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# Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions

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Volume 2009, Article ID 794178,21pages doi:10.1155/2009/794178

## Viscosity Method with Parallel Method for aGeneralized Equilibrium Problem and StrictPseudocontractions

1

2

### and Jen-Chih Yao

3

1College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China

2Department of Information Management, Cheng Shiu University, Kaohsiung, Taiwan 833, Taiwan

3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Taiwan

Correspondence should be addressed to Yeong-Cheng Liou,simplex liou@hotmail.com Received 5 August 2008; Accepted 4 January 2009

Recommended by Hichem Ben-El-Mechaiekh

We introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.

Copyrightq2009 Jian-Wen Peng et al. This 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.

### 1. Introduction

LetHbe a real Hilbert space with inner product ·,· and induced norm · ,and letCbe a nonempty-closed convex subset ofH. Letϕ :HR∪ {∞}be a function and letFbe a bifunction fromC×Cto Rsuch thatC∩domϕ /∅, where Ris the set of real numbers and domϕ {x∈ H : ϕx< ∞}. Flores-Baz´an 1introduced the following generalized equilibrium problem:

FindxCsuch thatFx, y ϕyϕx, ∀y∈C. 1.1 The set of solutions of 1.1 is denoted by GEPF, ϕ. Flores-Baz´an 1 provided some characterizations of the nonemptiness of the solution set for problem1.1in reflexive Banach spaces in the quasiconvex case. Bigi et al. 2 studied a dual problem associated with the problem1.1withCHRn.

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Letϕx δCx,∀x∈H. HereδCdenotes the indicator function of the setC; that is, δCx 0 ifxCandδCx ∞otherwise. Then the problem1.1becomes the following equilibrium problem:

FindingxCsuch thatFx, y≥0, ∀y∈C. 1.2 The set of solutions of1.2is denoted by EPF. The problem1.2includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative games, and the vector optimization problem. For more detail, please see 3–5 and the references therein.

IfFx, y gy−gxfor allx, yC, whereg:CRis a function, then the problem 1.1becomes a problem of findingxCwhich is a solution of the following minimization problem:

miny∈C

ϕy gy

. 1.3

The set of solutions of1.3is denoted by Argming, ϕ.

Ifϕ :HR∪ {∞}is replaced by a real-valued functionφ :CR, the problem 1.1reduces to the following mixed equilibrium problem introduced by Ceng and Yao6:

Find xCsuch thatFx, y φyφx≥0, ∀y∈C. 1.4 Recall that a mappingT :CCis said to be aκ-strict pseudocontraction7if there exists 0≤κ <1,such that

Tx−Ty2≤ x−y2κI−Tx−I−Ty2, ∀x, y∈C, 1.5 whereI denotes the identity operator onC. Whenκ0,T is said to be nonexpansive. Note that the class of strict pseudocontraction mappings strictly includes the class of nonexpansive mappings. We denote the set of fixed points ofSby FixS.

Ceng and Yao6, Yao et al. 8, and Peng and Yao9,10introduced some iterative schemes for finding a common element of the set of solutions of the mixed equilibrium problem 1.4 and the set of common fixed points of a family of finitely infinitely nonexpansive mappingsstrict pseudocontractionsin a Hilbert space and obtained some strong convergence theoremsweak convergence theorems. Some methods have been proposed to solve the problem 1.2; see, for instance, 3–5, 11–18 and the references therein. Recently, S. Takahashi and W. Takahashi 12 introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem1.2and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem. Su et al.13introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem 1.2 and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for anα-inverse strongly monotone mapping in a Hilbert space. Tada and Takahashi14 introduced two iterative schemes for finding

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a common element of the set of solutions of problem 1.2 and the set of fixed points of a nonexpansive mapping in a Hilbert space and obtained both strong convergence theorem and weak convergence theorem. Ceng et al.15introduced an iterative algorithm for finding a common element of the set of solutions of problem1.2and the set of fixed points of a strict pseudocontraction mapping. Chang et al.16introduced some iterative processes based on the extragradient method for finding the common element of the set of fixed points of a family of infinitely nonexpansive mappings, the set of problem1.2, and the set of solutions of a variational inequality problem for anα-inverse strongly monotone mapping. Colao et al.

17introduced an iterative method for finding a common element of the set of solutions of problem 1.2 and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space and proved the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. To the best of our knowledge, there is not any algorithms for solving problem1.1.

On the other hand, Marino and Xu19 and Zhou 20introduced and researched some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. Acedo and Xu21introduced some parallel and cyclic algorithms for finding a common fixed point of a family of finite strict pseudocontraction mappings and obtained both weak and strong convergence theorems for the sequences generated by the iterative schemes.

In the present paper, we introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.

### 2. Preliminaries

LetHbe a real Hilbert space with inner product ·,· and norm · . LetCbe a nonempty- closed convex subset ofH. Let symbols → anddenote strong and weak convergences, respectively. In a real Hilbert spaceH, it is well known that

λx 1−λy2λx2 1−λy2λ1λxy2 2.1 for allx, yHandλ∈0,1.

For anyxH, there exists a unique nearest point inC, denoted byPCx, such that x−PCx ≤ x−yfor allyC. The mappingPCis called the metric projection ofHonto C. We know thatPCis a nonexpansive mapping fromHontoC. It is also known thatPCxC and

xPCx, PCx−y

≥0 2.2

for allxHandyC.

For eachBH, we denote by convBthe convex hull ofB. A multivalued mapping G : B → 2H is said to be a KKM map if, for every finite subset {x1, x2, . . . , xn} ⊆ B, conv{x1, x2, . . . , xn}⊆

n1Gxi.

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We will use the following results in the sequel.

Lemma 2.1see22. LetBbe a nonempty subset of a Hausdorﬀtopological vector spaceXand let G:B → 2Xbe a KKM map. IfGxis closed for allxBand is compact for at least onexB, then

x∈BGx/∅.

For solving the generalized equilibrium problem, let us give the following assump- tions for the bifunctionF,ϕ,and the setC:

A1Fx, x 0 for allxC;

A2Fis monotone, that is,Fx, y Fy, x≤0 for anyx, yC;

A3for eachyC, xFx, yis weakly upper semicontinuous;

A4for eachxC, yFx, yis convex;

A5for eachxC, yFx, yis lower semicontinuous;

B1For eachxHandr >0, there exist a bounded subsetDxCandyxC∩domϕ such that for anyzC\Dx,

F z, yx

ϕ yx

1 r

yxz, zx

< ϕz; 2.3

B2Cis a bounded set.

Lemma 2.2. LetCbe a nonempty-closed convex subset ofH. LetF be a bifunction fromC×Cto Rsatisfying (A1)–(A4) and letϕ : HR∪ {∞}be a proper lower semicontinuous and convex function such thatC∩domϕ /∅. Forr >0 andxH,define a mappingSr :HCas follows:

Srx

zC:Fz, y ϕy 1

ry−z, zx ≥ϕz,∀y∈C

2.4

for allxH. Assume that either (B1) or (B2) holds. Then, the following conclusions hold:

1for eachxH, Srx/∅;

2Sris single-valued;

3Sris firmly nonexpansive, that is, for anyx, yH, Srx−Sry2

Srx−Sry, x−y

; 2.5

4FixSr GEPF, ϕ;

5GEPF, ϕis closed and convex.

Proof. Letx0be any given point inE. For eachyC, we define Gy

zC:Fz, y ϕy 1 r

yz, zx0

ϕz

. 2.6

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Note that for eachyC∩domϕ,Gyis nonempty sinceyGyand for eachyC\domϕ, Gy C. We will prove thatGis a KKM map onC∩domϕ. Suppose that there exists a finite subset{y1, y2, . . . , yn}ofC∩domϕandμi ≥0 for alli1,2, . . . , nwithn

i1μi 1 such that

zn

i1μiyi/∈Gyifor eachi1,2, . . . , n. Then we have

F z, y i

ϕ yi

ϕz 1 r

yiz, zx0

<0 2.7

for eachi1,2, . . . , n. ByA4and the convexity ofϕ, we have

0Fz,z ϕzϕz 1 r

zz, zx0

n

i1

μi

F z, y i

ϕ yi

ϕz 1

r n

i1

μi

yiz, zx0

<0,

2.8

which is a contradiction. Hence,Gis a KKM map onC∩domϕ. Note thatGywthe weak closure ofGyis a weakly closed subset ofCfor eachyC. Moreover, ifB2holds, then Gywis also weakly compact for eachyC. IfB1holds, then forx0E, there exists a bounded subsetDx0Candyx0C∩domϕsuch that for anyzC\Dx0,

F z, yx0

ϕ yx0

1 r

yx0z, zx0

< ϕz. 2.9

This shows that

G yx0

zC:F z, yx0

ϕ yx0 1

r

yx0z, zx0

ϕz

Dx0. 2.10

Hence, Gyx0w is weakly compact. Thus, in both cases, we can use Lemma 2.1and have

y∈C∩domϕGyw/∅.

Next, we will prove thatGyw Gyfor eachyC; that is,Gyis weakly closed.

LetzGywand letzmbe a sequence inGysuch thatzm z. Then,

F zm, y

ϕy 1 r

yzm, zmx0

ϕ zm

. 2.11

Since · 2is weakly lower semicontinuous, we can show that

lim sup

m→ ∞

yzm, zmx0

zy, x0z

. 2.12

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It follows fromA3and the weak lower semicontinuity ofϕthat ϕz≤lim inf

m→ ∞ ϕ zm

≤lim sup

m→ ∞

F zm, y

ϕy 1 r

yzm, zmx0

≤lim sup

m→ ∞

F zm, y

ϕy 1

rlim sup

m→ ∞

yzm, zmx0

Fz, y ϕy 1 r

zy, x0z .

2.13

This implies that zGy. Hence, Gy is weakly closed. Hence, Srx0

y∈CGy

y∈C∩domϕGy

y∈C∩domϕGyw/∅. Hence, from the arbitrariness ofx0, we conclude that Srx/∅, ∀x∈H.

We observe that Srx ⊆ domϕ. So by similar argument with that in the proof of Lemma 2.3 in9, we can easily show thatSris single-valued andSris a firmly nonexpansive- type map. Next, we claim that FixSr GEPF, ϕ. Indeed, we have the following:

u∈Fix Sr

⇐⇒uSru

⇐⇒Fu, y ϕy 1

ry−u, uu ≥ϕu, ∀y∈C

⇐⇒Fu, y ϕyϕu, ∀y∈C

⇐⇒u∈GEPF, ϕ.

2.14

At last, we claim that GEPF, ϕis a closed convex. Indeed, SinceSr is firmly nonexpansive, Sr is also nonexpansive. By23, Proposition 5.3, we know that GEPF, ϕ FixSris closed and convex.

Remark 2.3. It is easy to see thatLemma 2.2is a generalization of9, Lemma 2.3.

Lemma 2.4see24,25. Assume that{αn}is a sequence of nonnegative real numbers such that αn1≤ 1−γn

αnδn, 2.15

whereγnis a sequence in0,1andn}is a sequence such that i

n1

γn∞;

ii lim sup

n→ ∞

δn

γn0 or n1

δn<∞.

2.16

Then, limn→ ∞αn0.

Lemma 2.5. In a real Hilbert spaceH, there holds the following inequality:

xy2≤ x22y, xy 2.17

for allx, yH.

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### 3. Strong Convergence Theorems

In this section, we show a strong convergence of an iterative algorithm based on both viscosity approximation method and parallel method which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions in a Hilbert space.

We need the following assumptions for the parameters {γn},{rn},{αn},{ζ1n}, {ζn2 }, . . . ,{ζnN},and{βn}:

C1limn→ ∞αn0 and

n1αn∞;

C21>lim supn→ ∞βn≥lim infn→ ∞βn>0;

C3{γn} ⊂c, dfor somec, d∈ε,1and limn→ ∞n1γn|0;

C4lim infn→ ∞rn>0 and limn→ ∞|rn1rn|0;

C5limn→ ∞n1jζnj |0 for allj1,2, . . . , N.

Theorem 3.1. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a bifunction from C× C to R satisfying (A1)–(A5), and let ϕ : CR ∪ {∞} be a proper lower semicontinuous and convex function such that C∩domϕ /∅. Let N1 be an integer.

For each 1jN, let Tj : CC be anεj-strict pseudocontraction for some 0εj < 1 such thatΩ N

j1FixTj∩GEPF, ϕ/∅. Assume for eachn,nj }Nj1 is a finite sequence of positive numbers such that N

j1ζnj 1 for all n and infn≥1ζnj > 0 for all 0jN. Let ε max{εj : 1 ≤ jN}. Assume that either (B1) or (B2) holds. Let f be a contraction ofC into itself and let{xn},{un},and{yn}be sequences generated by

x1xC, F un, y

ϕy 1 rn

yun, unxn

ϕ un

, ∀y∈C, ynγnun 1−γnN

j1

ζnj Tjun, xn1 αnf xn

βnxn 1−αnβn yn

3.1

for everyn 1,2, . . ., where {γn},{rn},{αn},{ζ1n},{ζn2 }, . . . ,{ζnN},andn} are sequences of numbers satisfying the conditions (C1)–(C5). Then,{xn},{un},and{yn}converge strongly tow PΩfw.

Proof. We show thatPΩf is a contraction ofCinto itself. In fact, there existsa ∈0,1such thatfx−fy ≤axyfor allx, yC. So, we have

PΩfxPΩfyfx−fyaxy 3.2 for all x, yC.SinceH is complete, there exists a unique elementu0Csuch thatu0 PΩfu0.

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Letu ∈ Ωand let{Srn}be a sequence of mappings defined as inLemma 2.2. From unSrnxnC, we have

unuSrn xn

Srnu≤xnu. 3.3

We define a mappingWnby

WnxN

j1

ζnj Tjx, ∀x∈C. 3.4

By 21, Proposition 2.6, we know that Wn is an ε-strict pseudocontraction and FWn

N

j1FixTj. It follows from3.3,ynγnun 1−γnWnunanduWnusuch that ynu2γnunu2 1−γnWnunu2γn 1−γnunWnun2

γnunu2 1−γnunu2εunWnun2

γn 1−γnunWnun2 unu2 1−γn εγnunWnun2

unu2.

3.5 PutM0max{x1−u,1/1−afu−u}.It is obvious thatx1−u ≤M0.Suppose xnu ≤M0.From3.3,3.5, andxn1αnfxn βnxn 1−αnβnyn, we have

xn1nf xn

βnxn 1−αnβn

ynu

αnf xn

fuαnfunxnu 1−αnβnynu

αnaxnnfunxnu 1−αnβnunu

αnaxnnfuu 1−αnxnu 1−n

fuu

1−a

1−1−nxnu,

≤1−nM0

1−1−n

M0M0

3.6

for everyn 1,2, . . . .Therefore,{xn}is bounded. From3.3and3.5, we also obtain that {yn}and{un}are bounded.

Following26, defineBn:CCby

BnγnI 1−γn

Wn. 3.7

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As shown in26, eachBnis a nonexpansive mapping onC. SetM1 supn≥1{unWnun}, we have

yn1ynBn1 un1

Bn un

Bn1 un1

Bn1 unBn1 un

Bn un

un1unM1γn1γn 1−γn1Wn1 unWn un

un1unM1γn1γn 1−γn1N

j1

ζn1jζnj Tjun.

3.8

On the other hand, fromunTrnxnandun1Trn1xn1, we have

F un, y

ϕy 1 rn

yun, unxn

ϕ un

, ∀y∈C, 3.9

F un1, y

ϕy 1 rn1

yun1, un1xn1

ϕ un1

, ∀y∈C. 3.10

Puttingyun1in3.9andyunin3.10, we have

F un, un1

ϕ un1 1

rn

un1un, unxn

ϕ un , F un1, un

ϕ un 1

rn1

unun1, un1xn1

ϕ un1 .

3.11

So, from the monotonicity ofF, we get

un1un,unxn

rnun1xn1 rn1

≥0, 3.12

hence

un1un, unun1un1xnrn

rn1 un1xn1

≥0. 3.13

Without loss of generality, let us assume that there exists a real numberbsuch thatrn> b >0 for allnN.Then,

un1un2

un1un, xn1xn

1− rn

rn1 un1xn1

un1unxn1xn 1− rn

rn1

un1xn1 ,

3.14

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hence

un1unxn1xn 1

rn1rn1rnun1xn1

xn1xn1

brn1rnM2,

3.15

whereM2sup{unxn:n≥1}.

It follows from3.8and3.15that yn1ynxn1xn1

brn1rnM2M1γn1γn 1−γn1N

j1

ζjn1ζnj Tjun.

3.16

Define a sequence{vn}such that

xn1βnxn 1−βn

vn, ∀n≥1. 3.17

Then, we have

vn1vn xn2βn1xn1

1−βn1xn1βnxn

1−βn

αn1fxn1 1−αn1βn1yn1

1−βn1αnfxn 1−αnβnyn

1−βn

αn1

1−βn1f xn1

αn 1−βnf xn

yn1yn αn

1−βnynαn1 1−βn1yn1.

3.18

From3.18and3.16, we have vn1vnxn1xn

αn1

1−βn1 f xn1yn1 αn 1−βn

f xnyn yn1ynxn1xn

αn1

1−βn1 f xn1yn1 αn

1−βn f xnyn 1

brn1rnM2M1γn1γn 1−γn1N

j1

ζn1jζnj Tjun.

3.19

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It follows fromC1–C5that lim sup

n→ ∞ vn1vnxn1xn≤0. 3.20

Hence, by27, Lemma 2.2, we have limn→ ∞vnxn0. Consequently,

nlim→ ∞xn1xn lim

n→ ∞ 1−βnvnxn0. 3.21

Sincexn1 αnfxn βnxn 1−αnβnyn, we have xnynxn1xnxn1yn

xn1xnαnf xn

ynβnxnyn, 3.22

thus

xnyn≤ 1

1−βn xn1xnαnfxnyn. 3.23 It follows fromC1andC2that limn→ ∞xnyn0.

Sincexn1 αnfxn βnxn 1−αnβnyn,foru∈Ω, it follows from3.5and3.3 that

xn1u2αnf xn

βnxn 1−αnβn

ynu2

αnf xn

u2βnxnu2 1−αnβnynu2

αnf xn

u2βnxnu2

1−αnβnunu2 1−γn εγnunWnun2

αnf xn

u2 1−αnxnu2

1−αnβn 1−γn εγnunWnun2,

3.24

from which it follows that unWnun2αn

1−αnβn1−γnγnε f xn

u2xnu2 1

1−αnβn1−γnγnε xnu2xn1u2 .

αn

1−αnβn1−γnγnε f xn

u2xnu2 1

1−αnβn1−γnγnε xnuxn1uxn1xn.

3.25

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It follows fromC1–C3andxn1xn → 0 that

unWnun−→0. 3.26

Foru∈Ω, we have fromLemma 2.2, unu2SrnxnSrnu2

SrnxnSrnu, xnu

unu, xnu 1

2unu2xnu2xnun2 .

3.27

Hence,

unu2xnu2xnun2. 3.28

By3.24and3.28, we have xn1u2αnf xn

u2βnxnu2 1−αnβnunu2

αnf xn

u2βnxnu2 1−αnβnxnu2xnun2 . 3.29

Hence,

1−αnβnxnun2αnf xn

u2αnxnu2xnu2xn1u2

αnf xn

u2αnxnu2

xnuxn1uxnxn1.

3.30

It follows fromC1,C2, andxnxn1 → 0 that limn→ ∞xnun0.

Next, we show that

lim sup

n→ ∞

fu0u0, xnu0

≤0, 3.31

whereu0PΩfu0. To show this inequality, we can choose a subsequence{xni}of{xn}such that

ilim→ ∞

fu0u0, xniu0

lim sup

n→ ∞

fu0u0, xnu0

. 3.32

Since{xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly tow. Without loss of generality, we can assume that{xni} w.Fromxnun → 0,

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we obtain thatuni w. Fromxnyn → 0, we also obtain thatyni w. Since{uni} ⊂C andCis closed and convex, we obtainwC.

We first show thatwN

k1FixTk.To see this, we observe that we may assumeby passing to a further subsequence if necessaryζnkiζkasi → ∞fork1,2, . . . , N.It is easy to see thatζk>0 andN

k1ζk1. We also have

Wnix−→Wx asi−→ ∞∀x∈C, 3.33

whereW N

k1ζkTk. Note that by21, Proposition 2.6,W is anε-strict pseudocontraction

and FixW N

i1FixTi. Since

uniWuniuniWniuniWniuniWuni

uniWniuniN

k1

ζnkiζkTkuni, 3.34

it follows from3.26andζnkiζkthat

uniWuni−→0. 3.35 So by the demiclosedness principle21, Proposition 2.6ii, it follows thatw ∈ FixW N

i1FixTi.

We now showw∈GEPF, ϕ.ByunTrnxn,we know that

F un, y

ϕy 1 rn

yun, unxn

ϕ un

, ∀y∈C. 3.36

It follows fromA2that

ϕy 1 rn

yun, unxn

F y, un

ϕ un

, ∀y∈C. 3.37

Hence,

ϕy

yuni,unixni

rni

F y, uni

ϕ un

, ∀y∈C. 3.38

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It follows fromA4,A5and the weakly lower semicontinuity ofϕ,unixni/rni → 0,anduni wthat

Fy, w ϕwϕy, ∀y∈C. 3.39

Fortwith 0< t≤1 andyC∩domϕ, letytty 1−tw.SinceyC∩domϕand wC∩domϕ, we obtainytC∩domϕ,and henceFyt, w ϕwϕyt. So byA4and the convexity ofϕ, we have

0F yt, yt

ϕ yt

ϕ yt

tF yt, y

1−tF yt, w

tϕy 1tϕwϕ yt

t

F yt, y

ϕyϕ yt

.

3.40

Dividing byt, we get

F yt, y

ϕyϕ yt

≥0. 3.41

Lettingt → 0, it follows fromA3and the weakly lower semicontinuity ofϕthat

Fw, y ϕyϕw 3.42

for all yC∩domϕ. Observe that ifyC\domϕ, thenFw, y ϕyϕwholds.

Moreover, hencew∈GEPF, ϕ. This impliesw∈Ω.Therefore, we have lim sup

n→ ∞

fu0u0, xnu0

lim

i→ ∞

fu0u0, xniu0

fu0u0, wu0

≤0. 3.43

Finally, we show thatxnu0, whereu0PΩfu0. FromLemma 2.5, we have

xn1u02αn f xn

u0

βn xnu0

1−αnβn ynu02

βn xnu0

1−αnβn ynu02n

f xn

u0, xn1u0

≤ 1−αnβnynu0βnxnu02n

f xn

u0, xn1u0

≤ 1−αn

2xnu02n

f xn

f u0

, xn1u0

n

f u0

u0, xn1u0

≤ 1−αn

2xnu02naxnu0xn1u0n

f u0

u0, xn1u0

≤ 1−αn

2xnu02αna xnu02xn1u02n

f u0

u0, xn1u0

,

3.44

(15)

thus

xn1u02

1−21−n

1−n

xnu02 21−n

1−n

αn

21−axnu02 1 1−a

2f u0

−2u0, xn1u0 .

3.45

It follows fromC1,3.43,3.45, andLemma 2.4that limn→ ∞xnu0 0. From xnun → 0 andynxn → 0, we haveunu0 and ynu0. The proof is now complete.

Theorem 3.2. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a bifunction from C ×C to R satisfying (A1)–(A5), and let ϕ : HR ∪ {∞} be a proper lower semicontinuous and convex function such that C∩domϕ /∅. Let N1 be an integer.

For each 1jN, let Tj : CC be anεj-strict pseudocontraction for some 0εj < 1 such thatΩ N

j1FixTj∩GEPF, ϕ/∅. Assume for eachn,nj }Nj1 is a finite sequence of positive numbers such that N

j1ζnj 1 for all n and infn≥1ζnj > 0 for all 0jN. Let ε max{εj : 1 ≤ jN}. Assume that either (B1) or (B2) holds. Let vbe an arbitrary point in Cand let{xn}, {un},and{yn}be sequences generated by

x1xC, F un, y

ϕy 1 rn

yun, unxn

ϕ un

, ∀y∈C,

ynγnun 1−γn

N

j1

ζnj Tjun, xn1αnnxn 1−αnβn

yn

3.46

for every n 1,2, . . . ,wheren},{rn},{αn},{ζ1n},{ζn2 }, . . . ,{ζnN},andn}are sequences of numbers satisfying the conditions (C1)–(C5). Then,{xn}, {un},and{yn}converge strongly tow PΩv.

Proof. Letfx vfor allxC, byTheorem 3.1, we obtain the desired result.

### 4. Applications

By Theorems 3.1 and 3.2, we can obtain many new and interesting strong convergence theorems. Now, give some examples as follows: forj 1,2, . . . , N, letT1 T2 · · ·TN T, by Theorems3.1and3.2, respectively, we have the following results.

Theorem 4.1. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoRsatisfying (A1)–(A5), and letϕ : HR∪ {∞}be a proper lower semicontinuous and convex function such that C∩ domϕ /∅. Let T : CC be an ε-strict pseudocontraction for some 0ε <1 such that FixT∩GEPF, ϕ/∅. Assume that either (B1) or

(16)

(B2) holds. Letfbe a contraction ofCinto itself and let{xn}, {un},and{yn}be sequences generated by

x1xC, F un, y

ϕy 1 rn

yun, unxn

ϕ un

, ∀y∈C, yn γnun 1−γn

Tun, xn1 αnf xn

βnxn 1−αnβn yn

4.1

for everyn 1,2, . . . ,wheren}, {rn}, {αn},andn}are sequences of numbers satisfying the conditions (C1)–(C4). Then,{xn}, {un},and{yn}converge strongly towPFixT∩GEPF,ϕfw.

Theorem 4.2. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoRsatisfying (A1)–(A5), and letϕ : HR∪ {∞}be a proper lower semicontinuous and convex function such that C∩ domϕ /∅. Let T : CC be an ε-strict pseudocontraction for some 0ε < 1 such that FixT∩GEPF, ϕ/∅. Assume that either (B1) or (B2) holds. Letvbe an arbitrary point inC,and let{xn}, {un},and{yn}be sequences generated by

x1xC, F un, y

ϕy 1 rn

yun, unxn

ϕ un

, ∀y∈C, yn γnun 1−γn

Tun, xn1αnnxn 1−αnβn

yn

4.2

for everyn 1,2, . . . ,wheren}, {rn}, {αn},andn}are sequences of numbers satisfying the conditions (C1)–(C4). Then,{xn}, {un},and{yn}converge strongly towPFixT∩GEPF,ϕv.

We need the following two assumptions.

B3For eachxHandr >0, there exist a bounded subsetDxCandyxCsuch that for anyzC\Dx,

F z, yx

1 r

yxz, zx

<0. 4.3

B4For eachxHandr >0, there exist a bounded subsetDxCandyxC∩domϕ such that for anyzC\Dx,

g yx

ϕ yx

1 r

yxz, zx

< ϕz gz. 4.4

Let ϕx δCx,∀x ∈ H, by Theorems 3.1 and 3.2, respectively, we obtain the following results.

We introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of

Wangkeeree, A general iterative methods for variational inequality problems and mixed equilibrium problems and fixed point problems of strictly pseudocontractive mappings in

We introduce an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a

Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of

We propose new iterative schemes for finding the common element of the set of common fixed points of countable family of nonexpansive mappings, the set of solutions of the

Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods

Motivated by the ongoing research in this field, in this paper we suggest and analyze an iterative scheme for finding a common element of the set of fixed point of

We introduce a new iterative method for finding a common element of the set of solutions of a generalized equilibrium problem with a relaxed monotone mapping and the set of common