Approximation of Common Solutions to System of Mixed Equilibrium Problems, Variational Inequality Problem, and Strict Pseudo-Contractive Mappings

Volume 2011, Article ID 347204,30pages doi:10.1155/2011/347204

Research Article

Approximation of Common Solutions to

System of Mixed Equilibrium Problems,

Variational Inequality Problem, and Strict

Pseudo-Contractive Mappings

Poom Kumam

and Chaichana Jaiboon

1_{Department of Mathematics, Faculty of Science, King Mongkut’s University of}

Technology Thonburi (KMUTT), Bangkok 10140, Thailand

2_{Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand} 3_{Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of}

Technology Rattanakosin (RMUTR), Bangkok 10100, Thailand

Correspondence should be addressed to Chaichana Jaiboon,[email protected]

Received 3 October 2010; Accepted 5 March 2011

Academic Editor: Jong Kim

Copyrightq2011 P. Kumam and C. Jaiboon. 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.

We introduce an iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions mapping, the set of common solutions of a system of two mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse strongly monotone mappings. Strong convergence theorems are established in the framework of Hilbert spaces. Finally, we apply our results for solving convex feasibility problems in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

1. Introduction

Throughout this paper, we denote byNandRthe sets of positive integers and real numbers, respectively. LetHbe a real Hilbert space with inner product·,·and norm·, and letEbe a nonempty closed convex subset ofH. We denote weak convergence and strong convergence by notationsand →, respectively. Recall that a mappingf :E → Eis anα-contractionon

Eif there exists a constantα∈0,1such thatfx−fy ≤ αx−yfor allx, y∈E. Let

for example,1. Recall that a mappingS:E → Eis calledstrictly pseudo-contractionif there exists a constantk∈0,1such that

Sx−Sy2≤x−y2kI−Sx−I−Sy2, ∀x, y∈E, 1.1

where I denotes the identity operator on E. Note that if k 0, then Sis a nonexpansive mapping. The class of strict pseudo-contractions is one of the most important classes of mappings among nonlinear mappings. Within the past several decades, many authors have been devoted to the studies on the existence and convergence of fixed points for strict pseudo-contractions. In 1967, Browder and Petryshyn2introduced a convex combination method to study strict pseudo-contractions in Hilbert spaces. On the other hand, Marino and Xu3 and Zhou4developed some iterative scheme for finding a fixed point of a strict pseudo-contraction mapping. More precisely, takek∈0,1and define a mappingSkby

Skxkx 1−kSx, ∀x∈E, 1.2

where S is a strict pseudo-contraction. Under appropriate restrictions on k, it is proved that the mappingSk is nonexpansive. Therefore, the techniques of studying nonexpansive

mappings can be applied to study more general strict pseudo-contractions.

Let ϕ : E → R ∪ {∞} be a proper extended real-valued function and let φ be a bifunction ofE×EintoRsuch thatE∩domϕ /∅, whereRis the set of real numbers and domϕ{x∈E:ϕx<∞}. Ceng and Yao5considered the following mixed equilibrium problems for findingx∈Esuch that

φx, yϕy−ϕx≥0, ∀y∈E. 1.3

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

MEPφ, ϕx∈E:φx, yϕy−ϕx≥0,∀y∈E. 1.4

We see thatxis a solution of a problem1.3that implies thatx ∈domϕ {x∈E :

ϕx<∞}.

Special Examples

1Ifϕ0, then the mixed equilibrium problem1.3becomes to be the equilibrium problem which is to findx∈Esuch that

φx, y≥0, ∀y∈E. 1.5

The set of solutions of1.5is denoted by EPφ.

2Ifϕ0 andφx, y Bx, y−xfor allx, y∈E, whereB:E → His a nonlinear mapping, then problem1.5becomes to be the variational inequality problems which is to findx∈Esuch that

The set of solutions of 1.6 is denoted by VIE, B. The variational inequality has been extensively studied in the literature. See, for example,6–8and the references therein.

The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of1.3. Some authors have proposed some useful methods for solving the MEPφ, ϕ and EPφ; see, for instance 5, 9–27. In 1997, Combettes and Hirstoaga 10 introduced an iterative scheme of finding the best approximation to initial data when EPφ

is nonempty and proved a strong convergence theorem. Next, we recall some definitions.

Definition 1.1. LetB:E → Hbe nonlinear mappings. ThenBis called

1monotoneif

Bx−By, x−y≥0, ∀x, y∈E, 1.7

2ρ-strongly monotone if there exists a constantρ >0 such that

Bx−By, x−y≥ρx−y2, ∀x, y∈E, 1.8

3η-Lipschitz continuousif there exists a constantη >0 such that

_{Bx−By≤ηx−y, ∀x, y∈E, 1.9}

4β-inverse strongly monotoneif there exists a constantβ >0 such that

Bx−By, x−y≥βBx−By2, ∀x, y∈E. 1.10

Remark 1.2. It is obvious that anyβ-inverse strongly monotone mappingsBis monotone and

1/β-Lipschitz continuous.

5A set-valued mappingT :H → 2H_{is called amonotoneif, for allx, y ∈H,f ∈Tx}

andg ∈Tyimplyx−y, f−g ≥0.

6A monotone mappingT : H → 2H _{is amaximalif the graph of GTofT is not}

properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if forx, f∈H×H,x−y, f−g ≥0 for everyy, g∈GTimpliesf∈Tx.

LetBbe a monotone map ofEintoH,η-Lipschitz continuous mapping and letNEϑ

be thenormal conetoEwhenϑ∈E, that is,

and define a mappingTonEby

Tϑ

⎧ ⎨ ⎩

BϑNEϑ, ϑ∈E,

∅, ϑ /∈E.

1.12

ThenT is the maximal monotone and 0∈Tϑif and only ifϑ∈VIE, B; see28.

For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solution of variational inequalities forβ-inverse strongly monotone, Takahashi and Toyoda29first introduced the following iterative scheme:

x0 ∈E chosen arbitrary,

xn1αnxn 1−αnSPExn−λnBxn, ∀n≥0,

1.13

whereBis anβ-inverse strongly monotone,{αn}is a sequence in0, 1, and{λn}is a sequence

in0,2β. They showed that ifFS∩VIE, Bis nonempty, then the sequence{xn}generated

by1.13converges weakly to someq∈FS∩VIE, B.

Further, Y. Yao and J.-C. Yao30introduced the following iterative scheme:

x1x∈E chosen arbitrary,

ynPExn−λnBxn,

xn1αnxβnxnγnSPE

yn−λnByn

, ∀n≥1,

1.14

where Bis an β-inverse strongly monotone, {αn},{βn},{γn}are three sequences in 0, 1,

and{λn}is a sequence in0,2β. They showed that ifFS∩VIE, Bis nonempty, then the

sequence{xn}generated by1.14converges strongly to someq∈FS∩VIE, B.

A mapA:H → His said to be strongly positive if there exists a constantγ >0 such that

Ax, x ≥γx2, ∀x∈H. 1.15

A typical problem is to minimize a quadratic function over the set of the fixed points of some nonexpansive mapping on a real Hilbert spaceH:

min

x∈E

1

2Ax, x − x, b, 1.16

whereAis some linear,Eis the fixed point set of a nonexpansive mappingSonHandbis a point inH. LetAbe a strongly positive linear bounded map onHwith coefficientγ. In 2006, Marino and Xu31studied the following general iterative method:

They proved that if the sequencenof parameters appropriate conditions, then the sequence

xngenerated by1.17converges strongly toqPFSI−Aγfq. Recently, Plubtieng and Punpaeng32proposed the following iterative algorithm:

φun, y

1

rn

y−un, un−xn

≥0, ∀y∈H,

xn1nγfxn I−nASun.

1.18

They proved that if the sequences{n}and{rn}of parameters satisfy appropriate condition,

then both sequences {xn} and {un} converge to the unique solution q of the variational

inequality

A−γfq, x−q≥0, ∀x∈FS∩EPφ, 1.19

which is the optimality condition for the minimization problem

min

x∈FS∩EPφ

1

2Ax, x −hx, 1.20

wherehis a potential function forγfi.e.,hx γfxforx∈H.

On the other hand, for finding a common element of the set of fixed points of ak -strict pseudo-contraction mapping and the set of solutions of an equilibrium problem in a real Hilbert space, Liu33introduced the following iterative scheme:

φun, y

1

rn

y−un, un−xn

≥0, ∀y∈E,

ynβnun

1−βn

Sun,

xn1nγfxn I−nAun, ∀n≥1,

1.21

whereSis ak-strict pseudo-contraction mapping and{n},{βn}are sequences in0, 1. They

proved that under certain appropriate conditions over{n},{βn}, and {rn}, the sequences

{xn}and{un}converge strongly to someq ∈ FS∩EPφ, which solves some variational

inequality problems.

In 2008, Ceng and Yao5introduced an iterative scheme for finding a common fixed point of a finite family of nonexpansive mappings and the set of solutions of a problem1.3 in Hilbert spaces and obtained the strong convergence theorem which used the following condition:

H K : E → R isη-strongly convex with constantσ > 0 and its derivativeK is sequentially continuous from weak topology to strong topology. We note that the condition

Hfor the functionK : E → Ris a very strong condition. We also note that the condition

H does not cover the caseKx x2_{/2 and ηx, y x−y for eachx, y ∈ E×E.}

the variational inequality for an inverse strongly monotone mapping in Hilbert spaces. They obtained a strong convergence theorem except the conditionHfor the sequences generated by these processes.

In 2009, Qin et al.35introduced a general iterative scheme for finding a common element of the set of common solution of generalized equilibrium problems, the set of a common fixed point of a family of infinite nonexpansive mappings in Hilbert spaces. Let

{xn}be the sequence generated iterative by the following algorithm:

x1∈E, un∈E, vn∈E,

φ1un, u Cxn, u−un

1

ru−un, un−xn ≥0, ∀u∈E,

φ2vn, v Bxn, v−vn

1

sv−vn, vn−xn ≥0, ∀v∈E,

ynδnun 1−δnvn,

xn1nfxn βnxnγnWnyn, ∀n≥1.

1.22

They proved that under certain appropriate conditions imposed on{n},{βn},{γn}and{δn},

the sequence {xn}generated by 1.22converges strongly to q ∈ ∩∞_n1FTn∩EPφ1, C∩

EPφ2, B, whereqP∩∞

n1FTn∩EPφ1,C∩EPφ2,Bfq.

In the present paper, motivated and inspired by Qin et al. 35, Plubtieng and Punpaeng32, Peng and Yao17, R. Wangkeeree and R. Wangkeeree34, and Y. Yao and J.-C. Yao30, we introduce a new approximation iterative scheme for finding a common element of the set of fixed points of strict pseudo-contractions, the set of common solutions of the system of a mixed equilibrium problem, and the set of common solutions of the variational inequalities with inverse strongly monotone mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by these processes under some parameter controlling conditions. Moreover, we apply our results for solving convex feasibility problems in Hilbert spaces. The results in this paper extend and improve some well-known results in17,30,32,34,35.

2. Preliminaries

LetHbe a real Hilbert space andEbe a closed convex subset ofH. In a real Hilbert spaceH, it is well known that

λx 1−λy2λx2 1−λy2−λ1−λx−y2 2.1

for allx, y∈Handλ∈0,1.

For anyx∈H, there exists aunique nearest pointinE, denoted byPEx, such that

x−PEx ≤x−y, ∀y∈E. 2.2

It is well known thatPEis a firmly nonexpansive mapping ofHontoE, that is,

x−y, PEx−PEy

≥PEx−PEy2, ∀x, y∈H. 2.3

Further, for anyx∈Handz∈E,zPExif and only ifx−z, z−y ≥0, for ally∈E.

Moreover,PExis characterized by the following properties:PEx∈Eand

x−PEx, y−PEx

≤0, 2.4

x−y2≥ x−PEx2y−PEx2 2.5

for allx∈H, y∈E.

It is easy to see that the following is true:

u∈VIE, B⇐⇒uPEu−λBu, λ >0. 2.6

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1see36. LetE,·,·be an inner product space. Then, for allx, y, z∈Eandα, β, γ ∈

0,1withαβγ1, one has

_αxβyγz2

αx2βy2γz2−αβx−y2−αγx−z2−βγy−z2. 2.7

Lemma 2.2see31. Assume thatA is a strongly positive linear bounded operator onH with coefficientγ >0and0< ρ≤ A−1_{. ThenI−ρA ≤1−ργ.}

Lemma 2.3see4. LetEbe a nonempty closed convex subset of a real Hilbert spaceH and let S : E → Ebe ak-strict pseudo-contraction with a fixed point. ThenFS is closed and convex. DefineSk : E → EbySk kx 1−kSx for eachx ∈ E. ThenSkis nonexpansive such that

FSk FS.

Lemma 2.4see37. LetXbe a uniformly convex Banach spaces,Ebe a nonempty closed convex subset ofXandS:E → Ebe a nonexpansive mapping. ThenI−Sis demi-closed at zero.

Lemma 2.5see38. LetEbe a nonempty closed convex subset of strictly convex Banach spaceX. Let{Tn:n∈N}be a sequence of nonexpansive mappings onE. Suppose∩∞_n1FTnis nonempty. Let

δnbe a sequence of positive numbers with

_∞

n1δn1. Then a mappingSonEcan be defined by

Sx ∞

n1

δnTnx 2.8

In order to solve the mixed equilibrium problem, the following assumptions are given for the bifunctionφ,ϕand the setE:

A1φx, x 0 for allx∈E;

A2φis monotone, that is,φx, y φy, x≤0 for allx, y∈E;

A3for eachx, y, z∈E, limt→0φtz 1−tx, y≤φx, y;

A4for eachx∈E, y→φx, yis convex and lower semicontinuous;

A5for eachy∈E, x→φx, yis weakly upper semicontinuous;

B1for eachx∈Handr >0, there exist abounded subsetDx⊆Eandyx∈Esuch that

for anyz∈E\Dx,

φz, yx

ϕyx

−ϕz 1 r

yx−z, z−x

<0; 2.9

B2Eis a bounded set.

Lemma 2.6see39. LetEbe a nonempty closed convex subset ofH. Letφ :E×E → Rbe a bifunction satisfies (A1)–(A5) and letϕ : E → R ∪ {∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 andx ∈ H, define a mapping Trφ,ϕ:H → Eas follows:

Trφ,ϕx

z∈E:φz, yϕy−ϕz 1 r

y−z, z−x≥0,∀y∈E

2.10

for allz∈H. Then, the following holds:

ifor eachx∈H,Trφ,ϕx/∅; iiTrφ,ϕis single-valued;

iiiTrφ,ϕis firmly nonexpansive, that is, for anyx, y∈H,

Trφ,ϕx−Trφ,ϕy2≤

Trφ,ϕx−Trφ,ϕy, x−y

; 2.11

ivFTrφ,ϕ MEPφ, ϕ;

vMEPφ, ϕis closed and convex.

Remark 2.7. Ifϕ0, thenTrφ,ϕis rewritten asTrφ.

Remark 2.8. We remark thatLemma 2.6is not a consequence of Lemma 3.1 in5, because the condition of the sequential continuity from the weak topology to the strong topology for the derivativeKof the functionK:E → Rdoes not cover the caseKx x2/2.

Lemma 2.9see40. Let{xn}and{ln}be bounded sequences in a Banach spaceXand let{βn}be

a sequence in0,1with0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1. Supposexn1 1−βnlnβnxn

Lemma 2.10see41. Assume that{an}is a sequence of nonnegative real numbers such that

an1≤

1−n

anσn, n≥1, 2.12

where{n}is a sequence in0,1and{σn}is a sequence inRsuch that

1∞_n1n∞,

2lim sup_{n→ ∞}σn/n≤0or∞n1|σn|<∞.

Thenlimn→ ∞an0.

Lemma 2.11. LetHbe a real Hilbert space. Then for allx, y∈H,

xy2≤ x22y, xy. 2.13

3. Main Results

In this section, we will use the new approximation iterative method to prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudo-contractions, the set of common solutions of the system of a mixed equilibrium problem and the set of a common solutions of the variational inequalities with inverse strongly monotone mappings in a real Hilbert space.

Theorem 3.1. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. Letφ1andφ2be two bifunctions fromE×EtoRsatisfying (A1)–(A5) and letϕ:E → R ∪ {∞}be a proper lower semicontinuous and convex function. LetC :E → Hbe anξ-inverse strongly monotone mapping and B : E → H be anβ-inverse strongly monotone mapping. Letf : E → Ebe a contraction mapping with coefficientα0< α <1and letAbe a strongly positive linear bounded operator onH with coefficientγ >0and0< γ < γ/α. LetS:E → Ebe ak-strict pseudo-contraction with a fixed point. Define a mappingSk:E → EbySkxkx 1−kSx, for allx∈E. Assume that

Θ:FS∩VIE, C∩VIE, B∩MEPφ1, ϕ

∩MEPφ2, ϕ

/

∅. 3.1

Assume that either (B1) or (B2). Let{xn}be a sequence generated by the following iterative algorithm:

x1∈E, un∈E, vn∈E,

unTrφ1,ϕxn,

vnTsφ2,ϕxn,

znPE

un−μnCun

,

ynPEvn−λnBvn,

knanSkxnbnyncnzn,

xn1nγfxn βnxn

1−βn

I−nA

kn, ∀n≥1,

where{n},{βn},{an},{bn}, and{cn}are sequences in0,1and{λn},{μn}are positive sequences.

Assume that the control sequences satisfy the following restrictions:

C1anbncn 1,

C2limn→ ∞n0and∞n1n∞, C30<lim infn→ ∞βn≤lim supn→ ∞βn<1, C4limn→ ∞|λn1−λn|limn→ ∞|μn1−μn|0,

C5d≤λn≤2β,e≤μn≤2ξ, whered, eare two positive constants,

C6limn→ ∞ana,limn→ ∞bnbandlimn→ ∞cnc, for somea, b, c∈0,1.

Then, {xn} converges strongly to a point q ∈ Θwhich is the unique solution of the variational

inequality

A−γfq, x−q≥0, ∀x∈Θ 3.3

or equivalentqPΘI−Aγfq, wherePis a metric projection mapping formHontoΘ.

Proof. Sincen → 0, asn → ∞, we may assume, without loss of generality, thatn ≤ 1−

βnA−1_{for alln∈N. ByLemma 2.2, we know that if 0≤ρ≤ A}−1_{, thenI−ρA ≤1−ργ.}

We will assume thatI−A ≤ 1−γ. SinceAis a strongly positive bounded linear operator onH, we have

Asup{|Ax, x|:x∈H,x1}. 3.4

Observe that

1−βn

I−nA

x, x1−βn−nAx, x

≥1−βn−nA

≥0,

3.5

so this shows that1−βnI−nAis positive. It follows that

1−βn

I−nAsup1−βn

I−nA

x, x:x∈H,x1

sup1−βn−nAx, x:x∈H,x1

≤1−βn−nγ.

3.6

We divide the proof into seven steps.

Sincefbe a contraction ofHinto itself withα∈0,1. Then, we have

PΘI−Aγfx−PΘI−Aγfy≤I−Aγfx−I−Aγfy

≤I−Ax−yγfx−fy

≤1−γx−yγαx−y

1−γ−γαx−y, ∀x, y∈H.

3.7

Since 0<1−γ−γα<1, it follows thatPΘI−Aγfis a contraction ofHinto itself. Therefore the Banach Contraction Mapping Principle implies that there exists a unique elementq∈H

such thatqP_ΘI−Aγfq.

Step 2. We claim thatI−λnBis nonexpansive.

Indeed, from theβ-inverse strongly monotone mapping definition onBand condition

C5, we have

_{I−λnBx−I−λnBy}2

x−y−λnBx−By2

x−y2−2λn

x−y, Bx−Byλ2_nBx−By2

≤x−y2−2λnβBx−Byλ2nBx−By

2

x−y2λn

λn−2βBx−By2

≤x−y2,

3.8

whereλn ≤ 2β, for alln ∈ N implies that the mappingI−λnBis nonexpansive and so is,

I−μnC.

Step 3. We claim that{xn}is bounded.

Indeed, letp∈ΘandLemma 2.6, we obtain

pPE

p−λnBp

PE

p−μnCp

Tφ1,ϕ

r pT

φ2,ϕ

s p. 3.9

Note thatunTrφ1,ϕxn∈domϕandvnTsφ2,ϕxn∈domϕ, we have

un−pTrφ1,ϕxn−Trφ1,ϕp≤xn−p,

vn−pTsφ2,ϕxn−Tsφ2,ϕp≤xn−p.

SinceI−λnBandI−μnCare nonexpansive and from2.6, we have

_zn−pPE

un−μnCun

−PE

p−μnCp

≤un−μnCun

−p−μnCp

I−μnC

un−

I−μnC

p

≤un−p≤xn−p,

yn−pPEvn−λnBvn−PE

p−λnBp≤vn−p≤xn−p.

3.11

FromLemma 2.3, we have thatSkis nonexpansive withFSk FS. It follows that

kn−panSkxnbnyncnzn−p

≤anSkxn−pbnyn−pcnzn−p

≤anxn−pbnxn−pcnxn−pxn−p.

3.12

It follows that

xn1−pn

γfxn−Apβn

xn−p

1−βn

I−nA

kn−p

≤1−βn−nγkn−pβnxn−pnγfxn−Ap

≤1−βn−nγxn−pβnxn−pnγfxn−Ap

≤1−nγxn−pnγfxn−f

pnγf

p−Ap

≤1−nγxn−pnγαxn−pnγf

p−Ap

1−γ−αγnxn−p

γ−αγn

γfp−Ap γ−αγ

≤max

_xn−p,γfp−Ap γ−αγ

.

3.13

By simple induction, we have

_{xn−p≤maxx1−p,}γfp−Ap γ−αγ

, ∀n∈N. 3.14

Step 4. We claim that limn→ ∞xn1−xn0.

Observing that un Trφ1,ϕxn ∈ domϕ and un1 Trφ1,ϕxn1 ∈ domϕ, by the

nonexpansiveness ofTφ1,ϕ

r , we get

un1−unTrφ1,ϕxn1−Trφ1,ϕxn≤ xn1−xn. 3.15

Similarly, letvnTsφ2,ϕxn ∈domϕandvn1Tsφ2,ϕxn1∈domϕ, we have

vn1−vnTsφ2,ϕxn1−Tsφ2,ϕxn≤ xn1−xn. 3.16

FromznPEun−μnCunandyn PEvn−λnBvn, we compute

zn1−znPE

un1−μn1Cun1

−PE

un−μnCun

≤un1−μn1Cun1

−un−μnCun un1−μn1Cun1

−un−μn1Cun

μn−μn1

Cun

≤un1−μn1Cun1

−un−μn1Cunμn−μn1Cun

I−μn1C

un1−

I−μn1C

unμn−μn1Cun

≤ un1−unμn−μn1Cun

≤ xn1−xnμn−μn1Cun.

3.17

Similarly, we have

_{yn1−ynPEvn1−λn1Bvn1−PEvn−λnBvn}

≤ vn1−vn|λn−λn1|Bvn

≤ xn1−xn|λn−λn1|Bvn.

3.18

Observing that

kn anSkxnbnyncnzn,

kn1an1Skxn1bn1yn1cn1zn1,

3.19

we obtain

kn1−kn ≤an1Skxn1−Skxn|an1−an|Skxnbn1yn1−yn |bn1−bn|yncn1zn1−zn|cn1−cn|zn

≤an1xn1−xn|an1−an|Skxnbn1yn1−yn |bn1−bn|yncn1zn1−zn|cn1−cn|zn.

Substituting3.17and3.18into3.20, we have

kn1−kn ≤an1xn1−xn|an1−an|Skxnbn1{xn1−xn|λn−λn1|Bvn} cn1

xn1−xnμn−μn1Cun

|bn1−bn|yn|cn1−cn|zn

≤ xn1−xnM1

|an1−an||bn1−bn||cn1−cn||λn−λn1|μn−μn1,

3.21

whereM1 is an appropriate constant such that M1 max{sup_n≥1Skxn,yn,zn,Bvn,

Cun}.

Puttingxn1 1−βnlnβnxn, for alln≥1, we have

ln

xn1−βnxn

1−βn

nγfxn

1−βn

I−nA

kn

1−βn .

3.22

Then, we compute

ln1−ln

n1γfxn1

1−βn1

I−n1A

kn1

1−βn1

−nγfxn

1−βn

I−nA

kn

1−βn

n1

1−βn1

γfxn1− n

1−βn

γfxn kn1−kn

n

1−βn

Akn−

n1

1−βn1 Akn1

n1

1−βn1

γfxn1−Akn1

n

1−βn

Akn−γfxn

kn1−kn.

3.23

It follows from3.21and3.23, that

ln1−ln − xn1−xn

≤ n1

1−βn1

_γfxn1Akn1

n

1−βn

Aknγfxn

kn1−kn − xn1−xn

≤ n1

1−βn1

_γfxn1Akn1

n

1−βn

Aknγfxn

M1

|an1−an||bn1−bn||cn1−cn||λn−λn1|μn−μn1.

This together withC2,C3,C4, andC6imply that

lim sup

n→ ∞ ln1−ln − xn1−xn≤0. 3.25

Hence, byLemma 2.9, we obtainln−xn → 0 asn → ∞. It follows that

lim

n→ ∞xn1−xnnlim→ ∞

1−βn

ln−xn0. 3.26

So, we also get

lim

n→ ∞un1−unnlim→ ∞vn1−vnnlim→ ∞zn1−zn

lim

n→ ∞yn1−ynnlim→ ∞kn1−kn0.

3.27

Observe that

xn1−xnn

γfxn−Axn

1−βn−nγ

kn−xn. 3.28

By conditionC2and3.26, we have

limn→ ∞kn−xn0. 3.29

Step 5. We claim that the following statements hold:

s1limn→ ∞xn−vn0; s2limn→ ∞xn−un0; s3limn→ ∞xn−yn0; s4limn→ ∞xn−zn0.

Indeed, pick anyp∈Θ, to obtain

_un−p2 Tφ1,ϕ

r xn−Trφ1,ϕp

2

≤Tφ1,ϕ

r xn−Trφ1,ϕp, xn−p

un−p, xn−p

1

2

un−p2xn−p2− xn−un2

.

3.30

Therefore,

_un−p2_≤

Similarly, we have

_vn−p2_≤

xn−p2− xn−vn2. 3.32

Note that

_kn−p2

anSkxnbnyncnzn−p2

≤anSkxn−p2bnyn−p2cnzn−p2

≤anxn−p2bnvn−pcnun−p.

3.33

Substituting3.31and3.32into3.33, we obtain

kn−p2≤anxn−p2bnvn−pcnun−p

≤anxn−p2bn

xn−p2− xn−vn2

cn

xn−p2− xn−un2

xn−p2−bnxn−vn2−cnxn−un2.

3.34

FromLemma 2.1,3.2and3.34, we obtain

xn1−p2nγfxn−Ap βnxn−p 1−βnI−nAkn−p2

≤nγfxn−Ap2βnxn−p2

1−βn−nγkn−p2

≤nγfxn−Ap2βnxn−p2

1−βn−nγxn−p2−bnxn−vn2−cnxn−un2

nγfxn−Ap2

1−nγxn−p2−

1−βn−nγ

bnxn−vn2

−1−βn−nγ

cnxn−un2

≤nγfxn−Ap2xn−p2−

1−βn−nγ

bnxn−vn2

−1−βn−nγ

cnxn−un2.

3.35

It follows that

1−βn−nγ

cnxn−un2≤nγfxn−Ap2xn−p2−xn1−p2

≤nγfxn−Ap2xn1−xnxn−pxn1−p.

FromC2,C6, and3.26, we also have

lim

n→ ∞xn−un0. 3.37

Similarly, using3.35again, we have

1−βn−nγ

bnxn−vn2≤nγfxn−Ap2xn−p2−xn1−p2

≤nγfxn−Ap2xn1−xnxn−pxn1−p.

3.38

FromC2,C6, and3.26, we also have

lim

n→ ∞xn−vn0. 3.39

From3.37and3.39, we have

lim

n→ ∞un−vn0. 3.40

Forp∈Θ, we compute

_zn−p2

PEun−μnCun−PEp−μnCp2

≤un−μnCun−p−μnCp2

un−p−μnCun−Cp2

≤un−p2−2μn

un−p, Cun−Cp

μ2_nCun−Cp2

≤xn−p2μn

μn−2ξCun−Cp2

≤xn−p2−μn

2ξ−μnCun−Cp2.

3.41

Similarly, we have

_yn−p2_≤

xn−p2−λn

Substituting3.41and3.42into3.33, we also have

_kn−p2 _≤

anSkxn−p2bnyn−p2cnzn−p2

≤anxn−p2bn

xn−p2−λn

2β−λnBvn−Bp2

cn

xn−p2−μn

2ξ−μnCun−Cp2

xn−p2−bnλn

2β−λnBvn−Bp2−cnμn

2ξ−μnCun−Cp2.

3.43

On the other hand, we note that

xn1−p2

≤nγfxn−Ap2βnxn−p2

1−βn−nγkn−p2

≤nγfxn−Ap2βnxn−p2

1−βn−nγxn−p2−bnλn

2β−λnBvn−Bp2−cnμn

2ξ−μnCun−Cp2

nγfxn−Ap2

1−nγxn−p2−

1−βn−nγ

bnλn

2β−λnBvn−Bp2

−1−βn−nγ

cnμn

2ξ−μnCun−Cp2

≤nγfxn−Ap2xn−p2−

1−βn−nγ

bnλn

2β−λnBvn−Bp2

−1−βn−nγ

cnμn

2ξ−μnCun−Cp2.

3.44

It follows that

1−βn−nγ

cnμn

2ξ−μnCun−Cp2

≤nγfxn−Ap2xn−p2−xn1−p2

≤nγfxn−Ap2xn1−xnxn−pxn1−p.

3.45

FromC2,C5,C6, and3.26, we have

lim

n→ ∞Cun−Cp0. 3.46

Thanks to3.44, we also have

1−βn−nγ

bnλn

2β−λnBvn−Bp2

≤nγfxn−Ap2xn1−xnxn−pxn1−p.

FromC2,C5,C6, and3.26, we obtain

lim

n→ ∞Bvn−Bp0. 3.48

Observe that

_yn−p2

PEvn−λnBvn−PEp−λnBp2

≤I−λnBvn−I−λnBp, yn−p

1

2

I−λnBvn−I−λnBp2yn−p2− I−λnBvn−I−λnBp−

yn−p

2

≤ 1

2

vn−p2yn−p2−vn−yn

−λn

Bvn−Bp2

≤ 1

2

xn−p2yn−p2−vn−yn2−λ2nBvn−Bp22λn

vn−yn, Bvn−Bp

,

3.49

and hence

yn−p2≤xn−p2−vn−yn22λnvn−ynBvn−Bp. 3.50

Similarly, we can obtain that

_zn−p2_≤

xn−p2− un−zn22μnun−znCun−Cp. 3.51

Substituting3.50and3.51into3.33, we also have

kn−p2 ≤anSkxn−p2bnyn−p2cnzn−p2

≤anxn−p2bn

xn−p2−vn−yn22λnvn−ynBvn−Bp

cn

xn−p2− un−zn22μnun−znCun−Cp

xn−p2−bnvn−yn22bnλnvn−ynBvn−Bp

−cnun−zn22μnun−znCun−Cp.

On the other hand, we have

_xn1−p2_≤

nγfxn−Ap2βnxn−p2

1−βn−nγkn−p2

≤nγfxn−Ap2βnxn−p2

1−βn−nγxn−p2−bnvn−yn22bnλnvn−ynBvn−Bp

−cnun−zn22μnun−znCun−Cp

nγfxn−Ap2

1−nγxn−p2−

1−βn−nγ

bnvn−yn2

2bn

1−βn−nγ

λnvn−ynBvn−Bp−

1−βn−nγ

cnun−zn2

2cn

1−βn−nγ

μnun−znCun−Cp

≤nγfxn−Ap2xn−p2−

1−βn−nγ

bnvn−yn2

2bn

1−βn−nγ

λnvn−ynBvn−Bp−

1−βn−nγ

cnun−zn2

2cn

1−βn−nγ

μnun−znCun−Cp

3.53

and hence

1−βn−nγ

bnvn−yn2≤nγfxn−Ap2xn−p2−xn1−p2

2bn

1−βn−nγ

λnvn−ynBvn−Bp

2cn

1−βn−nγ

μnun−znCun−Cp

≤nγfxn−Ap2xn1−xnxn−pxn1−p

2bn

1−βn−nγ

λnvn−ynBvn−Bp

2cn

1−βn−nγ

μnun−znCun−Cp.

3.54

FromC2,C6,3.26,3.46, and3.48, we also have

lim

Similarly, using3.53again, we can prove

lim

n→ ∞un−zn0. 3.56

From3.39and3.55, we also have

lim

n→ ∞xn−yn0. 3.57

From3.37and3.56, we have

lim

n→ ∞xn−zn0. 3.58

Step 6. We claim that lim sup_{n→ ∞}A−γfq, q−xn ≤0, whereqPΘI−Aγfqis the

unique solution of the variational inequalityA−γfq, x−q ≥0, for allx∈Θ. To show this inequality, we choose a subsequence{xni}of{xn}such that

lim sup

n→ ∞

A−γfq, q−xn lim

i→ ∞

A−γfq, q−xni

. _3.59

Since{xni}is bounded, there exists a subsequence{xnij}of{xni}which converges weakly to z∈E. Without loss of generality, we can assume thatxni z. We claim thatz∈Θ.

a1First, we prove thatz∈FS∩VIE, C∩VIE, B. Assume also thatλn → λ∈d,2βandμn → μ∈e,2ξ.

Define a mappingΩ:E → Eby

ΩxaSkxbPE

1−μCxcPE1−λBx, ∀x∈E, 3.60

where limn→ ∞an a, limn→ ∞bn b, and limn→ ∞cn c, for some a, b, c ∈ 0,1. From

Lemma 2.5, we have thatΩis nonexpansive with

FΩ FSk∩FPE

Notice that

Ωxn−xn

≤ Ωxn−knkn−xn

≤aSkxnbPE1−λBxncPE

1−μCxn

−anSkxnbnyncnznkn−xn

≤ |a−an|SkxnbPEI−λBxn−bnPEI−λnBxn bnPEI−λnBxn−bnPEI−λnBvncPE

I−μCxn−cnPE

1−μnC

xn cnPE

I−μnB

xn−cnPE

I−μnC

unkn−xn

≤ |a−an|Skxn|b−bn|xn|bnλn−bλ|Bxn|c−cn|xncnμn−cμCxn bnxn−vncnxn−unkn−xn

≤K1

|a−an||b−bn||c−cn||bλ−bnλn|cμ−cnμn bnxn−vncnxn−unkn−xn,

3.62

whereK1is an appropriate constant such that

K1max

sup

n≥1

xn,sup n≥1

Bxn,sup n≥1

Cxn,sup n≥1

Skxn

. 3.63

FromC6,3.37,3.39, and3.29, we obtain

lim

n→ ∞xn−Ωxn0. 3.64

ByLemma 2.4, we havez∈FΩ, that is,z∈FS∩VIE, C∩VIE, B.

a2Now, we prove thatz∈FS∩MEPφ1, ϕ∩MEPφ2, ϕ.

Define a mappingQ:E → Eby

QxaSkxbTrφ1,ϕxcT φ2,ϕ

s x, ∀x∈E, 3.65

where limn→ ∞an a, limn→ ∞bn b, and limn→ ∞cn c, for some a, b, c ∈ 0,1. From

Lemma 2.5, we have thatQis nonexpansive with

FQ FSk∩FTφ1,ϕ

r

∩FTφ2,ϕ

s

FS∩MEPφ1, ϕ

∩φ2, ϕ

On the other hand, we have

Qxn−xn ≤ Qxn−knkn−xn

≤aSkxnbTrφ1,ϕxncTsφ2,ϕxn

−anSkxnbnyncnznkn−xn

≤ |a−an|Skxn|b|Trφ1,ϕxn|c|Tsφ2,ϕxn

|bn−b|PEI−λnBvn|cn−c|PE

I−μnC

un |b|PEI−λnBvn|c|PE

I−μnC

unkn−xn

≤K2|a−an||b−bn||c−cn|2|b|2|c| kn−xn,

3.67

whereK2is an appropriate constant such that

K2 max

sup

n≥1

Tφ1,ϕ

r xn,sup

n≥1

Tφ2,ϕ

s xn,sup

n≥1

Skxn,

sup

n≥1

_Tφ1,ϕ

r xnPEI−λnBvn

,sup

n≥1

_Tφ2,ϕ

s xnPE

I−μnC

un

.

3.68

FromC6and3.29, we obtain

lim

n→ ∞xn−Qxn0. 3.69

SincePΘI−Aγfqis a contraction with the coefficientα∈0,1, there exists a unique fixed point. We useqto denote the unique fixed point to the mappingP_ΘI−Aγfq, that is,q PΘI−Aγfq. Since{xni}is bounded, There exists a subsequence{xni}of{xn}

which converges weakly toz. Without loss of generality, we may assume that{xni} z. It

follows from3.69, that

lim

n→ ∞xni −Qxni0. 3.70

It follows from Lemma 2.4, we obtain that z ∈ FQ. Hencez ∈ Θ, whereΘ : FS∩

VIE, C∩VIE, B∩MEPφ1, ϕ∩MEPφ2, ϕ. From3.59and2.4, we arrive at

lim sup

n→ ∞

A−γfq, q−xn

lim sup

n→ ∞

A−γfq, q−xni

A−γfq, q−z≤0.

On the other hand, we have

A−γfq, q−xn1

A−γfq, xn−xn1

A−γfq, q−xn

≤A−γfqxn−xn1

A−γfq, q−xn

.

3.72

From3.26and3.71, we obtain that

lim sup

n→ ∞

A−γfq, q−xn1

≤0. _3.73

Step 7. We claim that limn→ ∞xn−q0.

Indeed, by3.2and using Lemmas2.2and2.11, we observe that

xn1−z2nγfxn βnxn 1−βnI−nAkn−q2

≤1−βn

1−βn

I−nA

1−βn

kn−q βn

xn−q

2

2n

γfxn−Aq, xn1−q

≤1−βn

1−βn

I−nA

1−βn

kn−q

2

βnxn−q2

2nγfxn−f

q, xn1−q2n

γfq−Aq, xn1−q

≤1−βn

1−βnI−nA

1−βn

kn−q

2βnxn−q2 2nγαxn−qxn1−q2n

γfq−Aq, xn1−q

≤ 1−βnI−nA 2

1−βn

kn−q2βnxn−q2

nγα

xn−q2xn1−q2

2n

γfq−Aq, xn1−q

≤ 1−βnI−nA 2

1−βn

_xn−q2

βnxn−q2

nγα

xn−q2xn1−q2

2n

γfq−Aq, xn1−q

≤

1−βn

−γn

2

1−βn βnnγα

_xn−q2

nγαxn1−q22n

γfq−Aq, xn1−q

≤

1−2γ−αγn

γ22

n

1−βn

xn−q2

nγαxn1−q22n

γfq−Aq, xn1−q

3.74

which implies that

xn1−z2≤

1−2

γ−αγn

1−αγn

xn−q2

n

1−αγn

γ22

n

1−βn

xn−q22n

γfq−Aq, xn1−q

. 3.75 Taking σn n

1−αγn

γ22

n

1−βn

_xn−q2

2n

γfq−Aq, xn1−q

,

n

2γ−αγn

1−αγn

.

3.76

Then we can rewrite3.75as

xn1−z2≤

1−n

xn−z2σn. 3.77

We have lim sup_{n→ ∞}σn/n ≤ 0. ApplyingLemma 2.10 to 3.77, we conclude that {xn}

converges strongly toqin norm. This completes the proof.

If the mappingSis nonexpansive, thenSkS0S. We can obtain the following result

fromTheorem 3.1immediately.

Corollary 3.2. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. Letφ1andφ2be two bifunction fromE×EtoRsatisfying (A1)–(A4) and letϕ:E → R ∪ {∞}be a proper lower semicontinuous and convex function. LetC :E → Hbe anξ-inverse strongly monotone mapping and B : E → H be anβ-inverse strongly monotone mapping. Letf : E → Ebe a contraction mapping with coefficientα0< α <1and letAbe a strongly positive linear bounded operator onH with coefficientγ >0and0< γ < γ/α. LetS:E → Ea nonexpansive mapping with a fixed point. Assume that

Θ:FS∩VIE, C∩VIE, B∩MEPφ1, ϕ

∩MEPφ2, ϕ

/

Assume that eitherB1orB2. Let{xn}be a sequence generated by the following iterative algorithm:

x1∈E, un∈E, vn∈E,

unTrφ1,ϕxn,

vnTsφ2,ϕxn,

znPE

un−μnCun

,

ynPEvn−λnBvn,

knanSxnbnyncnzn,

xn1nγfxn βnxn

1−βn

I−nA

kn, ∀n≥1,

3.79

where{n},{βn},{an},{bn}, and{cn}are sequences in0,1and{λn},{μn}are positive sequences.

Assume that the control sequences satisfy the following restrictions:

C1anbncn 1,

C2limn→ ∞n0and

_∞

n1n∞,

C30<lim infn→ ∞βn≤lim supn→ ∞βn<1, C4limn→ ∞|λn1−λn|limn→ ∞|μn1−μn|0,

C5d≤λn≤2β,e≤μn≤2ξ, whered, eare two positive constants,

C6limn→ ∞ana,limn→ ∞bnbandlimn→ ∞cnc, for somea, b, c∈0,1.

Then, {xn} converges strongly to a point q ∈ Θwhich is the unique solution of the variational

inequality

A−γfq, x−q≥0, ∀x∈Θ 3.80

or equivalentqPΘI−Aγfq, wherePis a metric projection mapping formHontoΘ.

Finally, we consider the following convex feasibility problemCFP:

finding anx∈

N

i1

Ci, 3.81

where N ≥ 1 is an integer and each Ci is assumed to be the of solutions of equilibrium

problem with the bifunction φi, i 1,2,3, . . . , N and the solution set of the variational

inequality problem. There is a considerable investigation on CEP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration42,43, computer tomography44, and radiation therapy treatment planning45.

The following result can be concluded fromTheorem 3.1easily.

Theorem 3.3. Let E be a nonempty closed convex subset of a real Hilbert space H. Let be aφi

semicontinuous and convex function. LetCi :E → Hbe anξi-inverse strongly monotone mapping

for eachi∈ {1,2,3, . . . , N}. Letf:E → Ebe a contraction mapping with coefficientα0< α <1

and letAbe a strongly positive linear bounded operator onHwith coefficientγ >0and0< γ < γ/α. LetS:E → Ebe ak-strict pseudo-contraction with a fixed point. Define a mappingSk:E → Eby

Skxkx 1−kSx, for allx∈E. Assume that

F:FS∩

_N

i1

VIE, Ci

∩

_N

i1

MEPφi, ϕ

/

∅. 3.82

Assume that eitherB1orB2. Let{xn}be a sequence generated by the following iterative algorithm:

x1∈E, un,i∈E,

φiun,i, ui ϕui−ϕun,i 1

ri

ui−un,i, un,i−xn ≥0, ∀ui ∈E, ∀i∈ {1,2,3, . . . , N},

knαn,0Skxn N

i1 αn,iPE

un,i−μn,iCiun,i

,

xn1nγfxn βnxn

1−βn

I−nA

kn, ∀n≥1,

3.83

whereαn,0, αn,1, αn,2, αn,3, . . . , αn,N∈0,1such thatNi0αn,i1,{μn,i}are positive sequences and

{n},{βn}are sequences in0,1. Assume that the control sequences satisfy the following restrictions:

C1limn→ ∞n0and∞n1n∞, C20<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1, C3limn→ ∞|μn1,i−μn,i|0, for each1≤i≤N,

C4ei≤μn,i≤2ξi, whereeiis some positive constant for each1≤i≤N, C5limn→ ∞αn,iαi∈0,1, for each1≤i≤N.

Then, {xn} converges strongly to a pointq ∈ F which is the unique solution of the variational

inequality

A−γfq, x−q≥0, ∀x∈ F 3.84

or equivalentqPFI−Aγfq, wherePis a metric projection mapping formHontoF.

Acknowledgments

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