accretive and pseudocontractive typesrdquo Journal ofMathematical Analysis and Applications vol 282 no 2 pp 756ndash765 2003

[14] L Ciric A Rafiq N Cakic and J S Ume ldquoImplicit Mannfixedpoint iterations for pseudo-contractivemappingsrdquoAppliedMathematics Letters vol 22 no 4 pp 581ndash584 2009

[15] C Moore and B V C Nnoli ldquoStrong convergence of averagedapproximants for Lipschitz pseudocontractive mapsrdquo Journal ofMathematical Analysis and Applications vol 260 no 1 pp 269ndash278 2001

[16] A Udomene ldquoPath convergence approximation of fixed pointsand variational solutions of Lipschitz pseudocontractions inBanach spacesrdquoNonlinear Analysis Theory Methods and Appli-cations vol 67 no 8 pp 2403ndash2414 2007

[17] H Zegeye N Shahzad and T Mekonen ldquoViscosity approx-imation methods for pseudocontractive mappings in Banachspacesrdquo Applied Mathematics and Computation vol 185 no 1pp 538ndash546 2007

[18] Q-B Zhang andC-Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008

[19] H Zhou ldquoStrong convergence of an explicit iterative algorithmfor continuous pseudo-contractions in Banach spacesrdquo Nonlin-ear Analysis Theory Methods and Applications vol 70 no 11pp 4039ndash4046 2009

[20] K Geobel and W A Kirk Topics in Metric Fixed PointTheory vol 28 Cambridge Studies in Advanced MathematicsCambridge University Press 1990

[21] X Lu H-K Xu and X Yin ldquoHybrid methods for a class ofmonotone variational inequalitiesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 3-4 pp 1032ndash1041 2009

[22] H K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 2 pp 1ndash17 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

[9] L-C Ceng and J-C Yao ldquoStrong convergence theorems forvariational inequalities and fixed point problems of asymptot-ically strict pseudocontractive mappings in the intermediatesenserdquo Acta Applicandae Mathematicae vol 115 no 2 pp 167ndash191 2011

[10] J C Yao and L C Zeng ldquoStrong convergence of averagedapproximants for asymptotically pseudocontractive mappingsin Banach spacesrdquo Journal of Nonlinear and Convex Analysisvol 8 pp 451ndash462 2007

[11] L C Ceng Q H Ansari and C F Wen ldquoImplicit relaxed andhybrid methods with regularization for minimization problemsand asymptotically strict pseudocontractive mappings in theintermediate senserdquo Abstract and Applied Analysis vol 2013Article ID 854297 14 pages 2013

[12] C E ChidumeMAbbas andBAli ldquoConvergence of theManniteration algorithm for a class of pseudocontractive mappingsrdquoApplied Mathematics and Computation vol 194 no 1 pp 1ndash62007

[13] C E Chidume and H Zegeye ldquoIterative solution of nonlinearequations of accretive and pseudocontractive typesrdquo Journal ofMathematical Analysis and Applications vol 282 no 2 pp 756ndash765 2003

[14] L Ciric A Rafiq N Cakic and J S Ume ldquoImplicit Mannfixedpoint iterations for pseudo-contractivemappingsrdquoAppliedMathematics Letters vol 22 no 4 pp 581ndash584 2009

[15] C Moore and B V C Nnoli ldquoStrong convergence of averagedapproximants for Lipschitz pseudocontractive mapsrdquo Journal ofMathematical Analysis and Applications vol 260 no 1 pp 269ndash278 2001

[16] A Udomene ldquoPath convergence approximation of fixed pointsand variational solutions of Lipschitz pseudocontractions inBanach spacesrdquoNonlinear Analysis Theory Methods and Appli-cations vol 67 no 8 pp 2403ndash2414 2007

[17] H Zegeye N Shahzad and T Mekonen ldquoViscosity approx-imation methods for pseudocontractive mappings in Banachspacesrdquo Applied Mathematics and Computation vol 185 no 1pp 538ndash546 2007

[18] Q-B Zhang andC-Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008

[19] H Zhou ldquoStrong convergence of an explicit iterative algorithmfor continuous pseudo-contractions in Banach spacesrdquo Nonlin-ear Analysis Theory Methods and Applications vol 70 no 11pp 4039ndash4046 2009

[20] K Geobel and W A Kirk Topics in Metric Fixed PointTheory vol 28 Cambridge Studies in Advanced MathematicsCambridge University Press 1990

[21] X Lu H-K Xu and X Yin ldquoHybrid methods for a class ofmonotone variational inequalitiesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 3-4 pp 1032ndash1041 2009

[22] H K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 2 pp 1ndash17 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of