A New Iterative Method for Equilibrium Problems and Fixed Point Problems
10
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 178053, 9 pages http://dx.doi.org/10.1155/2013/178053 Research Article A New Iterative Method for Equilibrium Problems and Fixed Point Problems Abdul Latif 1 and Mohammad Eslamian 2 1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Mazandaran University of Science and Technology, Behshahr, Iran Correspondence should be addressed to Abdul Latif; [email protected] Received 31 October 2013; Accepted 15 December 2013 Academic Editor: Ljubomir B. ´ Ciri´ c Copyright © 2013 A. Latif and M. Eslamian. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others. “Dedicated to Professor Miodrag Mateljevi’c on the occasion of his 65th birthday” 1. Introduction Let be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖⋅‖. Let be a nonempty closed convex subset of , and let Proj be a nearest point projection of into ; that is, for ∈, Proj is the unique point in with the property ‖ − Proj ‖ := inf {‖ − ‖ : ∈ }. It is well known that = Proj iff ⟨ − , − ⟩ ≥ 0 for all ∈. Let be a bifunction from × into R, such that (, ) = 0 for all ∈. Consider the Fan inequality [1]: find a point ⋆ ∈ such that ( ⋆ , ) ≥ 0, ∀ ∈ , (1) where (, ⋅) is convex and subdifferentiable on for every ∈. e set of solutions of this problem is denoted by Sol(, ). In fact, the Fan inequality can be formulated as an equilibrium problem. Such problems arise frequently in mathematics, physics, engineering, game theory, transporta- tion, economics, and network. Due to importance of the solutions of such problems, many researchers are working in this area and studying the existence of the solutions of such problems; for example, see, [2–4]. Further, if (, ) = ⟨, − ⟩ for every , ∈ , where is a mapping from into , then the Fan inequality problem (equilibrium prob- lem) becomes the classical variational inequality problem which is formulated as finding a point ⋆ ∈ such that ⟨ ⋆ ,− ⋆ ⟩≥ ∀ ∈ . (2) Variational inequalities were introduced and studied by Stampacchia [5]. It is well known that this area covers many branches of mathematics, such as partial differential equations, optimal control, optimization, mathematical pro- gramming, mechanics, and finance; see [6–11]. Here we recall some useful notions. A mapping :→ is said to be -Lipschitz on if there exists a constant ≥0 such that for each , ∈ , − ≤ − . (3) In particular, if ∈ [0, 1[, then is called a contraction on ; if =1, then is called a nonexpansive mapping on . e set of fixed points of is denoted by (). A family T := {() : 0 ≤ < ∞} of mappings on a closed convex subset of a Hilbert space is called a nonexpansive semigroup if it satisfies the following: (i) (0) = for all ∈; (ii) ( + ) = ()() for all , ≥ 0; (iii) for all ∈,
-
Upload
independent -
Category
Documents
-
view
0 -
download
0
Transcript of A New Iterative Method for Equilibrium Problems and Fixed Point Problems
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 178053 9 pageshttpdxdoiorg1011552013178053
Research ArticleA New Iterative Method for Equilibrium Problemsand Fixed Point Problems
Abdul Latif1 and Mohammad Eslamian2
1 Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Department of Mathematics Mazandaran University of Science and Technology Behshahr Iran
Correspondence should be addressed to Abdul Latif alatifkauedusa
Received 31 October 2013 Accepted 15 December 2013
Academic Editor Ljubomir B Ciric
Copyright copy 2013 A Latif and M Eslamian 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
Introducing a new iterative method we study the existence of a common element of the set of solutions of equilibrium problems fora family of monotone Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a realHilbert space We establish strong convergence theorems of the new iterative method for the solution of the variational inequalityproblem which is the optimality condition for the minimization problem Our results improve and generalize the correspondingrecent results of Anh (2012) Cianciaruso et al (2010) and many others
ldquoDedicated to Professor Miodrag Mateljevirsquoc on the occasion of his 65th birthdayrdquo
1 Introduction
Let 119867 be a real Hilbert space with inner product ⟨sdot sdot⟩ andnorm sdot Let 119862 be a nonempty closed convex subset of 119867and let Proj
119862be a nearest point projection of119867 into119862 that is
for 119909 isin 119867 Proj119862119909 is the unique point in 119862 with the property
119909 minus Proj119862119909 = inf119909 minus 119910 119910 isin 119862 It is well known that
119910 = Proj119862119909 iff ⟨119909 minus 119910 119910 minus 119911⟩ ge 0 for all 119911 isin 119862
Let 119891 be a bifunction from 119862 times 119862 into R such that119891(119909 119909) = 0 for all 119909 isin 119862 Consider the Fan inequality [1]find a point 119909⋆ isin 119862 such that
119891 (119909⋆ 119910) ge 0 forall119910 isin 119862 (1)
where 119891(119909 sdot) is convex and subdifferentiable on 119862 for every119909 isin 119862 The set of solutions of this problem is denoted bySol(119891 119862) In fact the Fan inequality can be formulated asan equilibrium problem Such problems arise frequently inmathematics physics engineering game theory transporta-tion economics and network Due to importance of thesolutions of such problems many researchers are workingin this area and studying the existence of the solutions ofsuch problems for example see [2ndash4] Further if 119891(119909 119910) =⟨119865119909 119910 minus 119909⟩ for every 119909 119910 isin 119862 where 119865 is a mapping from 119862
into 119867 then the Fan inequality problem (equilibrium prob-lem) becomes the classical variational inequality problemwhich is formulated as finding a point 119909⋆ isin 119862 such that
⟨119865119909⋆ 119910 minus 119909
⋆⟩ ge forall119910 isin 119862 (2)
Variational inequalities were introduced and studied byStampacchia [5] It is well known that this area coversmany branches of mathematics such as partial differentialequations optimal control optimization mathematical pro-gramming mechanics and finance see [6ndash11]
Here we recall some useful notionsA mapping 119879 119862 rarr 119867 is said to be 119871-Lipschitz on 119862 if
there exists a constant 119871 ge 0 such that for each 119909 119910 isin 1198621003817100381710038171003817119879119909 minus 119879119910
1003817100381710038171003817 le 1198711003817100381710038171003817119909 minus 119910
1003817100381710038171003817 (3)
In particular if 119871 isin [0 1[ then 119879 is called a contraction on 119862if 119871 = 1 then 119879 is called a nonexpansive mapping on 119862 Theset of fixed points of 119879 is denoted by 119865(119879)
A familyT = 119879(119904) 0 le 119904 lt infinofmappings on a closedconvex subset119862 of a Hilbert space119867 is called a nonexpansivesemigroup if it satisfies the following (i) 119879(0)119909 = 119909 for all119909 isin 119862 (ii)119879(119904+119905) = 119879(119904)119879(119905) for all 119904 119905 ge 0 (iii) for all 119909 isin 119862
2 Abstract and Applied Analysis
119904 rarr 119879(119904)119909 is continuous (iv) 119879(119904)119909 minus 119879(119904)119910 le 119909 minus 119910 forall 119909 119910 isin 119862 and 119904 ge 0
We use 119865(T) to denote the common fixed point set of thesemigroup T that is 119865(T) = 119909 isin 119862 119879(119904)119909 = 119909 forall119904 ge
0 It is well known that 119865(T) is closed and convex [12]A nonexpansive semigroup T on 119862 is said to be uniformlyasymptotically regular (in short uar) on 119862 if for all ℎ ge 0
and any bounded subset 119861 of 119862
lim119905rarrinfin
sup119909isin119861
119879 (ℎ) (119879 (119905) 119909) minus 119879 (119905) 119909 = 0 (4)
For each ℎ ge 0 define 120590119905(119909) = (1119905) int
119905
0119879(119904)119909119889119904 Then
lim119905rarrinfin
sup119909isin119861
1003817100381710038171003817119879 (ℎ) (120590119905 (119909)) minus 120590119905 (119909)1003817100381710038171003817 = 0 (5)
Provided that 119861 is closed bounded convex subset of 119862 It isknown that the set 120590
119905(119909) 119905 gt 0 is a uar nonexpansive
semigroup see [13] The other examples of uar operatorsemigroup can be found in [14]
A bifunction 119891 119862 times 119862 rarr R is said to be (i) stronglymonotone on 119862 with 120572 gt 0 if 119891(119909 119910) + 119891(119910 119909) le minus120572119909 minus 119910
2forall119909 119910 isin 119862 (ii)monotone on119862 if119891(119909 119910)+119891(119910 119909) le 0 forall119909 119910 isin
119862 (iii) pseudomonotone on 119862 if 119891(119909 119910) ge 0 rArr 119891(119910 119909) le 0forall119909 119910 isin 119862 (iv) Lipschitz-type continuous on 119862 with constants1198881 gt 0 and 1198882 gt 0 if 119891(119909 119910) + 119891(119910 119911) ge 119891(119909 119911) minus 1198881119909 minus 119910
2minus
1198882119910 minus 119911
2 forall119909 119910 119911 isin 119862Note that if 119879 is 119871-Lipschitz on 119862 then for each 119909 119910 isin
119862 the function 119891(119909 119910) = ⟨119865119909 119910 minus 119909⟩ is a Lipschitz-typecontinuous with constants 119888
1= 1198882= 1198712
An operator 119860 on119867 is called strongly positive if there is aconstant 120574 gt 0 such that
⟨119860119909 119909⟩ ge 1205741199092 forall119909 isin 119867 (6)
Recently iterative methods for nonexpansive mappingshave been applied to solve convex minimization problemsIn [15] Xu defined an iterative sequence 119909
119899 in 119862 which
converges strongly to the unique solution of theminimizationproblem under some suitable conditions A well-knowntypical problem is to minimize a quadratic function over theset of fixed points of a nonexpansive mapping 119879 on a realHilbert space119867
min119909isin119865(119879)
1
2⟨119860119909 119909⟩ minus ⟨119909 119887⟩ (7)
where 119887 is a given point in 119867 and 119860 is strongly positiveoperator
For solving the variational inequality problem MarinoandXu [16] introduced the following general iterative processfor nonexpansive mapping 119879 based on the viscosity approxi-mation method (see [17])
119909119899+1
= 119886119899120574ℎ (119909119899) + (119868 minus 119886
119899119860)119879119909
119899 forall119899 ge 0 (8)
where 119860 is strongly positive bounded linear operator on 119867ℎ is contraction on 119867 and 119886
119899 sub ]0 1[ They proved that
under some appropriate conditions on the parameters the
sequence 119909119899 generated by (8) converges strongly to the
unique solution 119909⋆ isin 119865(119879) of the variational inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin 119865 (119879) (9)
which is the optimality condition for the minimizationproblem
min119909isin119865(119879)
1
2⟨119860119909 119909⟩ minus 119892 (119909) (10)
where 119892 is a potential function for 120574ℎ (ie 1198921015840(119909) = 120574ℎ(119909)forall119909 isin 119867)
Iterative process for approximating common fixed pointsof a nonexpansive semigrouphas been investigated by variousauthors (see [13 14 18ndash21]) Recently Li et al [19] introducedthe following iterative procedure for the approximation ofcommon fixed points of a nonexpansive semigroup T =
119879(119904) 0 le 119904 lt infin on a closed convex subset 119862 of a Hilbertspace119867
119909119899 = 120572119899120574ℎ (119909119899) + (119868 minus 120572119899119860)1
119904119899
int
119904119899
0
119879 (119904) 119909119899119889119904 119899 ge 1 (11)
where 119860 is a strongly positive bounded linear operator on119867 and ℎ is a contraction on 119862 Imposing some appropriateconditions on the parameters they proved that the iterativesequence 119909
119899 generated by (11) converges strongly to the
unique solution 119909⋆
isin 119865(T) of the variational inequality⟨(120574ℎ minus 119860)119909
⋆ 119911 minus 119909
⋆⟩ le 0 forall119911 isin 119865(T)
For obtaining a common element of Sol(119891 119862) and the setof fixed points of a nonexpansive mapping 119879 S Takahashiand W Takahashi [9] first introduced an iterative scheme bythe viscosity approximation methodThey proved that undercertain conditions the iterative sequences converge stronglyto 119911 = Proj
119865(119879)capSol(119891119862)(ℎ(119911))During last few years iterative algorithms for finding a
common element of the set of solutions of Fan inequality andthe set of fixed points of nonexpansive mappings in a realHilbert space have been studied by many authors (see eg[2 4 22ndash28]) Recently Anh [22] studied the existence of acommon element of the set of fixed points of a nonexpansivemapping and the set of solutions of Fan inequality formonotone and Lipschitz-type continuous bifunctions Heintroduced the following new iterative process
119908119899 = argmin 120582119899119891 (119909119899 119908) +
1
2
1003817100381710038171003817119908 minus 119909119899
1003817100381710038171003817
2 119908 isin 119862
119911119899= argmin 120582
119899119891 (119908119899 119911) +
1
2
1003817100381710038171003817119911 minus 1199091198991003817100381710038171003817
2 119911 isin 119862
119909119899+1
= 120572119899ℎ (119909119899) + 120573119899119909119899+ 120574119899(120583119878 (119909
119899) + (1 minus 120583) 119911
119899)
forall119899 ge 0
(12)
where 120583 isin]0 1[119862 is nonempty closed convex subset of a realHilbert space 119867 119891 is monotone continuous and Lipschitz-type continuous bifunction ℎ is self-contraction on 119862 withconstant 119896 isin]0 1[ and 119878 is self nonexpansive mapping on119862 He proved that under some appropriate conditions over
Abstract and Applied Analysis 3
positive sequences 120572119899 120573119899 120574119899 and 120582
119899 the sequences
119909119899 119908119899 and 119911
119899 converge strongly to 119902 isin 119865(119878) cap Sol(119891 119862)
which is a solution of the variational inequality ⟨(119868 minus ℎ)119902 119909 minus119902⟩ ge 0 forall119909 isin 119865(119878) cap Sol(119891 119862)
In this paper we introduce a new iterative scheme basedon the viscosity method and study the existence of a commonelement of the set of solutions of equilibrium problems fora family of monotone Lipschitz-type continuous mappingsand the sets of fixed points of two nonexpansive semigroupsin a real Hilbert space We establish strong convergencetheorems of the new iterative scheme for the solution ofthe variational inequality problem which is the optimalitycondition for theminimization problemOur results improveand generalize the corresponding recent results of Anh [22]Cianciaruso et al [18] and many others
2 Preliminaries
In this section we collect some lemmas which are crucial forthe proofs of our results
Let 119909119899 be a sequence in 119867 and 119909 isin 119867 In the sequel119909119899 119909 denotes that 119909119899weakly converges to 119909 and 119909119899 rarr 119909
denotes that 119909119899 weakly converges to 119909
Lemma 1 Let 119867 be a real Hilbert space Then the followinginequality holds
1003817100381710038171003817119909 + 1199101003817100381710038171003817
2le 119909
2+ 2 ⟨119910 119909 + 119910⟩ forall119909 119910 isin 119867 (13)
Lemma 2 (see [15]) Assume that 119886119899 is a sequence of
nonnegative real numbers such that
119886119899+1
le (1 minus 120578119899) 119886119899 + 120578119899120575119899 119899 ge 0 (14)
where 120578119899 is a sequence in ]0 1[ and 120575
119899is a sequence inR such
that
(i) suminfin119899=1
120578119899= infin
(ii) lim sup119899rarrinfin
120575119899le 0 or suminfin
119899=1|120578119899120575119899| lt infin
Then lim119899rarrinfin119886119899 = 0
Lemma 3 (see [16]) Let 119860 be a strongly positive linearbounded self-adjoint operator on119867 with coefficient 120574 gt 0 and0 lt 120588 le 119860
minus1 Then 119868 minus 120588119860 le 1 minus 120588 120574
Lemma 4 (see [29]) Let 119867 be a Hilbert space and 119909119894isin 119867
(1 le 119894 le 119898) Then for any given 120582119894119898
119894=1sub ]0 1[ with sum119898
119894=1120582119894=
1 and for any positive integer 119896 119895 with 1 le 119896 lt 119895 le 119898
1003817100381710038171003817100381710038171003817100381710038171003817
119898
sum
119894=1
120582119894119909119894
1003817100381710038171003817100381710038171003817100381710038171003817
2
le
119898
sum
119894=1
120582119894
10038171003817100381710038171199091198941003817100381710038171003817
2minus 120582119896120582119895
10038171003817100381710038171003817119909119896minus 119909119895
10038171003817100381710038171003817
2
(15)
Lemma 5 (see [30]) Let 119905119899 be a sequence of real numbers
such that there exists a subsequence 119899119894 of 119899 such that 119905
119899119894
lt
119905119899119894+1
for all 119894 isin N Then there exists a nondecreasing sequence120591(119899) sub N such that 120591(119899) rarr infin and the following propertiesare satisfied by all (sufficiently large) numbers 119899 isin N
119905120591(119899)
le 119905120591(119899)+1
119905119899le 119905120591(119899)+1
(16)
In fact
120591 (119899) = max 119896 le 119899 119905119896lt 119905119896+1 (17)
Lemma 6 (see [2]) Let 119862 be a nonempty closed convex subsetof a real Hilbert space 119867 and let 119891 119862 times 119862 rarr R bea pseudomonotone and Lipschitz-type continuous bifunctionwith constants 119888
1 1198882ge 0 For each 119909 isin 119862 let 119891(119909 sdot) be convex
and subdifferentiable on 119862 Let 1199090isin 119862 and let 119909
119899 119911119899 and
119908119899 be sequences generated by
119908119899= argmin 120582
119899119891 (119909119899 119908) +
1
2
1003817100381710038171003817119908 minus 119909119899
1003817100381710038171003817
2 119908 isin 119862
119911119899= argmin 120582
119899119891 (119908119899 119911) +
1
2
1003817100381710038171003817119911 minus 1199091198991003817100381710038171003817
2 119911 isin 119862
(18)
Then for each 119909⋆ isin Sol(119891 119862)
1003817100381710038171003817119911119899 minus 119909⋆1003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus (1 minus 2120582
1198991198881)1003817100381710038171003817119909119899 minus 119908119899
1003817100381710038171003817
2
minus (1 minus 2120582119899 1198882)1003817100381710038171003817119908119899 minus 119911119899
1003817100381710038171003817
2 forall119899 ge 0
(19)
3 Main Results
In this section we prove the main strong convergence resultwhich solves the problem of finding a common elementof three sets 119865(T) 119865(S) and Sol(119891
119894 119862) for finite family
of monotone continuous and Lipschitz-type continuousbifunctions 119891
119894in a real Hilbert space119867
Theorem7 Let119862 be a nonempty closed convex subset of a realHilbert space 119867 and let 119891
119894 119862 times 119862 rarr R (119894 = 1 2 119898)
be a finite family of monotone continuous and Lipschitz-type continuous bifunctions with constants 1198881119894 and 1198882119894 LetT = 119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119878119900119897(119891119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a
119896-contraction self-mapping of 119862 and 119860 is a strongly positivebounded linear self-adjoint operator on119867 with coefficient 120574 lt1 and 0 lt 120574 lt 120574119896 Let 1199090 isin 119862 and let 119909119899 119908119899119894 and 119911119899119894be sequences generated by
119908119899119894= argmin 120582
119899119894119891119894(119909119899 119908) +
1
2
1003817100381710038171003817119908 minus 119909119899
1003817100381710038171003817
2 119908 isin 119862
119894 isin 1 2 119898
119911119899119894= argmin 120582
119899119894119891119894(119908119899119894 119911) +
1
2
1003817100381710038171003817119911 minus 1199091198991003817100381710038171003817
2 119911 isin 119862
119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(20)
4 Abstract and Applied Analysis
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 119905
119899 120572119899 120573119899119894 120574119899
120578119899 120582119899119894 and 120579
119899 satisfy the following conditions
(i) lim119899rarrinfin
119905119899= infin
(ii) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub]0 1119871[ where 119871 =max2119888
1119894 21198882119894 1 le
119894 le 119898(iv) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119878119900119897(119891119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (21)
Proof Since 119865(T) 119865(S) and Sol(119891119894 119862) are closed andconvex Proj
Ωis well-definedWe claim that Proj
Ω(119868minus119860+120574ℎ)
is a contraction from 119862 into itself Indeed for each 119909 119910 isin 119862we have
1003817100381710038171003817ProjΩ (119868 minus 119860 + 120574ℎ) (119909) minus ProjΩ(119868 minus 119860 + 120574ℎ) (119910)
1003817100381710038171003817
le1003817100381710038171003817(119868 minus 119860 + 120574ℎ) (119909) minus (119868 minus 119860 + 120574ℎ) (119910)
1003817100381710038171003817
le1003817100381710038171003817(119868 minus 119860) 119909 minus (119868 minus 119860) 119910
1003817100381710038171003817 + 1205741003817100381710038171003817ℎ119909 minus ℎ119910
1003817100381710038171003817
le (1 minus 120574)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 + 1205741198961003817100381710038171003817119909 minus 119910
1003817100381710038171003817
le (1 minus (120574 minus 120574119896))1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(22)
Therefore by the Banach contraction principle thereexists a unique element 119909⋆ isin 119862 such that 119909⋆ = Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ We show that 119909119899 is bounded Since lim119899rarrinfin120579119899 = 0
we can assume with no loss of generality that 120579119899 isin (0 119860
minus1)
for all 119899 ge 0 By Lemma 6 for each 1 le 119894 le 119898 we have
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817 le
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 (23)
This implies that
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le
1003817100381710038171003817100381710038171003817100381710038171003817
120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899 minus 119909⋆
1003817100381710038171003817100381710038171003817100381710038171003817
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 +
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
+ 120574119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817 + 120578119899
1003817100381710038171003817119878 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 +
119898
sum
119894=1
1205731198991198941003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
+ 120574119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120578119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
=1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
(24)
It follows from Lemma 3 that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
=1003817100381710038171003817120579119899 (120574ℎ (119909119899) minus 119860119909
⋆) + (119868 minus 120579
119899119860) (119910119899minus 119909⋆)1003817100381710038171003817
le 120579119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817 +
1003817100381710038171003817119868 minus 1205791198991198601003817100381710038171003817
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le 120579119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817 + (1 minus 120579119899120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le 1205791198991205741003817100381710038171003817ℎ (119909119899) minus ℎ (119909
⋆)1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
+ (1 minus 120579119899120574)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
le 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
+ (1 minus 120579119899120574)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
le (1 minus 120579119899(120574 minus 120574119896))
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
= (1 minus 120579119899(120574 minus 120574119896))
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
+ 120579119899(120574 minus 120574119896)
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
120574 minus 120574119896
le max1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817120574ℎ119909⋆minusA119909⋆1003817100381710038171003817
120574 minus 120574119896
(25)
This implies that 119909119899 is bounded and so are 119911119899119894 ℎ(119909119899)119879(119905119899)119909119899 and 119878(119905
119899)119909119899 Next we show that lim
119899rarrinfin119909119899minus
119879(119905119899)119909119899 = lim
119899rarrinfin119909119899minus 119878(119905119899)119909119899 = 0 Indeed by Lemma 6
for each 1 le 119894 le 119898 we have
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus (1 minus 2120582
1198991198941198881119894)1003817100381710038171003817119909119899 minus 119908119899119894
1003817100381710038171003817
2
minus (1 minus 21205821198991198941198882119894)1003817100381710038171003817119908119899119894 minus 119911119899119894
1003817100381710038171003817
2
(26)
Applying Lemma 4 and inequality (26) for 119896 isin 1 2 119898
we have that
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
2
=
1003817100381710038171003817100381710038171003817100381710038171003817
120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899 minus 119909⋆
1003817100381710038171003817100381710038171003817100381710038171003817
2
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
2
+ 120574119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
2+ 120578119899
1003817100381710038171003817119878 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
2
minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus 120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
Abstract and Applied Analysis 5
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+ 120574119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 120578119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus 120573119899119896 (1 minus 21205821198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2minus 120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
=1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus 120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus 1205721198991205781198991003817100381710038171003817119909119899 minus 119878 (119905119899) 119909119899
1003817100381710038171003817
2
(27)
We now compute1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2
=1003817100381710038171003817120579119899 (120574ℎ (119909119899) minus 119860119909
⋆) + (119868 minus 120579
119899119860) (119910119899minus 119909⋆)1003817100381710038171003817
2
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119910119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
minus (1 minus 120579119899120574)2120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
(28)
Therefore
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899(1 minus 120579
119899120574)
times1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579
2
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2
(29)
In order to prove that 119909119899rarr 119909⋆ as 119899 rarr infin we consider the
following two cases
Case 1 Assume that 119909119899minus 119909⋆ is a monotone sequence
In other words for large enough 1198990 119909119899minus 119909⋆119899ge1198990
iseither nondecreasing or nonincreasing Since 119909119899 minus 119909
⋆ is
bounded it is convergent Since lim119899rarrinfin
120579119899= 0 and ℎ(119909
119899)
and 119909119899 are bounded from (29) we have
lim119899rarrinfin
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2= 0 (30)
and by assumption we get
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817 = 0 1 le 119896 le 119898 (31)
By similar argument we can obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199081198991198961003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
= lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817 = 0
(32)
Further for all ℎ ge 0 and 119899 ge 0 we see that1003817100381710038171003817119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817 +1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 119909119899
1003817100381710038171003817
+1003817100381710038171003817119879 (ℎ) 119879 (119905119899) 119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le 21003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817
+ sup119909isin119909119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 1199091198991003817100381710038171003817
(33)
Since 119879(119905) is uar nonexpansive semigroup andlim119899rarrinfin
119905119899= infin we have
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (ℎ) 1199091198991003817100381710038171003817 = 0 (34)
Similarly for all ℎ ge 0 and 119899 ge 0 we obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (ℎ) 1199091198991003817100381710038171003817 = 0 (35)
Next we show that
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩ le 0 (36)
To show this inequality we choose a subsequence 119909119899119894
of 119909119899
such that
lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
(37)
Since 119909119899119894
is bounded there exists a subsequence 119909119899119894119895
of119909119899119894
which converges weakly to 119909 Without loss of generalitywe can assume that 119909119899
119895
119909 Consider
100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119879 (119905) 119909119899
119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817
le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119909119899119895
minus 119909100381710038171003817100381710038171003817
(38)
Thus we have
lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 119879 (119905) 11990910038171003817100381710038171003817le lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 11990910038171003817100381710038171003817 (39)
By the Opial property of the Hilbert space 119867 we obtain119879(119905)119909 = 119909 for all 119905 ge 0 Similarly we have that 119878(119905)119909 = 119909
for all 119905 ge 0 This implies that 119909 isin 119865(T)⋂119865(S) Now weshow that 119909 isin Sol(119891119894 119862) For each 1 le 119894 le 119898 since 119891119894(119909 sdot) isconvex on 119862 for each 119909 isin 119862 we see that
119908119899119894= argmin 120582
119899119894119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (40)
6 Abstract and Applied Analysis
if and only if
0 isin 1205972(119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2) (119908119899119894) + 119873119862(119908119899119894) (41)
where119873119862(119909) is the (outward) normal cone of119862 at 119909 isin 119862This
follows that
0 = 120582119899119894V + 119908119899119894minus 119909119899+ 119906119899 (42)
where V isin 1205972119891119894(119909119899 119908119899119894) and 119906119899 isin 119873119862(119908119899119894) By the definition
of the normal cone119873119862 we have
⟨119908119899119894minus 119909119899 119910 minus 119908
119899119894⟩ ge 120582119899119894⟨V 119908119899119894minus 119910⟩ forall119910 isin 119862 (43)
Since 119891119894(119909119899 sdot) is subdifferentiable on 119862 by the well-known
Moreau-Rockafellar theorem [31] (also see [6]) for V isin
1205972119891119894(119909119899 119908119899119894) we have
119891119894(119909119899 119910) minus 119891
119894(119909119899 119908119899119894) ge ⟨V 119910 minus 119908
119899119894⟩ forall119910 isin 119862 (44)
Combining this with (43) we have
120582119899119894 (119891119894 (119909119899 119910) minus 119891119894 (119909119899 119908119899119894)) ge ⟨119908119899119894 minus 119909119899 119908119899119894 minus 119910⟩
forall119910 isin 119862
(45)
In particular we have
120582119899119895119894(119891119894(119909119899119895
119910) minus 119891119894(119909119899119895
119908119899119894)) ge ⟨119908
119899119895119894minus 119909119899119895
119908119899119895119894minus 119910⟩
forall119910 isin 119862
(46)
Since 119909119899119895
119909 it follows from (32) that 119908119899119895119894 119909 And thus
we have
119891119894(119909 119910) ge 0 forall119910 isin 119862 (47)
This implies that 119909 isin Sol(119891119894 119862) and hence 119909 isin Ω Since 119909⋆ =
ProjΩ(119868 minus 119860 + 120574ℎ)119909
⋆ and 119909 isin Ω we have
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
= lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= ⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909⟩ le 0
(48)
From Lemma 1 it follows that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1003817100381710038171003817(119868 minus 120579119899119860) (119910119899 minus 119909
⋆)1003817100381710038171003817
2
+ 2120579119899⟨120574ℎ (119909
119899) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899120574 ⟨ℎ (119909
119899) minus ℎ (119909
⋆) 119909119899+1
minus 119909⋆⟩
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899119896120574
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 120579119899119896120574 (1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2)
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le ((1 minus 120579119899120574)2+ 120579119899119896120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 1205791198991205741198961003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899 ⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1 minus 119909
⋆⟩
(49)
This implies that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1 minus 2120579119899120574 + (120579119899120574)
2+ 120579119899120574119896
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896
⟨120574ℎ (119909⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
= (1 minus2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+
(120579119899120574)2
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
times (
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩)
= (1 minus 120578119899)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 120578119899120575119899
(50)
Abstract and Applied Analysis 7
where
119872 = sup 1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2 119899 ge 0 120578
119899=2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896
120575119899=
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ119909⋆minus 119860119909⋆ 119909119899+1
minus 119909⋆⟩
(51)
It is easy to see that 120578119899 rarr 0 suminfin119899=1
120578119899 = infin andlim sup
119899rarrinfin120575119899le 0 Hence by Lemma 2 the sequence 119909
119899
converges strongly to 119909⋆ From (32) we have that 119908119899119894 and
119911119899119894 converge strongly to 119909⋆
Case 2 Assume that 119909119899minus119909⋆ is not a monotone sequence
Then we can define an integer sequence 120591(119899) for all 119899 ge 1198990
(for some large enough 1198990) by
120591 (119899) = max 119896 isin N 119896 le 119899 1003817100381710038171003817119909119896 minus 119909
⋆1003817100381710038171003817 lt1003817100381710038171003817119909119896+1 minus 119909
⋆1003817100381710038171003817
(52)
Clearly 120591 is a nondecreasing sequence such that 120591(119899) rarr infin
as 119899 rarr infin and for all 119899 ge 1198990
1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817 lt
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817 (53)
From (33) we obtain that
lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119911120591(119899)1198941003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119908120591(119899)1198941003817100381710038171003817
= lim119899rarrinfin
10038171003817100381710038171003817119909120591(119899) minus 119879 (119905120591
119899
) 119909120591(119899)
10038171003817100381710038171003817= 0
(54)
Following an argument similar to that in Case 1 we have
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817
2le (1 minus 120578
120591(119899))1003817100381710038171003817119909120591(119899) minus 119909
⋆1003817100381710038171003817
2+ 120578120591(119899)
120575120591(119899)
(55)
where 120578120591(119899)
rarr 0 suminfin119899=1
120578120591(119899)
= infin and lim sup119899rarrinfin
120575120591(119899)
le 0Hence by Lemma 2 we obtain lim
119899rarrinfin119909120591(119899)
minus 119909⋆ = 0 and
lim119899rarrinfin
119909120591(119899)+1
minus 119909⋆ = 0 Now Lemma 5 implies that
0 le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817 le max 1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le1003817100381710038171003817119909120591(119899)+1 minus 119909
⋆1003817100381710038171003817
(56)
Therefore 119909119899 converges strongly to 119909
⋆= Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ This completes the proof
4 Application
In this section we consider a particular Fan inequalitycorresponding to the function 119891 defined by the following forevery 119909 119910 isin 119862
119891 (119909 119910) = ⟨119865 (119909) 119910 minus 119909⟩ (57)
where 119865 119862 rarr 119867 Then we obtain the classical variationalinequality as follows
Find 119911 isin 119862 such that ⟨119865 (119911) 119910 minus 119911⟩ ge 0 forall119910 isin 119862 (58)
The set of solutions of this problem is denoted by 119881119868(119865 119862)In that particular case the solution 119910
119899of the minimization
problem
argmin 120582119899 119891 (119909119899 119910) +1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (59)
can be expressed as
119910119899= 119875119862 (119909119899 minus 120582119899119865 (119909119899)) (60)
Let 119865 be 119871-Lipschitz continuous on 119862 Then
119891 (119909 119910) + 119891 (119910 119911) minus 119891 (119909 119911) = ⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
119909 119910 119911 isin 119862
(61)
Therefore1003816100381610038161003816⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
1003816100381610038161003816 le 1198711003817100381710038171003817119909 minus 119910
1003817100381710038171003817
1003817100381710038171003817119910 minus 1199111003817100381710038171003817
le119871
2(1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2+1003817100381710038171003817119910 minus 119911
1003817100381710038171003817
2)
(62)
hence 119891 satisfies Lipschitz-type continuous condition with1198881 = 1198882 = 1198712
Using Theorem 7 we obtain the following convergencetheorem
Theorem 8 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le 119898119865119894is monotone and 119871-Lipschitz continuous on 119862 Let T =
119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909
0isin 119862 and let 119909
119899 119908119899119894 and 119911
119899119894 be
sequences generated by
119908119899119894 = 119875119862 (119909119899 minus 120582119899119894119865119894 (119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899= 120572119899119909119899+
119898
sum
119894=1
120573119899119894119911119899119894+ 120574119899119879 (119905119899) 119909119899+ 120578119899119878 (119905119899) 119909119899
119909119899+1 = 120579119899120574ℎ (119909119899) + (119868 minus 120579119899119860)119910119899 forall119899 ge 0
(63)
where 120572119899 + sum
119898
119894=1120573119899119894 + 120574119899 + 120578119899 = 1 and 120572119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579119899 satisfy the following conditions(i) lim
119899rarrinfin119905119899= infin
(ii) 120579119899 sub ]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub ]0 1119871[
(iv) 120572119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf119899120572119899120574119899 gt 0 lim inf119899120572119899120578119899 gt 0 andlim inf119899120573119899119894(1 minus 21205821198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (64)
8 Abstract and Applied Analysis
In [32] Baillon proved a strong mean convergence the-orem for nonexpansive mappings and it was generalizedin [33] It follows from the above proof that Theorems 7 isvalid for nonexpansivemappingsThuswe have the followingmean ergodic theorems for nonexpansive mappings in aHilbert space
Theorem 9 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le
119898 119865119894 is monotone and 119871-Lipschitz continuous on 119862 Let119879 and 119878 be two nonexpansive mappings on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(119879)⋂119865(119878) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909119899 119908119899119894 and 119911119899119894 be sequences generatedby 1199090isin 119862 and by
119908119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879119899119909119899 + 120578119899119878
119899119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(65)
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 120572
119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579
119899 satisfy the following conditions
(i) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(ii) 120582119899119894 sub [119886 119887] sub ]0 1119871[ where 119871 = max21198881119894 21198882119894
1 le 119894 le 119898(iii) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (66)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Therefore theauthors acknowledge with thanks DSR for technical andfinancial support
References
[1] K Fan ldquoA minimax inequality and applicationsrdquo in InequalityIII pp 103ndash113 Academic Press New York NY USA 1972
[2] P N Anh ldquoA hybrid extragradient method extended to fixedpoint problems and equilibrium problemsrdquo Optimization vol62 no 2 pp 271ndash283 2013
[3] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994
[4] L-C Ceng N Hadjisavvas and N-C Wong ldquoStrong conver-gence theorem by a hybrid extragradient-like approximationmethod for variational inequalities and fixed point problemsrdquoJournal of Global Optimization vol 46 no 4 pp 635ndash646 2010
[5] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964
[6] P Daniele F Giannessi and A Maugeri Equilibrium Problemsand Variational Models vol 68 Kluwer Academic NorwellMass USA 2003
[7] J-W Peng and J-C Yao ldquoSome new extragradient-likemethodsfor generalized equilibrium problems fixed point problemsand variational inequality problemsrdquo Optimization Methods ampSoftware vol 25 no 4ndash6 pp 677ndash698 2010
[8] D R Sahu N-C Wong and J-C Yao ldquoStrong convergencetheorems for semigroups of asymptotically nonexpansive map-pings inBanach spacesrdquoAbstract andAppliedAnalysis vol 2013Article ID 202095 8 pages 2013
[9] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[10] S Wang and B Guo ldquoNew iterative scheme with nonexpansivemappings for equilibrium problems and variational inequalityproblems in Hilbert spacesrdquo Journal of Computational andApplied Mathematics vol 233 no 10 pp 2620ndash2630 2010
[11] Y Yao Y-C Liou and Y-J Wu ldquoAn extragradient method formixed equilibrium problems and fixed point problemsrdquo FixedPoint Theory and Applications vol 2009 Article ID 632819 15pages 2009
[12] F E Browder ldquoNonexpansive nonlinear operators in a Banachspacerdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 54 pp 1041ndash1044 1965
[13] R Chen and Y Song ldquoConvergence to common fixed pointof nonexpansive semigroupsrdquo Journal of Computational andApplied Mathematics vol 200 no 2 pp 566ndash575 2007
[14] A Aleyner and Y Censor ldquoBest approximation to commonfixed points of a semigroup of nonexpansive operatorsrdquo Journalof Nonlinear and Convex Analysis vol 6 no 1 pp 137ndash151 2005
[15] H K Xu ldquoAn iterative approach to quadratic optimizationrdquoJournal of Optimization Theory and Applications vol 116 no 3pp 659ndash678 2003
[16] G Marino and H-K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006
[17] AMoudafi ldquoViscosity approximationmethods for fixed-pointsproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 241 no 1 pp 46ndash55 2000
[18] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[19] S Li L Li and Y Su ldquoGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert spacerdquo Nonlin-ear Analysis Theory Methods amp Applications vol 70 no 9 pp3065ndash3071 2009
Abstract and Applied Analysis 9
[20] D R Sahu N C Wong and J C Yao ldquoA unified hybriditerative method for solving variational inequalities involvinggeneralized pseudocontractive mappingsrdquo SIAM Journal onControl and Optimization vol 50 no 4 pp 2335ndash2354 2012
[21] N Shioji and W Takahashi ldquoStrong convergence theoremsfor asymptotically nonexpansive semigroups in Hilbert spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 34 no1 pp 87ndash99 1998
[22] P N Anh ldquoStrong convergence theorems for nonexpansivemappings and Ky Fan inequalitiesrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 303ndash320 2012
[23] L-C Ceng Q H Ansari and J-C Yao ldquoHybrid proximal-type and hybrid shrinking projection algorithms for equilib-rium problems maximal monotone operators and relativelynonexpansive mappingsrdquo Numerical Functional Analysis andOptimization vol 31 no 7-9 pp 763ndash797 2010
[24] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009
[25] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011
[26] L-C Ceng S-M Guu and J-C Yao ldquoHybrid iterative methodfor finding common solutions of generalizedmixed equilibriumand fixed point problemsrdquo Fixed PointTheory and Applicationsvol 2012 article 92 2012
[27] M Eslamian ldquoConvergence theorems for nonspreading map-pings andnonexpansivemultivaluedmappings and equilibriumproblemsrdquo Optimization Letters vol 7 no 3 pp 547ndash557 2013
[28] M Eslamian ldquoHybrid method for equilibrium problems andfixed point problems of finite families of nonexpansive semi-groupsrdquo RACSAM vol 107 pp 299ndash307 2013
[29] M Eslamian and A Abkar ldquoOne-step iterative process fora finite family of multivalued mappingsrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 105ndash111 2011
[30] P-E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[31] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976
[32] J-B Baillon ldquoUn theoreme de type ergodique pour les contrac-tions non lineaires dans un espace de Hilbertrdquo Comptes Rendusde lrsquoAcademie des Sciences vol 280 no 22 pp A1511ndashA15141975
[33] W Kaczor T Kuczumow and S Reich ldquoA mean ergodictheorem for nonlinear semigroups which are asymptoticallynonexpansive in the intermediate senserdquo Journal of Mathemat-ical Analysis and Applications vol 246 no 1 pp 1ndash27 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
119904 rarr 119879(119904)119909 is continuous (iv) 119879(119904)119909 minus 119879(119904)119910 le 119909 minus 119910 forall 119909 119910 isin 119862 and 119904 ge 0
We use 119865(T) to denote the common fixed point set of thesemigroup T that is 119865(T) = 119909 isin 119862 119879(119904)119909 = 119909 forall119904 ge
0 It is well known that 119865(T) is closed and convex [12]A nonexpansive semigroup T on 119862 is said to be uniformlyasymptotically regular (in short uar) on 119862 if for all ℎ ge 0
and any bounded subset 119861 of 119862
lim119905rarrinfin
sup119909isin119861
119879 (ℎ) (119879 (119905) 119909) minus 119879 (119905) 119909 = 0 (4)
For each ℎ ge 0 define 120590119905(119909) = (1119905) int
119905
0119879(119904)119909119889119904 Then
lim119905rarrinfin
sup119909isin119861
1003817100381710038171003817119879 (ℎ) (120590119905 (119909)) minus 120590119905 (119909)1003817100381710038171003817 = 0 (5)
Provided that 119861 is closed bounded convex subset of 119862 It isknown that the set 120590
119905(119909) 119905 gt 0 is a uar nonexpansive
semigroup see [13] The other examples of uar operatorsemigroup can be found in [14]
A bifunction 119891 119862 times 119862 rarr R is said to be (i) stronglymonotone on 119862 with 120572 gt 0 if 119891(119909 119910) + 119891(119910 119909) le minus120572119909 minus 119910
2forall119909 119910 isin 119862 (ii)monotone on119862 if119891(119909 119910)+119891(119910 119909) le 0 forall119909 119910 isin
119862 (iii) pseudomonotone on 119862 if 119891(119909 119910) ge 0 rArr 119891(119910 119909) le 0forall119909 119910 isin 119862 (iv) Lipschitz-type continuous on 119862 with constants1198881 gt 0 and 1198882 gt 0 if 119891(119909 119910) + 119891(119910 119911) ge 119891(119909 119911) minus 1198881119909 minus 119910
2minus
1198882119910 minus 119911
2 forall119909 119910 119911 isin 119862Note that if 119879 is 119871-Lipschitz on 119862 then for each 119909 119910 isin
119862 the function 119891(119909 119910) = ⟨119865119909 119910 minus 119909⟩ is a Lipschitz-typecontinuous with constants 119888
1= 1198882= 1198712
An operator 119860 on119867 is called strongly positive if there is aconstant 120574 gt 0 such that
⟨119860119909 119909⟩ ge 1205741199092 forall119909 isin 119867 (6)
Recently iterative methods for nonexpansive mappingshave been applied to solve convex minimization problemsIn [15] Xu defined an iterative sequence 119909
119899 in 119862 which
converges strongly to the unique solution of theminimizationproblem under some suitable conditions A well-knowntypical problem is to minimize a quadratic function over theset of fixed points of a nonexpansive mapping 119879 on a realHilbert space119867
min119909isin119865(119879)
1
2⟨119860119909 119909⟩ minus ⟨119909 119887⟩ (7)
where 119887 is a given point in 119867 and 119860 is strongly positiveoperator
For solving the variational inequality problem MarinoandXu [16] introduced the following general iterative processfor nonexpansive mapping 119879 based on the viscosity approxi-mation method (see [17])
119909119899+1
= 119886119899120574ℎ (119909119899) + (119868 minus 119886
119899119860)119879119909
119899 forall119899 ge 0 (8)
where 119860 is strongly positive bounded linear operator on 119867ℎ is contraction on 119867 and 119886
119899 sub ]0 1[ They proved that
under some appropriate conditions on the parameters the
sequence 119909119899 generated by (8) converges strongly to the
unique solution 119909⋆ isin 119865(119879) of the variational inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin 119865 (119879) (9)
which is the optimality condition for the minimizationproblem
min119909isin119865(119879)
1
2⟨119860119909 119909⟩ minus 119892 (119909) (10)
where 119892 is a potential function for 120574ℎ (ie 1198921015840(119909) = 120574ℎ(119909)forall119909 isin 119867)
Iterative process for approximating common fixed pointsof a nonexpansive semigrouphas been investigated by variousauthors (see [13 14 18ndash21]) Recently Li et al [19] introducedthe following iterative procedure for the approximation ofcommon fixed points of a nonexpansive semigroup T =
119879(119904) 0 le 119904 lt infin on a closed convex subset 119862 of a Hilbertspace119867
119909119899 = 120572119899120574ℎ (119909119899) + (119868 minus 120572119899119860)1
119904119899
int
119904119899
0
119879 (119904) 119909119899119889119904 119899 ge 1 (11)
where 119860 is a strongly positive bounded linear operator on119867 and ℎ is a contraction on 119862 Imposing some appropriateconditions on the parameters they proved that the iterativesequence 119909
119899 generated by (11) converges strongly to the
unique solution 119909⋆
isin 119865(T) of the variational inequality⟨(120574ℎ minus 119860)119909
⋆ 119911 minus 119909
⋆⟩ le 0 forall119911 isin 119865(T)
For obtaining a common element of Sol(119891 119862) and the setof fixed points of a nonexpansive mapping 119879 S Takahashiand W Takahashi [9] first introduced an iterative scheme bythe viscosity approximation methodThey proved that undercertain conditions the iterative sequences converge stronglyto 119911 = Proj
119865(119879)capSol(119891119862)(ℎ(119911))During last few years iterative algorithms for finding a
common element of the set of solutions of Fan inequality andthe set of fixed points of nonexpansive mappings in a realHilbert space have been studied by many authors (see eg[2 4 22ndash28]) Recently Anh [22] studied the existence of acommon element of the set of fixed points of a nonexpansivemapping and the set of solutions of Fan inequality formonotone and Lipschitz-type continuous bifunctions Heintroduced the following new iterative process
119908119899 = argmin 120582119899119891 (119909119899 119908) +
1
2
1003817100381710038171003817119908 minus 119909119899
1003817100381710038171003817
2 119908 isin 119862
119911119899= argmin 120582
119899119891 (119908119899 119911) +
1
2
1003817100381710038171003817119911 minus 1199091198991003817100381710038171003817
2 119911 isin 119862
119909119899+1
= 120572119899ℎ (119909119899) + 120573119899119909119899+ 120574119899(120583119878 (119909
119899) + (1 minus 120583) 119911
119899)
forall119899 ge 0
(12)
where 120583 isin]0 1[119862 is nonempty closed convex subset of a realHilbert space 119867 119891 is monotone continuous and Lipschitz-type continuous bifunction ℎ is self-contraction on 119862 withconstant 119896 isin]0 1[ and 119878 is self nonexpansive mapping on119862 He proved that under some appropriate conditions over
Abstract and Applied Analysis 3
positive sequences 120572119899 120573119899 120574119899 and 120582
119899 the sequences
119909119899 119908119899 and 119911
119899 converge strongly to 119902 isin 119865(119878) cap Sol(119891 119862)
which is a solution of the variational inequality ⟨(119868 minus ℎ)119902 119909 minus119902⟩ ge 0 forall119909 isin 119865(119878) cap Sol(119891 119862)
In this paper we introduce a new iterative scheme basedon the viscosity method and study the existence of a commonelement of the set of solutions of equilibrium problems fora family of monotone Lipschitz-type continuous mappingsand the sets of fixed points of two nonexpansive semigroupsin a real Hilbert space We establish strong convergencetheorems of the new iterative scheme for the solution ofthe variational inequality problem which is the optimalitycondition for theminimization problemOur results improveand generalize the corresponding recent results of Anh [22]Cianciaruso et al [18] and many others
2 Preliminaries
In this section we collect some lemmas which are crucial forthe proofs of our results
Let 119909119899 be a sequence in 119867 and 119909 isin 119867 In the sequel119909119899 119909 denotes that 119909119899weakly converges to 119909 and 119909119899 rarr 119909
denotes that 119909119899 weakly converges to 119909
Lemma 1 Let 119867 be a real Hilbert space Then the followinginequality holds
1003817100381710038171003817119909 + 1199101003817100381710038171003817
2le 119909
2+ 2 ⟨119910 119909 + 119910⟩ forall119909 119910 isin 119867 (13)
Lemma 2 (see [15]) Assume that 119886119899 is a sequence of
nonnegative real numbers such that
119886119899+1
le (1 minus 120578119899) 119886119899 + 120578119899120575119899 119899 ge 0 (14)
where 120578119899 is a sequence in ]0 1[ and 120575
119899is a sequence inR such
that
(i) suminfin119899=1
120578119899= infin
(ii) lim sup119899rarrinfin
120575119899le 0 or suminfin
119899=1|120578119899120575119899| lt infin
Then lim119899rarrinfin119886119899 = 0
Lemma 3 (see [16]) Let 119860 be a strongly positive linearbounded self-adjoint operator on119867 with coefficient 120574 gt 0 and0 lt 120588 le 119860
minus1 Then 119868 minus 120588119860 le 1 minus 120588 120574
Lemma 4 (see [29]) Let 119867 be a Hilbert space and 119909119894isin 119867
(1 le 119894 le 119898) Then for any given 120582119894119898
119894=1sub ]0 1[ with sum119898
119894=1120582119894=
1 and for any positive integer 119896 119895 with 1 le 119896 lt 119895 le 119898
1003817100381710038171003817100381710038171003817100381710038171003817
119898
sum
119894=1
120582119894119909119894
1003817100381710038171003817100381710038171003817100381710038171003817
2
le
119898
sum
119894=1
120582119894
10038171003817100381710038171199091198941003817100381710038171003817
2minus 120582119896120582119895
10038171003817100381710038171003817119909119896minus 119909119895
10038171003817100381710038171003817
2
(15)
Lemma 5 (see [30]) Let 119905119899 be a sequence of real numbers
such that there exists a subsequence 119899119894 of 119899 such that 119905
119899119894
lt
119905119899119894+1
for all 119894 isin N Then there exists a nondecreasing sequence120591(119899) sub N such that 120591(119899) rarr infin and the following propertiesare satisfied by all (sufficiently large) numbers 119899 isin N
119905120591(119899)
le 119905120591(119899)+1
119905119899le 119905120591(119899)+1
(16)
In fact
120591 (119899) = max 119896 le 119899 119905119896lt 119905119896+1 (17)
Lemma 6 (see [2]) Let 119862 be a nonempty closed convex subsetof a real Hilbert space 119867 and let 119891 119862 times 119862 rarr R bea pseudomonotone and Lipschitz-type continuous bifunctionwith constants 119888
1 1198882ge 0 For each 119909 isin 119862 let 119891(119909 sdot) be convex
and subdifferentiable on 119862 Let 1199090isin 119862 and let 119909
119899 119911119899 and
119908119899 be sequences generated by
119908119899= argmin 120582
119899119891 (119909119899 119908) +
1
2
1003817100381710038171003817119908 minus 119909119899
1003817100381710038171003817
2 119908 isin 119862
119911119899= argmin 120582
119899119891 (119908119899 119911) +
1
2
1003817100381710038171003817119911 minus 1199091198991003817100381710038171003817
2 119911 isin 119862
(18)
Then for each 119909⋆ isin Sol(119891 119862)
1003817100381710038171003817119911119899 minus 119909⋆1003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus (1 minus 2120582
1198991198881)1003817100381710038171003817119909119899 minus 119908119899
1003817100381710038171003817
2
minus (1 minus 2120582119899 1198882)1003817100381710038171003817119908119899 minus 119911119899
1003817100381710038171003817
2 forall119899 ge 0
(19)
3 Main Results
In this section we prove the main strong convergence resultwhich solves the problem of finding a common elementof three sets 119865(T) 119865(S) and Sol(119891
119894 119862) for finite family
of monotone continuous and Lipschitz-type continuousbifunctions 119891
119894in a real Hilbert space119867
Theorem7 Let119862 be a nonempty closed convex subset of a realHilbert space 119867 and let 119891
119894 119862 times 119862 rarr R (119894 = 1 2 119898)
be a finite family of monotone continuous and Lipschitz-type continuous bifunctions with constants 1198881119894 and 1198882119894 LetT = 119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119878119900119897(119891119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a
119896-contraction self-mapping of 119862 and 119860 is a strongly positivebounded linear self-adjoint operator on119867 with coefficient 120574 lt1 and 0 lt 120574 lt 120574119896 Let 1199090 isin 119862 and let 119909119899 119908119899119894 and 119911119899119894be sequences generated by
119908119899119894= argmin 120582
119899119894119891119894(119909119899 119908) +
1
2
1003817100381710038171003817119908 minus 119909119899
1003817100381710038171003817
2 119908 isin 119862
119894 isin 1 2 119898
119911119899119894= argmin 120582
119899119894119891119894(119908119899119894 119911) +
1
2
1003817100381710038171003817119911 minus 1199091198991003817100381710038171003817
2 119911 isin 119862
119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(20)
4 Abstract and Applied Analysis
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 119905
119899 120572119899 120573119899119894 120574119899
120578119899 120582119899119894 and 120579
119899 satisfy the following conditions
(i) lim119899rarrinfin
119905119899= infin
(ii) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub]0 1119871[ where 119871 =max2119888
1119894 21198882119894 1 le
119894 le 119898(iv) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119878119900119897(119891119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (21)
Proof Since 119865(T) 119865(S) and Sol(119891119894 119862) are closed andconvex Proj
Ωis well-definedWe claim that Proj
Ω(119868minus119860+120574ℎ)
is a contraction from 119862 into itself Indeed for each 119909 119910 isin 119862we have
1003817100381710038171003817ProjΩ (119868 minus 119860 + 120574ℎ) (119909) minus ProjΩ(119868 minus 119860 + 120574ℎ) (119910)
1003817100381710038171003817
le1003817100381710038171003817(119868 minus 119860 + 120574ℎ) (119909) minus (119868 minus 119860 + 120574ℎ) (119910)
1003817100381710038171003817
le1003817100381710038171003817(119868 minus 119860) 119909 minus (119868 minus 119860) 119910
1003817100381710038171003817 + 1205741003817100381710038171003817ℎ119909 minus ℎ119910
1003817100381710038171003817
le (1 minus 120574)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 + 1205741198961003817100381710038171003817119909 minus 119910
1003817100381710038171003817
le (1 minus (120574 minus 120574119896))1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(22)
Therefore by the Banach contraction principle thereexists a unique element 119909⋆ isin 119862 such that 119909⋆ = Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ We show that 119909119899 is bounded Since lim119899rarrinfin120579119899 = 0
we can assume with no loss of generality that 120579119899 isin (0 119860
minus1)
for all 119899 ge 0 By Lemma 6 for each 1 le 119894 le 119898 we have
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817 le
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 (23)
This implies that
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le
1003817100381710038171003817100381710038171003817100381710038171003817
120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899 minus 119909⋆
1003817100381710038171003817100381710038171003817100381710038171003817
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 +
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
+ 120574119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817 + 120578119899
1003817100381710038171003817119878 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 +
119898
sum
119894=1
1205731198991198941003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
+ 120574119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120578119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
=1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
(24)
It follows from Lemma 3 that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
=1003817100381710038171003817120579119899 (120574ℎ (119909119899) minus 119860119909
⋆) + (119868 minus 120579
119899119860) (119910119899minus 119909⋆)1003817100381710038171003817
le 120579119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817 +
1003817100381710038171003817119868 minus 1205791198991198601003817100381710038171003817
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le 120579119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817 + (1 minus 120579119899120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le 1205791198991205741003817100381710038171003817ℎ (119909119899) minus ℎ (119909
⋆)1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
+ (1 minus 120579119899120574)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
le 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
+ (1 minus 120579119899120574)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
le (1 minus 120579119899(120574 minus 120574119896))
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
= (1 minus 120579119899(120574 minus 120574119896))
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
+ 120579119899(120574 minus 120574119896)
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
120574 minus 120574119896
le max1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817120574ℎ119909⋆minusA119909⋆1003817100381710038171003817
120574 minus 120574119896
(25)
This implies that 119909119899 is bounded and so are 119911119899119894 ℎ(119909119899)119879(119905119899)119909119899 and 119878(119905
119899)119909119899 Next we show that lim
119899rarrinfin119909119899minus
119879(119905119899)119909119899 = lim
119899rarrinfin119909119899minus 119878(119905119899)119909119899 = 0 Indeed by Lemma 6
for each 1 le 119894 le 119898 we have
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus (1 minus 2120582
1198991198941198881119894)1003817100381710038171003817119909119899 minus 119908119899119894
1003817100381710038171003817
2
minus (1 minus 21205821198991198941198882119894)1003817100381710038171003817119908119899119894 minus 119911119899119894
1003817100381710038171003817
2
(26)
Applying Lemma 4 and inequality (26) for 119896 isin 1 2 119898
we have that
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
2
=
1003817100381710038171003817100381710038171003817100381710038171003817
120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899 minus 119909⋆
1003817100381710038171003817100381710038171003817100381710038171003817
2
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
2
+ 120574119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
2+ 120578119899
1003817100381710038171003817119878 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
2
minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus 120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
Abstract and Applied Analysis 5
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+ 120574119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 120578119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus 120573119899119896 (1 minus 21205821198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2minus 120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
=1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus 120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus 1205721198991205781198991003817100381710038171003817119909119899 minus 119878 (119905119899) 119909119899
1003817100381710038171003817
2
(27)
We now compute1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2
=1003817100381710038171003817120579119899 (120574ℎ (119909119899) minus 119860119909
⋆) + (119868 minus 120579
119899119860) (119910119899minus 119909⋆)1003817100381710038171003817
2
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119910119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
minus (1 minus 120579119899120574)2120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
(28)
Therefore
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899(1 minus 120579
119899120574)
times1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579
2
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2
(29)
In order to prove that 119909119899rarr 119909⋆ as 119899 rarr infin we consider the
following two cases
Case 1 Assume that 119909119899minus 119909⋆ is a monotone sequence
In other words for large enough 1198990 119909119899minus 119909⋆119899ge1198990
iseither nondecreasing or nonincreasing Since 119909119899 minus 119909
⋆ is
bounded it is convergent Since lim119899rarrinfin
120579119899= 0 and ℎ(119909
119899)
and 119909119899 are bounded from (29) we have
lim119899rarrinfin
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2= 0 (30)
and by assumption we get
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817 = 0 1 le 119896 le 119898 (31)
By similar argument we can obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199081198991198961003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
= lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817 = 0
(32)
Further for all ℎ ge 0 and 119899 ge 0 we see that1003817100381710038171003817119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817 +1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 119909119899
1003817100381710038171003817
+1003817100381710038171003817119879 (ℎ) 119879 (119905119899) 119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le 21003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817
+ sup119909isin119909119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 1199091198991003817100381710038171003817
(33)
Since 119879(119905) is uar nonexpansive semigroup andlim119899rarrinfin
119905119899= infin we have
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (ℎ) 1199091198991003817100381710038171003817 = 0 (34)
Similarly for all ℎ ge 0 and 119899 ge 0 we obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (ℎ) 1199091198991003817100381710038171003817 = 0 (35)
Next we show that
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩ le 0 (36)
To show this inequality we choose a subsequence 119909119899119894
of 119909119899
such that
lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
(37)
Since 119909119899119894
is bounded there exists a subsequence 119909119899119894119895
of119909119899119894
which converges weakly to 119909 Without loss of generalitywe can assume that 119909119899
119895
119909 Consider
100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119879 (119905) 119909119899
119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817
le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119909119899119895
minus 119909100381710038171003817100381710038171003817
(38)
Thus we have
lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 119879 (119905) 11990910038171003817100381710038171003817le lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 11990910038171003817100381710038171003817 (39)
By the Opial property of the Hilbert space 119867 we obtain119879(119905)119909 = 119909 for all 119905 ge 0 Similarly we have that 119878(119905)119909 = 119909
for all 119905 ge 0 This implies that 119909 isin 119865(T)⋂119865(S) Now weshow that 119909 isin Sol(119891119894 119862) For each 1 le 119894 le 119898 since 119891119894(119909 sdot) isconvex on 119862 for each 119909 isin 119862 we see that
119908119899119894= argmin 120582
119899119894119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (40)
6 Abstract and Applied Analysis
if and only if
0 isin 1205972(119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2) (119908119899119894) + 119873119862(119908119899119894) (41)
where119873119862(119909) is the (outward) normal cone of119862 at 119909 isin 119862This
follows that
0 = 120582119899119894V + 119908119899119894minus 119909119899+ 119906119899 (42)
where V isin 1205972119891119894(119909119899 119908119899119894) and 119906119899 isin 119873119862(119908119899119894) By the definition
of the normal cone119873119862 we have
⟨119908119899119894minus 119909119899 119910 minus 119908
119899119894⟩ ge 120582119899119894⟨V 119908119899119894minus 119910⟩ forall119910 isin 119862 (43)
Since 119891119894(119909119899 sdot) is subdifferentiable on 119862 by the well-known
Moreau-Rockafellar theorem [31] (also see [6]) for V isin
1205972119891119894(119909119899 119908119899119894) we have
119891119894(119909119899 119910) minus 119891
119894(119909119899 119908119899119894) ge ⟨V 119910 minus 119908
119899119894⟩ forall119910 isin 119862 (44)
Combining this with (43) we have
120582119899119894 (119891119894 (119909119899 119910) minus 119891119894 (119909119899 119908119899119894)) ge ⟨119908119899119894 minus 119909119899 119908119899119894 minus 119910⟩
forall119910 isin 119862
(45)
In particular we have
120582119899119895119894(119891119894(119909119899119895
119910) minus 119891119894(119909119899119895
119908119899119894)) ge ⟨119908
119899119895119894minus 119909119899119895
119908119899119895119894minus 119910⟩
forall119910 isin 119862
(46)
Since 119909119899119895
119909 it follows from (32) that 119908119899119895119894 119909 And thus
we have
119891119894(119909 119910) ge 0 forall119910 isin 119862 (47)
This implies that 119909 isin Sol(119891119894 119862) and hence 119909 isin Ω Since 119909⋆ =
ProjΩ(119868 minus 119860 + 120574ℎ)119909
⋆ and 119909 isin Ω we have
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
= lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= ⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909⟩ le 0
(48)
From Lemma 1 it follows that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1003817100381710038171003817(119868 minus 120579119899119860) (119910119899 minus 119909
⋆)1003817100381710038171003817
2
+ 2120579119899⟨120574ℎ (119909
119899) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899120574 ⟨ℎ (119909
119899) minus ℎ (119909
⋆) 119909119899+1
minus 119909⋆⟩
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899119896120574
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 120579119899119896120574 (1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2)
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le ((1 minus 120579119899120574)2+ 120579119899119896120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 1205791198991205741198961003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899 ⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1 minus 119909
⋆⟩
(49)
This implies that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1 minus 2120579119899120574 + (120579119899120574)
2+ 120579119899120574119896
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896
⟨120574ℎ (119909⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
= (1 minus2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+
(120579119899120574)2
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
times (
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩)
= (1 minus 120578119899)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 120578119899120575119899
(50)
Abstract and Applied Analysis 7
where
119872 = sup 1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2 119899 ge 0 120578
119899=2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896
120575119899=
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ119909⋆minus 119860119909⋆ 119909119899+1
minus 119909⋆⟩
(51)
It is easy to see that 120578119899 rarr 0 suminfin119899=1
120578119899 = infin andlim sup
119899rarrinfin120575119899le 0 Hence by Lemma 2 the sequence 119909
119899
converges strongly to 119909⋆ From (32) we have that 119908119899119894 and
119911119899119894 converge strongly to 119909⋆
Case 2 Assume that 119909119899minus119909⋆ is not a monotone sequence
Then we can define an integer sequence 120591(119899) for all 119899 ge 1198990
(for some large enough 1198990) by
120591 (119899) = max 119896 isin N 119896 le 119899 1003817100381710038171003817119909119896 minus 119909
⋆1003817100381710038171003817 lt1003817100381710038171003817119909119896+1 minus 119909
⋆1003817100381710038171003817
(52)
Clearly 120591 is a nondecreasing sequence such that 120591(119899) rarr infin
as 119899 rarr infin and for all 119899 ge 1198990
1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817 lt
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817 (53)
From (33) we obtain that
lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119911120591(119899)1198941003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119908120591(119899)1198941003817100381710038171003817
= lim119899rarrinfin
10038171003817100381710038171003817119909120591(119899) minus 119879 (119905120591
119899
) 119909120591(119899)
10038171003817100381710038171003817= 0
(54)
Following an argument similar to that in Case 1 we have
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817
2le (1 minus 120578
120591(119899))1003817100381710038171003817119909120591(119899) minus 119909
⋆1003817100381710038171003817
2+ 120578120591(119899)
120575120591(119899)
(55)
where 120578120591(119899)
rarr 0 suminfin119899=1
120578120591(119899)
= infin and lim sup119899rarrinfin
120575120591(119899)
le 0Hence by Lemma 2 we obtain lim
119899rarrinfin119909120591(119899)
minus 119909⋆ = 0 and
lim119899rarrinfin
119909120591(119899)+1
minus 119909⋆ = 0 Now Lemma 5 implies that
0 le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817 le max 1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le1003817100381710038171003817119909120591(119899)+1 minus 119909
⋆1003817100381710038171003817
(56)
Therefore 119909119899 converges strongly to 119909
⋆= Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ This completes the proof
4 Application
In this section we consider a particular Fan inequalitycorresponding to the function 119891 defined by the following forevery 119909 119910 isin 119862
119891 (119909 119910) = ⟨119865 (119909) 119910 minus 119909⟩ (57)
where 119865 119862 rarr 119867 Then we obtain the classical variationalinequality as follows
Find 119911 isin 119862 such that ⟨119865 (119911) 119910 minus 119911⟩ ge 0 forall119910 isin 119862 (58)
The set of solutions of this problem is denoted by 119881119868(119865 119862)In that particular case the solution 119910
119899of the minimization
problem
argmin 120582119899 119891 (119909119899 119910) +1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (59)
can be expressed as
119910119899= 119875119862 (119909119899 minus 120582119899119865 (119909119899)) (60)
Let 119865 be 119871-Lipschitz continuous on 119862 Then
119891 (119909 119910) + 119891 (119910 119911) minus 119891 (119909 119911) = ⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
119909 119910 119911 isin 119862
(61)
Therefore1003816100381610038161003816⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
1003816100381610038161003816 le 1198711003817100381710038171003817119909 minus 119910
1003817100381710038171003817
1003817100381710038171003817119910 minus 1199111003817100381710038171003817
le119871
2(1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2+1003817100381710038171003817119910 minus 119911
1003817100381710038171003817
2)
(62)
hence 119891 satisfies Lipschitz-type continuous condition with1198881 = 1198882 = 1198712
Using Theorem 7 we obtain the following convergencetheorem
Theorem 8 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le 119898119865119894is monotone and 119871-Lipschitz continuous on 119862 Let T =
119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909
0isin 119862 and let 119909
119899 119908119899119894 and 119911
119899119894 be
sequences generated by
119908119899119894 = 119875119862 (119909119899 minus 120582119899119894119865119894 (119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899= 120572119899119909119899+
119898
sum
119894=1
120573119899119894119911119899119894+ 120574119899119879 (119905119899) 119909119899+ 120578119899119878 (119905119899) 119909119899
119909119899+1 = 120579119899120574ℎ (119909119899) + (119868 minus 120579119899119860)119910119899 forall119899 ge 0
(63)
where 120572119899 + sum
119898
119894=1120573119899119894 + 120574119899 + 120578119899 = 1 and 120572119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579119899 satisfy the following conditions(i) lim
119899rarrinfin119905119899= infin
(ii) 120579119899 sub ]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub ]0 1119871[
(iv) 120572119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf119899120572119899120574119899 gt 0 lim inf119899120572119899120578119899 gt 0 andlim inf119899120573119899119894(1 minus 21205821198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (64)
8 Abstract and Applied Analysis
In [32] Baillon proved a strong mean convergence the-orem for nonexpansive mappings and it was generalizedin [33] It follows from the above proof that Theorems 7 isvalid for nonexpansivemappingsThuswe have the followingmean ergodic theorems for nonexpansive mappings in aHilbert space
Theorem 9 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le
119898 119865119894 is monotone and 119871-Lipschitz continuous on 119862 Let119879 and 119878 be two nonexpansive mappings on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(119879)⋂119865(119878) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909119899 119908119899119894 and 119911119899119894 be sequences generatedby 1199090isin 119862 and by
119908119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879119899119909119899 + 120578119899119878
119899119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(65)
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 120572
119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579
119899 satisfy the following conditions
(i) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(ii) 120582119899119894 sub [119886 119887] sub ]0 1119871[ where 119871 = max21198881119894 21198882119894
1 le 119894 le 119898(iii) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (66)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Therefore theauthors acknowledge with thanks DSR for technical andfinancial support
References
[1] K Fan ldquoA minimax inequality and applicationsrdquo in InequalityIII pp 103ndash113 Academic Press New York NY USA 1972
[2] P N Anh ldquoA hybrid extragradient method extended to fixedpoint problems and equilibrium problemsrdquo Optimization vol62 no 2 pp 271ndash283 2013
[3] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994
[4] L-C Ceng N Hadjisavvas and N-C Wong ldquoStrong conver-gence theorem by a hybrid extragradient-like approximationmethod for variational inequalities and fixed point problemsrdquoJournal of Global Optimization vol 46 no 4 pp 635ndash646 2010
[5] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964
[6] P Daniele F Giannessi and A Maugeri Equilibrium Problemsand Variational Models vol 68 Kluwer Academic NorwellMass USA 2003
[7] J-W Peng and J-C Yao ldquoSome new extragradient-likemethodsfor generalized equilibrium problems fixed point problemsand variational inequality problemsrdquo Optimization Methods ampSoftware vol 25 no 4ndash6 pp 677ndash698 2010
[8] D R Sahu N-C Wong and J-C Yao ldquoStrong convergencetheorems for semigroups of asymptotically nonexpansive map-pings inBanach spacesrdquoAbstract andAppliedAnalysis vol 2013Article ID 202095 8 pages 2013
[9] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[10] S Wang and B Guo ldquoNew iterative scheme with nonexpansivemappings for equilibrium problems and variational inequalityproblems in Hilbert spacesrdquo Journal of Computational andApplied Mathematics vol 233 no 10 pp 2620ndash2630 2010
[11] Y Yao Y-C Liou and Y-J Wu ldquoAn extragradient method formixed equilibrium problems and fixed point problemsrdquo FixedPoint Theory and Applications vol 2009 Article ID 632819 15pages 2009
[12] F E Browder ldquoNonexpansive nonlinear operators in a Banachspacerdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 54 pp 1041ndash1044 1965
[13] R Chen and Y Song ldquoConvergence to common fixed pointof nonexpansive semigroupsrdquo Journal of Computational andApplied Mathematics vol 200 no 2 pp 566ndash575 2007
[14] A Aleyner and Y Censor ldquoBest approximation to commonfixed points of a semigroup of nonexpansive operatorsrdquo Journalof Nonlinear and Convex Analysis vol 6 no 1 pp 137ndash151 2005
[15] H K Xu ldquoAn iterative approach to quadratic optimizationrdquoJournal of Optimization Theory and Applications vol 116 no 3pp 659ndash678 2003
[16] G Marino and H-K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006
[17] AMoudafi ldquoViscosity approximationmethods for fixed-pointsproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 241 no 1 pp 46ndash55 2000
[18] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[19] S Li L Li and Y Su ldquoGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert spacerdquo Nonlin-ear Analysis Theory Methods amp Applications vol 70 no 9 pp3065ndash3071 2009
Abstract and Applied Analysis 9
[20] D R Sahu N C Wong and J C Yao ldquoA unified hybriditerative method for solving variational inequalities involvinggeneralized pseudocontractive mappingsrdquo SIAM Journal onControl and Optimization vol 50 no 4 pp 2335ndash2354 2012
[21] N Shioji and W Takahashi ldquoStrong convergence theoremsfor asymptotically nonexpansive semigroups in Hilbert spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 34 no1 pp 87ndash99 1998
[22] P N Anh ldquoStrong convergence theorems for nonexpansivemappings and Ky Fan inequalitiesrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 303ndash320 2012
[23] L-C Ceng Q H Ansari and J-C Yao ldquoHybrid proximal-type and hybrid shrinking projection algorithms for equilib-rium problems maximal monotone operators and relativelynonexpansive mappingsrdquo Numerical Functional Analysis andOptimization vol 31 no 7-9 pp 763ndash797 2010
[24] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009
[25] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011
[26] L-C Ceng S-M Guu and J-C Yao ldquoHybrid iterative methodfor finding common solutions of generalizedmixed equilibriumand fixed point problemsrdquo Fixed PointTheory and Applicationsvol 2012 article 92 2012
[27] M Eslamian ldquoConvergence theorems for nonspreading map-pings andnonexpansivemultivaluedmappings and equilibriumproblemsrdquo Optimization Letters vol 7 no 3 pp 547ndash557 2013
[28] M Eslamian ldquoHybrid method for equilibrium problems andfixed point problems of finite families of nonexpansive semi-groupsrdquo RACSAM vol 107 pp 299ndash307 2013
[29] M Eslamian and A Abkar ldquoOne-step iterative process fora finite family of multivalued mappingsrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 105ndash111 2011
[30] P-E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[31] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976
[32] J-B Baillon ldquoUn theoreme de type ergodique pour les contrac-tions non lineaires dans un espace de Hilbertrdquo Comptes Rendusde lrsquoAcademie des Sciences vol 280 no 22 pp A1511ndashA15141975
[33] W Kaczor T Kuczumow and S Reich ldquoA mean ergodictheorem for nonlinear semigroups which are asymptoticallynonexpansive in the intermediate senserdquo Journal of Mathemat-ical Analysis and Applications vol 246 no 1 pp 1ndash27 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
positive sequences 120572119899 120573119899 120574119899 and 120582
119899 the sequences
119909119899 119908119899 and 119911
119899 converge strongly to 119902 isin 119865(119878) cap Sol(119891 119862)
which is a solution of the variational inequality ⟨(119868 minus ℎ)119902 119909 minus119902⟩ ge 0 forall119909 isin 119865(119878) cap Sol(119891 119862)
In this paper we introduce a new iterative scheme basedon the viscosity method and study the existence of a commonelement of the set of solutions of equilibrium problems fora family of monotone Lipschitz-type continuous mappingsand the sets of fixed points of two nonexpansive semigroupsin a real Hilbert space We establish strong convergencetheorems of the new iterative scheme for the solution ofthe variational inequality problem which is the optimalitycondition for theminimization problemOur results improveand generalize the corresponding recent results of Anh [22]Cianciaruso et al [18] and many others
2 Preliminaries
In this section we collect some lemmas which are crucial forthe proofs of our results
Let 119909119899 be a sequence in 119867 and 119909 isin 119867 In the sequel119909119899 119909 denotes that 119909119899weakly converges to 119909 and 119909119899 rarr 119909
denotes that 119909119899 weakly converges to 119909
Lemma 1 Let 119867 be a real Hilbert space Then the followinginequality holds
1003817100381710038171003817119909 + 1199101003817100381710038171003817
2le 119909
2+ 2 ⟨119910 119909 + 119910⟩ forall119909 119910 isin 119867 (13)
Lemma 2 (see [15]) Assume that 119886119899 is a sequence of
nonnegative real numbers such that
119886119899+1
le (1 minus 120578119899) 119886119899 + 120578119899120575119899 119899 ge 0 (14)
where 120578119899 is a sequence in ]0 1[ and 120575
119899is a sequence inR such
that
(i) suminfin119899=1
120578119899= infin
(ii) lim sup119899rarrinfin
120575119899le 0 or suminfin
119899=1|120578119899120575119899| lt infin
Then lim119899rarrinfin119886119899 = 0
Lemma 3 (see [16]) Let 119860 be a strongly positive linearbounded self-adjoint operator on119867 with coefficient 120574 gt 0 and0 lt 120588 le 119860
minus1 Then 119868 minus 120588119860 le 1 minus 120588 120574
Lemma 4 (see [29]) Let 119867 be a Hilbert space and 119909119894isin 119867
(1 le 119894 le 119898) Then for any given 120582119894119898
119894=1sub ]0 1[ with sum119898
119894=1120582119894=
1 and for any positive integer 119896 119895 with 1 le 119896 lt 119895 le 119898
1003817100381710038171003817100381710038171003817100381710038171003817
119898
sum
119894=1
120582119894119909119894
1003817100381710038171003817100381710038171003817100381710038171003817
2
le
119898
sum
119894=1
120582119894
10038171003817100381710038171199091198941003817100381710038171003817
2minus 120582119896120582119895
10038171003817100381710038171003817119909119896minus 119909119895
10038171003817100381710038171003817
2
(15)
Lemma 5 (see [30]) Let 119905119899 be a sequence of real numbers
such that there exists a subsequence 119899119894 of 119899 such that 119905
119899119894
lt
119905119899119894+1
for all 119894 isin N Then there exists a nondecreasing sequence120591(119899) sub N such that 120591(119899) rarr infin and the following propertiesare satisfied by all (sufficiently large) numbers 119899 isin N
119905120591(119899)
le 119905120591(119899)+1
119905119899le 119905120591(119899)+1
(16)
In fact
120591 (119899) = max 119896 le 119899 119905119896lt 119905119896+1 (17)
Lemma 6 (see [2]) Let 119862 be a nonempty closed convex subsetof a real Hilbert space 119867 and let 119891 119862 times 119862 rarr R bea pseudomonotone and Lipschitz-type continuous bifunctionwith constants 119888
1 1198882ge 0 For each 119909 isin 119862 let 119891(119909 sdot) be convex
and subdifferentiable on 119862 Let 1199090isin 119862 and let 119909
119899 119911119899 and
119908119899 be sequences generated by
119908119899= argmin 120582
119899119891 (119909119899 119908) +
1
2
1003817100381710038171003817119908 minus 119909119899
1003817100381710038171003817
2 119908 isin 119862
119911119899= argmin 120582
119899119891 (119908119899 119911) +
1
2
1003817100381710038171003817119911 minus 1199091198991003817100381710038171003817
2 119911 isin 119862
(18)
Then for each 119909⋆ isin Sol(119891 119862)
1003817100381710038171003817119911119899 minus 119909⋆1003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus (1 minus 2120582
1198991198881)1003817100381710038171003817119909119899 minus 119908119899
1003817100381710038171003817
2
minus (1 minus 2120582119899 1198882)1003817100381710038171003817119908119899 minus 119911119899
1003817100381710038171003817
2 forall119899 ge 0
(19)
3 Main Results
In this section we prove the main strong convergence resultwhich solves the problem of finding a common elementof three sets 119865(T) 119865(S) and Sol(119891
119894 119862) for finite family
of monotone continuous and Lipschitz-type continuousbifunctions 119891
119894in a real Hilbert space119867
Theorem7 Let119862 be a nonempty closed convex subset of a realHilbert space 119867 and let 119891
119894 119862 times 119862 rarr R (119894 = 1 2 119898)
be a finite family of monotone continuous and Lipschitz-type continuous bifunctions with constants 1198881119894 and 1198882119894 LetT = 119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119878119900119897(119891119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a
119896-contraction self-mapping of 119862 and 119860 is a strongly positivebounded linear self-adjoint operator on119867 with coefficient 120574 lt1 and 0 lt 120574 lt 120574119896 Let 1199090 isin 119862 and let 119909119899 119908119899119894 and 119911119899119894be sequences generated by
119908119899119894= argmin 120582
119899119894119891119894(119909119899 119908) +
1
2
1003817100381710038171003817119908 minus 119909119899
1003817100381710038171003817
2 119908 isin 119862
119894 isin 1 2 119898
119911119899119894= argmin 120582
119899119894119891119894(119908119899119894 119911) +
1
2
1003817100381710038171003817119911 minus 1199091198991003817100381710038171003817
2 119911 isin 119862
119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(20)
4 Abstract and Applied Analysis
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 119905
119899 120572119899 120573119899119894 120574119899
120578119899 120582119899119894 and 120579
119899 satisfy the following conditions
(i) lim119899rarrinfin
119905119899= infin
(ii) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub]0 1119871[ where 119871 =max2119888
1119894 21198882119894 1 le
119894 le 119898(iv) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119878119900119897(119891119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (21)
Proof Since 119865(T) 119865(S) and Sol(119891119894 119862) are closed andconvex Proj
Ωis well-definedWe claim that Proj
Ω(119868minus119860+120574ℎ)
is a contraction from 119862 into itself Indeed for each 119909 119910 isin 119862we have
1003817100381710038171003817ProjΩ (119868 minus 119860 + 120574ℎ) (119909) minus ProjΩ(119868 minus 119860 + 120574ℎ) (119910)
1003817100381710038171003817
le1003817100381710038171003817(119868 minus 119860 + 120574ℎ) (119909) minus (119868 minus 119860 + 120574ℎ) (119910)
1003817100381710038171003817
le1003817100381710038171003817(119868 minus 119860) 119909 minus (119868 minus 119860) 119910
1003817100381710038171003817 + 1205741003817100381710038171003817ℎ119909 minus ℎ119910
1003817100381710038171003817
le (1 minus 120574)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 + 1205741198961003817100381710038171003817119909 minus 119910
1003817100381710038171003817
le (1 minus (120574 minus 120574119896))1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(22)
Therefore by the Banach contraction principle thereexists a unique element 119909⋆ isin 119862 such that 119909⋆ = Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ We show that 119909119899 is bounded Since lim119899rarrinfin120579119899 = 0
we can assume with no loss of generality that 120579119899 isin (0 119860
minus1)
for all 119899 ge 0 By Lemma 6 for each 1 le 119894 le 119898 we have
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817 le
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 (23)
This implies that
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le
1003817100381710038171003817100381710038171003817100381710038171003817
120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899 minus 119909⋆
1003817100381710038171003817100381710038171003817100381710038171003817
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 +
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
+ 120574119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817 + 120578119899
1003817100381710038171003817119878 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 +
119898
sum
119894=1
1205731198991198941003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
+ 120574119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120578119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
=1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
(24)
It follows from Lemma 3 that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
=1003817100381710038171003817120579119899 (120574ℎ (119909119899) minus 119860119909
⋆) + (119868 minus 120579
119899119860) (119910119899minus 119909⋆)1003817100381710038171003817
le 120579119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817 +
1003817100381710038171003817119868 minus 1205791198991198601003817100381710038171003817
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le 120579119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817 + (1 minus 120579119899120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le 1205791198991205741003817100381710038171003817ℎ (119909119899) minus ℎ (119909
⋆)1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
+ (1 minus 120579119899120574)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
le 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
+ (1 minus 120579119899120574)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
le (1 minus 120579119899(120574 minus 120574119896))
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
= (1 minus 120579119899(120574 minus 120574119896))
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
+ 120579119899(120574 minus 120574119896)
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
120574 minus 120574119896
le max1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817120574ℎ119909⋆minusA119909⋆1003817100381710038171003817
120574 minus 120574119896
(25)
This implies that 119909119899 is bounded and so are 119911119899119894 ℎ(119909119899)119879(119905119899)119909119899 and 119878(119905
119899)119909119899 Next we show that lim
119899rarrinfin119909119899minus
119879(119905119899)119909119899 = lim
119899rarrinfin119909119899minus 119878(119905119899)119909119899 = 0 Indeed by Lemma 6
for each 1 le 119894 le 119898 we have
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus (1 minus 2120582
1198991198941198881119894)1003817100381710038171003817119909119899 minus 119908119899119894
1003817100381710038171003817
2
minus (1 minus 21205821198991198941198882119894)1003817100381710038171003817119908119899119894 minus 119911119899119894
1003817100381710038171003817
2
(26)
Applying Lemma 4 and inequality (26) for 119896 isin 1 2 119898
we have that
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
2
=
1003817100381710038171003817100381710038171003817100381710038171003817
120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899 minus 119909⋆
1003817100381710038171003817100381710038171003817100381710038171003817
2
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
2
+ 120574119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
2+ 120578119899
1003817100381710038171003817119878 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
2
minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus 120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
Abstract and Applied Analysis 5
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+ 120574119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 120578119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus 120573119899119896 (1 minus 21205821198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2minus 120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
=1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus 120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus 1205721198991205781198991003817100381710038171003817119909119899 minus 119878 (119905119899) 119909119899
1003817100381710038171003817
2
(27)
We now compute1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2
=1003817100381710038171003817120579119899 (120574ℎ (119909119899) minus 119860119909
⋆) + (119868 minus 120579
119899119860) (119910119899minus 119909⋆)1003817100381710038171003817
2
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119910119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
minus (1 minus 120579119899120574)2120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
(28)
Therefore
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899(1 minus 120579
119899120574)
times1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579
2
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2
(29)
In order to prove that 119909119899rarr 119909⋆ as 119899 rarr infin we consider the
following two cases
Case 1 Assume that 119909119899minus 119909⋆ is a monotone sequence
In other words for large enough 1198990 119909119899minus 119909⋆119899ge1198990
iseither nondecreasing or nonincreasing Since 119909119899 minus 119909
⋆ is
bounded it is convergent Since lim119899rarrinfin
120579119899= 0 and ℎ(119909
119899)
and 119909119899 are bounded from (29) we have
lim119899rarrinfin
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2= 0 (30)
and by assumption we get
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817 = 0 1 le 119896 le 119898 (31)
By similar argument we can obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199081198991198961003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
= lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817 = 0
(32)
Further for all ℎ ge 0 and 119899 ge 0 we see that1003817100381710038171003817119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817 +1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 119909119899
1003817100381710038171003817
+1003817100381710038171003817119879 (ℎ) 119879 (119905119899) 119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le 21003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817
+ sup119909isin119909119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 1199091198991003817100381710038171003817
(33)
Since 119879(119905) is uar nonexpansive semigroup andlim119899rarrinfin
119905119899= infin we have
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (ℎ) 1199091198991003817100381710038171003817 = 0 (34)
Similarly for all ℎ ge 0 and 119899 ge 0 we obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (ℎ) 1199091198991003817100381710038171003817 = 0 (35)
Next we show that
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩ le 0 (36)
To show this inequality we choose a subsequence 119909119899119894
of 119909119899
such that
lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
(37)
Since 119909119899119894
is bounded there exists a subsequence 119909119899119894119895
of119909119899119894
which converges weakly to 119909 Without loss of generalitywe can assume that 119909119899
119895
119909 Consider
100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119879 (119905) 119909119899
119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817
le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119909119899119895
minus 119909100381710038171003817100381710038171003817
(38)
Thus we have
lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 119879 (119905) 11990910038171003817100381710038171003817le lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 11990910038171003817100381710038171003817 (39)
By the Opial property of the Hilbert space 119867 we obtain119879(119905)119909 = 119909 for all 119905 ge 0 Similarly we have that 119878(119905)119909 = 119909
for all 119905 ge 0 This implies that 119909 isin 119865(T)⋂119865(S) Now weshow that 119909 isin Sol(119891119894 119862) For each 1 le 119894 le 119898 since 119891119894(119909 sdot) isconvex on 119862 for each 119909 isin 119862 we see that
119908119899119894= argmin 120582
119899119894119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (40)
6 Abstract and Applied Analysis
if and only if
0 isin 1205972(119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2) (119908119899119894) + 119873119862(119908119899119894) (41)
where119873119862(119909) is the (outward) normal cone of119862 at 119909 isin 119862This
follows that
0 = 120582119899119894V + 119908119899119894minus 119909119899+ 119906119899 (42)
where V isin 1205972119891119894(119909119899 119908119899119894) and 119906119899 isin 119873119862(119908119899119894) By the definition
of the normal cone119873119862 we have
⟨119908119899119894minus 119909119899 119910 minus 119908
119899119894⟩ ge 120582119899119894⟨V 119908119899119894minus 119910⟩ forall119910 isin 119862 (43)
Since 119891119894(119909119899 sdot) is subdifferentiable on 119862 by the well-known
Moreau-Rockafellar theorem [31] (also see [6]) for V isin
1205972119891119894(119909119899 119908119899119894) we have
119891119894(119909119899 119910) minus 119891
119894(119909119899 119908119899119894) ge ⟨V 119910 minus 119908
119899119894⟩ forall119910 isin 119862 (44)
Combining this with (43) we have
120582119899119894 (119891119894 (119909119899 119910) minus 119891119894 (119909119899 119908119899119894)) ge ⟨119908119899119894 minus 119909119899 119908119899119894 minus 119910⟩
forall119910 isin 119862
(45)
In particular we have
120582119899119895119894(119891119894(119909119899119895
119910) minus 119891119894(119909119899119895
119908119899119894)) ge ⟨119908
119899119895119894minus 119909119899119895
119908119899119895119894minus 119910⟩
forall119910 isin 119862
(46)
Since 119909119899119895
119909 it follows from (32) that 119908119899119895119894 119909 And thus
we have
119891119894(119909 119910) ge 0 forall119910 isin 119862 (47)
This implies that 119909 isin Sol(119891119894 119862) and hence 119909 isin Ω Since 119909⋆ =
ProjΩ(119868 minus 119860 + 120574ℎ)119909
⋆ and 119909 isin Ω we have
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
= lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= ⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909⟩ le 0
(48)
From Lemma 1 it follows that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1003817100381710038171003817(119868 minus 120579119899119860) (119910119899 minus 119909
⋆)1003817100381710038171003817
2
+ 2120579119899⟨120574ℎ (119909
119899) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899120574 ⟨ℎ (119909
119899) minus ℎ (119909
⋆) 119909119899+1
minus 119909⋆⟩
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899119896120574
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 120579119899119896120574 (1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2)
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le ((1 minus 120579119899120574)2+ 120579119899119896120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 1205791198991205741198961003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899 ⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1 minus 119909
⋆⟩
(49)
This implies that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1 minus 2120579119899120574 + (120579119899120574)
2+ 120579119899120574119896
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896
⟨120574ℎ (119909⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
= (1 minus2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+
(120579119899120574)2
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
times (
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩)
= (1 minus 120578119899)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 120578119899120575119899
(50)
Abstract and Applied Analysis 7
where
119872 = sup 1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2 119899 ge 0 120578
119899=2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896
120575119899=
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ119909⋆minus 119860119909⋆ 119909119899+1
minus 119909⋆⟩
(51)
It is easy to see that 120578119899 rarr 0 suminfin119899=1
120578119899 = infin andlim sup
119899rarrinfin120575119899le 0 Hence by Lemma 2 the sequence 119909
119899
converges strongly to 119909⋆ From (32) we have that 119908119899119894 and
119911119899119894 converge strongly to 119909⋆
Case 2 Assume that 119909119899minus119909⋆ is not a monotone sequence
Then we can define an integer sequence 120591(119899) for all 119899 ge 1198990
(for some large enough 1198990) by
120591 (119899) = max 119896 isin N 119896 le 119899 1003817100381710038171003817119909119896 minus 119909
⋆1003817100381710038171003817 lt1003817100381710038171003817119909119896+1 minus 119909
⋆1003817100381710038171003817
(52)
Clearly 120591 is a nondecreasing sequence such that 120591(119899) rarr infin
as 119899 rarr infin and for all 119899 ge 1198990
1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817 lt
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817 (53)
From (33) we obtain that
lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119911120591(119899)1198941003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119908120591(119899)1198941003817100381710038171003817
= lim119899rarrinfin
10038171003817100381710038171003817119909120591(119899) minus 119879 (119905120591
119899
) 119909120591(119899)
10038171003817100381710038171003817= 0
(54)
Following an argument similar to that in Case 1 we have
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817
2le (1 minus 120578
120591(119899))1003817100381710038171003817119909120591(119899) minus 119909
⋆1003817100381710038171003817
2+ 120578120591(119899)
120575120591(119899)
(55)
where 120578120591(119899)
rarr 0 suminfin119899=1
120578120591(119899)
= infin and lim sup119899rarrinfin
120575120591(119899)
le 0Hence by Lemma 2 we obtain lim
119899rarrinfin119909120591(119899)
minus 119909⋆ = 0 and
lim119899rarrinfin
119909120591(119899)+1
minus 119909⋆ = 0 Now Lemma 5 implies that
0 le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817 le max 1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le1003817100381710038171003817119909120591(119899)+1 minus 119909
⋆1003817100381710038171003817
(56)
Therefore 119909119899 converges strongly to 119909
⋆= Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ This completes the proof
4 Application
In this section we consider a particular Fan inequalitycorresponding to the function 119891 defined by the following forevery 119909 119910 isin 119862
119891 (119909 119910) = ⟨119865 (119909) 119910 minus 119909⟩ (57)
where 119865 119862 rarr 119867 Then we obtain the classical variationalinequality as follows
Find 119911 isin 119862 such that ⟨119865 (119911) 119910 minus 119911⟩ ge 0 forall119910 isin 119862 (58)
The set of solutions of this problem is denoted by 119881119868(119865 119862)In that particular case the solution 119910
119899of the minimization
problem
argmin 120582119899 119891 (119909119899 119910) +1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (59)
can be expressed as
119910119899= 119875119862 (119909119899 minus 120582119899119865 (119909119899)) (60)
Let 119865 be 119871-Lipschitz continuous on 119862 Then
119891 (119909 119910) + 119891 (119910 119911) minus 119891 (119909 119911) = ⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
119909 119910 119911 isin 119862
(61)
Therefore1003816100381610038161003816⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
1003816100381610038161003816 le 1198711003817100381710038171003817119909 minus 119910
1003817100381710038171003817
1003817100381710038171003817119910 minus 1199111003817100381710038171003817
le119871
2(1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2+1003817100381710038171003817119910 minus 119911
1003817100381710038171003817
2)
(62)
hence 119891 satisfies Lipschitz-type continuous condition with1198881 = 1198882 = 1198712
Using Theorem 7 we obtain the following convergencetheorem
Theorem 8 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le 119898119865119894is monotone and 119871-Lipschitz continuous on 119862 Let T =
119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909
0isin 119862 and let 119909
119899 119908119899119894 and 119911
119899119894 be
sequences generated by
119908119899119894 = 119875119862 (119909119899 minus 120582119899119894119865119894 (119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899= 120572119899119909119899+
119898
sum
119894=1
120573119899119894119911119899119894+ 120574119899119879 (119905119899) 119909119899+ 120578119899119878 (119905119899) 119909119899
119909119899+1 = 120579119899120574ℎ (119909119899) + (119868 minus 120579119899119860)119910119899 forall119899 ge 0
(63)
where 120572119899 + sum
119898
119894=1120573119899119894 + 120574119899 + 120578119899 = 1 and 120572119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579119899 satisfy the following conditions(i) lim
119899rarrinfin119905119899= infin
(ii) 120579119899 sub ]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub ]0 1119871[
(iv) 120572119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf119899120572119899120574119899 gt 0 lim inf119899120572119899120578119899 gt 0 andlim inf119899120573119899119894(1 minus 21205821198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (64)
8 Abstract and Applied Analysis
In [32] Baillon proved a strong mean convergence the-orem for nonexpansive mappings and it was generalizedin [33] It follows from the above proof that Theorems 7 isvalid for nonexpansivemappingsThuswe have the followingmean ergodic theorems for nonexpansive mappings in aHilbert space
Theorem 9 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le
119898 119865119894 is monotone and 119871-Lipschitz continuous on 119862 Let119879 and 119878 be two nonexpansive mappings on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(119879)⋂119865(119878) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909119899 119908119899119894 and 119911119899119894 be sequences generatedby 1199090isin 119862 and by
119908119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879119899119909119899 + 120578119899119878
119899119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(65)
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 120572
119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579
119899 satisfy the following conditions
(i) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(ii) 120582119899119894 sub [119886 119887] sub ]0 1119871[ where 119871 = max21198881119894 21198882119894
1 le 119894 le 119898(iii) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (66)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Therefore theauthors acknowledge with thanks DSR for technical andfinancial support
References
[1] K Fan ldquoA minimax inequality and applicationsrdquo in InequalityIII pp 103ndash113 Academic Press New York NY USA 1972
[2] P N Anh ldquoA hybrid extragradient method extended to fixedpoint problems and equilibrium problemsrdquo Optimization vol62 no 2 pp 271ndash283 2013
[3] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994
[4] L-C Ceng N Hadjisavvas and N-C Wong ldquoStrong conver-gence theorem by a hybrid extragradient-like approximationmethod for variational inequalities and fixed point problemsrdquoJournal of Global Optimization vol 46 no 4 pp 635ndash646 2010
[5] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964
[6] P Daniele F Giannessi and A Maugeri Equilibrium Problemsand Variational Models vol 68 Kluwer Academic NorwellMass USA 2003
[7] J-W Peng and J-C Yao ldquoSome new extragradient-likemethodsfor generalized equilibrium problems fixed point problemsand variational inequality problemsrdquo Optimization Methods ampSoftware vol 25 no 4ndash6 pp 677ndash698 2010
[8] D R Sahu N-C Wong and J-C Yao ldquoStrong convergencetheorems for semigroups of asymptotically nonexpansive map-pings inBanach spacesrdquoAbstract andAppliedAnalysis vol 2013Article ID 202095 8 pages 2013
[9] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[10] S Wang and B Guo ldquoNew iterative scheme with nonexpansivemappings for equilibrium problems and variational inequalityproblems in Hilbert spacesrdquo Journal of Computational andApplied Mathematics vol 233 no 10 pp 2620ndash2630 2010
[11] Y Yao Y-C Liou and Y-J Wu ldquoAn extragradient method formixed equilibrium problems and fixed point problemsrdquo FixedPoint Theory and Applications vol 2009 Article ID 632819 15pages 2009
[12] F E Browder ldquoNonexpansive nonlinear operators in a Banachspacerdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 54 pp 1041ndash1044 1965
[13] R Chen and Y Song ldquoConvergence to common fixed pointof nonexpansive semigroupsrdquo Journal of Computational andApplied Mathematics vol 200 no 2 pp 566ndash575 2007
[14] A Aleyner and Y Censor ldquoBest approximation to commonfixed points of a semigroup of nonexpansive operatorsrdquo Journalof Nonlinear and Convex Analysis vol 6 no 1 pp 137ndash151 2005
[15] H K Xu ldquoAn iterative approach to quadratic optimizationrdquoJournal of Optimization Theory and Applications vol 116 no 3pp 659ndash678 2003
[16] G Marino and H-K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006
[17] AMoudafi ldquoViscosity approximationmethods for fixed-pointsproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 241 no 1 pp 46ndash55 2000
[18] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[19] S Li L Li and Y Su ldquoGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert spacerdquo Nonlin-ear Analysis Theory Methods amp Applications vol 70 no 9 pp3065ndash3071 2009
Abstract and Applied Analysis 9
[20] D R Sahu N C Wong and J C Yao ldquoA unified hybriditerative method for solving variational inequalities involvinggeneralized pseudocontractive mappingsrdquo SIAM Journal onControl and Optimization vol 50 no 4 pp 2335ndash2354 2012
[21] N Shioji and W Takahashi ldquoStrong convergence theoremsfor asymptotically nonexpansive semigroups in Hilbert spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 34 no1 pp 87ndash99 1998
[22] P N Anh ldquoStrong convergence theorems for nonexpansivemappings and Ky Fan inequalitiesrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 303ndash320 2012
[23] L-C Ceng Q H Ansari and J-C Yao ldquoHybrid proximal-type and hybrid shrinking projection algorithms for equilib-rium problems maximal monotone operators and relativelynonexpansive mappingsrdquo Numerical Functional Analysis andOptimization vol 31 no 7-9 pp 763ndash797 2010
[24] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009
[25] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011
[26] L-C Ceng S-M Guu and J-C Yao ldquoHybrid iterative methodfor finding common solutions of generalizedmixed equilibriumand fixed point problemsrdquo Fixed PointTheory and Applicationsvol 2012 article 92 2012
[27] M Eslamian ldquoConvergence theorems for nonspreading map-pings andnonexpansivemultivaluedmappings and equilibriumproblemsrdquo Optimization Letters vol 7 no 3 pp 547ndash557 2013
[28] M Eslamian ldquoHybrid method for equilibrium problems andfixed point problems of finite families of nonexpansive semi-groupsrdquo RACSAM vol 107 pp 299ndash307 2013
[29] M Eslamian and A Abkar ldquoOne-step iterative process fora finite family of multivalued mappingsrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 105ndash111 2011
[30] P-E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[31] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976
[32] J-B Baillon ldquoUn theoreme de type ergodique pour les contrac-tions non lineaires dans un espace de Hilbertrdquo Comptes Rendusde lrsquoAcademie des Sciences vol 280 no 22 pp A1511ndashA15141975
[33] W Kaczor T Kuczumow and S Reich ldquoA mean ergodictheorem for nonlinear semigroups which are asymptoticallynonexpansive in the intermediate senserdquo Journal of Mathemat-ical Analysis and Applications vol 246 no 1 pp 1ndash27 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 119905
119899 120572119899 120573119899119894 120574119899
120578119899 120582119899119894 and 120579
119899 satisfy the following conditions
(i) lim119899rarrinfin
119905119899= infin
(ii) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub]0 1119871[ where 119871 =max2119888
1119894 21198882119894 1 le
119894 le 119898(iv) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119878119900119897(119891119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (21)
Proof Since 119865(T) 119865(S) and Sol(119891119894 119862) are closed andconvex Proj
Ωis well-definedWe claim that Proj
Ω(119868minus119860+120574ℎ)
is a contraction from 119862 into itself Indeed for each 119909 119910 isin 119862we have
1003817100381710038171003817ProjΩ (119868 minus 119860 + 120574ℎ) (119909) minus ProjΩ(119868 minus 119860 + 120574ℎ) (119910)
1003817100381710038171003817
le1003817100381710038171003817(119868 minus 119860 + 120574ℎ) (119909) minus (119868 minus 119860 + 120574ℎ) (119910)
1003817100381710038171003817
le1003817100381710038171003817(119868 minus 119860) 119909 minus (119868 minus 119860) 119910
1003817100381710038171003817 + 1205741003817100381710038171003817ℎ119909 minus ℎ119910
1003817100381710038171003817
le (1 minus 120574)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 + 1205741198961003817100381710038171003817119909 minus 119910
1003817100381710038171003817
le (1 minus (120574 minus 120574119896))1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(22)
Therefore by the Banach contraction principle thereexists a unique element 119909⋆ isin 119862 such that 119909⋆ = Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ We show that 119909119899 is bounded Since lim119899rarrinfin120579119899 = 0
we can assume with no loss of generality that 120579119899 isin (0 119860
minus1)
for all 119899 ge 0 By Lemma 6 for each 1 le 119894 le 119898 we have
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817 le
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 (23)
This implies that
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le
1003817100381710038171003817100381710038171003817100381710038171003817
120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899 minus 119909⋆
1003817100381710038171003817100381710038171003817100381710038171003817
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 +
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
+ 120574119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817 + 120578119899
1003817100381710038171003817119878 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 +
119898
sum
119894=1
1205731198991198941003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
+ 120574119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120578119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
=1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
(24)
It follows from Lemma 3 that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
=1003817100381710038171003817120579119899 (120574ℎ (119909119899) minus 119860119909
⋆) + (119868 minus 120579
119899119860) (119910119899minus 119909⋆)1003817100381710038171003817
le 120579119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817 +
1003817100381710038171003817119868 minus 1205791198991198601003817100381710038171003817
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le 120579119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817 + (1 minus 120579119899120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le 1205791198991205741003817100381710038171003817ℎ (119909119899) minus ℎ (119909
⋆)1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
+ (1 minus 120579119899120574)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
le 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
+ (1 minus 120579119899120574)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
le (1 minus 120579119899(120574 minus 120574119896))
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579119899
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
= (1 minus 120579119899(120574 minus 120574119896))
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
+ 120579119899(120574 minus 120574119896)
1003817100381710038171003817120574ℎ (119909⋆) minus 119860119909
⋆1003817100381710038171003817
120574 minus 120574119896
le max1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817120574ℎ119909⋆minusA119909⋆1003817100381710038171003817
120574 minus 120574119896
(25)
This implies that 119909119899 is bounded and so are 119911119899119894 ℎ(119909119899)119879(119905119899)119909119899 and 119878(119905
119899)119909119899 Next we show that lim
119899rarrinfin119909119899minus
119879(119905119899)119909119899 = lim
119899rarrinfin119909119899minus 119878(119905119899)119909119899 = 0 Indeed by Lemma 6
for each 1 le 119894 le 119898 we have
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus (1 minus 2120582
1198991198941198881119894)1003817100381710038171003817119909119899 minus 119908119899119894
1003817100381710038171003817
2
minus (1 minus 21205821198991198941198882119894)1003817100381710038171003817119908119899119894 minus 119911119899119894
1003817100381710038171003817
2
(26)
Applying Lemma 4 and inequality (26) for 119896 isin 1 2 119898
we have that
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
2
=
1003817100381710038171003817100381710038171003817100381710038171003817
120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879 (119905119899) 119909119899 + 120578119899119878 (119905119899) 119909119899 minus 119909⋆
1003817100381710038171003817100381710038171003817100381710038171003817
2
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119911119899119894 minus 119909⋆1003817100381710038171003817
2
+ 120574119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
2+ 120578119899
1003817100381710038171003817119878 (119905119899) 119909119899 minus 119909⋆1003817100381710038171003817
2
minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus 120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
Abstract and Applied Analysis 5
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+ 120574119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 120578119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus 120573119899119896 (1 minus 21205821198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2minus 120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
=1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus 120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus 1205721198991205781198991003817100381710038171003817119909119899 minus 119878 (119905119899) 119909119899
1003817100381710038171003817
2
(27)
We now compute1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2
=1003817100381710038171003817120579119899 (120574ℎ (119909119899) minus 119860119909
⋆) + (119868 minus 120579
119899119860) (119910119899minus 119909⋆)1003817100381710038171003817
2
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119910119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
minus (1 minus 120579119899120574)2120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
(28)
Therefore
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899(1 minus 120579
119899120574)
times1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579
2
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2
(29)
In order to prove that 119909119899rarr 119909⋆ as 119899 rarr infin we consider the
following two cases
Case 1 Assume that 119909119899minus 119909⋆ is a monotone sequence
In other words for large enough 1198990 119909119899minus 119909⋆119899ge1198990
iseither nondecreasing or nonincreasing Since 119909119899 minus 119909
⋆ is
bounded it is convergent Since lim119899rarrinfin
120579119899= 0 and ℎ(119909
119899)
and 119909119899 are bounded from (29) we have
lim119899rarrinfin
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2= 0 (30)
and by assumption we get
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817 = 0 1 le 119896 le 119898 (31)
By similar argument we can obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199081198991198961003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
= lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817 = 0
(32)
Further for all ℎ ge 0 and 119899 ge 0 we see that1003817100381710038171003817119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817 +1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 119909119899
1003817100381710038171003817
+1003817100381710038171003817119879 (ℎ) 119879 (119905119899) 119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le 21003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817
+ sup119909isin119909119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 1199091198991003817100381710038171003817
(33)
Since 119879(119905) is uar nonexpansive semigroup andlim119899rarrinfin
119905119899= infin we have
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (ℎ) 1199091198991003817100381710038171003817 = 0 (34)
Similarly for all ℎ ge 0 and 119899 ge 0 we obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (ℎ) 1199091198991003817100381710038171003817 = 0 (35)
Next we show that
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩ le 0 (36)
To show this inequality we choose a subsequence 119909119899119894
of 119909119899
such that
lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
(37)
Since 119909119899119894
is bounded there exists a subsequence 119909119899119894119895
of119909119899119894
which converges weakly to 119909 Without loss of generalitywe can assume that 119909119899
119895
119909 Consider
100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119879 (119905) 119909119899
119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817
le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119909119899119895
minus 119909100381710038171003817100381710038171003817
(38)
Thus we have
lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 119879 (119905) 11990910038171003817100381710038171003817le lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 11990910038171003817100381710038171003817 (39)
By the Opial property of the Hilbert space 119867 we obtain119879(119905)119909 = 119909 for all 119905 ge 0 Similarly we have that 119878(119905)119909 = 119909
for all 119905 ge 0 This implies that 119909 isin 119865(T)⋂119865(S) Now weshow that 119909 isin Sol(119891119894 119862) For each 1 le 119894 le 119898 since 119891119894(119909 sdot) isconvex on 119862 for each 119909 isin 119862 we see that
119908119899119894= argmin 120582
119899119894119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (40)
6 Abstract and Applied Analysis
if and only if
0 isin 1205972(119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2) (119908119899119894) + 119873119862(119908119899119894) (41)
where119873119862(119909) is the (outward) normal cone of119862 at 119909 isin 119862This
follows that
0 = 120582119899119894V + 119908119899119894minus 119909119899+ 119906119899 (42)
where V isin 1205972119891119894(119909119899 119908119899119894) and 119906119899 isin 119873119862(119908119899119894) By the definition
of the normal cone119873119862 we have
⟨119908119899119894minus 119909119899 119910 minus 119908
119899119894⟩ ge 120582119899119894⟨V 119908119899119894minus 119910⟩ forall119910 isin 119862 (43)
Since 119891119894(119909119899 sdot) is subdifferentiable on 119862 by the well-known
Moreau-Rockafellar theorem [31] (also see [6]) for V isin
1205972119891119894(119909119899 119908119899119894) we have
119891119894(119909119899 119910) minus 119891
119894(119909119899 119908119899119894) ge ⟨V 119910 minus 119908
119899119894⟩ forall119910 isin 119862 (44)
Combining this with (43) we have
120582119899119894 (119891119894 (119909119899 119910) minus 119891119894 (119909119899 119908119899119894)) ge ⟨119908119899119894 minus 119909119899 119908119899119894 minus 119910⟩
forall119910 isin 119862
(45)
In particular we have
120582119899119895119894(119891119894(119909119899119895
119910) minus 119891119894(119909119899119895
119908119899119894)) ge ⟨119908
119899119895119894minus 119909119899119895
119908119899119895119894minus 119910⟩
forall119910 isin 119862
(46)
Since 119909119899119895
119909 it follows from (32) that 119908119899119895119894 119909 And thus
we have
119891119894(119909 119910) ge 0 forall119910 isin 119862 (47)
This implies that 119909 isin Sol(119891119894 119862) and hence 119909 isin Ω Since 119909⋆ =
ProjΩ(119868 minus 119860 + 120574ℎ)119909
⋆ and 119909 isin Ω we have
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
= lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= ⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909⟩ le 0
(48)
From Lemma 1 it follows that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1003817100381710038171003817(119868 minus 120579119899119860) (119910119899 minus 119909
⋆)1003817100381710038171003817
2
+ 2120579119899⟨120574ℎ (119909
119899) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899120574 ⟨ℎ (119909
119899) minus ℎ (119909
⋆) 119909119899+1
minus 119909⋆⟩
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899119896120574
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 120579119899119896120574 (1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2)
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le ((1 minus 120579119899120574)2+ 120579119899119896120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 1205791198991205741198961003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899 ⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1 minus 119909
⋆⟩
(49)
This implies that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1 minus 2120579119899120574 + (120579119899120574)
2+ 120579119899120574119896
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896
⟨120574ℎ (119909⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
= (1 minus2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+
(120579119899120574)2
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
times (
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩)
= (1 minus 120578119899)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 120578119899120575119899
(50)
Abstract and Applied Analysis 7
where
119872 = sup 1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2 119899 ge 0 120578
119899=2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896
120575119899=
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ119909⋆minus 119860119909⋆ 119909119899+1
minus 119909⋆⟩
(51)
It is easy to see that 120578119899 rarr 0 suminfin119899=1
120578119899 = infin andlim sup
119899rarrinfin120575119899le 0 Hence by Lemma 2 the sequence 119909
119899
converges strongly to 119909⋆ From (32) we have that 119908119899119894 and
119911119899119894 converge strongly to 119909⋆
Case 2 Assume that 119909119899minus119909⋆ is not a monotone sequence
Then we can define an integer sequence 120591(119899) for all 119899 ge 1198990
(for some large enough 1198990) by
120591 (119899) = max 119896 isin N 119896 le 119899 1003817100381710038171003817119909119896 minus 119909
⋆1003817100381710038171003817 lt1003817100381710038171003817119909119896+1 minus 119909
⋆1003817100381710038171003817
(52)
Clearly 120591 is a nondecreasing sequence such that 120591(119899) rarr infin
as 119899 rarr infin and for all 119899 ge 1198990
1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817 lt
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817 (53)
From (33) we obtain that
lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119911120591(119899)1198941003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119908120591(119899)1198941003817100381710038171003817
= lim119899rarrinfin
10038171003817100381710038171003817119909120591(119899) minus 119879 (119905120591
119899
) 119909120591(119899)
10038171003817100381710038171003817= 0
(54)
Following an argument similar to that in Case 1 we have
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817
2le (1 minus 120578
120591(119899))1003817100381710038171003817119909120591(119899) minus 119909
⋆1003817100381710038171003817
2+ 120578120591(119899)
120575120591(119899)
(55)
where 120578120591(119899)
rarr 0 suminfin119899=1
120578120591(119899)
= infin and lim sup119899rarrinfin
120575120591(119899)
le 0Hence by Lemma 2 we obtain lim
119899rarrinfin119909120591(119899)
minus 119909⋆ = 0 and
lim119899rarrinfin
119909120591(119899)+1
minus 119909⋆ = 0 Now Lemma 5 implies that
0 le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817 le max 1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le1003817100381710038171003817119909120591(119899)+1 minus 119909
⋆1003817100381710038171003817
(56)
Therefore 119909119899 converges strongly to 119909
⋆= Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ This completes the proof
4 Application
In this section we consider a particular Fan inequalitycorresponding to the function 119891 defined by the following forevery 119909 119910 isin 119862
119891 (119909 119910) = ⟨119865 (119909) 119910 minus 119909⟩ (57)
where 119865 119862 rarr 119867 Then we obtain the classical variationalinequality as follows
Find 119911 isin 119862 such that ⟨119865 (119911) 119910 minus 119911⟩ ge 0 forall119910 isin 119862 (58)
The set of solutions of this problem is denoted by 119881119868(119865 119862)In that particular case the solution 119910
119899of the minimization
problem
argmin 120582119899 119891 (119909119899 119910) +1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (59)
can be expressed as
119910119899= 119875119862 (119909119899 minus 120582119899119865 (119909119899)) (60)
Let 119865 be 119871-Lipschitz continuous on 119862 Then
119891 (119909 119910) + 119891 (119910 119911) minus 119891 (119909 119911) = ⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
119909 119910 119911 isin 119862
(61)
Therefore1003816100381610038161003816⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
1003816100381610038161003816 le 1198711003817100381710038171003817119909 minus 119910
1003817100381710038171003817
1003817100381710038171003817119910 minus 1199111003817100381710038171003817
le119871
2(1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2+1003817100381710038171003817119910 minus 119911
1003817100381710038171003817
2)
(62)
hence 119891 satisfies Lipschitz-type continuous condition with1198881 = 1198882 = 1198712
Using Theorem 7 we obtain the following convergencetheorem
Theorem 8 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le 119898119865119894is monotone and 119871-Lipschitz continuous on 119862 Let T =
119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909
0isin 119862 and let 119909
119899 119908119899119894 and 119911
119899119894 be
sequences generated by
119908119899119894 = 119875119862 (119909119899 minus 120582119899119894119865119894 (119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899= 120572119899119909119899+
119898
sum
119894=1
120573119899119894119911119899119894+ 120574119899119879 (119905119899) 119909119899+ 120578119899119878 (119905119899) 119909119899
119909119899+1 = 120579119899120574ℎ (119909119899) + (119868 minus 120579119899119860)119910119899 forall119899 ge 0
(63)
where 120572119899 + sum
119898
119894=1120573119899119894 + 120574119899 + 120578119899 = 1 and 120572119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579119899 satisfy the following conditions(i) lim
119899rarrinfin119905119899= infin
(ii) 120579119899 sub ]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub ]0 1119871[
(iv) 120572119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf119899120572119899120574119899 gt 0 lim inf119899120572119899120578119899 gt 0 andlim inf119899120573119899119894(1 minus 21205821198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (64)
8 Abstract and Applied Analysis
In [32] Baillon proved a strong mean convergence the-orem for nonexpansive mappings and it was generalizedin [33] It follows from the above proof that Theorems 7 isvalid for nonexpansivemappingsThuswe have the followingmean ergodic theorems for nonexpansive mappings in aHilbert space
Theorem 9 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le
119898 119865119894 is monotone and 119871-Lipschitz continuous on 119862 Let119879 and 119878 be two nonexpansive mappings on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(119879)⋂119865(119878) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909119899 119908119899119894 and 119911119899119894 be sequences generatedby 1199090isin 119862 and by
119908119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879119899119909119899 + 120578119899119878
119899119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(65)
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 120572
119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579
119899 satisfy the following conditions
(i) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(ii) 120582119899119894 sub [119886 119887] sub ]0 1119871[ where 119871 = max21198881119894 21198882119894
1 le 119894 le 119898(iii) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (66)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Therefore theauthors acknowledge with thanks DSR for technical andfinancial support
References
[1] K Fan ldquoA minimax inequality and applicationsrdquo in InequalityIII pp 103ndash113 Academic Press New York NY USA 1972
[2] P N Anh ldquoA hybrid extragradient method extended to fixedpoint problems and equilibrium problemsrdquo Optimization vol62 no 2 pp 271ndash283 2013
[3] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994
[4] L-C Ceng N Hadjisavvas and N-C Wong ldquoStrong conver-gence theorem by a hybrid extragradient-like approximationmethod for variational inequalities and fixed point problemsrdquoJournal of Global Optimization vol 46 no 4 pp 635ndash646 2010
[5] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964
[6] P Daniele F Giannessi and A Maugeri Equilibrium Problemsand Variational Models vol 68 Kluwer Academic NorwellMass USA 2003
[7] J-W Peng and J-C Yao ldquoSome new extragradient-likemethodsfor generalized equilibrium problems fixed point problemsand variational inequality problemsrdquo Optimization Methods ampSoftware vol 25 no 4ndash6 pp 677ndash698 2010
[8] D R Sahu N-C Wong and J-C Yao ldquoStrong convergencetheorems for semigroups of asymptotically nonexpansive map-pings inBanach spacesrdquoAbstract andAppliedAnalysis vol 2013Article ID 202095 8 pages 2013
[9] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[10] S Wang and B Guo ldquoNew iterative scheme with nonexpansivemappings for equilibrium problems and variational inequalityproblems in Hilbert spacesrdquo Journal of Computational andApplied Mathematics vol 233 no 10 pp 2620ndash2630 2010
[11] Y Yao Y-C Liou and Y-J Wu ldquoAn extragradient method formixed equilibrium problems and fixed point problemsrdquo FixedPoint Theory and Applications vol 2009 Article ID 632819 15pages 2009
[12] F E Browder ldquoNonexpansive nonlinear operators in a Banachspacerdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 54 pp 1041ndash1044 1965
[13] R Chen and Y Song ldquoConvergence to common fixed pointof nonexpansive semigroupsrdquo Journal of Computational andApplied Mathematics vol 200 no 2 pp 566ndash575 2007
[14] A Aleyner and Y Censor ldquoBest approximation to commonfixed points of a semigroup of nonexpansive operatorsrdquo Journalof Nonlinear and Convex Analysis vol 6 no 1 pp 137ndash151 2005
[15] H K Xu ldquoAn iterative approach to quadratic optimizationrdquoJournal of Optimization Theory and Applications vol 116 no 3pp 659ndash678 2003
[16] G Marino and H-K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006
[17] AMoudafi ldquoViscosity approximationmethods for fixed-pointsproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 241 no 1 pp 46ndash55 2000
[18] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[19] S Li L Li and Y Su ldquoGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert spacerdquo Nonlin-ear Analysis Theory Methods amp Applications vol 70 no 9 pp3065ndash3071 2009
Abstract and Applied Analysis 9
[20] D R Sahu N C Wong and J C Yao ldquoA unified hybriditerative method for solving variational inequalities involvinggeneralized pseudocontractive mappingsrdquo SIAM Journal onControl and Optimization vol 50 no 4 pp 2335ndash2354 2012
[21] N Shioji and W Takahashi ldquoStrong convergence theoremsfor asymptotically nonexpansive semigroups in Hilbert spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 34 no1 pp 87ndash99 1998
[22] P N Anh ldquoStrong convergence theorems for nonexpansivemappings and Ky Fan inequalitiesrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 303ndash320 2012
[23] L-C Ceng Q H Ansari and J-C Yao ldquoHybrid proximal-type and hybrid shrinking projection algorithms for equilib-rium problems maximal monotone operators and relativelynonexpansive mappingsrdquo Numerical Functional Analysis andOptimization vol 31 no 7-9 pp 763ndash797 2010
[24] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009
[25] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011
[26] L-C Ceng S-M Guu and J-C Yao ldquoHybrid iterative methodfor finding common solutions of generalizedmixed equilibriumand fixed point problemsrdquo Fixed PointTheory and Applicationsvol 2012 article 92 2012
[27] M Eslamian ldquoConvergence theorems for nonspreading map-pings andnonexpansivemultivaluedmappings and equilibriumproblemsrdquo Optimization Letters vol 7 no 3 pp 547ndash557 2013
[28] M Eslamian ldquoHybrid method for equilibrium problems andfixed point problems of finite families of nonexpansive semi-groupsrdquo RACSAM vol 107 pp 299ndash307 2013
[29] M Eslamian and A Abkar ldquoOne-step iterative process fora finite family of multivalued mappingsrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 105ndash111 2011
[30] P-E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[31] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976
[32] J-B Baillon ldquoUn theoreme de type ergodique pour les contrac-tions non lineaires dans un espace de Hilbertrdquo Comptes Rendusde lrsquoAcademie des Sciences vol 280 no 22 pp A1511ndashA15141975
[33] W Kaczor T Kuczumow and S Reich ldquoA mean ergodictheorem for nonlinear semigroups which are asymptoticallynonexpansive in the intermediate senserdquo Journal of Mathemat-ical Analysis and Applications vol 246 no 1 pp 1ndash27 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
le 120572119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+
119898
sum
119894=1
120573119899119894
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2+ 120574119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 120578119899
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus 120573119899119896 (1 minus 21205821198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2minus 120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
=1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus 120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus 120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2minus 120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus 1205721198991205781198991003817100381710038171003817119909119899 minus 119878 (119905119899) 119909119899
1003817100381710038171003817
2
(27)
We now compute1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2
=1003817100381710038171003817120579119899 (120574ℎ (119909119899) minus 119860119909
⋆) + (119868 minus 120579
119899119860) (119910119899minus 119909⋆)1003817100381710038171003817
2
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119910119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119910119899 minus 119909⋆1003817100381710038171003817
le 1205792
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2+ (1 minus 120579
119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899(1 minus 120579
119899120574)1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
minus (1 minus 120579119899120574)2120573119899119896(1 minus 2120582
1198991198961198881119896)1003817100381710038171003817119909119899 minus 119908119899119896
1003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120574119899
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
2
minus (1 minus 120579119899120574)2120572119899120578119899
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817
2
(28)
Therefore
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2
le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2minus1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899(1 minus 120579
119899120574)
times1003817100381710038171003817120574ℎ (119909119899) minus 119860119909
⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817 + 120579
2
119899
1003817100381710038171003817120574ℎ (119909119899) minus 119860119909⋆1003817100381710038171003817
2
(29)
In order to prove that 119909119899rarr 119909⋆ as 119899 rarr infin we consider the
following two cases
Case 1 Assume that 119909119899minus 119909⋆ is a monotone sequence
In other words for large enough 1198990 119909119899minus 119909⋆119899ge1198990
iseither nondecreasing or nonincreasing Since 119909119899 minus 119909
⋆ is
bounded it is convergent Since lim119899rarrinfin
120579119899= 0 and ℎ(119909
119899)
and 119909119899 are bounded from (29) we have
lim119899rarrinfin
(1 minus 120579119899120574)2120572119899120573119899119896
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817
2= 0 (30)
and by assumption we get
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199111198991198961003817100381710038171003817 = 0 1 le 119896 le 119898 (31)
By similar argument we can obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199081198991198961003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (119905119899) 1199091198991003817100381710038171003817
= lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (119905119899) 1199091198991003817100381710038171003817 = 0
(32)
Further for all ℎ ge 0 and 119899 ge 0 we see that1003817100381710038171003817119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817 +1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 119909119899
1003817100381710038171003817
+1003817100381710038171003817119879 (ℎ) 119879 (119905119899) 119909119899 minus 119879 (ℎ) 119909119899
1003817100381710038171003817
le 21003817100381710038171003817119909119899 minus 119879 (119905119899) 119909119899
1003817100381710038171003817
+ sup119909isin119909119899
1003817100381710038171003817119879 (119905119899) 119909119899 minus 119879 (ℎ) 119879 (119905119899) 1199091198991003817100381710038171003817
(33)
Since 119879(119905) is uar nonexpansive semigroup andlim119899rarrinfin
119905119899= infin we have
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119879 (ℎ) 1199091198991003817100381710038171003817 = 0 (34)
Similarly for all ℎ ge 0 and 119899 ge 0 we obtain that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 119878 (ℎ) 1199091198991003817100381710038171003817 = 0 (35)
Next we show that
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩ le 0 (36)
To show this inequality we choose a subsequence 119909119899119894
of 119909119899
such that
lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
(37)
Since 119909119899119894
is bounded there exists a subsequence 119909119899119894119895
of119909119899119894
which converges weakly to 119909 Without loss of generalitywe can assume that 119909119899
119895
119909 Consider
100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119879 (119905) 119909119899
119895
minus 119879 (119905) 119909100381710038171003817100381710038171003817
le100381710038171003817100381710038171003817119909119899119895
minus 119879 (119905) 119909119899119895
100381710038171003817100381710038171003817+100381710038171003817100381710038171003817119909119899119895
minus 119909100381710038171003817100381710038171003817
(38)
Thus we have
lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 119879 (119905) 11990910038171003817100381710038171003817le lim sup119899rarrinfin
10038171003817100381710038171003817119909119899119894
minus 11990910038171003817100381710038171003817 (39)
By the Opial property of the Hilbert space 119867 we obtain119879(119905)119909 = 119909 for all 119905 ge 0 Similarly we have that 119878(119905)119909 = 119909
for all 119905 ge 0 This implies that 119909 isin 119865(T)⋂119865(S) Now weshow that 119909 isin Sol(119891119894 119862) For each 1 le 119894 le 119898 since 119891119894(119909 sdot) isconvex on 119862 for each 119909 isin 119862 we see that
119908119899119894= argmin 120582
119899119894119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (40)
6 Abstract and Applied Analysis
if and only if
0 isin 1205972(119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2) (119908119899119894) + 119873119862(119908119899119894) (41)
where119873119862(119909) is the (outward) normal cone of119862 at 119909 isin 119862This
follows that
0 = 120582119899119894V + 119908119899119894minus 119909119899+ 119906119899 (42)
where V isin 1205972119891119894(119909119899 119908119899119894) and 119906119899 isin 119873119862(119908119899119894) By the definition
of the normal cone119873119862 we have
⟨119908119899119894minus 119909119899 119910 minus 119908
119899119894⟩ ge 120582119899119894⟨V 119908119899119894minus 119910⟩ forall119910 isin 119862 (43)
Since 119891119894(119909119899 sdot) is subdifferentiable on 119862 by the well-known
Moreau-Rockafellar theorem [31] (also see [6]) for V isin
1205972119891119894(119909119899 119908119899119894) we have
119891119894(119909119899 119910) minus 119891
119894(119909119899 119908119899119894) ge ⟨V 119910 minus 119908
119899119894⟩ forall119910 isin 119862 (44)
Combining this with (43) we have
120582119899119894 (119891119894 (119909119899 119910) minus 119891119894 (119909119899 119908119899119894)) ge ⟨119908119899119894 minus 119909119899 119908119899119894 minus 119910⟩
forall119910 isin 119862
(45)
In particular we have
120582119899119895119894(119891119894(119909119899119895
119910) minus 119891119894(119909119899119895
119908119899119894)) ge ⟨119908
119899119895119894minus 119909119899119895
119908119899119895119894minus 119910⟩
forall119910 isin 119862
(46)
Since 119909119899119895
119909 it follows from (32) that 119908119899119895119894 119909 And thus
we have
119891119894(119909 119910) ge 0 forall119910 isin 119862 (47)
This implies that 119909 isin Sol(119891119894 119862) and hence 119909 isin Ω Since 119909⋆ =
ProjΩ(119868 minus 119860 + 120574ℎ)119909
⋆ and 119909 isin Ω we have
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
= lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= ⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909⟩ le 0
(48)
From Lemma 1 it follows that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1003817100381710038171003817(119868 minus 120579119899119860) (119910119899 minus 119909
⋆)1003817100381710038171003817
2
+ 2120579119899⟨120574ℎ (119909
119899) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899120574 ⟨ℎ (119909
119899) minus ℎ (119909
⋆) 119909119899+1
minus 119909⋆⟩
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899119896120574
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 120579119899119896120574 (1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2)
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le ((1 minus 120579119899120574)2+ 120579119899119896120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 1205791198991205741198961003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899 ⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1 minus 119909
⋆⟩
(49)
This implies that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1 minus 2120579119899120574 + (120579119899120574)
2+ 120579119899120574119896
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896
⟨120574ℎ (119909⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
= (1 minus2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+
(120579119899120574)2
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
times (
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩)
= (1 minus 120578119899)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 120578119899120575119899
(50)
Abstract and Applied Analysis 7
where
119872 = sup 1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2 119899 ge 0 120578
119899=2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896
120575119899=
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ119909⋆minus 119860119909⋆ 119909119899+1
minus 119909⋆⟩
(51)
It is easy to see that 120578119899 rarr 0 suminfin119899=1
120578119899 = infin andlim sup
119899rarrinfin120575119899le 0 Hence by Lemma 2 the sequence 119909
119899
converges strongly to 119909⋆ From (32) we have that 119908119899119894 and
119911119899119894 converge strongly to 119909⋆
Case 2 Assume that 119909119899minus119909⋆ is not a monotone sequence
Then we can define an integer sequence 120591(119899) for all 119899 ge 1198990
(for some large enough 1198990) by
120591 (119899) = max 119896 isin N 119896 le 119899 1003817100381710038171003817119909119896 minus 119909
⋆1003817100381710038171003817 lt1003817100381710038171003817119909119896+1 minus 119909
⋆1003817100381710038171003817
(52)
Clearly 120591 is a nondecreasing sequence such that 120591(119899) rarr infin
as 119899 rarr infin and for all 119899 ge 1198990
1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817 lt
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817 (53)
From (33) we obtain that
lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119911120591(119899)1198941003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119908120591(119899)1198941003817100381710038171003817
= lim119899rarrinfin
10038171003817100381710038171003817119909120591(119899) minus 119879 (119905120591
119899
) 119909120591(119899)
10038171003817100381710038171003817= 0
(54)
Following an argument similar to that in Case 1 we have
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817
2le (1 minus 120578
120591(119899))1003817100381710038171003817119909120591(119899) minus 119909
⋆1003817100381710038171003817
2+ 120578120591(119899)
120575120591(119899)
(55)
where 120578120591(119899)
rarr 0 suminfin119899=1
120578120591(119899)
= infin and lim sup119899rarrinfin
120575120591(119899)
le 0Hence by Lemma 2 we obtain lim
119899rarrinfin119909120591(119899)
minus 119909⋆ = 0 and
lim119899rarrinfin
119909120591(119899)+1
minus 119909⋆ = 0 Now Lemma 5 implies that
0 le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817 le max 1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le1003817100381710038171003817119909120591(119899)+1 minus 119909
⋆1003817100381710038171003817
(56)
Therefore 119909119899 converges strongly to 119909
⋆= Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ This completes the proof
4 Application
In this section we consider a particular Fan inequalitycorresponding to the function 119891 defined by the following forevery 119909 119910 isin 119862
119891 (119909 119910) = ⟨119865 (119909) 119910 minus 119909⟩ (57)
where 119865 119862 rarr 119867 Then we obtain the classical variationalinequality as follows
Find 119911 isin 119862 such that ⟨119865 (119911) 119910 minus 119911⟩ ge 0 forall119910 isin 119862 (58)
The set of solutions of this problem is denoted by 119881119868(119865 119862)In that particular case the solution 119910
119899of the minimization
problem
argmin 120582119899 119891 (119909119899 119910) +1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (59)
can be expressed as
119910119899= 119875119862 (119909119899 minus 120582119899119865 (119909119899)) (60)
Let 119865 be 119871-Lipschitz continuous on 119862 Then
119891 (119909 119910) + 119891 (119910 119911) minus 119891 (119909 119911) = ⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
119909 119910 119911 isin 119862
(61)
Therefore1003816100381610038161003816⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
1003816100381610038161003816 le 1198711003817100381710038171003817119909 minus 119910
1003817100381710038171003817
1003817100381710038171003817119910 minus 1199111003817100381710038171003817
le119871
2(1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2+1003817100381710038171003817119910 minus 119911
1003817100381710038171003817
2)
(62)
hence 119891 satisfies Lipschitz-type continuous condition with1198881 = 1198882 = 1198712
Using Theorem 7 we obtain the following convergencetheorem
Theorem 8 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le 119898119865119894is monotone and 119871-Lipschitz continuous on 119862 Let T =
119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909
0isin 119862 and let 119909
119899 119908119899119894 and 119911
119899119894 be
sequences generated by
119908119899119894 = 119875119862 (119909119899 minus 120582119899119894119865119894 (119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899= 120572119899119909119899+
119898
sum
119894=1
120573119899119894119911119899119894+ 120574119899119879 (119905119899) 119909119899+ 120578119899119878 (119905119899) 119909119899
119909119899+1 = 120579119899120574ℎ (119909119899) + (119868 minus 120579119899119860)119910119899 forall119899 ge 0
(63)
where 120572119899 + sum
119898
119894=1120573119899119894 + 120574119899 + 120578119899 = 1 and 120572119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579119899 satisfy the following conditions(i) lim
119899rarrinfin119905119899= infin
(ii) 120579119899 sub ]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub ]0 1119871[
(iv) 120572119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf119899120572119899120574119899 gt 0 lim inf119899120572119899120578119899 gt 0 andlim inf119899120573119899119894(1 minus 21205821198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (64)
8 Abstract and Applied Analysis
In [32] Baillon proved a strong mean convergence the-orem for nonexpansive mappings and it was generalizedin [33] It follows from the above proof that Theorems 7 isvalid for nonexpansivemappingsThuswe have the followingmean ergodic theorems for nonexpansive mappings in aHilbert space
Theorem 9 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le
119898 119865119894 is monotone and 119871-Lipschitz continuous on 119862 Let119879 and 119878 be two nonexpansive mappings on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(119879)⋂119865(119878) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909119899 119908119899119894 and 119911119899119894 be sequences generatedby 1199090isin 119862 and by
119908119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879119899119909119899 + 120578119899119878
119899119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(65)
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 120572
119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579
119899 satisfy the following conditions
(i) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(ii) 120582119899119894 sub [119886 119887] sub ]0 1119871[ where 119871 = max21198881119894 21198882119894
1 le 119894 le 119898(iii) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (66)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Therefore theauthors acknowledge with thanks DSR for technical andfinancial support
References
[1] K Fan ldquoA minimax inequality and applicationsrdquo in InequalityIII pp 103ndash113 Academic Press New York NY USA 1972
[2] P N Anh ldquoA hybrid extragradient method extended to fixedpoint problems and equilibrium problemsrdquo Optimization vol62 no 2 pp 271ndash283 2013
[3] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994
[4] L-C Ceng N Hadjisavvas and N-C Wong ldquoStrong conver-gence theorem by a hybrid extragradient-like approximationmethod for variational inequalities and fixed point problemsrdquoJournal of Global Optimization vol 46 no 4 pp 635ndash646 2010
[5] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964
[6] P Daniele F Giannessi and A Maugeri Equilibrium Problemsand Variational Models vol 68 Kluwer Academic NorwellMass USA 2003
[7] J-W Peng and J-C Yao ldquoSome new extragradient-likemethodsfor generalized equilibrium problems fixed point problemsand variational inequality problemsrdquo Optimization Methods ampSoftware vol 25 no 4ndash6 pp 677ndash698 2010
[8] D R Sahu N-C Wong and J-C Yao ldquoStrong convergencetheorems for semigroups of asymptotically nonexpansive map-pings inBanach spacesrdquoAbstract andAppliedAnalysis vol 2013Article ID 202095 8 pages 2013
[9] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[10] S Wang and B Guo ldquoNew iterative scheme with nonexpansivemappings for equilibrium problems and variational inequalityproblems in Hilbert spacesrdquo Journal of Computational andApplied Mathematics vol 233 no 10 pp 2620ndash2630 2010
[11] Y Yao Y-C Liou and Y-J Wu ldquoAn extragradient method formixed equilibrium problems and fixed point problemsrdquo FixedPoint Theory and Applications vol 2009 Article ID 632819 15pages 2009
[12] F E Browder ldquoNonexpansive nonlinear operators in a Banachspacerdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 54 pp 1041ndash1044 1965
[13] R Chen and Y Song ldquoConvergence to common fixed pointof nonexpansive semigroupsrdquo Journal of Computational andApplied Mathematics vol 200 no 2 pp 566ndash575 2007
[14] A Aleyner and Y Censor ldquoBest approximation to commonfixed points of a semigroup of nonexpansive operatorsrdquo Journalof Nonlinear and Convex Analysis vol 6 no 1 pp 137ndash151 2005
[15] H K Xu ldquoAn iterative approach to quadratic optimizationrdquoJournal of Optimization Theory and Applications vol 116 no 3pp 659ndash678 2003
[16] G Marino and H-K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006
[17] AMoudafi ldquoViscosity approximationmethods for fixed-pointsproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 241 no 1 pp 46ndash55 2000
[18] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[19] S Li L Li and Y Su ldquoGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert spacerdquo Nonlin-ear Analysis Theory Methods amp Applications vol 70 no 9 pp3065ndash3071 2009
Abstract and Applied Analysis 9
[20] D R Sahu N C Wong and J C Yao ldquoA unified hybriditerative method for solving variational inequalities involvinggeneralized pseudocontractive mappingsrdquo SIAM Journal onControl and Optimization vol 50 no 4 pp 2335ndash2354 2012
[21] N Shioji and W Takahashi ldquoStrong convergence theoremsfor asymptotically nonexpansive semigroups in Hilbert spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 34 no1 pp 87ndash99 1998
[22] P N Anh ldquoStrong convergence theorems for nonexpansivemappings and Ky Fan inequalitiesrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 303ndash320 2012
[23] L-C Ceng Q H Ansari and J-C Yao ldquoHybrid proximal-type and hybrid shrinking projection algorithms for equilib-rium problems maximal monotone operators and relativelynonexpansive mappingsrdquo Numerical Functional Analysis andOptimization vol 31 no 7-9 pp 763ndash797 2010
[24] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009
[25] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011
[26] L-C Ceng S-M Guu and J-C Yao ldquoHybrid iterative methodfor finding common solutions of generalizedmixed equilibriumand fixed point problemsrdquo Fixed PointTheory and Applicationsvol 2012 article 92 2012
[27] M Eslamian ldquoConvergence theorems for nonspreading map-pings andnonexpansivemultivaluedmappings and equilibriumproblemsrdquo Optimization Letters vol 7 no 3 pp 547ndash557 2013
[28] M Eslamian ldquoHybrid method for equilibrium problems andfixed point problems of finite families of nonexpansive semi-groupsrdquo RACSAM vol 107 pp 299ndash307 2013
[29] M Eslamian and A Abkar ldquoOne-step iterative process fora finite family of multivalued mappingsrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 105ndash111 2011
[30] P-E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[31] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976
[32] J-B Baillon ldquoUn theoreme de type ergodique pour les contrac-tions non lineaires dans un espace de Hilbertrdquo Comptes Rendusde lrsquoAcademie des Sciences vol 280 no 22 pp A1511ndashA15141975
[33] W Kaczor T Kuczumow and S Reich ldquoA mean ergodictheorem for nonlinear semigroups which are asymptoticallynonexpansive in the intermediate senserdquo Journal of Mathemat-ical Analysis and Applications vol 246 no 1 pp 1ndash27 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
if and only if
0 isin 1205972(119891119894(119909119899 119910) +
1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2) (119908119899119894) + 119873119862(119908119899119894) (41)
where119873119862(119909) is the (outward) normal cone of119862 at 119909 isin 119862This
follows that
0 = 120582119899119894V + 119908119899119894minus 119909119899+ 119906119899 (42)
where V isin 1205972119891119894(119909119899 119908119899119894) and 119906119899 isin 119873119862(119908119899119894) By the definition
of the normal cone119873119862 we have
⟨119908119899119894minus 119909119899 119910 minus 119908
119899119894⟩ ge 120582119899119894⟨V 119908119899119894minus 119910⟩ forall119910 isin 119862 (43)
Since 119891119894(119909119899 sdot) is subdifferentiable on 119862 by the well-known
Moreau-Rockafellar theorem [31] (also see [6]) for V isin
1205972119891119894(119909119899 119908119899119894) we have
119891119894(119909119899 119910) minus 119891
119894(119909119899 119908119899119894) ge ⟨V 119910 minus 119908
119899119894⟩ forall119910 isin 119862 (44)
Combining this with (43) we have
120582119899119894 (119891119894 (119909119899 119910) minus 119891119894 (119909119899 119908119899119894)) ge ⟨119908119899119894 minus 119909119899 119908119899119894 minus 119910⟩
forall119910 isin 119862
(45)
In particular we have
120582119899119895119894(119891119894(119909119899119895
119910) minus 119891119894(119909119899119895
119908119899119894)) ge ⟨119908
119899119895119894minus 119909119899119895
119908119899119895119894minus 119910⟩
forall119910 isin 119862
(46)
Since 119909119899119895
119909 it follows from (32) that 119908119899119895119894 119909 And thus
we have
119891119894(119909 119910) ge 0 forall119910 isin 119862 (47)
This implies that 119909 isin Sol(119891119894 119862) and hence 119909 isin Ω Since 119909⋆ =
ProjΩ(119868 minus 119860 + 120574ℎ)119909
⋆ and 119909 isin Ω we have
lim sup119899rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899⟩
= lim119894rarrinfin
⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909119899119894
⟩
= ⟨(119860 minus 120574ℎ) 119909⋆ 119909⋆minus 119909⟩ le 0
(48)
From Lemma 1 it follows that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1003817100381710038171003817(119868 minus 120579119899119860) (119910119899 minus 119909
⋆)1003817100381710038171003817
2
+ 2120579119899⟨120574ℎ (119909
119899) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 2120579119899120574 ⟨ℎ (119909
119899) minus ℎ (119909
⋆) 119909119899+1
minus 119909⋆⟩
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899119896120574
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus 120579119899120574)21003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2
+ 120579119899119896120574 (1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+1003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2)
+ 2120579119899⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le ((1 minus 120579119899120574)2+ 120579119899119896120574)
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+ 1205791198991205741198961003817100381710038171003817119909119899+1 minus 119909
⋆1003817100381710038171003817
2+ 2120579119899 ⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1 minus 119909
⋆⟩
(49)
This implies that
1003817100381710038171003817119909119899+1 minus 119909⋆1003817100381710038171003817
2
le1 minus 2120579119899120574 + (120579119899120574)
2+ 120579119899120574119896
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896
⟨120574ℎ (119909⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
= (1 minus2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+
(120579119899120574)2
1 minus 120579119899120574119896
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2
+2120579119899
1 minus 120579119899120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩
le (1 minus2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+2 (120574 minus 120574119896) 120579119899
1 minus 120579119899120574119896
times (
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ (119909
⋆) minus 119860119909
⋆ 119909119899+1
minus 119909⋆⟩)
= (1 minus 120578119899)1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817
2+ 120578119899120575119899
(50)
Abstract and Applied Analysis 7
where
119872 = sup 1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2 119899 ge 0 120578
119899=2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896
120575119899=
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ119909⋆minus 119860119909⋆ 119909119899+1
minus 119909⋆⟩
(51)
It is easy to see that 120578119899 rarr 0 suminfin119899=1
120578119899 = infin andlim sup
119899rarrinfin120575119899le 0 Hence by Lemma 2 the sequence 119909
119899
converges strongly to 119909⋆ From (32) we have that 119908119899119894 and
119911119899119894 converge strongly to 119909⋆
Case 2 Assume that 119909119899minus119909⋆ is not a monotone sequence
Then we can define an integer sequence 120591(119899) for all 119899 ge 1198990
(for some large enough 1198990) by
120591 (119899) = max 119896 isin N 119896 le 119899 1003817100381710038171003817119909119896 minus 119909
⋆1003817100381710038171003817 lt1003817100381710038171003817119909119896+1 minus 119909
⋆1003817100381710038171003817
(52)
Clearly 120591 is a nondecreasing sequence such that 120591(119899) rarr infin
as 119899 rarr infin and for all 119899 ge 1198990
1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817 lt
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817 (53)
From (33) we obtain that
lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119911120591(119899)1198941003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119908120591(119899)1198941003817100381710038171003817
= lim119899rarrinfin
10038171003817100381710038171003817119909120591(119899) minus 119879 (119905120591
119899
) 119909120591(119899)
10038171003817100381710038171003817= 0
(54)
Following an argument similar to that in Case 1 we have
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817
2le (1 minus 120578
120591(119899))1003817100381710038171003817119909120591(119899) minus 119909
⋆1003817100381710038171003817
2+ 120578120591(119899)
120575120591(119899)
(55)
where 120578120591(119899)
rarr 0 suminfin119899=1
120578120591(119899)
= infin and lim sup119899rarrinfin
120575120591(119899)
le 0Hence by Lemma 2 we obtain lim
119899rarrinfin119909120591(119899)
minus 119909⋆ = 0 and
lim119899rarrinfin
119909120591(119899)+1
minus 119909⋆ = 0 Now Lemma 5 implies that
0 le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817 le max 1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le1003817100381710038171003817119909120591(119899)+1 minus 119909
⋆1003817100381710038171003817
(56)
Therefore 119909119899 converges strongly to 119909
⋆= Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ This completes the proof
4 Application
In this section we consider a particular Fan inequalitycorresponding to the function 119891 defined by the following forevery 119909 119910 isin 119862
119891 (119909 119910) = ⟨119865 (119909) 119910 minus 119909⟩ (57)
where 119865 119862 rarr 119867 Then we obtain the classical variationalinequality as follows
Find 119911 isin 119862 such that ⟨119865 (119911) 119910 minus 119911⟩ ge 0 forall119910 isin 119862 (58)
The set of solutions of this problem is denoted by 119881119868(119865 119862)In that particular case the solution 119910
119899of the minimization
problem
argmin 120582119899 119891 (119909119899 119910) +1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (59)
can be expressed as
119910119899= 119875119862 (119909119899 minus 120582119899119865 (119909119899)) (60)
Let 119865 be 119871-Lipschitz continuous on 119862 Then
119891 (119909 119910) + 119891 (119910 119911) minus 119891 (119909 119911) = ⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
119909 119910 119911 isin 119862
(61)
Therefore1003816100381610038161003816⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
1003816100381610038161003816 le 1198711003817100381710038171003817119909 minus 119910
1003817100381710038171003817
1003817100381710038171003817119910 minus 1199111003817100381710038171003817
le119871
2(1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2+1003817100381710038171003817119910 minus 119911
1003817100381710038171003817
2)
(62)
hence 119891 satisfies Lipschitz-type continuous condition with1198881 = 1198882 = 1198712
Using Theorem 7 we obtain the following convergencetheorem
Theorem 8 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le 119898119865119894is monotone and 119871-Lipschitz continuous on 119862 Let T =
119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909
0isin 119862 and let 119909
119899 119908119899119894 and 119911
119899119894 be
sequences generated by
119908119899119894 = 119875119862 (119909119899 minus 120582119899119894119865119894 (119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899= 120572119899119909119899+
119898
sum
119894=1
120573119899119894119911119899119894+ 120574119899119879 (119905119899) 119909119899+ 120578119899119878 (119905119899) 119909119899
119909119899+1 = 120579119899120574ℎ (119909119899) + (119868 minus 120579119899119860)119910119899 forall119899 ge 0
(63)
where 120572119899 + sum
119898
119894=1120573119899119894 + 120574119899 + 120578119899 = 1 and 120572119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579119899 satisfy the following conditions(i) lim
119899rarrinfin119905119899= infin
(ii) 120579119899 sub ]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub ]0 1119871[
(iv) 120572119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf119899120572119899120574119899 gt 0 lim inf119899120572119899120578119899 gt 0 andlim inf119899120573119899119894(1 minus 21205821198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (64)
8 Abstract and Applied Analysis
In [32] Baillon proved a strong mean convergence the-orem for nonexpansive mappings and it was generalizedin [33] It follows from the above proof that Theorems 7 isvalid for nonexpansivemappingsThuswe have the followingmean ergodic theorems for nonexpansive mappings in aHilbert space
Theorem 9 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le
119898 119865119894 is monotone and 119871-Lipschitz continuous on 119862 Let119879 and 119878 be two nonexpansive mappings on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(119879)⋂119865(119878) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909119899 119908119899119894 and 119911119899119894 be sequences generatedby 1199090isin 119862 and by
119908119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879119899119909119899 + 120578119899119878
119899119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(65)
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 120572
119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579
119899 satisfy the following conditions
(i) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(ii) 120582119899119894 sub [119886 119887] sub ]0 1119871[ where 119871 = max21198881119894 21198882119894
1 le 119894 le 119898(iii) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (66)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Therefore theauthors acknowledge with thanks DSR for technical andfinancial support
References
[1] K Fan ldquoA minimax inequality and applicationsrdquo in InequalityIII pp 103ndash113 Academic Press New York NY USA 1972
[2] P N Anh ldquoA hybrid extragradient method extended to fixedpoint problems and equilibrium problemsrdquo Optimization vol62 no 2 pp 271ndash283 2013
[3] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994
[4] L-C Ceng N Hadjisavvas and N-C Wong ldquoStrong conver-gence theorem by a hybrid extragradient-like approximationmethod for variational inequalities and fixed point problemsrdquoJournal of Global Optimization vol 46 no 4 pp 635ndash646 2010
[5] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964
[6] P Daniele F Giannessi and A Maugeri Equilibrium Problemsand Variational Models vol 68 Kluwer Academic NorwellMass USA 2003
[7] J-W Peng and J-C Yao ldquoSome new extragradient-likemethodsfor generalized equilibrium problems fixed point problemsand variational inequality problemsrdquo Optimization Methods ampSoftware vol 25 no 4ndash6 pp 677ndash698 2010
[8] D R Sahu N-C Wong and J-C Yao ldquoStrong convergencetheorems for semigroups of asymptotically nonexpansive map-pings inBanach spacesrdquoAbstract andAppliedAnalysis vol 2013Article ID 202095 8 pages 2013
[9] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[10] S Wang and B Guo ldquoNew iterative scheme with nonexpansivemappings for equilibrium problems and variational inequalityproblems in Hilbert spacesrdquo Journal of Computational andApplied Mathematics vol 233 no 10 pp 2620ndash2630 2010
[11] Y Yao Y-C Liou and Y-J Wu ldquoAn extragradient method formixed equilibrium problems and fixed point problemsrdquo FixedPoint Theory and Applications vol 2009 Article ID 632819 15pages 2009
[12] F E Browder ldquoNonexpansive nonlinear operators in a Banachspacerdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 54 pp 1041ndash1044 1965
[13] R Chen and Y Song ldquoConvergence to common fixed pointof nonexpansive semigroupsrdquo Journal of Computational andApplied Mathematics vol 200 no 2 pp 566ndash575 2007
[14] A Aleyner and Y Censor ldquoBest approximation to commonfixed points of a semigroup of nonexpansive operatorsrdquo Journalof Nonlinear and Convex Analysis vol 6 no 1 pp 137ndash151 2005
[15] H K Xu ldquoAn iterative approach to quadratic optimizationrdquoJournal of Optimization Theory and Applications vol 116 no 3pp 659ndash678 2003
[16] G Marino and H-K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006
[17] AMoudafi ldquoViscosity approximationmethods for fixed-pointsproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 241 no 1 pp 46ndash55 2000
[18] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[19] S Li L Li and Y Su ldquoGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert spacerdquo Nonlin-ear Analysis Theory Methods amp Applications vol 70 no 9 pp3065ndash3071 2009
Abstract and Applied Analysis 9
[20] D R Sahu N C Wong and J C Yao ldquoA unified hybriditerative method for solving variational inequalities involvinggeneralized pseudocontractive mappingsrdquo SIAM Journal onControl and Optimization vol 50 no 4 pp 2335ndash2354 2012
[21] N Shioji and W Takahashi ldquoStrong convergence theoremsfor asymptotically nonexpansive semigroups in Hilbert spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 34 no1 pp 87ndash99 1998
[22] P N Anh ldquoStrong convergence theorems for nonexpansivemappings and Ky Fan inequalitiesrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 303ndash320 2012
[23] L-C Ceng Q H Ansari and J-C Yao ldquoHybrid proximal-type and hybrid shrinking projection algorithms for equilib-rium problems maximal monotone operators and relativelynonexpansive mappingsrdquo Numerical Functional Analysis andOptimization vol 31 no 7-9 pp 763ndash797 2010
[24] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009
[25] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011
[26] L-C Ceng S-M Guu and J-C Yao ldquoHybrid iterative methodfor finding common solutions of generalizedmixed equilibriumand fixed point problemsrdquo Fixed PointTheory and Applicationsvol 2012 article 92 2012
[27] M Eslamian ldquoConvergence theorems for nonspreading map-pings andnonexpansivemultivaluedmappings and equilibriumproblemsrdquo Optimization Letters vol 7 no 3 pp 547ndash557 2013
[28] M Eslamian ldquoHybrid method for equilibrium problems andfixed point problems of finite families of nonexpansive semi-groupsrdquo RACSAM vol 107 pp 299ndash307 2013
[29] M Eslamian and A Abkar ldquoOne-step iterative process fora finite family of multivalued mappingsrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 105ndash111 2011
[30] P-E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[31] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976
[32] J-B Baillon ldquoUn theoreme de type ergodique pour les contrac-tions non lineaires dans un espace de Hilbertrdquo Comptes Rendusde lrsquoAcademie des Sciences vol 280 no 22 pp A1511ndashA15141975
[33] W Kaczor T Kuczumow and S Reich ldquoA mean ergodictheorem for nonlinear semigroups which are asymptoticallynonexpansive in the intermediate senserdquo Journal of Mathemat-ical Analysis and Applications vol 246 no 1 pp 1ndash27 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
where
119872 = sup 1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
2 119899 ge 0 120578
119899=2 (120574 minus 120574119896) 120579
119899
1 minus 120579119899120574119896
120575119899=
(1205791198991205742)119872
2 (120574 minus 120574119896)+
1
120574 minus 120574119896⟨120574ℎ119909⋆minus 119860119909⋆ 119909119899+1
minus 119909⋆⟩
(51)
It is easy to see that 120578119899 rarr 0 suminfin119899=1
120578119899 = infin andlim sup
119899rarrinfin120575119899le 0 Hence by Lemma 2 the sequence 119909
119899
converges strongly to 119909⋆ From (32) we have that 119908119899119894 and
119911119899119894 converge strongly to 119909⋆
Case 2 Assume that 119909119899minus119909⋆ is not a monotone sequence
Then we can define an integer sequence 120591(119899) for all 119899 ge 1198990
(for some large enough 1198990) by
120591 (119899) = max 119896 isin N 119896 le 119899 1003817100381710038171003817119909119896 minus 119909
⋆1003817100381710038171003817 lt1003817100381710038171003817119909119896+1 minus 119909
⋆1003817100381710038171003817
(52)
Clearly 120591 is a nondecreasing sequence such that 120591(119899) rarr infin
as 119899 rarr infin and for all 119899 ge 1198990
1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817 lt
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817 (53)
From (33) we obtain that
lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119911120591(119899)1198941003817100381710038171003817 = lim119899rarrinfin
1003817100381710038171003817119909120591(119899) minus 119908120591(119899)1198941003817100381710038171003817
= lim119899rarrinfin
10038171003817100381710038171003817119909120591(119899) minus 119879 (119905120591
119899
) 119909120591(119899)
10038171003817100381710038171003817= 0
(54)
Following an argument similar to that in Case 1 we have
1003817100381710038171003817119909120591(119899)+1 minus 119909⋆1003817100381710038171003817
2le (1 minus 120578
120591(119899))1003817100381710038171003817119909120591(119899) minus 119909
⋆1003817100381710038171003817
2+ 120578120591(119899)
120575120591(119899)
(55)
where 120578120591(119899)
rarr 0 suminfin119899=1
120578120591(119899)
= infin and lim sup119899rarrinfin
120575120591(119899)
le 0Hence by Lemma 2 we obtain lim
119899rarrinfin119909120591(119899)
minus 119909⋆ = 0 and
lim119899rarrinfin
119909120591(119899)+1
minus 119909⋆ = 0 Now Lemma 5 implies that
0 le1003817100381710038171003817119909119899 minus 119909
⋆1003817100381710038171003817 le max 1003817100381710038171003817119909120591(119899) minus 119909⋆1003817100381710038171003817
1003817100381710038171003817119909119899 minus 119909⋆1003817100381710038171003817
le1003817100381710038171003817119909120591(119899)+1 minus 119909
⋆1003817100381710038171003817
(56)
Therefore 119909119899 converges strongly to 119909
⋆= Proj
Ω(119868 minus 119860 +
120574ℎ)119909⋆ This completes the proof
4 Application
In this section we consider a particular Fan inequalitycorresponding to the function 119891 defined by the following forevery 119909 119910 isin 119862
119891 (119909 119910) = ⟨119865 (119909) 119910 minus 119909⟩ (57)
where 119865 119862 rarr 119867 Then we obtain the classical variationalinequality as follows
Find 119911 isin 119862 such that ⟨119865 (119911) 119910 minus 119911⟩ ge 0 forall119910 isin 119862 (58)
The set of solutions of this problem is denoted by 119881119868(119865 119862)In that particular case the solution 119910
119899of the minimization
problem
argmin 120582119899 119891 (119909119899 119910) +1
2
1003817100381710038171003817119910 minus 1199091198991003817100381710038171003817
2 119910 isin 119862 (59)
can be expressed as
119910119899= 119875119862 (119909119899 minus 120582119899119865 (119909119899)) (60)
Let 119865 be 119871-Lipschitz continuous on 119862 Then
119891 (119909 119910) + 119891 (119910 119911) minus 119891 (119909 119911) = ⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
119909 119910 119911 isin 119862
(61)
Therefore1003816100381610038161003816⟨119865 (119909) minus 119865 (119910) 119910 minus 119911⟩
1003816100381610038161003816 le 1198711003817100381710038171003817119909 minus 119910
1003817100381710038171003817
1003817100381710038171003817119910 minus 1199111003817100381710038171003817
le119871
2(1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2+1003817100381710038171003817119910 minus 119911
1003817100381710038171003817
2)
(62)
hence 119891 satisfies Lipschitz-type continuous condition with1198881 = 1198882 = 1198712
Using Theorem 7 we obtain the following convergencetheorem
Theorem 8 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le 119898119865119894is monotone and 119871-Lipschitz continuous on 119862 Let T =
119879(119906) 119906 ge 0 and S = 119878(119906) 119906 ge 0 be twouar nonexpansive self-mapping semigroups on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909
0isin 119862 and let 119909
119899 119908119899119894 and 119911
119899119894 be
sequences generated by
119908119899119894 = 119875119862 (119909119899 minus 120582119899119894119865119894 (119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899= 120572119899119909119899+
119898
sum
119894=1
120573119899119894119911119899119894+ 120574119899119879 (119905119899) 119909119899+ 120578119899119878 (119905119899) 119909119899
119909119899+1 = 120579119899120574ℎ (119909119899) + (119868 minus 120579119899119860)119910119899 forall119899 ge 0
(63)
where 120572119899 + sum
119898
119894=1120573119899119894 + 120574119899 + 120578119899 = 1 and 120572119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579119899 satisfy the following conditions(i) lim
119899rarrinfin119905119899= infin
(ii) 120579119899 sub ]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(iii) 120582119899119894 sub [119886 119887] sub ]0 1119871[
(iv) 120572119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf119899120572119899120574119899 gt 0 lim inf119899120572119899120578119899 gt 0 andlim inf119899120573119899119894(1 minus 21205821198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (64)
8 Abstract and Applied Analysis
In [32] Baillon proved a strong mean convergence the-orem for nonexpansive mappings and it was generalizedin [33] It follows from the above proof that Theorems 7 isvalid for nonexpansivemappingsThuswe have the followingmean ergodic theorems for nonexpansive mappings in aHilbert space
Theorem 9 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le
119898 119865119894 is monotone and 119871-Lipschitz continuous on 119862 Let119879 and 119878 be two nonexpansive mappings on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(119879)⋂119865(119878) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909119899 119908119899119894 and 119911119899119894 be sequences generatedby 1199090isin 119862 and by
119908119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879119899119909119899 + 120578119899119878
119899119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(65)
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 120572
119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579
119899 satisfy the following conditions
(i) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(ii) 120582119899119894 sub [119886 119887] sub ]0 1119871[ where 119871 = max21198881119894 21198882119894
1 le 119894 le 119898(iii) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (66)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Therefore theauthors acknowledge with thanks DSR for technical andfinancial support
References
[1] K Fan ldquoA minimax inequality and applicationsrdquo in InequalityIII pp 103ndash113 Academic Press New York NY USA 1972
[2] P N Anh ldquoA hybrid extragradient method extended to fixedpoint problems and equilibrium problemsrdquo Optimization vol62 no 2 pp 271ndash283 2013
[3] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994
[4] L-C Ceng N Hadjisavvas and N-C Wong ldquoStrong conver-gence theorem by a hybrid extragradient-like approximationmethod for variational inequalities and fixed point problemsrdquoJournal of Global Optimization vol 46 no 4 pp 635ndash646 2010
[5] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964
[6] P Daniele F Giannessi and A Maugeri Equilibrium Problemsand Variational Models vol 68 Kluwer Academic NorwellMass USA 2003
[7] J-W Peng and J-C Yao ldquoSome new extragradient-likemethodsfor generalized equilibrium problems fixed point problemsand variational inequality problemsrdquo Optimization Methods ampSoftware vol 25 no 4ndash6 pp 677ndash698 2010
[8] D R Sahu N-C Wong and J-C Yao ldquoStrong convergencetheorems for semigroups of asymptotically nonexpansive map-pings inBanach spacesrdquoAbstract andAppliedAnalysis vol 2013Article ID 202095 8 pages 2013
[9] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[10] S Wang and B Guo ldquoNew iterative scheme with nonexpansivemappings for equilibrium problems and variational inequalityproblems in Hilbert spacesrdquo Journal of Computational andApplied Mathematics vol 233 no 10 pp 2620ndash2630 2010
[11] Y Yao Y-C Liou and Y-J Wu ldquoAn extragradient method formixed equilibrium problems and fixed point problemsrdquo FixedPoint Theory and Applications vol 2009 Article ID 632819 15pages 2009
[12] F E Browder ldquoNonexpansive nonlinear operators in a Banachspacerdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 54 pp 1041ndash1044 1965
[13] R Chen and Y Song ldquoConvergence to common fixed pointof nonexpansive semigroupsrdquo Journal of Computational andApplied Mathematics vol 200 no 2 pp 566ndash575 2007
[14] A Aleyner and Y Censor ldquoBest approximation to commonfixed points of a semigroup of nonexpansive operatorsrdquo Journalof Nonlinear and Convex Analysis vol 6 no 1 pp 137ndash151 2005
[15] H K Xu ldquoAn iterative approach to quadratic optimizationrdquoJournal of Optimization Theory and Applications vol 116 no 3pp 659ndash678 2003
[16] G Marino and H-K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006
[17] AMoudafi ldquoViscosity approximationmethods for fixed-pointsproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 241 no 1 pp 46ndash55 2000
[18] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[19] S Li L Li and Y Su ldquoGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert spacerdquo Nonlin-ear Analysis Theory Methods amp Applications vol 70 no 9 pp3065ndash3071 2009
Abstract and Applied Analysis 9
[20] D R Sahu N C Wong and J C Yao ldquoA unified hybriditerative method for solving variational inequalities involvinggeneralized pseudocontractive mappingsrdquo SIAM Journal onControl and Optimization vol 50 no 4 pp 2335ndash2354 2012
[21] N Shioji and W Takahashi ldquoStrong convergence theoremsfor asymptotically nonexpansive semigroups in Hilbert spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 34 no1 pp 87ndash99 1998
[22] P N Anh ldquoStrong convergence theorems for nonexpansivemappings and Ky Fan inequalitiesrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 303ndash320 2012
[23] L-C Ceng Q H Ansari and J-C Yao ldquoHybrid proximal-type and hybrid shrinking projection algorithms for equilib-rium problems maximal monotone operators and relativelynonexpansive mappingsrdquo Numerical Functional Analysis andOptimization vol 31 no 7-9 pp 763ndash797 2010
[24] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009
[25] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011
[26] L-C Ceng S-M Guu and J-C Yao ldquoHybrid iterative methodfor finding common solutions of generalizedmixed equilibriumand fixed point problemsrdquo Fixed PointTheory and Applicationsvol 2012 article 92 2012
[27] M Eslamian ldquoConvergence theorems for nonspreading map-pings andnonexpansivemultivaluedmappings and equilibriumproblemsrdquo Optimization Letters vol 7 no 3 pp 547ndash557 2013
[28] M Eslamian ldquoHybrid method for equilibrium problems andfixed point problems of finite families of nonexpansive semi-groupsrdquo RACSAM vol 107 pp 299ndash307 2013
[29] M Eslamian and A Abkar ldquoOne-step iterative process fora finite family of multivalued mappingsrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 105ndash111 2011
[30] P-E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[31] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976
[32] J-B Baillon ldquoUn theoreme de type ergodique pour les contrac-tions non lineaires dans un espace de Hilbertrdquo Comptes Rendusde lrsquoAcademie des Sciences vol 280 no 22 pp A1511ndashA15141975
[33] W Kaczor T Kuczumow and S Reich ldquoA mean ergodictheorem for nonlinear semigroups which are asymptoticallynonexpansive in the intermediate senserdquo Journal of Mathemat-ical Analysis and Applications vol 246 no 1 pp 1ndash27 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
In [32] Baillon proved a strong mean convergence the-orem for nonexpansive mappings and it was generalizedin [33] It follows from the above proof that Theorems 7 isvalid for nonexpansivemappingsThuswe have the followingmean ergodic theorems for nonexpansive mappings in aHilbert space
Theorem 9 Let 119862 be a nonempty closed convex subset ofa real Hilbert space 119867 and let 119865119894 119862 rarr 119867 (119894 =
1 2 119898) be functions such that for each 1 le 119894 le
119898 119865119894 is monotone and 119871-Lipschitz continuous on 119862 Let119879 and 119878 be two nonexpansive mappings on 119862 such thatΩ = ⋂
119898
119894=1119881119868(119865119894 119862)⋂119865(119879)⋂119865(119878) = 0 Assume that ℎ is a 119896-
contraction of 119862 into itself and119860 is a strongly positive boundedlinear self-adjoint operator on 119867 with coefficient 120574 lt 1 and0 lt 120574 lt 120574119896 Let 119909119899 119908119899119894 and 119911119899119894 be sequences generatedby 1199090isin 119862 and by
119908119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119909119899)) 119894 isin 1 2 119898
119911119899119894= 119875119862(119909119899minus 120582119899119894119865119894(119908119899119894)) 119894 isin 1 2 119898
119910119899 = 120572119899119909119899 +
119898
sum
119894=1
120573119899119894119911119899119894 + 120574119899119879119899119909119899 + 120578119899119878
119899119909119899
119909119899+1
= 120579119899120574ℎ (119909119899) + (119868 minus 120579
119899119860)119910119899 forall119899 ge 0
(65)
where 120572119899+ sum119898
119894=1120573119899119894+ 120574119899+ 120578119899= 1 and 120572
119899 120573119899119894 120574119899 120578119899
120582119899119894 and 120579
119899 satisfy the following conditions
(i) 120579119899 sub]0 1[ lim
119899rarrinfin120579119899= 0 and suminfin
119899=1120579119899= infin
(ii) 120582119899119894 sub [119886 119887] sub ]0 1119871[ where 119871 = max21198881119894 21198882119894
1 le 119894 le 119898(iii) 120572
119899 120573119899119894 120574119899 120578119899 sub ]0 1[ lim inf
119899120572119899120573119899119894
gt 0lim inf
119899120572119899120574119899
gt 0 lim inf119899120572119899120578119899
gt 0 andlim inf
119899120573119899119894(1 minus 2120582
1198991198941198881119894) gt 0 for each 1 le 119894 le 119898
Then the sequence 119909119899 converges strongly to 119909
⋆isin
⋂119898
119894=1119881119868(119865119894 119862)⋂119865(T)⋂119865(S) which solves the variational
inequality
⟨(119860 minus 120574ℎ) 119909⋆ 119909 minus 119909
⋆⟩ ge 0 forall119909 isin Ω (66)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Therefore theauthors acknowledge with thanks DSR for technical andfinancial support
References
[1] K Fan ldquoA minimax inequality and applicationsrdquo in InequalityIII pp 103ndash113 Academic Press New York NY USA 1972
[2] P N Anh ldquoA hybrid extragradient method extended to fixedpoint problems and equilibrium problemsrdquo Optimization vol62 no 2 pp 271ndash283 2013
[3] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994
[4] L-C Ceng N Hadjisavvas and N-C Wong ldquoStrong conver-gence theorem by a hybrid extragradient-like approximationmethod for variational inequalities and fixed point problemsrdquoJournal of Global Optimization vol 46 no 4 pp 635ndash646 2010
[5] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964
[6] P Daniele F Giannessi and A Maugeri Equilibrium Problemsand Variational Models vol 68 Kluwer Academic NorwellMass USA 2003
[7] J-W Peng and J-C Yao ldquoSome new extragradient-likemethodsfor generalized equilibrium problems fixed point problemsand variational inequality problemsrdquo Optimization Methods ampSoftware vol 25 no 4ndash6 pp 677ndash698 2010
[8] D R Sahu N-C Wong and J-C Yao ldquoStrong convergencetheorems for semigroups of asymptotically nonexpansive map-pings inBanach spacesrdquoAbstract andAppliedAnalysis vol 2013Article ID 202095 8 pages 2013
[9] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[10] S Wang and B Guo ldquoNew iterative scheme with nonexpansivemappings for equilibrium problems and variational inequalityproblems in Hilbert spacesrdquo Journal of Computational andApplied Mathematics vol 233 no 10 pp 2620ndash2630 2010
[11] Y Yao Y-C Liou and Y-J Wu ldquoAn extragradient method formixed equilibrium problems and fixed point problemsrdquo FixedPoint Theory and Applications vol 2009 Article ID 632819 15pages 2009
[12] F E Browder ldquoNonexpansive nonlinear operators in a Banachspacerdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 54 pp 1041ndash1044 1965
[13] R Chen and Y Song ldquoConvergence to common fixed pointof nonexpansive semigroupsrdquo Journal of Computational andApplied Mathematics vol 200 no 2 pp 566ndash575 2007
[14] A Aleyner and Y Censor ldquoBest approximation to commonfixed points of a semigroup of nonexpansive operatorsrdquo Journalof Nonlinear and Convex Analysis vol 6 no 1 pp 137ndash151 2005
[15] H K Xu ldquoAn iterative approach to quadratic optimizationrdquoJournal of Optimization Theory and Applications vol 116 no 3pp 659ndash678 2003
[16] G Marino and H-K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006
[17] AMoudafi ldquoViscosity approximationmethods for fixed-pointsproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 241 no 1 pp 46ndash55 2000
[18] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[19] S Li L Li and Y Su ldquoGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert spacerdquo Nonlin-ear Analysis Theory Methods amp Applications vol 70 no 9 pp3065ndash3071 2009
Abstract and Applied Analysis 9
[20] D R Sahu N C Wong and J C Yao ldquoA unified hybriditerative method for solving variational inequalities involvinggeneralized pseudocontractive mappingsrdquo SIAM Journal onControl and Optimization vol 50 no 4 pp 2335ndash2354 2012
[21] N Shioji and W Takahashi ldquoStrong convergence theoremsfor asymptotically nonexpansive semigroups in Hilbert spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 34 no1 pp 87ndash99 1998
[22] P N Anh ldquoStrong convergence theorems for nonexpansivemappings and Ky Fan inequalitiesrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 303ndash320 2012
[23] L-C Ceng Q H Ansari and J-C Yao ldquoHybrid proximal-type and hybrid shrinking projection algorithms for equilib-rium problems maximal monotone operators and relativelynonexpansive mappingsrdquo Numerical Functional Analysis andOptimization vol 31 no 7-9 pp 763ndash797 2010
[24] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009
[25] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011
[26] L-C Ceng S-M Guu and J-C Yao ldquoHybrid iterative methodfor finding common solutions of generalizedmixed equilibriumand fixed point problemsrdquo Fixed PointTheory and Applicationsvol 2012 article 92 2012
[27] M Eslamian ldquoConvergence theorems for nonspreading map-pings andnonexpansivemultivaluedmappings and equilibriumproblemsrdquo Optimization Letters vol 7 no 3 pp 547ndash557 2013
[28] M Eslamian ldquoHybrid method for equilibrium problems andfixed point problems of finite families of nonexpansive semi-groupsrdquo RACSAM vol 107 pp 299ndash307 2013
[29] M Eslamian and A Abkar ldquoOne-step iterative process fora finite family of multivalued mappingsrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 105ndash111 2011
[30] P-E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[31] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976
[32] J-B Baillon ldquoUn theoreme de type ergodique pour les contrac-tions non lineaires dans un espace de Hilbertrdquo Comptes Rendusde lrsquoAcademie des Sciences vol 280 no 22 pp A1511ndashA15141975
[33] W Kaczor T Kuczumow and S Reich ldquoA mean ergodictheorem for nonlinear semigroups which are asymptoticallynonexpansive in the intermediate senserdquo Journal of Mathemat-ical Analysis and Applications vol 246 no 1 pp 1ndash27 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
[20] D R Sahu N C Wong and J C Yao ldquoA unified hybriditerative method for solving variational inequalities involvinggeneralized pseudocontractive mappingsrdquo SIAM Journal onControl and Optimization vol 50 no 4 pp 2335ndash2354 2012
[21] N Shioji and W Takahashi ldquoStrong convergence theoremsfor asymptotically nonexpansive semigroups in Hilbert spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 34 no1 pp 87ndash99 1998
[22] P N Anh ldquoStrong convergence theorems for nonexpansivemappings and Ky Fan inequalitiesrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 303ndash320 2012
[23] L-C Ceng Q H Ansari and J-C Yao ldquoHybrid proximal-type and hybrid shrinking projection algorithms for equilib-rium problems maximal monotone operators and relativelynonexpansive mappingsrdquo Numerical Functional Analysis andOptimization vol 31 no 7-9 pp 763ndash797 2010
[24] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009
[25] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011
[26] L-C Ceng S-M Guu and J-C Yao ldquoHybrid iterative methodfor finding common solutions of generalizedmixed equilibriumand fixed point problemsrdquo Fixed PointTheory and Applicationsvol 2012 article 92 2012
[27] M Eslamian ldquoConvergence theorems for nonspreading map-pings andnonexpansivemultivaluedmappings and equilibriumproblemsrdquo Optimization Letters vol 7 no 3 pp 547ndash557 2013
[28] M Eslamian ldquoHybrid method for equilibrium problems andfixed point problems of finite families of nonexpansive semi-groupsrdquo RACSAM vol 107 pp 299ndash307 2013
[29] M Eslamian and A Abkar ldquoOne-step iterative process fora finite family of multivalued mappingsrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 105ndash111 2011
[30] P-E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[31] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976
[32] J-B Baillon ldquoUn theoreme de type ergodique pour les contrac-tions non lineaires dans un espace de Hilbertrdquo Comptes Rendusde lrsquoAcademie des Sciences vol 280 no 22 pp A1511ndashA15141975
[33] W Kaczor T Kuczumow and S Reich ldquoA mean ergodictheorem for nonlinear semigroups which are asymptoticallynonexpansive in the intermediate senserdquo Journal of Mathemat-ical Analysis and Applications vol 246 no 1 pp 1ndash27 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
