research article common fixed point results for six...

Research ArticleCommon Fixed Point Results for Six Mappings via IntegralContractions with Applications

Mian Bahadur Zada1 Muhammad Sarwar1 and Nayyar Mehmood2

1Department of Mathematics University of Malakand Chakdara Pakistan2Department of Mathematics and Statistics International Islamic University Sector H-10 Islamabad Pakistan

Correspondence should be addressed to Mian Bahadur Zada mbzmathgmailcomand Muhammad Sarwar sarwarswatigmailcom

Received 3 July 2016 Accepted 20 September 2016

Academic Editor Remi Leandre

Copyright copy 2016 Mian Bahadur Zada et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Common fixed point theorems for six self-mappings under integral type inequality satisfying (EA) and (CLR) properties in thecontext of complex valued metric space (not necessarily complete) are established The derived results are new even for ordinarymetric spaces We prove existence result for optimal unique solution of the system of functional equations used in dynamicalprogramming with complex domain

1 Introduction and Preliminaries

Metric fixed point theory is the most impressive and activebranch of modern mathematics that has vast applications inapplied functional and numerical analysis Banach contrac-tion principle [1] is one of the best known results in thistheory This principle can be considered as the launch ofmetric fixed point theory that guarantees the existence anduniqueness of fixed points of mappings In the followingyears various efforts have been done to further generalizeBanach contraction principle in different direction for asingle map

The exploration of commonfixed point theory is an activefield of research activity since 1976 The work of Jungck [2]is considered as major achievement in the field of commonfixed point theory Jungck presented the notion of commutingmaps to introduce the common fixed point results for twoself-maps on complete metric space To improve commonfixed point theorems researchers began to utilize weakerconditions than commuting mappings such as weakly com-muting maps compatible mappings compatible mappingsof type (A) compatible mappings of type (B) compatiblemappings of type (P) and compatible mappings of type

(C) In the study of common fixed point results of weaklycompatible mappings we often require the assumption of thecontinuity of mappings or the completeness of underlyingspace As a consequence a natural question arises as towhether there exist common fixed point theorems which donot enforce such conditions Regarding this Aamri and ElMoutawakil [3] relaxed these conditions by introducing thenotion of (EA) property and it was marked that (EA) prop-erty does not require the condition of continuity of mappingsand completeness of the underlying space However (EA)property tolerates the condition of closeness of the rangesubspaces of the involved mappings In 2011 the new notionof Common Limit in the range property (shortly (CLR)property) was given by Sintunavarat and Kumam [4] thatdoes not enforce the above-mentioned conditions Moreoverthe significance of (CLR) property reveals that closeness ofrange subspaces is not essential Using these two importantnotions many fixed point theorems were established [3ndash6]

One of the most pleasant generalizations of Banachprinciple is the Branciari [7] fixed point theorem for asingle mapping satisfying an integral type inequality Afterthat serval researchers ([8ndash11] etc) generalize the result ofBranciari in ordinary metric spaces

Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2016 Article ID 7480469 13 pageshttpdxdoiorg10115520167480469

2 International Journal of Analysis

On the other hand Azam et al [12] studied complex val-ued metric space and proved common fixed point theoremsfor two self-mappings satisfying a rational type inequalityManro et al [13] generalized the theorem of Branciari [7] fortwo self-maps under contractive condition of integral typesatisfying property (EA) and (CLR) property in the settingof complex valued metric spaces Bahadur Zada et al [6]generalized the results of [13] for four self-maps in the contextof complex valued metric spaces

The aim of this paper is to prove common fixed pointtheorems for six self-maps satisfying integral type contrac-tive condition using property (EA) and (CLR) property incomplex valuedmetric spaces which extends and generalizesmany results of the existing literature

Throughout the paper C+ = 119911 isin C 119911 ≿ (0 0) optstand for sup or inf 119885 and 119884 are Banach spaces Ω sube 119885is the state space 119863 sube 119884 is the decision space Φ = 120601 120601 [0infin[rarr [0infin[ is a Lebesgue integrable mapping whichis summable on each compact subset of [0infin[ nonnegativeand nondecreasing such that for each 120576 gt 0 int120576

0120601(119905)119889119905 gt 0

and Φlowast = 120593 R119899 rarr C is a complex valued Lebesgueintegrable mapping which is summable and nonvanishingon each measurable subset of R119899 such that for each 120576 ≻ 0int1205760120593(119905)119889119905 ≻ 0

Definition 1 (see [12]) Let C be the set of complex numbersand 119911 119908 isin C Define a partial order ≾ on C as follows

119911 ≾ 119908 iff Re (119911) le Re (119908) Im (119911) le Im (119908) 119911 ≺ 119908 iff Re (119911) lt Re (119908) Im (119911) lt Im (119908) (1)

Note that(i) 1198961 1198962 isin 119877 and 1198961 le 1198962 rArr 1198961119911 ≾ 1198962119911 for all 119911 isin C(ii) 0 ≾ 119911 ≾ 119908 rArr |119911| lt |119908| for all 119911 119908 isin C(iii) 119911 ≾ 119908 and 119908 ≺ 119908lowast rArr 119911 ≺ 119908lowast for all 119911 119908 119908lowast isin C

Definition 2 (see [14]) The ldquomaxrdquo function for the partialorder relation ldquo≾rdquo is defined by the following

(1) max1199081 1199082 = 1199082 hArr 1199081 ≾ 1199082(2) If 1199081 ≾ max1199082 1199083 then 1199081 ≾ 1199082 or 1199081 ≾ 1199083(3) max1199081 1199082 = 1199082 hArr 1199081 ≾ 1199082 or |1199081| le |1199082|

Definition 3 (see [12]) Let 119883 be a nonempty set and 119889 119883 times119883 rarr C be the mapping satisfying the following axioms(1) 0 ≾ 119889(1199111 1199112) for all 1199111 1199112 isin 119883 and 119889(1199111 1199112) = 0 if

and only if 1199111 = 1199112(2) 119889(1199111 1199112) = 119889(1199112 1199111) for all 1199111 1199112 isin 119883(3) 119889(1199111 1199112) ≾ 119889(1199111 1199113) + 119889(1199113 1199112) for all 1199111 1199112 1199113 isin 119883Then pair (119883 119889) is called a complex valued metric space

Example 4 Let 1199111 1199112 isin C and define the mapping 119889 C timesCrarr C by

119889 (1199111 1199112) = 0 if 1199111 = 1199112120580 if 1199111 = 1199112 (2)

Then (C 119889) is a complex valued metric space

Definition 5 (see [12]) Let 119911119899 be a sequence in complexvalued metric (119883 119889) and 119911 isin 119883 Then 119911 is called the limitof 119911119899 if for every 119908 isin C with 0 ≺ 119908 there is 1198990 isin 119873 suchthat 119889(119911119899 119911) ≺ 119908 for all 119899 gt 1198990 and one writes lim119899rarrinfin119911119899 = 119911Lemma 6 (see [12]) Any sequence 119911119899 in complex valuedmetric space (119883 119889) converges to 119911 if and only if |119889(119911119899 119911)| rarr 0as 119899 rarr infin

Definition 7 (see [4]) Let 119883 be a nonempty set and 119870 119871 119883 rarr 119883 be two self-maps Then

(i) 119911 isin 119883 is called a fixed point of 119871 if 119871119911 = 119911(ii) 119911 isin 119883 is called a coincidence point of119870 and 119871 if119870119911 =119871119911(iii) 119911 isin 119883 is called a common fixed point of 119870 and 119871 if119870119911 = 119871119911 = 119911Jungck [2] initiated the concept of commuting maps in

the following way

Definition 8 Two self-maps 119870 and 119871 of nonempty set 119883 arecommuting if 119871119870119911 = 119870119871119911 for all 119911 isin 119883

Jungck [15] initiated the concept of weakly compatiblemaps in ordinary metric spaces while Bhatt et al [16] refinedthis notion in the complex valued metric space in thefollowing way

Definition 9 Two self-maps 119870 and 119871 on complex valuedmetric space 119883 are weakly compatible if there exists point119911 isin 119883 such that119870119871119911 = 119871119870119911 whenever119870119911 = 119871119911

Aamri and El Moutawakil [3] initiated the concept of(EA) property in ordinary metric spaces while Verma andPathak [14] defined this concept in complex valued metricspace as follows

Definition 10 Two self-maps 119870 and 119871 on a complex valuedmetric space119883 satisfy property (EA) if there exists sequence119911119899 in119883 such that

lim119903rarrinfin

119871119911119899 = lim119903rarrinfin

119870119911119899 = 119911 for some 119911 isin 119883 (3)

Sintunavarat and Kumam [4] introduced the notion of(CLR) property in ordinary metric spaces in a similar modeVerma and Pathak [14] defined this notion in a complexvalued metric space in the following way

Definition 11 Two self-maps 119870 and 119871 on a complex valuedmetric space119883 satisfy (CLR119870) if there exists sequence 119911119899 in119883 such that

lim119899rarrinfin

119871119911119899 = lim119903rarrinfin

119870119911119899 = 119870119911 for some 119911 isin 119883 (4)

International Journal of Analysis 3

Remark 12 (see [6]) Let 120593 isin Φlowast such that Re(120593) Im(120593) isinΦ and 119911119899 is a sequence in C+ converges to 119911 and thenlim119899rarrinfin int1199111198990 120593(119904)119889119904 = int1199110 120593(119904)119889119904Lemma 13 (see [6]) Let 120593 isin Φlowast such that Re(120593) Im(120593) isin Φand 119911119899 is a sequence in C+ and then lim119899rarrinfin int1199111198990 120593(119904)119889119904 = 0if and only if 119911119899 rarr (0 0) as 119899 rarr infin2 Main Results

Let Ψ be the class of all functions 120595 C+ rarr C+ that satisfythe following properties

(1) 120595 is nondecreasing on C+(2) 120595 is upper semicontinuous on C+(3) 120595(0) = 0 and 120595(119911) ≺ 119911 for every 119911 ≻ 0Now we present our first result

Theorem 14 Let (119883 119889) be a complex valued metric space and119870 119871119872119873 119877 119878 119883 rarr 119883 be six self-mappings satisfying thefollowing conditions

(1) One of pairs (119870119873119877) and (119871119872119878) satisfies property(119864119860) such that 119870(119883) sube 119872119878(119883) and 119871(119883) sube 119873119877(119883)(2) forall1199111 1199112 isin 119883int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (5)

where 120595 isin Ψ 120593 isin Φlowast andΔ 1 (1199111 1199112) = 119889 (1198721198781199112 1198711199112) 1 + 119889 (1198731198771199111 1198701199111)1 + 119889 (11987311987711991111198721198781199112)

Δ 2 (1199111 1199112) = 119889 (1198731198771199111 1198701199111) 1 + 119889 (1198721198781199112 1198711199112)1 + 119889 (11987311987711991111198721198781199112)

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(6)

If one of119872119878(119883) and 119873119877(119883) is closed subspace of 119883 suchthat pairs (119870119873119877) and (119871119872119878) are weakly compatible theneach pair of pairs (119870119873119877) and (119871119872119878) has a coincidence pointin 119883 Moreover if (119870 119878) (119871 119877) (119872119878 119877) and (119873119877 119878) arecommuting pairs then 119870 119871119872119873 119877 and 119878 have a uniquecommon fixed point in119883Proof Let pair (119870119873119877) satisfy (EA) property so there existssequence 119911119899 in119883 such that

lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119911 for some 119911 isin 119883 (7)

Since119870(119883) sube 119872119878(119883) there exists 119908119899 in119883 such that119870119911119899 =119872119878119908119899 and thus from (7) we get

lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119872119878119908119899 = 119911 (8)

We assert that lim119899rarrinfin119871119908119899 = 119911 If lim119899rarrinfin119871119908119899 = 119908 = 119911then upon putting 1199111 = 119911119899 and 1199112 = 119908119899 in condition (2) ofTheorem 14 we have

int119889(119870119911119899 119871119908119899)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899 119908119899)0

120593 (119905) 119889119905 1 le 119895 le 3) (9)

where

Δ 1 (119911119899 119908119899) = 119889 (119872119878119908119899 119871119908119899) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119908119899)

Δ 2 (119911119899 119908119899) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119908119899 119871119908119899)1 + 119889 (119873119877119911119899119872119878119908119899)

Δ 3 (119911119899 119908119899) = max 119889 (119873119877119911119899119872119878119908119899) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119908119899 119871119908119899) 12 [119889 (119870119911119899119872119878119908119899) + 119889 (119871119908119899 119873119877119911119899)]

(10)

Taking upper limit as 119899 rarr infin in (9) we have

Δ 1 (119911119899 119908119899) 997888rarr 119889 (119911 119908) Δ 2 (119911119899 119908119899) 997888rarr 0Δ 3 (119911119899 119908119899) 997888rarr 119889 (119911 119908) int119889(119911119908)0

120593 (119905) 119889119905 = lim sup119899rarrinfin

int119889(119870119911119899 119871119908119899)0

120593 (119905) 119889119905

≾ lim sup119899rarrinfin

120595(maxintΔ 119895(119911119899119908119899)0

120593 (119905) 119889119905 1 le 119895 le 3)

≾ 120595(lim sup119899rarrinfin

maxintΔ 119895(119911119899119908119899)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119911119908)0

120593 (119905) 119889119905 0 int119889(119911119908)0

120593 (119905) 119889119905)

= 120595(int119889(119911119908)0

120593 (119905) 119889119905) ≺ int119889(119911119908)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119911119908)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119911119908)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(11)

which contradict with our assumption thus 119911 = 119908 andlim119899rarrinfin119871119908119899 = 119911 Therefore (8) becomes

lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119871119908119899 = lim119899rarrinfin

119872119878119908119899 = 119911 (12)


Also since119872119878(119883) is closed subspace of119883 there exists 119906 isin 119883such that119872119878119906 = 119911 and using (12) we get

lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119871119908119899 = lim119899rarrinfin

119872119878119908119899 = 119911= 119872119878119906 (13)

Now we claim that 119871119906 = 119872119878119906 To support the claim let 119871119906 =119872119878119906 Then using condition (2) of Theorem 14 with 1199111 = 119911119899and 1199112 = 119906 one can get

int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (14)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(15)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119911119871119906)0

120593 (119905) 119889119905 0 int119889(119911119871119906)0

120593 (119905) 119889119905)

= 120595(int119889(119911119871119906)0

120593 (119905) 119889119905) ≺ int119889(119911119871119906)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119911119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119911119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(16)

which is a contradiction Thus 119871119906 = 119911 and hence

119871119906 = 119872119878119906 = 119911 (17)

Since 119871(119883) sube 119873119877(119883) there exists V isin 119883 such that 119871119906 = 119873119877Vand it follows from (17) that

119871119906 = 119872119878119906 = 119873119877V = 119911 (18)

We show that 119870V = 119873119877V Let on contrary 119870V = 119873119877V thenusing condition (2) ofTheorem 14 with 1199111 = V and 1199112 = 119906 wehave

int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (19)

where

Δ 1 (V 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877V 119870V)1 + 119889 (119873119877V119872119878119906) = 0Δ 2 (V 119906) = 119889 (119873119877V 119870V) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877V119872119878119906) = 119889 (119911119870V)

Δ 3 (V 119906) = max 119889 (119873119877V119872119878119906) 119889 (119873119877V 119870V) 119889 (119872119878119906 119871119906) 12 [119889 (119870V119872119878119906) + 119889 (119871119906119873119877V)]= 119889 (119911 119870V)

(20)

Therefore

int119889(119870V119911)0

120593 (119905) 119889119905

≾ 120595(max0 int119889(119911119870V)0

120593 (119905) 119889119905 int119889(119911119870V)0

120593 (119905) 119889119905)

≾ 120595(int119889(119911119870V)0

120593 (119905) 119889119905) ≺ int119889(119911119870V)0

120593 (119905) 119889119905

(21)

which is a contradiction to our assumption that 119870V = 119873119877VThus 119870V = 119873119877V and hence from (18) we get

119870V = 119871119906 = 119872119878119906 = 119873119877V = 119911 (22)

Now using the weak compatibility of pairs (119870119873119877) (119871119872119878)and (22) we have

119870V = 119873119877V 997904rArr 119873119877119870V = 119870119873119877V 997904rArr 119870119911 = 119873119877119911 (23)

119871119906 = 119872119878119906 997904rArr 119872119878119871119906 = 119871119872119878119906 997904rArr 119871119911 = 119872119878119911 (24)

Hence 119911 is the coincident point of each pair (119870119873119877) and(119871119872119878)Next we have to show that 119911 is the common fixed point

of 119870 119871119872119873 119877 and 119878 For this we claim that 119870119911 = 119911


If 119870119911 = 119911 then upon putting 1199111 = 119911 1199112 = 119906 in condition(2) of Theorem 14 and using (22) and (23) we have

int119889(119870119911119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (25)

where

Δ 1 (119911 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119870119911)1 + 119889 (119873119877119911119872119878119906) = 0Δ 2 (119911 119906) = 119889 (119873119877119911119870119911) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119872119878119906) = 0Δ 3 (119911 119906) = max 119889 (119873119877119911119872119878119906) 119889 (119873119877119911119870119911) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119872119878119906) + 119889 (119871119906119873119877119911)]= 119889 (119870119911 119911)

(26)

Therefore

int119889(119870119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119870119911119911)0

120593 (119905) 119889119905)

≾ 120595(int119889(119870119911119911)0

120593 (119905) 119889119905)

≺ int119889(119870119911119911)0

120593 (119905) 119889119905

(27)

which is impossible Thus 119870119911 = 119911 and hence in view of (23)we get

119870119911 = 119873119877119911 = 119911 (28)

Similarly we can show that

119871119911 = 119872119878119911 = 119911 (29)

Hence from (28) and (29) we get

119870119911 = 119871119911 = 119872119878119911 = 119873119877119911 = 119911 (30)

Now by commuting conditions of pairs (119870 119878) and(119873119877 119878) and using (28) and (30) we have119870(119878119911) = 119878(119870119911) = 119878119911and119873119877(119878119911) = 119878(119873119877119911) = 119878119911 from here it follows that

119870 (119878119911) = 119873119877 (119878119911) = 119878119911 (31)

Also by commuting conditions of pairs (119871 119877) and(119872119878 119877) and taking (29) and (30) we have 119871(119877119911) = 119877(119871119911) =119877119911 and119872119878(119877119911) = 119877(119872119878119911) = 119877119911 from here it follows that

119871 (119877119911) = 119872119878 (119877119911) = 119877119911 (32)

Further assume the 119878119911 = 119911 Then upon putting 1199111 =119878119911 1199112 = 119911 in condition (2) of Theorem 14 and using (29)and (31) we have

int119889(119870119878119911119871119911)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119878119911119911)0

120593 (119905) 119889119905 1 le 119895 le 3) (33)

where

Δ 1 (119878119911 119911) = 119889 (119872119878119911 119871119911) 1 + 119889 (119873119877119878119911119870119878119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 2 (119878119911 119911) = 119889 (119873119877119878119911119870119878119911) 1 + 119889 (119872119878119911 119871119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 3 (119878119911 119911) = max 119889 (119873119877119878119911119872119878119911) 119889 (119873119877119878119911119870119878119911) 119889 (119872119878119911 119871119911) 12 [119889 (119870119878119911119872119878119911) + 119889 (119871119911119873119877119878119911)]= max 119889 (119878119911 119911) 119889 (119878119911 119878119911) 119889 (119911 119911) 12 [119889 (119878119911 119911) + 119889 (119911 119878119911)] = 119889 (119878119911 119911)

(34)

Therefore

int119889(119878119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int(119878119911119911)0

120593 (119905) 119889119905)

≺ int(119878119911119911)0

120593 (119905) 119889119905(35)

which is a contradiction thus 119878119911 = 119911 Also119872119911 = 119911 as119872119878119911 =119911 so from (30) it follows that

119870119911 = 119871119911 = 119872119911 = 119878119911 = 119873119877119911 = 119911 (36)

Similarly using condition (2) of Theorem 14 with 1199111 = 119911and 1199112 = 119877119911 and taking (28) and (32) one can easily obtainthat 119877119911 = 119911 Also119873119911 = 119911 as119873119877119911 = 119911 Hence from (36) weget

119870119911 = 119871119911 = 119872119911 = 119873119911 = 119877119911 = 119878119911 = 119911 (37)

That is 119911 is a common fixed point of 119870 119871119872119873 119877 and 119878 in119883Similarly if (119871119872119878) satisfies property (EA) and119873119877(119883) is

closed subspace of 119883 then we can prove that 119911 is a common


fixed point of119870 119871119872119873 119877 and 119878 in119883 in the same argumentsas above

Uniqueness For the uniqueness of common fixed point let119911lowast = 119911 be another fixed point of 119870 119871119872119873 119877 and 119878 Thenusing condition (2) of Theorem 14 we have

int119889(119911119911lowast)

0

120593 (119905) 119889119905 = int119889(119870119911119871119911lowast)

0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119911lowast)

0

120593 (119905) 119889119905 1 le 119895 le 3) (38)

where

Δ 1 (119911 119911lowast) = 119889 (119872119878119911lowast 119871119911lowast) 1 + 119889 (119873119877119911119870119911)1 + 119889 (119873119877119911119872119878119911lowast) = 0

Δ 2 (119911 119911lowast) = 119889 (119873119877119911119870119911) 1 + 119889 (119872119878119911lowast 119871119911lowast)

1 + 119889 (119873119877119911119872119878119911lowast) = 0

Δ 3 (119911 119911lowast) = max119889 (119873119877119911119872119878119911lowast) 119889 (119873119877119911119870119911)

119889 (119872119878119911lowast 119871119911lowast) 119889 (119870119911119872119878119911lowast) + 119889 (119871119911lowast 119873119877119911)2 = 119889 (119911 119911lowast)

(39)

Thus

int119889(119911119911lowast)

0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119911119911lowast)

0

120593 (119905) 119889119905)

≺ int119889(119911119911lowast)

0

120593 (119905) 119889119905(40)

which is a contradiction hence 119911 is a unique common fixedpoint of 119870 119871119872119873 119877 and 119878 in119883

Now we present some corollaries their proofs are easilyfollowed fromTheorem 14 so we omit the proofs

Corollary 15 Let (119883 119889) be a complex valuedmetric space and119870119872119873 119877 119878 119883 rarr 119883 be five self-mappings satisfying thefollowing conditions


120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (41)

where 120595 isin Ψ 120593 isin Φlowast and

Δ 1 (1199111 1199112) = 119889 (1198721198781199112 1198701199112) 1 + 119889 (1198731198771199111 1198701199111)1 + 119889 (11987311987711991111198721198781199112)

Δ 2 (1199111 1199112) = 119889 (1198731198771199111 1198701199111) 1 + 119889 (1198721198781199112 1198701199112)1 + 119889 (11987311987711991111198721198781199112)

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198701199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198701199112 1198731198771199111)]

(42)

If one of119872119878(119883) and 119873119877(119883) is closed subspace of 119883 suchthat pairs (119870119873119877) and (119870119872119878) are weakly compatible theneach pair of pairs (119870119873119877) and (119870119872119878) has a coincidencepoint in 119883 Moreover if (119870 119878) (119870 119877) (119872119878 119877) and (119873119877 119878)are commuting pairs then 119870119872119873 119877 and 119878 have a uniquecommon fixed point in 119883Corollary 16 Let (119883 119889) be a complex valued metric spaceand 119870 119871 119877 119878 119883 rarr 119883 be four self-mappings satisfying thefollowing conditions

(1) One of the pairs (119870 119878) and (119871 119877) satisfies property(119864119860) such that 119870(119883) sube 119877(119883) and 119871(119883) sube 119878(119883)(2) forall1199111 1199112 isin 119883

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (43)


Δ 1 (1199111 1199112) = 119889 (1198771199112 1198711199112) 1 + 119889 (1198781199111 1198701199111)1 + 119889 (1198781199111 1198771199112)

Δ 2 (1199111 1199112) = 119889 (1198781199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198781199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198781199111 1198771199112) 119889 (1198781199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198781199111)]

(44)

If one of 119877(119883) and 119878(119883) is closed subspace of119883 then pairs(119870 119878) and (119871 119877) have a coincidence point in 119883 Moreover if(119870 119878) and (119871 119877) are weakly compatible then 119870 119871 119877 and 119878have a unique common fixed point in 119883


Corollary 17 Let (119883 119889) be a complex valuedmetric space and119870 119871 119877 119883 rarr 119883 be three self-mappings satisfying the followingconditions

(1) One of the pairs (119870 119877) and (119871 119877) satisfies property(119864119860) such that 119870(119883) sube 119877(119883) and 119871(119883) sube 119877(119883)(2) forall1199111 1199112 isin 119883int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (45)

where 120595 isin Ψ 120593 isin Φlowast andΔ 1 (1199111 1199112) = 119889 (1198771199112 1198711199112) 1 + 119889 (1198771199111 1198701199111)1 + 119889 (1198771199111 1198771199112)

Δ 2 (1199111 1199112) = 119889 (1198771199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198771199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198771199111 1198771199112) 119889 (1198771199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198771199111)]

(46)

If119877(119883) is closed subspace of119883 then pairs (119870 119877) and (119871 119877)have a coincidence point in 119883 Moreover if (119870 119877) and (119871 119877)areweakly compatible then119870 119871 and119877 have a unique commonfixed point in 119883Corollary 18 Let (119883 119889) be a complex valuedmetric space and119870 119871 119883 rarr 119883 be two self-mappings satisfying the followingconditions

(1) Pair (119870 119871) satisfies property (119864119860)(2) forall1199111 1199112 isin 119883int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (47)


Δ 2 (1199111 1199112) = 119889 (1198711199111 1198701199111) 1 + 119889 (1198701199112 1198711199112)1 + 119889 (1198711199111 1198701199112)

Δ 3 (1199111 1199112) = max 119889 (1198711199111 1198701199112) 119889 (1198711199111 1198701199111) 119889 (1198701199112 1198711199112) 12 [119889 (1198701199111 1198701199112) + 119889 (1198711199112 1198711199111)]

(48)

If119870(119883) is closed subspace of119883 then pair (119870 119871) has a coin-cidence point in 119883 Moreover if (119870 119871) is weakly compatiblethen mappings 119870 and 119871 have a unique common fixed point in119883

Similar to the arguments of Theorem 14 we conclude thefollowing result and omit their proof


(1) One of pairs (119870119873119877) and (119871119872119878) satisfies property(119864119860) such that 119870(119883) sube 119872119878(119883) and 119871(119883) sube 119873119877(119883)(2) forall1199111 1199112 isin 119883

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (49)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(50)

If one of119872119878(119883) and 119873119877(119883) is closed subspace of 119883 suchthat pairs (119870119873119877) and (119871119872119878) are weakly compatible theneach pair of pairs (119870119873119877) and (119871119872119878) has a coincidence pointin 119883 Moreover if (119870 119878) (119871 119877) (119872119878 119877) and (119873119877 119878) arecommuting pairs then 119870 119871119872119873 119877 and 119878 have a uniquecommon fixed point in 119883Theorem 20 Let (119883 119889) be a complex valued metric space and119870 119871119872119873 119877 119878 119883 rarr 119883 be six self-mappings satisfyingcondition (2) of Theorem 14 and either pair (119870119873119877) satisfies(119862119871119877119870) property or pair (119871119872119878) satisfies (119862119871119877119871) property suchthat 119870(119883) sube 119872119878(119883) and 119871(119883) sube 119873119877(119883) If pairs (119870119873119877)and (119871119872119878) are weakly compatible then each pair of pairs(119870119873119877) and (119871119872119878) has a coincidence point in119883 Moreover if(119870 119878) (119871 119877) (119872119878 119877) and (119873119877 119878) are commuting pairs then119870 119871119872119873 119877 and 119878 have a unique common fixed point in 119883Proof Suppose that pair (119870119873119877) satisfies (CLR119870) propertythen there exists sequence 119911119899 in119883 such that

lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119870119905 for some 119905 isin 119883 (51)

Since 119870(119883) sube 119872119878(119883) there exists 119906 isin 119883 such that 119870119905 =119872119878119906We claim that 119871119906 = 119872119878119906 To support the claim let 119871119906 =119872119878119906Then on using condition (2) ofTheorem 14 with setting


1199111 = 119911119899 and 1199112 = 119906 we have

int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (52)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(53)

Taking upper limit as 119899 rarr infin in (52) and using (51) we get

Δ 1 (119911119899 119906) 997888rarr 119889 (119870119905 119871119906) Δ 2 (119911119899 119906) 997888rarr 0Δ 3 (119911119899 119906) 997888rarr 119889 (119871119906119870119905) int119889(119870119905119871119906)0


int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119870119905119871119906)0

120593 (119905) 119889119905 0

int119889(119871119906119870119905)0

120593 (119905) 119889119905) = 120595(int119889(119871119906119870119905)0

120593 (119905) 119889119905)

≺ int119889(119871119906119870119905)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(54)


119871119906 = 119872119878119906 = 119870119905 (55)

Also since 119871(119883) sube 119873119877(119883) there exists V isin 119883 such that 119871119906 =119873119877V Thus (55) becomes

119871119906 = 119872119878119906 = 119873119877V = 119870119905 (56)

Now we assert that 119870V = 119873119877V Let on contrary 119870V = 119873119877Vthen setting 1199111 = V and 1199112 = 119906 in condition (2) ofTheorem 14we get

int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (57)

where

Δ 1 (V 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877V 119870V)1 + 119889 (119873119877V119872119878119906) Δ 2 (V 119906) = 119889 (119873119877V 119870V) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877V119872119878119906) Δ 3 (V 119906) = max 119889 (119873119877V119872119878119906) 119889 (119873119877V 119870V) 119889 (119872119878119906 119871119906) 12 [119889 (119870V119872119878119906) + 119889 (119871119906119873119877V)]

(58)

Using (56) we have

int119889(119870V119870119905)0

120593 (119905) 119889119905 ≾ 120595(max0 int119889(119870119905119870V)0

120593 (119905) 119889119905

int119889(119870119905119870V)0

120593 (119905) 119889119905) ≾ 120595(int119889(119870119905119870V)0

120593 (119905) 119889119905)

≺ int119889(119870119905119870V)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870V119870119905)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119870V)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(59)

which is impossible Thus119870V = 119870119905 and hence

119870V = 119873119877V = 119870119905 (60)

Therefore from (56) and (60) we get

119870V = 119871119906 = 119872119878119906 = 119873119877V = 119870119905 = 119911 (say) (61)

Finally following the lines in the proof of Theorem 14 wecan show that 119911 is the coincident point of pairs (119870119873119877) and(119871119872119878) and is a unique common fixed point of the mappings119870 119871119872119873 119877 and 119878

Similar to the arguments ofTheorem 20 we conclude thefollowing results and omit their proofs



(1) Either pair (119870119873119877) satisfies (119862119871119877119870) property or pair(119871119872119878) satisfies (119862119871119877119871) property such that 119870(119883) sube119872119878(119883) and 119871(119883) sube 119873119877(119883)(2) forall1199111 1199112 isin 119883

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (62)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(63)

If pairs (119870119873119877) and (119871119872119878) are weakly compatible then eachpair of pairs (119870119873119877) and (119871119872119878) has a coincidence pointin 119883 Moreover if (119870 119878) (119871 119877) (119872119878 119877) and (119873119877 119878) arecommuting pairs then 119870 119871119872119873 119877 and 119878 have a uniquecommon fixed point in119883Corollary 22 Let (119883 119889) be a metric space and 119870 119871119872119873 119877 119878 119883 rarr 119883 be six self-mappings satisfying the followingconditions


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 le 120572intΔ 3(1199111 1199112)0

120593 (119905) 119889119905 (64)

where 0 le 120572 lt 1 120593 isin Φ and

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(65)

If pairs (119870119873119877) and (119871119872119878) are weakly compatiblethen each pair (119870119873119877) and (119871119872119878) has a coincidence pointin 119883 Moreover if (119870 119878) (119871 119877) (119872119878 119877) and (119873119877 119878) arecommuting pairs then 119870 119871119872119873 119877 and 119878 have a uniquecommon fixed point in119883

Similarly to Theorem 14 one can derive variant of corol-laries fromTheorems 19 20 and 21

Remark 23 Theconclusions ofTheorems 14 19 20 and 21 arestill valid if we replace Δ 3 with Δlowast3 whereΔlowast3 (1199111 1199112) = max 119889 (119873119877119909119872119878119910) 119889 (119873119877119909119870119905) 119889 (119872119878119910 119871119910) 119889 (119870119905119872119878119910) 119889 (119871119910119873119877119909) (66)

Remark 24 Theorems 14 and 20 and Corollary 15 extendsTheorem 21 of [11] in complex valued metric space Corol-lary 16 generalizes the results of [8ndash11] in complex valuedmetric space Moreover the real valued metric space versionof our main results generalizes the results of [8ndash11]

To supportTheorem21 we present the following example

Example 25 Let119883 = 119911 = 119909 + 120580119910 119909 119910 isin [0 1) be a complexvalued metric space with metric 119889 119883 times 119883 rarr C defined by

119889 (1199111 1199112) = 10038161003816100381610038161199111 minus 11991121003816100381610038161003816 119890119894120579 for a given 120579 isin [0 1205872 ] (67)

Define self-maps 119870 119871119872119873 119877 and 119878 on 119883 by 119870119911 = 0 119871119911 =0119872119911 = 1199112119873119911 = 1199114 119877119911 = 1199113 and 119878119911 = 1199116Then

119872119878119911 = 119872(1199116) =11991112

119873119877119911 = 119873(1199113) =11991112

(68)

Also we define 120593 R2 rarr C by 120593(119909 119910) = 2+ 0 120580 and 120595 C+ rarrC+ by 120595(119911) = 1199112

Clearly 119870(119883) = 0 sube 119872119878(119883) = 119911 = 119909 + 120580119910 119909 119910 isin[0 112) and 119871(119883) sube 119873119877(119883)Now we construct sequence 119911119899 = 119909119899 + 120580119910119899 = 1(119899 + 1) +120580(119899 + 1) in119883 such that

lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119870( 1119899 + 1 +

120580119899 + 1) = 0

lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119873119877( 1119899 + 1 +

120580119899 + 1)

= lim119899rarrinfin

112 (

1119899 + 1 +

120580119899 + 1) = 0

(69)

that is there exists sequence 119911119899 in119883 such that

lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 0 = 119870119911 for 119911 = 0 + 0 120580 isin 119883 (70)

Hence (119870119873119877) satisfies (CLR119870) property


Next check the following condition

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)

Thus from (71) (73) and (74) and by using the value of120595 wehave

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)

Also pairs (119870119873119877) and (119871119872119878) are weakly compatible and(119870 119878) (119871 119877) (119872119878 119877) and (119873119877 119878) are commuting pairs

Hence from Theorem 21 0 is a unique common fixed pointof 119870 119871119872119873 119877 and 119878

3 Applications

Many researchers study the applications of common fixedpoint theorems in complex valued metric spaces see forinstance [17 18] and the references therein On the otherhand Liu et al [19] and Sarwar et al [20] study the existenceand uniqueness of common solution for the system offunctional equations arising in dynamic programming withreal domain We apply Corollary 22 for the existence anduniqueness of a common solution for the following system offunctional equations arising in dynamic programming withcomplex domain (see [21])

1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)

where 119911 and 119908 signify the state and decision vectors respec-tively 119901119894(119911) denotes the optimal return functions with initialstate 119911 120591119894 Ω times 119863 rarr Ω Θ119894 Ω times 119863 times C rarr R forall119894 isin1 2 3 4 5 6 and 119906 V Ω times 119863 rarr C

Let 119862(Ω) be the space of all continuous real valuedfunctions on possibly complex domain Ω with metric

119889 (ℎ 119896) = sup119911isinΩ

|ℎ (119911) minus 119896 (119911)| forallℎ 119896 isin 119862 (Ω) (78)

We prove the following result

Theorem 26 Let 119906 V andΘ119894 Ωtimes119863timesCrarr R 119894 = 1 2 6be bounded functions and let119870 119871119872119873 119877 119878 119862(Ω) rarr 119862(Ω)be six operators defined as


119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)

for all ℎ119894 isin 119862(Ω) and 119911 isin Ω Assume that the following condi-tions hold

(i) There exist ℎ119899 isin 119862(Ω) such that lim119899rarrinfin119870ℎ119899 =lim119899rarrinfin119873119877ℎ119899 = 119870ℎlowast for some ℎlowast isin 119862(Ω)

(ii) 119870(119862(Ω)) sube 119872119878(119862(Ω)) such that pairs (119870119873119877) and(119871119872119878) are weakly compatible(iii) Pairs (119870 119878) (119871 119877) (119872119878 119877) and (119873119877 119878) are commut-

ing(iv) For ℎ1 ℎ2 isin 119862(Ω)

int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where

Δ 3 (ℎ1 ℎ2) = max 1003816100381610038161003816119873119877ℎ1 minus119872119878ℎ21003816100381610038161003816 1003816100381610038161003816119873119877ℎ1 minus 119870ℎ11003816100381610038161003816 1003816100381610038161003816119872119878ℎ2 minus 119871ℎ21003816100381610038161003816 12 1003816100381610038161003816119870ℎ1 minus119872119878ℎ21003816100381610038161003816 + 1003816100381610038161003816119871ℎ2 minus 119873119877ℎ11003816100381610038161003816

(81)

where ℎ1 isin 119862(Ω) 0 le 120572 lt 1 and 120601 R+ rarr R+ is anonnegative summable Lebesgue integrable function such that

int1205760

120601 (119904) 119889119904 gt 0 (82)

for each 120576 gt 0Then the system of functional equations (77) hasa unique bounded solution

Proof Notice that the system of functional equations (77)has a unique bounded solution if and only if the system ofoperators (79) have a unique common fixed point Now since119906 V andΘ119894 are bounded there exists positive number 120582 suchthat

sup |119906 (119911 119908)| |V (119911 119908)| 1003816100381610038161003816Θ119894 (119911 119908 119908lowast)1003816100381610038161003816 (119911 119908 119908lowast)isin Ω times 119863 times C 119894 = 1 2 6 le 120582 (83)

Now by using properties of the theory of integration anddefinition of 120601 we conclude that for each positive number120582 there exists positive 120575(120582) such that

intΓ

120601 (119904) 119889119904 le 120582 (84)

for all Γ sube [0 2120582] with 119898(Γ) le 120575(120582) where 119898(Γ) is theLebesgue measure of Γ

Now we consider two possible cases

Case 1 Suppose that opt119908isin119863 = sup119908isin119863 Let 119911 isin Ω and ℎ1 ℎ2 isin119862(Ω) then for 120575(120582) gt 0 there exist 1199081 1199082 isin 119863 such that

119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)

From inequalities (85) and (88) it follows that

119870ℎ1 (119911) minus 119871ℎ2 (119911) lt Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))minus Θ2 ((119911 1199081 ℎ2 (1205912 (119911 1199081)))) + 120575 (120582)le 1003816100381610038161003816Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))minus Θ2 ((119911 1199081 ℎ2 (1205912 (119911 1199081))))1003816100381610038161003816 + 120575 (120582)

(89)

which gives

119870ℎ1 (119911) minus 119871ℎ2 (119911) lt max 1003816100381610038161003816Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))minus Θ2 ((119911 1199081 ℎ2 (1205912 (119911 1199081))))1003816100381610038161003816 + 120575 (120582) 1003816100381610038161003816Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082)))minus Θ2 ((119911 1199082 ℎ2 (1205912 (119911 1199082))))1003816100381610038161003816 + 120575 (120582)

(90)

Similarly using inequalities (86) and (87) we obtain


(91)



1003816100381610038161003816119870ℎ1 (119911) minus 119871ℎ2 (119911)1003816100381610038161003816 lt max 1003816100381610038161003816Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))minus Θ2 ((119911 1199081 ℎ2 (1205912 (119911 1199081))))1003816100381610038161003816 + 120575 (120582) 1003816100381610038161003816Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082)))minus Θ2 ((119911 1199082 ℎ2 (1205912 (119911 1199082))))1003816100381610038161003816 + 120575 (120582) lt max 119860+ 120575 (120582) 119861 + 120575 (120582)

(92)

where 119860 = |Θ1(119911 1199081 ℎ1(1205911(119911 1199081))) minus Θ2((119911 1199081 ℎ2(1205912(1199111199081))))| and 119861 = |Θ1(119911 1199082 ℎ1(1205911(119911 1199082))) minusΘ2((119911 1199082 ℎ2(1205912(1199111199082))))|Case 2 Suppose that opt119908isin119863 = inf119908isin119863 By following theprocedure in Case 1 one can check that (92) holds

Now from (310) we have

int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)

And by condition (iv) of Theorem 26 we get

int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt 120572intmax|119873119877ℎ1minus119872119878ℎ2||119873119877ℎ1minus119870ℎ1||119872119878ℎ2minus119871ℎ2|(12)|119870ℎ1minus119872119878ℎ2|+|119871ℎ2minus119873119877ℎ1|

0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)

and using (84) we get

int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)

Since above inequality is true for each 119911 isin Ω and 120582 gt 0 istaken arbitrarily we deduce that

int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where

Δ 3 (ℎ1 ℎ2) = max 119889 (119873119877ℎ1119872119878ℎ2) 119889 (119873119877ℎ1 119870ℎ1) 119889 (119872119878ℎ2 119871ℎ2) 12 119889 (119870ℎ1119872119878ℎ2) + 119889 (119871ℎ2 119873119877ℎ1)

(97)

Also from condition (i) of Theorem 26 pair (119870119873119877) satis-fies (CLR) property Thus all hypothesis of Corollary 22 aresatisfied Consequently operators (79) have a unique com-mon fixed point that is system (77) of functional equationshas a unique bounded solution

Competing Interests

The authors declare that they have no competing interestsregarding this manuscript

Authorsrsquo Contributions

All authors read and approved the final version

References

[1] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 no 1 pp 133ndash181 1922

[2] G Jungck ldquoCommuting maps and fixed pointsrdquoThe AmericanMathematical Monthly vol 83 no 4 pp 261ndash263 1976

[3] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[4] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[5] A-F Roldan-Lopez-de-Hierro andW Sintunavarat ldquoCommonfixed point theorems in fuzzy metric spaces using the CLRgpropertyrdquo Fuzzy Sets and Systems vol 282 pp 131ndash142 2016

[6] M Bahadur Zada M Sarwar N Rahman and M ImdadldquoCommon fixed point results involving contractive condition


of integral type in complex valued metric spacesrdquo Journal ofNonlinear Science and its Applications vol 9 no 5 pp 2900ndash2913 2016

[7] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[8] I AltunD Turkoglu andB E Rhoades ldquoFixed points ofweaklycompatible maps satisfying a general contractive condition ofintegral typerdquo Fixed Point Theory and Applications vol 2007Article ID 17301 9 pages 2007

[9] I Altun ldquoCommon fixed point theorem for maps satisfying ageneral contractive condition of integral typerdquo Acta Universi-tatis Apulensis vol 22 pp 195ndash206 2010

[10] J Kumar ldquoCommon fixed point theorems of weakly compatiblemaps satisfying (EA) and (CLR) propertyrdquo International Jour-nal of Pure and AppliedMathematics vol 88 no 3 pp 363ndash3762013

[11] Z Liu Y Han S M Kang and J S Ume ldquoCommon fixedpoint theorems for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Fixed Point Theory andApplications vol 2014 article 132 2014

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] SManro S B Jeong and SM Kang ldquoFixed point theorems formappings satisfying a general contractive condition of integraltyperdquo International Journal of Mathematical Analysis vol 7 no57 pp 2811ndash2819 2013

[14] R K Verma and H K Pathak ldquoCommon fixed point theoremsusing property (EA) in complex-valued metric spacesrdquo ThaiJournal of Mathematics vol 11 no 2 pp 347ndash355 2013

[15] G Jungck ldquoCommon fixed points for noncontinuous nonselfmaps on nonmetric spacesrdquo Far East Journal of MathematicalSciences vol 4 no 2 pp 199ndash215 1996

[16] S Bhatt S Chaukiyal and R C Dimri ldquoA common fixedpoint theorem for weakly compatible maps in complex-valuedmetric spacesrdquo Intenational Journal of Mathematical Sciencesand Appllications vol 1 no 3 pp 1385ndash1389 2011

[17] W Sintunavarat Y J Cho and P Kumam ldquoUrysohn integralequations approach by common fixed points in complex-valuedmetric spacesrdquo Advances in Difference Equations vol 2013article 49 14 pages 2013

[18] W SintunavaratM B Zada andM Sarwar ldquoCommon solutionof Urysohn integral equations with the help of common fixedpoint results in complex valuedmetric spacesrdquoRevista de la RealAcademia de Ciencias Exactas Fisicas y Naturales Serie A Inpress

[19] Z Liu X Zou S M Kang and J S Ume ldquoCommon fixedpoints for a pair of mappings satisfying contractive conditionsof integral typerdquo Journal of Inequalities and Applications vol2014 article 394 19 pages 2014

[20] M Sarwar M B Zada and I M Erhan ldquoCommon fixed pointtheorems of integral type contraction on metric spaces andits applications to system of functional equationsrdquo Fixed PointTheory and Applications vol 2015 article 217 2015

[21] M L Agranovskii Complex Analysis and Dynamical SystemsAmericanMathematical Society Providence RI USA Bar-IlanUniversity Ramat Gan Israel 2004

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


On the other hand Azam et al [12] studied complex val-ued metric space and proved common fixed point theoremsfor two self-mappings satisfying a rational type inequalityManro et al [13] generalized the theorem of Branciari [7] fortwo self-maps under contractive condition of integral typesatisfying property (EA) and (CLR) property in the settingof complex valued metric spaces Bahadur Zada et al [6]generalized the results of [13] for four self-maps in the contextof complex valued metric spaces

The aim of this paper is to prove common fixed pointtheorems for six self-maps satisfying integral type contrac-tive condition using property (EA) and (CLR) property incomplex valuedmetric spaces which extends and generalizesmany results of the existing literature

Throughout the paper C+ = 119911 isin C 119911 ≿ (0 0) optstand for sup or inf 119885 and 119884 are Banach spaces Ω sube 119885is the state space 119863 sube 119884 is the decision space Φ = 120601 120601 [0infin[rarr [0infin[ is a Lebesgue integrable mapping whichis summable on each compact subset of [0infin[ nonnegativeand nondecreasing such that for each 120576 gt 0 int120576

0120601(119905)119889119905 gt 0

and Φlowast = 120593 R119899 rarr C is a complex valued Lebesgueintegrable mapping which is summable and nonvanishingon each measurable subset of R119899 such that for each 120576 ≻ 0int1205760120593(119905)119889119905 ≻ 0

Definition 1 (see [12]) Let C be the set of complex numbersand 119911 119908 isin C Define a partial order ≾ on C as follows

119911 ≾ 119908 iff Re (119911) le Re (119908) Im (119911) le Im (119908) 119911 ≺ 119908 iff Re (119911) lt Re (119908) Im (119911) lt Im (119908) (1)

Note that(i) 1198961 1198962 isin 119877 and 1198961 le 1198962 rArr 1198961119911 ≾ 1198962119911 for all 119911 isin C(ii) 0 ≾ 119911 ≾ 119908 rArr |119911| lt |119908| for all 119911 119908 isin C(iii) 119911 ≾ 119908 and 119908 ≺ 119908lowast rArr 119911 ≺ 119908lowast for all 119911 119908 119908lowast isin C

Definition 2 (see [14]) The ldquomaxrdquo function for the partialorder relation ldquo≾rdquo is defined by the following

(1) max1199081 1199082 = 1199082 hArr 1199081 ≾ 1199082(2) If 1199081 ≾ max1199082 1199083 then 1199081 ≾ 1199082 or 1199081 ≾ 1199083(3) max1199081 1199082 = 1199082 hArr 1199081 ≾ 1199082 or |1199081| le |1199082|

Definition 3 (see [12]) Let 119883 be a nonempty set and 119889 119883 times119883 rarr C be the mapping satisfying the following axioms(1) 0 ≾ 119889(1199111 1199112) for all 1199111 1199112 isin 119883 and 119889(1199111 1199112) = 0 if

and only if 1199111 = 1199112(2) 119889(1199111 1199112) = 119889(1199112 1199111) for all 1199111 1199112 isin 119883(3) 119889(1199111 1199112) ≾ 119889(1199111 1199113) + 119889(1199113 1199112) for all 1199111 1199112 1199113 isin 119883Then pair (119883 119889) is called a complex valued metric space

Example 4 Let 1199111 1199112 isin C and define the mapping 119889 C timesCrarr C by

119889 (1199111 1199112) = 0 if 1199111 = 1199112120580 if 1199111 = 1199112 (2)

Then (C 119889) is a complex valued metric space

Definition 5 (see [12]) Let 119911119899 be a sequence in complexvalued metric (119883 119889) and 119911 isin 119883 Then 119911 is called the limitof 119911119899 if for every 119908 isin C with 0 ≺ 119908 there is 1198990 isin 119873 suchthat 119889(119911119899 119911) ≺ 119908 for all 119899 gt 1198990 and one writes lim119899rarrinfin119911119899 = 119911Lemma 6 (see [12]) Any sequence 119911119899 in complex valuedmetric space (119883 119889) converges to 119911 if and only if |119889(119911119899 119911)| rarr 0as 119899 rarr infin

Definition 7 (see [4]) Let 119883 be a nonempty set and 119870 119871 119883 rarr 119883 be two self-maps Then

(i) 119911 isin 119883 is called a fixed point of 119871 if 119871119911 = 119911(ii) 119911 isin 119883 is called a coincidence point of119870 and 119871 if119870119911 =119871119911(iii) 119911 isin 119883 is called a common fixed point of 119870 and 119871 if119870119911 = 119871119911 = 119911Jungck [2] initiated the concept of commuting maps in

the following way

Definition 8 Two self-maps 119870 and 119871 of nonempty set 119883 arecommuting if 119871119870119911 = 119870119871119911 for all 119911 isin 119883

Jungck [15] initiated the concept of weakly compatiblemaps in ordinary metric spaces while Bhatt et al [16] refinedthis notion in the complex valued metric space in thefollowing way

Definition 9 Two self-maps 119870 and 119871 on complex valuedmetric space 119883 are weakly compatible if there exists point119911 isin 119883 such that119870119871119911 = 119871119870119911 whenever119870119911 = 119871119911

Aamri and El Moutawakil [3] initiated the concept of(EA) property in ordinary metric spaces while Verma andPathak [14] defined this concept in complex valued metricspace as follows

Definition 10 Two self-maps 119870 and 119871 on a complex valuedmetric space119883 satisfy property (EA) if there exists sequence119911119899 in119883 such that

lim119903rarrinfin

119871119911119899 = lim119903rarrinfin

119870119911119899 = 119911 for some 119911 isin 119883 (3)

Sintunavarat and Kumam [4] introduced the notion of(CLR) property in ordinary metric spaces in a similar modeVerma and Pathak [14] defined this notion in a complexvalued metric space in the following way

Definition 11 Two self-maps 119870 and 119871 on a complex valuedmetric space119883 satisfy (CLR119870) if there exists sequence 119911119899 in119883 such that

lim119899rarrinfin

119871119911119899 = lim119903rarrinfin

119870119911119899 = 119870119911 for some 119911 isin 119883 (4)







120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (5)


Δ 2 (1199111 1199112) = 119889 (1198731198771199111 1198701199111) 1 + 119889 (1198721198781199112 1198711199112)1 + 119889 (11987311987711991111198721198781199112)

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(6)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119911 for some 119911 isin 119883 (7)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119872119878119908119899 = 119911 (8)


int119889(119870119911119899 119871119908119899)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899 119908119899)0

120593 (119905) 119889119905 1 le 119895 le 3) (9)

where

Δ 1 (119911119899 119908119899) = 119889 (119872119878119908119899 119871119908119899) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119908119899)

Δ 2 (119911119899 119908119899) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119908119899 119871119908119899)1 + 119889 (119873119877119911119899119872119878119908119899)

Δ 3 (119911119899 119908119899) = max 119889 (119873119877119911119899119872119878119908119899) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119908119899 119871119908119899) 12 [119889 (119870119911119899119872119878119908119899) + 119889 (119871119908119899 119873119877119911119899)]

(10)




int119889(119870119911119899 119871119908119899)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899119908119899)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899119908119899)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119911119908)0

120593 (119905) 119889119905 0 int119889(119911119908)0

120593 (119905) 119889119905)

= 120595(int119889(119911119908)0

120593 (119905) 119889119905) ≺ int119889(119911119908)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119911119908)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119911119908)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(11)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119871119908119899 = lim119899rarrinfin

119872119878119908119899 = 119911 (12)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119871119908119899 = lim119899rarrinfin

119872119878119908119899 = 119911= 119872119878119906 (13)


int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (14)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(15)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119911119871119906)0

120593 (119905) 119889119905 0 int119889(119911119871119906)0

120593 (119905) 119889119905)

= 120595(int119889(119911119871119906)0

120593 (119905) 119889119905) ≺ int119889(119911119871119906)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119911119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119911119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(16)


119871119906 = 119872119878119906 = 119911 (17)


119871119906 = 119872119878119906 = 119873119877V = 119911 (18)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (19)

where

Δ 1 (V 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877V 119870V)1 + 119889 (119873119877V119872119878119906) = 0Δ 2 (V 119906) = 119889 (119873119877V 119870V) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877V119872119878119906) = 119889 (119911119870V)

Δ 3 (V 119906) = max 119889 (119873119877V119872119878119906) 119889 (119873119877V 119870V) 119889 (119872119878119906 119871119906) 12 [119889 (119870V119872119878119906) + 119889 (119871119906119873119877V)]= 119889 (119911 119870V)

(20)

Therefore

int119889(119870V119911)0

120593 (119905) 119889119905

≾ 120595(max0 int119889(119911119870V)0

120593 (119905) 119889119905 int119889(119911119870V)0

120593 (119905) 119889119905)

≾ 120595(int119889(119911119870V)0

120593 (119905) 119889119905) ≺ int119889(119911119870V)0

120593 (119905) 119889119905

(21)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119911 (22)


119870V = 119873119877V 997904rArr 119873119877119870V = 119870119873119877V 997904rArr 119870119911 = 119873119877119911 (23)

119871119906 = 119872119878119906 997904rArr 119872119878119871119906 = 119871119872119878119906 997904rArr 119871119911 = 119872119878119911 (24)





int119889(119870119911119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (25)

where

Δ 1 (119911 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119870119911)1 + 119889 (119873119877119911119872119878119906) = 0Δ 2 (119911 119906) = 119889 (119873119877119911119870119911) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119872119878119906) = 0Δ 3 (119911 119906) = max 119889 (119873119877119911119872119878119906) 119889 (119873119877119911119870119911) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119872119878119906) + 119889 (119871119906119873119877119911)]= 119889 (119870119911 119911)

(26)

Therefore

int119889(119870119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119870119911119911)0

120593 (119905) 119889119905)

≾ 120595(int119889(119870119911119911)0

120593 (119905) 119889119905)

≺ int119889(119870119911119911)0

120593 (119905) 119889119905

(27)


119870119911 = 119873119877119911 = 119911 (28)


119871119911 = 119872119878119911 = 119911 (29)


119870119911 = 119871119911 = 119872119878119911 = 119873119877119911 = 119911 (30)


119870 (119878119911) = 119873119877 (119878119911) = 119878119911 (31)


119871 (119877119911) = 119872119878 (119877119911) = 119877119911 (32)


int119889(119870119878119911119871119911)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119878119911119911)0

120593 (119905) 119889119905 1 le 119895 le 3) (33)

where

Δ 1 (119878119911 119911) = 119889 (119872119878119911 119871119911) 1 + 119889 (119873119877119878119911119870119878119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 2 (119878119911 119911) = 119889 (119873119877119878119911119870119878119911) 1 + 119889 (119872119878119911 119871119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 3 (119878119911 119911) = max 119889 (119873119877119878119911119872119878119911) 119889 (119873119877119878119911119870119878119911) 119889 (119872119878119911 119871119911) 12 [119889 (119870119878119911119872119878119911) + 119889 (119871119911119873119877119878119911)]= max 119889 (119878119911 119911) 119889 (119878119911 119878119911) 119889 (119911 119911) 12 [119889 (119878119911 119911) + 119889 (119911 119878119911)] = 119889 (119878119911 119911)

(34)

Therefore

int119889(119878119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int(119878119911119911)0

120593 (119905) 119889119905)

≺ int(119878119911119911)0

120593 (119905) 119889119905(35)


119870119911 = 119871119911 = 119872119911 = 119878119911 = 119873119877119911 = 119911 (36)


119870119911 = 119871119911 = 119872119911 = 119873119911 = 119877119911 = 119878119911 = 119911 (37)






int119889(119911119911lowast)

0

120593 (119905) 119889119905 = int119889(119870119911119871119911lowast)

0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119911lowast)

0

120593 (119905) 119889119905 1 le 119895 le 3) (38)

where



1 + 119889 (119873119877119911119872119878119911lowast) = 0



(39)

Thus

int119889(119911119911lowast)

0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119911119911lowast)

0

120593 (119905) 119889119905)

≺ int119889(119911119911lowast)

0

120593 (119905) 119889119905(40)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (41)


Δ 1 (1199111 1199112) = 119889 (1198721198781199112 1198701199112) 1 + 119889 (1198731198771199111 1198701199111)1 + 119889 (11987311987711991111198721198781199112)

Δ 2 (1199111 1199112) = 119889 (1198731198771199111 1198701199111) 1 + 119889 (1198721198781199112 1198701199112)1 + 119889 (11987311987711991111198721198781199112)

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198701199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198701199112 1198731198771199111)]

(42)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (43)


Δ 1 (1199111 1199112) = 119889 (1198771199112 1198711199112) 1 + 119889 (1198781199111 1198701199111)1 + 119889 (1198781199111 1198771199112)

Δ 2 (1199111 1199112) = 119889 (1198781199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198781199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198781199111 1198771199112) 119889 (1198781199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198781199111)]

(44)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (45)


Δ 2 (1199111 1199112) = 119889 (1198771199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198771199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198771199111 1198771199112) 119889 (1198771199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198771199111)]

(46)



120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (47)


Δ 2 (1199111 1199112) = 119889 (1198711199111 1198701199111) 1 + 119889 (1198701199112 1198711199112)1 + 119889 (1198711199111 1198701199112)

Δ 3 (1199111 1199112) = max 119889 (1198711199111 1198701199112) 119889 (1198711199111 1198701199111) 119889 (1198701199112 1198711199112) 12 [119889 (1198701199111 1198701199112) + 119889 (1198711199112 1198711199111)]

(48)





int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (49)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(50)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119870119905 for some 119905 isin 119883 (51)



1199111 = 119911119899 and 1199112 = 119906 we have

int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (52)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(53)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119870119905119871119906)0

120593 (119905) 119889119905 0

int119889(119871119906119870119905)0

120593 (119905) 119889119905) = 120595(int119889(119871119906119870119905)0

120593 (119905) 119889119905)

≺ int119889(119871119906119870119905)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(54)


119871119906 = 119872119878119906 = 119870119905 (55)


119871119906 = 119872119878119906 = 119873119877V = 119870119905 (56)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (57)

where


(58)

Using (56) we have

int119889(119870V119870119905)0

120593 (119905) 119889119905 ≾ 120595(max0 int119889(119870119905119870V)0

120593 (119905) 119889119905

int119889(119870119905119870V)0

120593 (119905) 119889119905) ≾ 120595(int119889(119870119905119870V)0

120593 (119905) 119889119905)

≺ int119889(119870119905119870V)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870V119870119905)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119870V)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(59)


119870V = 119873119877V = 119870119905 (60)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119870119905 = 119911 (say) (61)






int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (62)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(63)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 le 120572intΔ 3(1199111 1199112)0

120593 (119905) 119889119905 (64)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(65)









119872119878119911 = 119872(1199116) =11991112

119873119877119911 = 119873(1199113) =11991112

(68)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119870( 1119899 + 1 +

120580119899 + 1) = 0

lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119873119877( 1119899 + 1 +

120580119899 + 1)


112 (

1119899 + 1 +

120580119899 + 1) = 0

(69)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 0 = 119870119911 for 119911 = 0 + 0 120580 isin 119883 (70)




int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)



3 Applications


1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)



119889 (ℎ 119896) = sup119911isinΩ





119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (5)


Δ 2 (1199111 1199112) = 119889 (1198731198771199111 1198701199111) 1 + 119889 (1198721198781199112 1198711199112)1 + 119889 (11987311987711991111198721198781199112)

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(6)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119911 for some 119911 isin 119883 (7)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119872119878119908119899 = 119911 (8)


int119889(119870119911119899 119871119908119899)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899 119908119899)0

120593 (119905) 119889119905 1 le 119895 le 3) (9)

where

Δ 1 (119911119899 119908119899) = 119889 (119872119878119908119899 119871119908119899) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119908119899)

Δ 2 (119911119899 119908119899) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119908119899 119871119908119899)1 + 119889 (119873119877119911119899119872119878119908119899)

Δ 3 (119911119899 119908119899) = max 119889 (119873119877119911119899119872119878119908119899) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119908119899 119871119908119899) 12 [119889 (119870119911119899119872119878119908119899) + 119889 (119871119908119899 119873119877119911119899)]

(10)




int119889(119870119911119899 119871119908119899)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899119908119899)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899119908119899)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119911119908)0

120593 (119905) 119889119905 0 int119889(119911119908)0

120593 (119905) 119889119905)

= 120595(int119889(119911119908)0

120593 (119905) 119889119905) ≺ int119889(119911119908)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119911119908)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119911119908)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(11)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119871119908119899 = lim119899rarrinfin

119872119878119908119899 = 119911 (12)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119871119908119899 = lim119899rarrinfin

119872119878119908119899 = 119911= 119872119878119906 (13)


int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (14)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(15)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119911119871119906)0

120593 (119905) 119889119905 0 int119889(119911119871119906)0

120593 (119905) 119889119905)

= 120595(int119889(119911119871119906)0

120593 (119905) 119889119905) ≺ int119889(119911119871119906)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119911119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119911119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(16)


119871119906 = 119872119878119906 = 119911 (17)


119871119906 = 119872119878119906 = 119873119877V = 119911 (18)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (19)

where

Δ 1 (V 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877V 119870V)1 + 119889 (119873119877V119872119878119906) = 0Δ 2 (V 119906) = 119889 (119873119877V 119870V) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877V119872119878119906) = 119889 (119911119870V)

Δ 3 (V 119906) = max 119889 (119873119877V119872119878119906) 119889 (119873119877V 119870V) 119889 (119872119878119906 119871119906) 12 [119889 (119870V119872119878119906) + 119889 (119871119906119873119877V)]= 119889 (119911 119870V)

(20)

Therefore

int119889(119870V119911)0

120593 (119905) 119889119905

≾ 120595(max0 int119889(119911119870V)0

120593 (119905) 119889119905 int119889(119911119870V)0

120593 (119905) 119889119905)

≾ 120595(int119889(119911119870V)0

120593 (119905) 119889119905) ≺ int119889(119911119870V)0

120593 (119905) 119889119905

(21)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119911 (22)


119870V = 119873119877V 997904rArr 119873119877119870V = 119870119873119877V 997904rArr 119870119911 = 119873119877119911 (23)

119871119906 = 119872119878119906 997904rArr 119872119878119871119906 = 119871119872119878119906 997904rArr 119871119911 = 119872119878119911 (24)





int119889(119870119911119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (25)

where

Δ 1 (119911 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119870119911)1 + 119889 (119873119877119911119872119878119906) = 0Δ 2 (119911 119906) = 119889 (119873119877119911119870119911) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119872119878119906) = 0Δ 3 (119911 119906) = max 119889 (119873119877119911119872119878119906) 119889 (119873119877119911119870119911) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119872119878119906) + 119889 (119871119906119873119877119911)]= 119889 (119870119911 119911)

(26)

Therefore

int119889(119870119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119870119911119911)0

120593 (119905) 119889119905)

≾ 120595(int119889(119870119911119911)0

120593 (119905) 119889119905)

≺ int119889(119870119911119911)0

120593 (119905) 119889119905

(27)


119870119911 = 119873119877119911 = 119911 (28)


119871119911 = 119872119878119911 = 119911 (29)


119870119911 = 119871119911 = 119872119878119911 = 119873119877119911 = 119911 (30)


119870 (119878119911) = 119873119877 (119878119911) = 119878119911 (31)


119871 (119877119911) = 119872119878 (119877119911) = 119877119911 (32)


int119889(119870119878119911119871119911)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119878119911119911)0

120593 (119905) 119889119905 1 le 119895 le 3) (33)

where

Δ 1 (119878119911 119911) = 119889 (119872119878119911 119871119911) 1 + 119889 (119873119877119878119911119870119878119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 2 (119878119911 119911) = 119889 (119873119877119878119911119870119878119911) 1 + 119889 (119872119878119911 119871119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 3 (119878119911 119911) = max 119889 (119873119877119878119911119872119878119911) 119889 (119873119877119878119911119870119878119911) 119889 (119872119878119911 119871119911) 12 [119889 (119870119878119911119872119878119911) + 119889 (119871119911119873119877119878119911)]= max 119889 (119878119911 119911) 119889 (119878119911 119878119911) 119889 (119911 119911) 12 [119889 (119878119911 119911) + 119889 (119911 119878119911)] = 119889 (119878119911 119911)

(34)

Therefore

int119889(119878119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int(119878119911119911)0

120593 (119905) 119889119905)

≺ int(119878119911119911)0

120593 (119905) 119889119905(35)


119870119911 = 119871119911 = 119872119911 = 119878119911 = 119873119877119911 = 119911 (36)


119870119911 = 119871119911 = 119872119911 = 119873119911 = 119877119911 = 119878119911 = 119911 (37)






int119889(119911119911lowast)

0

120593 (119905) 119889119905 = int119889(119870119911119871119911lowast)

0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119911lowast)

0

120593 (119905) 119889119905 1 le 119895 le 3) (38)

where



1 + 119889 (119873119877119911119872119878119911lowast) = 0



(39)

Thus

int119889(119911119911lowast)

0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119911119911lowast)

0

120593 (119905) 119889119905)

≺ int119889(119911119911lowast)

0

120593 (119905) 119889119905(40)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (41)


Δ 1 (1199111 1199112) = 119889 (1198721198781199112 1198701199112) 1 + 119889 (1198731198771199111 1198701199111)1 + 119889 (11987311987711991111198721198781199112)

Δ 2 (1199111 1199112) = 119889 (1198731198771199111 1198701199111) 1 + 119889 (1198721198781199112 1198701199112)1 + 119889 (11987311987711991111198721198781199112)

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198701199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198701199112 1198731198771199111)]

(42)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (43)


Δ 1 (1199111 1199112) = 119889 (1198771199112 1198711199112) 1 + 119889 (1198781199111 1198701199111)1 + 119889 (1198781199111 1198771199112)

Δ 2 (1199111 1199112) = 119889 (1198781199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198781199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198781199111 1198771199112) 119889 (1198781199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198781199111)]

(44)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (45)


Δ 2 (1199111 1199112) = 119889 (1198771199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198771199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198771199111 1198771199112) 119889 (1198771199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198771199111)]

(46)



120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (47)


Δ 2 (1199111 1199112) = 119889 (1198711199111 1198701199111) 1 + 119889 (1198701199112 1198711199112)1 + 119889 (1198711199111 1198701199112)

Δ 3 (1199111 1199112) = max 119889 (1198711199111 1198701199112) 119889 (1198711199111 1198701199111) 119889 (1198701199112 1198711199112) 12 [119889 (1198701199111 1198701199112) + 119889 (1198711199112 1198711199111)]

(48)





int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (49)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(50)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119870119905 for some 119905 isin 119883 (51)



1199111 = 119911119899 and 1199112 = 119906 we have

int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (52)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(53)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119870119905119871119906)0

120593 (119905) 119889119905 0

int119889(119871119906119870119905)0

120593 (119905) 119889119905) = 120595(int119889(119871119906119870119905)0

120593 (119905) 119889119905)

≺ int119889(119871119906119870119905)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(54)


119871119906 = 119872119878119906 = 119870119905 (55)


119871119906 = 119872119878119906 = 119873119877V = 119870119905 (56)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (57)

where


(58)

Using (56) we have

int119889(119870V119870119905)0

120593 (119905) 119889119905 ≾ 120595(max0 int119889(119870119905119870V)0

120593 (119905) 119889119905

int119889(119870119905119870V)0

120593 (119905) 119889119905) ≾ 120595(int119889(119870119905119870V)0

120593 (119905) 119889119905)

≺ int119889(119870119905119870V)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870V119870119905)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119870V)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(59)


119870V = 119873119877V = 119870119905 (60)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119870119905 = 119911 (say) (61)






int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (62)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(63)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 le 120572intΔ 3(1199111 1199112)0

120593 (119905) 119889119905 (64)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(65)









119872119878119911 = 119872(1199116) =11991112

119873119877119911 = 119873(1199113) =11991112

(68)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119870( 1119899 + 1 +

120580119899 + 1) = 0

lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119873119877( 1119899 + 1 +

120580119899 + 1)


112 (

1119899 + 1 +

120580119899 + 1) = 0

(69)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 0 = 119870119911 for 119911 = 0 + 0 120580 isin 119883 (70)




int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)



3 Applications


1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)



119889 (ℎ 119896) = sup119911isinΩ





119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119871119908119899 = lim119899rarrinfin

119872119878119908119899 = 119911= 119872119878119906 (13)


int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (14)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(15)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119911119871119906)0

120593 (119905) 119889119905 0 int119889(119911119871119906)0

120593 (119905) 119889119905)

= 120595(int119889(119911119871119906)0

120593 (119905) 119889119905) ≺ int119889(119911119871119906)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119911119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119911119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(16)


119871119906 = 119872119878119906 = 119911 (17)


119871119906 = 119872119878119906 = 119873119877V = 119911 (18)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (19)

where

Δ 1 (V 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877V 119870V)1 + 119889 (119873119877V119872119878119906) = 0Δ 2 (V 119906) = 119889 (119873119877V 119870V) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877V119872119878119906) = 119889 (119911119870V)

Δ 3 (V 119906) = max 119889 (119873119877V119872119878119906) 119889 (119873119877V 119870V) 119889 (119872119878119906 119871119906) 12 [119889 (119870V119872119878119906) + 119889 (119871119906119873119877V)]= 119889 (119911 119870V)

(20)

Therefore

int119889(119870V119911)0

120593 (119905) 119889119905

≾ 120595(max0 int119889(119911119870V)0

120593 (119905) 119889119905 int119889(119911119870V)0

120593 (119905) 119889119905)

≾ 120595(int119889(119911119870V)0

120593 (119905) 119889119905) ≺ int119889(119911119870V)0

120593 (119905) 119889119905

(21)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119911 (22)


119870V = 119873119877V 997904rArr 119873119877119870V = 119870119873119877V 997904rArr 119870119911 = 119873119877119911 (23)

119871119906 = 119872119878119906 997904rArr 119872119878119871119906 = 119871119872119878119906 997904rArr 119871119911 = 119872119878119911 (24)





int119889(119870119911119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (25)

where

Δ 1 (119911 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119870119911)1 + 119889 (119873119877119911119872119878119906) = 0Δ 2 (119911 119906) = 119889 (119873119877119911119870119911) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119872119878119906) = 0Δ 3 (119911 119906) = max 119889 (119873119877119911119872119878119906) 119889 (119873119877119911119870119911) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119872119878119906) + 119889 (119871119906119873119877119911)]= 119889 (119870119911 119911)

(26)

Therefore

int119889(119870119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119870119911119911)0

120593 (119905) 119889119905)

≾ 120595(int119889(119870119911119911)0

120593 (119905) 119889119905)

≺ int119889(119870119911119911)0

120593 (119905) 119889119905

(27)


119870119911 = 119873119877119911 = 119911 (28)


119871119911 = 119872119878119911 = 119911 (29)


119870119911 = 119871119911 = 119872119878119911 = 119873119877119911 = 119911 (30)


119870 (119878119911) = 119873119877 (119878119911) = 119878119911 (31)


119871 (119877119911) = 119872119878 (119877119911) = 119877119911 (32)


int119889(119870119878119911119871119911)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119878119911119911)0

120593 (119905) 119889119905 1 le 119895 le 3) (33)

where

Δ 1 (119878119911 119911) = 119889 (119872119878119911 119871119911) 1 + 119889 (119873119877119878119911119870119878119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 2 (119878119911 119911) = 119889 (119873119877119878119911119870119878119911) 1 + 119889 (119872119878119911 119871119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 3 (119878119911 119911) = max 119889 (119873119877119878119911119872119878119911) 119889 (119873119877119878119911119870119878119911) 119889 (119872119878119911 119871119911) 12 [119889 (119870119878119911119872119878119911) + 119889 (119871119911119873119877119878119911)]= max 119889 (119878119911 119911) 119889 (119878119911 119878119911) 119889 (119911 119911) 12 [119889 (119878119911 119911) + 119889 (119911 119878119911)] = 119889 (119878119911 119911)

(34)

Therefore

int119889(119878119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int(119878119911119911)0

120593 (119905) 119889119905)

≺ int(119878119911119911)0

120593 (119905) 119889119905(35)


119870119911 = 119871119911 = 119872119911 = 119878119911 = 119873119877119911 = 119911 (36)


119870119911 = 119871119911 = 119872119911 = 119873119911 = 119877119911 = 119878119911 = 119911 (37)






int119889(119911119911lowast)

0

120593 (119905) 119889119905 = int119889(119870119911119871119911lowast)

0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119911lowast)

0

120593 (119905) 119889119905 1 le 119895 le 3) (38)

where



1 + 119889 (119873119877119911119872119878119911lowast) = 0



(39)

Thus

int119889(119911119911lowast)

0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119911119911lowast)

0

120593 (119905) 119889119905)

≺ int119889(119911119911lowast)

0

120593 (119905) 119889119905(40)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (41)


Δ 1 (1199111 1199112) = 119889 (1198721198781199112 1198701199112) 1 + 119889 (1198731198771199111 1198701199111)1 + 119889 (11987311987711991111198721198781199112)

Δ 2 (1199111 1199112) = 119889 (1198731198771199111 1198701199111) 1 + 119889 (1198721198781199112 1198701199112)1 + 119889 (11987311987711991111198721198781199112)

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198701199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198701199112 1198731198771199111)]

(42)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (43)


Δ 1 (1199111 1199112) = 119889 (1198771199112 1198711199112) 1 + 119889 (1198781199111 1198701199111)1 + 119889 (1198781199111 1198771199112)

Δ 2 (1199111 1199112) = 119889 (1198781199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198781199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198781199111 1198771199112) 119889 (1198781199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198781199111)]

(44)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (45)


Δ 2 (1199111 1199112) = 119889 (1198771199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198771199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198771199111 1198771199112) 119889 (1198771199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198771199111)]

(46)



120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (47)


Δ 2 (1199111 1199112) = 119889 (1198711199111 1198701199111) 1 + 119889 (1198701199112 1198711199112)1 + 119889 (1198711199111 1198701199112)

Δ 3 (1199111 1199112) = max 119889 (1198711199111 1198701199112) 119889 (1198711199111 1198701199111) 119889 (1198701199112 1198711199112) 12 [119889 (1198701199111 1198701199112) + 119889 (1198711199112 1198711199111)]

(48)





int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (49)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(50)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119870119905 for some 119905 isin 119883 (51)



1199111 = 119911119899 and 1199112 = 119906 we have

int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (52)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(53)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119870119905119871119906)0

120593 (119905) 119889119905 0

int119889(119871119906119870119905)0

120593 (119905) 119889119905) = 120595(int119889(119871119906119870119905)0

120593 (119905) 119889119905)

≺ int119889(119871119906119870119905)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(54)


119871119906 = 119872119878119906 = 119870119905 (55)


119871119906 = 119872119878119906 = 119873119877V = 119870119905 (56)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (57)

where


(58)

Using (56) we have

int119889(119870V119870119905)0

120593 (119905) 119889119905 ≾ 120595(max0 int119889(119870119905119870V)0

120593 (119905) 119889119905

int119889(119870119905119870V)0

120593 (119905) 119889119905) ≾ 120595(int119889(119870119905119870V)0

120593 (119905) 119889119905)

≺ int119889(119870119905119870V)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870V119870119905)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119870V)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(59)


119870V = 119873119877V = 119870119905 (60)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119870119905 = 119911 (say) (61)






int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (62)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(63)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 le 120572intΔ 3(1199111 1199112)0

120593 (119905) 119889119905 (64)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(65)









119872119878119911 = 119872(1199116) =11991112

119873119877119911 = 119873(1199113) =11991112

(68)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119870( 1119899 + 1 +

120580119899 + 1) = 0

lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119873119877( 1119899 + 1 +

120580119899 + 1)


112 (

1119899 + 1 +

120580119899 + 1) = 0

(69)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 0 = 119870119911 for 119911 = 0 + 0 120580 isin 119883 (70)




int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)



3 Applications


1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)



119889 (ℎ 119896) = sup119911isinΩ





119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int119889(119870119911119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (25)

where

Δ 1 (119911 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119870119911)1 + 119889 (119873119877119911119872119878119906) = 0Δ 2 (119911 119906) = 119889 (119873119877119911119870119911) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119872119878119906) = 0Δ 3 (119911 119906) = max 119889 (119873119877119911119872119878119906) 119889 (119873119877119911119870119911) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119872119878119906) + 119889 (119871119906119873119877119911)]= 119889 (119870119911 119911)

(26)

Therefore

int119889(119870119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119870119911119911)0

120593 (119905) 119889119905)

≾ 120595(int119889(119870119911119911)0

120593 (119905) 119889119905)

≺ int119889(119870119911119911)0

120593 (119905) 119889119905

(27)


119870119911 = 119873119877119911 = 119911 (28)


119871119911 = 119872119878119911 = 119911 (29)


119870119911 = 119871119911 = 119872119878119911 = 119873119877119911 = 119911 (30)


119870 (119878119911) = 119873119877 (119878119911) = 119878119911 (31)


119871 (119877119911) = 119872119878 (119877119911) = 119877119911 (32)


int119889(119870119878119911119871119911)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119878119911119911)0

120593 (119905) 119889119905 1 le 119895 le 3) (33)

where

Δ 1 (119878119911 119911) = 119889 (119872119878119911 119871119911) 1 + 119889 (119873119877119878119911119870119878119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 2 (119878119911 119911) = 119889 (119873119877119878119911119870119878119911) 1 + 119889 (119872119878119911 119871119911)1 + 119889 (119873119877119878119911119872119878119911) = 0

Δ 3 (119878119911 119911) = max 119889 (119873119877119878119911119872119878119911) 119889 (119873119877119878119911119870119878119911) 119889 (119872119878119911 119871119911) 12 [119889 (119870119878119911119872119878119911) + 119889 (119871119911119873119877119878119911)]= max 119889 (119878119911 119911) 119889 (119878119911 119878119911) 119889 (119911 119911) 12 [119889 (119878119911 119911) + 119889 (119911 119878119911)] = 119889 (119878119911 119911)

(34)

Therefore

int119889(119878119911119911)0

120593 (119905) 119889119905 ≾ 120595(max0 0 int(119878119911119911)0

120593 (119905) 119889119905)

≺ int(119878119911119911)0

120593 (119905) 119889119905(35)


119870119911 = 119871119911 = 119872119911 = 119878119911 = 119873119877119911 = 119911 (36)


119870119911 = 119871119911 = 119872119911 = 119873119911 = 119877119911 = 119878119911 = 119911 (37)






int119889(119911119911lowast)

0

120593 (119905) 119889119905 = int119889(119870119911119871119911lowast)

0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119911lowast)

0

120593 (119905) 119889119905 1 le 119895 le 3) (38)

where



1 + 119889 (119873119877119911119872119878119911lowast) = 0



(39)

Thus

int119889(119911119911lowast)

0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119911119911lowast)

0

120593 (119905) 119889119905)

≺ int119889(119911119911lowast)

0

120593 (119905) 119889119905(40)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (41)


Δ 1 (1199111 1199112) = 119889 (1198721198781199112 1198701199112) 1 + 119889 (1198731198771199111 1198701199111)1 + 119889 (11987311987711991111198721198781199112)

Δ 2 (1199111 1199112) = 119889 (1198731198771199111 1198701199111) 1 + 119889 (1198721198781199112 1198701199112)1 + 119889 (11987311987711991111198721198781199112)

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198701199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198701199112 1198731198771199111)]

(42)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (43)


Δ 1 (1199111 1199112) = 119889 (1198771199112 1198711199112) 1 + 119889 (1198781199111 1198701199111)1 + 119889 (1198781199111 1198771199112)

Δ 2 (1199111 1199112) = 119889 (1198781199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198781199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198781199111 1198771199112) 119889 (1198781199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198781199111)]

(44)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (45)


Δ 2 (1199111 1199112) = 119889 (1198771199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198771199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198771199111 1198771199112) 119889 (1198771199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198771199111)]

(46)



120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (47)


Δ 2 (1199111 1199112) = 119889 (1198711199111 1198701199111) 1 + 119889 (1198701199112 1198711199112)1 + 119889 (1198711199111 1198701199112)

Δ 3 (1199111 1199112) = max 119889 (1198711199111 1198701199112) 119889 (1198711199111 1198701199111) 119889 (1198701199112 1198711199112) 12 [119889 (1198701199111 1198701199112) + 119889 (1198711199112 1198711199111)]

(48)





int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (49)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(50)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119870119905 for some 119905 isin 119883 (51)



1199111 = 119911119899 and 1199112 = 119906 we have

int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (52)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(53)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119870119905119871119906)0

120593 (119905) 119889119905 0

int119889(119871119906119870119905)0

120593 (119905) 119889119905) = 120595(int119889(119871119906119870119905)0

120593 (119905) 119889119905)

≺ int119889(119871119906119870119905)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(54)


119871119906 = 119872119878119906 = 119870119905 (55)


119871119906 = 119872119878119906 = 119873119877V = 119870119905 (56)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (57)

where


(58)

Using (56) we have

int119889(119870V119870119905)0

120593 (119905) 119889119905 ≾ 120595(max0 int119889(119870119905119870V)0

120593 (119905) 119889119905

int119889(119870119905119870V)0

120593 (119905) 119889119905) ≾ 120595(int119889(119870119905119870V)0

120593 (119905) 119889119905)

≺ int119889(119870119905119870V)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870V119870119905)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119870V)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(59)


119870V = 119873119877V = 119870119905 (60)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119870119905 = 119911 (say) (61)






int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (62)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(63)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 le 120572intΔ 3(1199111 1199112)0

120593 (119905) 119889119905 (64)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(65)









119872119878119911 = 119872(1199116) =11991112

119873119877119911 = 119873(1199113) =11991112

(68)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119870( 1119899 + 1 +

120580119899 + 1) = 0

lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119873119877( 1119899 + 1 +

120580119899 + 1)


112 (

1119899 + 1 +

120580119899 + 1) = 0

(69)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 0 = 119870119911 for 119911 = 0 + 0 120580 isin 119883 (70)




int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)



3 Applications


1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)



119889 (ℎ 119896) = sup119911isinΩ





119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












int119889(119911119911lowast)

0

120593 (119905) 119889119905 = int119889(119870119911119871119911lowast)

0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119911lowast)

0

120593 (119905) 119889119905 1 le 119895 le 3) (38)

where



1 + 119889 (119873119877119911119872119878119911lowast) = 0



(39)

Thus

int119889(119911119911lowast)

0

120593 (119905) 119889119905 ≾ 120595(max0 0 int119889(119911119911lowast)

0

120593 (119905) 119889119905)

≺ int119889(119911119911lowast)

0

120593 (119905) 119889119905(40)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (41)


Δ 1 (1199111 1199112) = 119889 (1198721198781199112 1198701199112) 1 + 119889 (1198731198771199111 1198701199111)1 + 119889 (11987311987711991111198721198781199112)

Δ 2 (1199111 1199112) = 119889 (1198731198771199111 1198701199111) 1 + 119889 (1198721198781199112 1198701199112)1 + 119889 (11987311987711991111198721198781199112)

Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198701199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198701199112 1198731198771199111)]

(42)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (43)


Δ 1 (1199111 1199112) = 119889 (1198771199112 1198711199112) 1 + 119889 (1198781199111 1198701199111)1 + 119889 (1198781199111 1198771199112)

Δ 2 (1199111 1199112) = 119889 (1198781199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198781199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198781199111 1198771199112) 119889 (1198781199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198781199111)]

(44)





120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (45)


Δ 2 (1199111 1199112) = 119889 (1198771199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198771199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198771199111 1198771199112) 119889 (1198771199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198771199111)]

(46)



120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (47)


Δ 2 (1199111 1199112) = 119889 (1198711199111 1198701199111) 1 + 119889 (1198701199112 1198711199112)1 + 119889 (1198711199111 1198701199112)

Δ 3 (1199111 1199112) = max 119889 (1198711199111 1198701199112) 119889 (1198711199111 1198701199111) 119889 (1198701199112 1198711199112) 12 [119889 (1198701199111 1198701199112) + 119889 (1198711199112 1198711199111)]

(48)





int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (49)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(50)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119870119905 for some 119905 isin 119883 (51)



1199111 = 119911119899 and 1199112 = 119906 we have

int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (52)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(53)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119870119905119871119906)0

120593 (119905) 119889119905 0

int119889(119871119906119870119905)0

120593 (119905) 119889119905) = 120595(int119889(119871119906119870119905)0

120593 (119905) 119889119905)

≺ int119889(119871119906119870119905)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(54)


119871119906 = 119872119878119906 = 119870119905 (55)


119871119906 = 119872119878119906 = 119873119877V = 119870119905 (56)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (57)

where


(58)

Using (56) we have

int119889(119870V119870119905)0

120593 (119905) 119889119905 ≾ 120595(max0 int119889(119870119905119870V)0

120593 (119905) 119889119905

int119889(119870119905119870V)0

120593 (119905) 119889119905) ≾ 120595(int119889(119870119905119870V)0

120593 (119905) 119889119905)

≺ int119889(119870119905119870V)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870V119870119905)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119870V)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(59)


119870V = 119873119877V = 119870119905 (60)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119870119905 = 119911 (say) (61)






int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (62)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(63)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 le 120572intΔ 3(1199111 1199112)0

120593 (119905) 119889119905 (64)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(65)









119872119878119911 = 119872(1199116) =11991112

119873119877119911 = 119873(1199113) =11991112

(68)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119870( 1119899 + 1 +

120580119899 + 1) = 0

lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119873119877( 1119899 + 1 +

120580119899 + 1)


112 (

1119899 + 1 +

120580119899 + 1) = 0

(69)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 0 = 119870119911 for 119911 = 0 + 0 120580 isin 119883 (70)




int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)



3 Applications


1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)



119889 (ℎ 119896) = sup119911isinΩ





119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (45)


Δ 2 (1199111 1199112) = 119889 (1198771199111 1198701199111) 1 + 119889 (1198771199112 1198711199112)1 + 119889 (1198771199111 1198771199112)

Δ 3 (1199111 1199112) = max 119889 (1198771199111 1198771199112) 119889 (1198771199111 1198701199111) 119889 (1198771199112 1198711199112) 12 [119889 (1198701199111 1198771199112) + 119889 (1198711199112 1198771199111)]

(46)



120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(1199111 1199112)0

120593 (119905) 119889119905 1 le 119895 le 3) (47)


Δ 2 (1199111 1199112) = 119889 (1198711199111 1198701199111) 1 + 119889 (1198701199112 1198711199112)1 + 119889 (1198711199111 1198701199112)

Δ 3 (1199111 1199112) = max 119889 (1198711199111 1198701199112) 119889 (1198711199111 1198701199111) 119889 (1198701199112 1198711199112) 12 [119889 (1198701199111 1198701199112) + 119889 (1198711199112 1198711199111)]

(48)





int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (49)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(50)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 119870119905 for some 119905 isin 119883 (51)



1199111 = 119911119899 and 1199112 = 119906 we have

int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (52)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(53)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119870119905119871119906)0

120593 (119905) 119889119905 0

int119889(119871119906119870119905)0

120593 (119905) 119889119905) = 120595(int119889(119871119906119870119905)0

120593 (119905) 119889119905)

≺ int119889(119871119906119870119905)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(54)


119871119906 = 119872119878119906 = 119870119905 (55)


119871119906 = 119872119878119906 = 119873119877V = 119870119905 (56)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (57)

where


(58)

Using (56) we have

int119889(119870V119870119905)0

120593 (119905) 119889119905 ≾ 120595(max0 int119889(119870119905119870V)0

120593 (119905) 119889119905

int119889(119870119905119870V)0

120593 (119905) 119889119905) ≾ 120595(int119889(119870119905119870V)0

120593 (119905) 119889119905)

≺ int119889(119870119905119870V)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870V119870119905)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119870V)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(59)


119870V = 119873119877V = 119870119905 (60)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119870119905 = 119911 (say) (61)






int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (62)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(63)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 le 120572intΔ 3(1199111 1199112)0

120593 (119905) 119889119905 (64)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(65)









119872119878119911 = 119872(1199116) =11991112

119873119877119911 = 119873(1199113) =11991112

(68)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119870( 1119899 + 1 +

120580119899 + 1) = 0

lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119873119877( 1119899 + 1 +

120580119899 + 1)


112 (

1119899 + 1 +

120580119899 + 1) = 0

(69)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 0 = 119870119911 for 119911 = 0 + 0 120580 isin 119883 (70)




int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)



3 Applications


1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)



119889 (ℎ 119896) = sup119911isinΩ





119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1199111 = 119911119899 and 1199112 = 119906 we have

int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(119911119899119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (52)

where

Δ 1 (119911119899 119906) = 119889 (119872119878119906 119871119906) 1 + 119889 (119873119877119911119899 119870119911119899)1 + 119889 (119873119877119911119899119872119878119906)

Δ 2 (119911119899 119906) = 119889 (119873119877119911119899 119870119911119899) 1 + 119889 (119872119878119906 119871119906)1 + 119889 (119873119877119911119899119872119878119906)

Δ 3 (119911119899 119906) = max 119889 (119873119877119911119899119872119878119906) 119889 (119873119877119911119899 119870119911119899) 119889 (119872119878119906 119871119906) 12 [119889 (119870119911119899119872119878119906) + 119889 (119871119906119873119877119911119899)]

(53)




int119889(119870119911119899 119871119906)0

120593 (119905) 119889119905


120595(maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)


maxintΔ 119895(119911119899 119906)0

120593 (119905) 119889119905 1 le 119895 le 3)

= 120595(maxint119889(119870119905119871119906)0

120593 (119905) 119889119905 0

int119889(119871119906119870119905)0

120593 (119905) 119889119905) = 120595(int119889(119871119906119870119905)0

120593 (119905) 119889119905)

≺ int119889(119871119906119870119905)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119871119906)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(54)


119871119906 = 119872119878119906 = 119870119905 (55)


119871119906 = 119872119878119906 = 119873119877V = 119870119905 (56)


int119889(119870V119871119906)0

120593 (119905) 119889119905

≾ 120595(maxintΔ 119895(V119906)0

120593 (119905) 119889119905 1 le 119895 le 3) (57)

where


(58)

Using (56) we have

int119889(119870V119870119905)0

120593 (119905) 119889119905 ≾ 120595(max0 int119889(119870119905119870V)0

120593 (119905) 119889119905

int119889(119870119905119870V)0

120593 (119905) 119889119905) ≾ 120595(int119889(119870119905119870V)0

120593 (119905) 119889119905)

≺ int119889(119870119905119870V)0

120593 (119905) 119889119905 997904rArr100381610038161003816100381610038161003816100381610038161003816int119889(119870V119870119905)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816 lt100381610038161003816100381610038161003816100381610038161003816int119889(119870119905119870V)

0

120593 (119905) 119889119905100381610038161003816100381610038161003816100381610038161003816

(59)


119870V = 119873119877V = 119870119905 (60)


119870V = 119871119906 = 119872119878119906 = 119873119877V = 119870119905 = 119911 (say) (61)






int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (62)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(63)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 le 120572intΔ 3(1199111 1199112)0

120593 (119905) 119889119905 (64)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(65)









119872119878119911 = 119872(1199116) =11991112

119873119877119911 = 119873(1199113) =11991112

(68)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119870( 1119899 + 1 +

120580119899 + 1) = 0

lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119873119877( 1119899 + 1 +

120580119899 + 1)


112 (

1119899 + 1 +

120580119899 + 1) = 0

(69)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 0 = 119870119911 for 119911 = 0 + 0 120580 isin 119883 (70)




int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)



3 Applications


1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)



119889 (ℎ 119896) = sup119911isinΩ





119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ 3(1199111 1199112)0

120593 (119905) 119889119905) (62)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(63)



int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 le 120572intΔ 3(1199111 1199112)0

120593 (119905) 119889119905 (64)


Δ 3 (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(65)









119872119878119911 = 119872(1199116) =11991112

119873119877119911 = 119873(1199113) =11991112

(68)



lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119870( 1119899 + 1 +

120580119899 + 1) = 0

lim119899rarrinfin

119873119877119911119899 = lim119899rarrinfin

119873119877( 1119899 + 1 +

120580119899 + 1)


112 (

1119899 + 1 +

120580119899 + 1) = 0

(69)


lim119899rarrinfin

119870119911119899 = lim119899rarrinfin

119873119877119911119899 = 0 = 119870119911 for 119911 = 0 + 0 120580 isin 119883 (70)




int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)



3 Applications


1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)



119889 (ℎ 119896) = sup119911isinΩ





119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905)= 120595 (2119905|Δ(1199111 1199112)) = Δ (1199111 1199112)

(71)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)] = max 1003816100381610038161003816100381610038161003816

119911112minus 119911212

1003816100381610038161003816100381610038161003816 11989011989412057910038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816 119890119894120579

12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(72)

Since

0 ≾ max 1003816100381610038161003816100381610038161003816119911112 minus

1199112121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

10038161003816100381610038161003816100381610038161199112121003816100381610038161003816100381610038161003816

sdot 119890119894120579 12 10038161003816100381610038161003816100381610038161199111361003816100381610038161003816100381610038161003816 119890119894120579 +

10038161003816100381610038161003816100381610038161199111121003816100381610038161003816100381610038161003816 119890119894120579

(73)

therefore

int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(74)


int119889(1198701199111 1198711199112)0

120593 (119905) 119889119905 ≾ 120595(intΔ(11991111199112)0

120593 (119905) 119889119905) (75)

where

Δ (1199111 1199112) = max 119889 (11987311987711991111198721198781199112) 119889 (1198731198771199111 1198701199111) 119889 (1198721198781199112 1198711199112) 12 [119889 (11987011991111198721198781199112) + 119889 (1198711199112 1198731198771199111)]

(76)



3 Applications


1199011 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 1199011 (1205911 (119911 119908)))forall119911 isin Ω

1199012 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 1199012 (1205912 (119911 119908)))forall119911 isin Ω

1199013 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 1199013 (1205913 (119911 119908)))forall119911 isin Ω

1199014 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 1199014 (1205914 (119911 119908)))forall119911 isin Ω

1199015 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 1199015 (1205915 (119911 119908)))forall119911 isin Ω

1199016 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 1199016 (1205916 (119911 119908)))forall119911 isin Ω

(77)



119889 (ℎ 119896) = sup119911isinΩ





119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119870ℎ1 (119911) = opt119908isin119863

119906 (119911 119908) + Θ1 (119911 119908 ℎ1 (1205911 (119911 119908)))forall119911 isin Ω

119871ℎ2 (119911) = opt119908isin119863

119906 (119911 119908) + Θ2 (119911 119908 ℎ2 (1205912 (119911 119908)))forall119911 isin Ω

119872ℎ3 (119911) = opt119908isin119863

V (119911 119908) + Θ3 (119911 119908 ℎ3 (1205913 (119911 119908)))forall119911 isin Ω

119873ℎ4 (119911) = opt119908isin119863

V (119911 119908) + Θ4 (119911 119908 ℎ4 (1205914 (119911 119908)))forall119911 isin Ω

119877ℎ5 (119911) = opt119908isin119863

V (119911 119908) + Θ5 (119911 119908 ℎ5 (1205915 (119911 119908)))forall119911 isin Ω

119878ℎ6 (119911) = opt119908isin119863

V (119911 119908) + Θ6 (119911 119908 ℎ6 (1205916 (119911 119908)))forall119911 isin Ω

(79)





int|Θ1(119911119908ℎ1(120591(119911119908)))minusΘ2(119911119908ℎ2(120591(119911119908)))|0

120593 (119905) 119889119905

le 120572intΔ 3(ℎ1 ℎ2)0

120593 (119905) 119889119905(80)

where


(81)


int1205760

120601 (119904) 119889119904 gt 0 (82)





intΓ

120601 (119904) 119889119904 le 120582 (84)




119870ℎ1 (119911) lt 119906 (119911 1199081) + Θ1 (119911 1199081 ℎ1 (1205911 (119911 1199081)))+ 120575 (120582) (85)

119871ℎ2 (119911) lt 119906 (119911 1199082) + Θ2 (119911 1199082 ℎ2 (1205912 (119911 1199082)))+ 120575 (120582) (86)

119870ℎ1 (119911) ge 119906 (119911 1199082) + Θ1 (119911 1199082 ℎ1 (1205911 (119911 1199082))) (87)

119871ℎ2 (119911) ge 119906 (119911 1199081) + Θ2 (119911 1199081 ℎ2 (1205912 (119911 1199081))) (88)



(89)

which gives


(90)



(91)




(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(92)



int|119870ℎ1(119911)minus119871ℎ2(119911)|0

120601 (119905) 119889119905 lt intmax119860+120575(120582)119861+120575(120582)

0

120601 (119905) 119889119905

= maxint119860+120575(120582)0

120601 (119905) 119889119905 int119861+120575(120582)0

120601 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 + int119860+120575(120582)119860

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+ int119861+120575(120582)119861

120593 (119905) 119889119905

= maxint1198600

120593 (119905) 119889119905 int1198610

120593 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905

(93)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905

+maxint119860+120575(120582)119860

120593 (119905) 119889119905 int119861+120575(120582)119861

120593 (119905) 119889119905 (94)


int|119870ℎ1(119911)minus119871ℎ2(119911)|0


0

120601 (119905) 119889119905 + 120582 (95)


int119889(119870ℎ1 119871ℎ2)0

120601 (119905) 119889119905 le 120572intΔ 3(ℎ1 ℎ2)0

120601 (119905) 119889119905 (96)

where


(97)


Competing Interests




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article common fixed point results for six...

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