A New Hybrid Algorithm for Variational Inclusions, Generalized Equilibrium Problems, and a Finite Family of Quasi-Nonexpansive Mappings

Fixed Point Theory and Applications Volume 2009, Article ID 350979,20pages doi:10.1155/2009/350979

Research Article

A New Hybrid Algorithm for Variational

Inclusions, Generalized Equilibrium Problems, and

a Finite Family of Quasi-Nonexpansive Mappings

Prasit Cholamjiak and Suthep Suantai

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Correspondence should be addressed to Suthep Suantai,[email protected]

Received 12 June 2009; Accepted 28 September 2009

Recommended by Naseer Shahzad

We proposed in this paper a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasi-nonexpansive mappings, the set of solutions of variational inclusion, and the set of solutions of generalized equilibrium problems. Some strong convergence results were derived by using the concept ofW-mappings for a finite family of quasi-nonexpansive mappings. Strong convergence results are derived under suitable conditions in Hilbert spaces.

Copyrightq2009 P. Cholamjiak and S. Suantai. 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 inducted norm · , and letCbe a nonempty closed and convex subset ofH. Then, a mappingT:C → Cis said to be

1nonexpansiveifTx−Ty ≤ x−y, for allx, y∈C;

2quasi-nonexpansiveifTx−p ≤ x−p, for allx∈Candp∈FT;

3L-Lipschitzianif there exists a constantL >0 such thatTx−Ty ≤Lx−y, for all

x, y∈C. We denoted byFTthe set of fixed points ofT.

In 1953, Mann1introduced the following iterative procedure to approximate a fixed point of a nonexpansive mappingT in a Hilbert spaceH:

xn1αnxn 1−αnTxn, ∀n∈N, 1.1

However, we note that Mann’s iteration process1.1has only weak convergence, in general; for instance, see2,3.

Many authors attempt to modify the process 1.1 so that strong convergence is guaranteed that has recently been made. Nakajo and Takahashi4proposed the following modification which is the so-called CQ method and proved the following strong convergence theorem for a nonexpansive mappingTin a Hilbert spaceH.

Theorem 1.1see4. LetCbe a nonempty closed convex subset of a Hilbert spaceHand letTbe a nonexpansive mapping ofCinto itself such thatFT/∅. Suppose thatx1x∈Cand{xn}is given by

ynαnxn 1−αnTxn,

Cn

z∈C:yn−z≤ xn−z

,

Qn {z∈C:xn−z, x−xn ≥0},

xn1PCn∩Qnx, ∀n∈N,

1.2

where0≤αn≤a <1. Then,{xn}converges strongly toz0PFTx.

Letϕ : H → R∪ {∞}be a function and letF be a bifunction fromC×CtoRsuch thatC∩domϕ /∅, whereRis the set of real numbers and domϕ{x∈H:ϕx<∞}. The generalized equilibrium problem is to findx∈Csuch that

Fx, y ϕy−ϕx≥0, ∀y∈C. 1.3

The set of solutions of1.3is denoted by GEPF, ϕ; see also5–7.

Ifϕ :H → R∪ {∞}is replaced by a real-valued functionφ: C → R, problem1.3

reduces to the following mixed equilibrium problem introduced by Ceng and Yao8: find

x∈Csuch that

Fx, y φy−φx≥0, ∀y∈C. 1.4

Letϕx δCx, for allx∈H. HereδC denotes the indicator function of the setC; that is, δCx 0 ifx ∈ CandδCx ∞otherwise. Then problem1.3reduces to the following

equilibrium problem: findx∈Csuch that

Fx, y ≥0, ∀y∈C. 1.5

The set of solutions 1.5 is denoted by EPF. Problem 1.5 includes, 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; see9–12and the reference cited therein.

Theorem 1.2 see13. LetH be a real Hilbert space, let Cbe a closed convex subset ofH, let F : C×C → Rbe a bifunction, and letT :C → Cbe a nonexpansive mapping such thatFT∩ EPF/∅. For an initial pointx1x∈C, let a sequence{xn}be generated by

Fun, y

1

rn

y−un, un−xn ≥0 ∀y∈C,

ynαnxn 1−αnTun,

Cn

z∈C:yn−z≤ xn−z

,

Qn{z∈C:xn−z, xn−x ≤0},

xn1PCn∩Qnx, ∀n∈N,

1.6

where0≤αn≤a <1andlim infn→ ∞rn>0. Then,{xn}converges strongly toPFT∩EPFx.

LetA : H → H be a single-valued nonlinear mapping and letM : H → 2H _{be a}

set-valued mapping. The variational inclusion is to findx∈Hsuch that

θ∈Ax Mx, 1.7

whereθis the zero vector inH. The set of solutions of problem1.7is denoted byIA, M. Recall that a mapping A : H → H is called α-inverse strongly monotone if there exists a constantα >0 such that

Ax−Ay, x−y ≥αAx−Ay2, ∀x, y∈H. 1.8

A set-valued mappingM:H → 2H_{is calledmonotoneif for allx, y ∈H,f ∈Mx,}

and g ∈ Myimply x−y, f −g ≥ 0. A monotone mapping Mismaximal if its graph

GM : {f, x ∈H×H : f ∈Mx}ofMis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingMis maximal if and only if forx, f∈H×H,x−y, f −g ≥0 for ally, g∈GMimplyf ∈Mx. We define the resolvent operatorJM,λassociated withMandλas follows:

JM,λx IλM−1x, x∈H, λ >0. 1.9

It is known that the resolvent operator JM,λ is single-valued, nonexpansive, and

1-inverse strongly monotone; see14, and that a solution of problem1.7is a fixed point of the operatorJM,λI −λAfor allλ > 0; see also15. If 0 < λ < 2α, it is easy to see that JM,λI−λAis a nonexpansive mapping; consequently,IA, Mis closed and convex.

The equilibrium problems, generalized equilibrium problems, variational inequality problems, and variational inclusions have been intensively studied by many authors; for instance, see8,16–43.

equilibrium problem, and the set of solutions of a variational inclusion with set-valued maximal monotone and inverse strongly monotone mappings in the framework of Hilbert spaces.

2. Preliminaries and Lemmas

LetCbe a closed convex subset of a real Hilbert spaceHwith norm · and inner product

·,·. For eachx ∈ H, there exists a unique nearest point in C, denoted by PCx, such that

x−PCxminy∈Cx−y.PCis called themetric projectionofHon toC. It is also known that

forx∈Handz∈C,zPCxis equivalent tox−z, y−z ≤0 for ally∈C. Furthermore

y−PCx2x−PCx2≤x−y2 2.1

for allx∈H,y∈C; see also4,44. In a real Hilbert space, we also know that

λx 1−λy2λx2 1−λy2−λ1−λx−y2 2.2

for allx, y∈Handλ∈0,1.

Lemma 2.1 see45. LetCbe a nonempty closed convex subset of a Hilbert spaceH. Then for pointsw, x, y∈Hand a real numbera∈R, the set

D:z∈C:y−z2≤ x−z2w, zais closed and convex. 2.3

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

A1Fx, x 0 for allx∈C;

A2Fis monotone, that is,Fx, y Fy, x≤0 for allx, y∈C;

A3for eachy∈C, x→Fx, yis weakly upper semicontinuous;

A4for eachx∈C, y→Fx, yis convex;

A5for eachx∈C,y→Fx, yis lower semicontinuous;

B1for eachx∈Handr >0, there exists a bounded subsetDx⊆Candyx∈C∩domϕ

such that for anyz∈C\Dx,

Fz, yx

ϕyx

1

r

yx−z, z−x < ϕz; 2.4

B2Cis a bounded set.

semicontinuous and convex function such thatC∩domϕ /∅. Forr >0andx∈H, define a mapping Sr :H → Cas follows:

Srx

z∈C:Fz, yϕy 1 r

y−z, z−x ≥ϕz, ∀y∈C

. 2.5

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

1for eachx∈H,Srx/∅;

2Sris single-valued;

3Sris firmly nonexpansive, that is, for anyx, y∈H,

Srx−Sry2≤

Srx−Sr

y, x−y ; 2.6

4FSr GEPF, ϕ;

5GEPF, ϕis closed and convex.

Lemma 2.3see14. LetM:H → 2H_{be a maximal monotone mapping and letA:H → Hbe} a Lipshitz continuous mapping. Then the mappingS MA :H → 2H _{is a maximal monotone} mapping.

Lemma 2.4. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetT be a quasi-nonexpansive andL-Lipschitz mapping ofCinto itself. Then,FTis closed and convex.

Proof. SinceT isL-Lipschitz, it is easy to show thatFTis closed.

Letx, y∈FTandztx 1−tywheret∈0,1. From2.2, we have

z−Tz2tx−Tz2 1−ty−Tz2−t1−tx−y2

≤tx−z2 1−ty−z2−t1−tx−y2

t1−t2x−y2 1−tt2x−y2−t1−tx−y2 0,

2.7

which impliesz∈FT; consequently,FTis convex. This completes the proof.

Lemma 2.5see46. In a strictly convex Banach spaceX, if

xyλx 1−λy 2.8

In 1999, Atsushiba and Takahashi47introduced the concept of theW-mapping as follows:

U1β1T1

1−β1

I,

U2β2T2U1

1−β2

I,

.. .

UN−1βN−1TN−1UN−2

1−βN−1

I,

WUNβNTNUN−1

1−βN

I,

2.9

where{Ti}Ni1 is a finite mapping ofCinto itself and βi ∈ 0,1 for alli 1,2, . . . , N with

_N

i1βi1.

Such a mapping W is called the W-mapping generated by T1, T2, . . . , TN and β1, β2, . . . , βN; see also48–50. Throughout this paper, we denoteF:Ni1FTi.

Next, we prove some useful lemmas concerning theW-mapping.

Lemma 2.6. LetCbe a nonempty closed convex subset of a strictly convex Banach spaceX. Let{Ti}Ni1

be a finite family of quasi-nonexpansive andLi-Lipschitz mappings ofC into itself such thatF :

_N

i1FTi/∅and letβ1, β2, . . . , βNbe real numbers such that0< βi<1for alli1,2, . . . , N−1,0< βN ≤ 1, and

_N

i1βi 1. LetWbe theW-mapping generated byT1, T2, . . . , TN andβ1, β2, . . . , βN. Then, the followings hold:

iWis quasi-nonexpansive and Lipschitz;

iiFW _iN₁FTi.

Proof. iFor eachx∈Candz∈F, we observe that

T1x−z ≤ x−z. 2.10

Letk∈ {2,3, . . . , N}, then

Ukx−zβkTkUk−1x

1−βk

x−z

≤βkUk−1x−z

1−βk

x−z.

2.11

Hence,

Wx−zUNx−z

≤βNUN−1x−z

1−βN

x−z

≤βN

βN−1UN−2x−z

1−βN−1

x−z1−βN

≤βN

βN−1

βN−2UN−3x−z

1−βN−2

x−z1−βN−1

x−z

1−βN

x−z

.. .

≤βN

βN−1

βN−2· · ·

β2

β1T1x−z

1−β1

x−z1−β2

x−z

· · ·1−βN−2

x−z1−βN−1

x−z1−βN

x−z

≤βN

βN−1

βN−2· · ·

β2

β1x−z1−β1

x−z1−β2

x−z

· · ·1−βN−2

x−z1−βN−1

x−z1−βN

x−z

βN

βN−1

βN−2· · ·

β3

β2x−z1−β2

x−z1−β3

x−z

· · ·1−βN−2

x−z1−βN−1

x−z1−βN

x−z

x−z.

2.12

This shows thatWis a quasi-nonexpansive mapping.

Next, we claim thatWis a Lipschitz mapping. Note thatTi isLi-Lipschitz for alli

1,2, . . . , N. For eachx, y∈C, we observe

U1x−U1yβ1T1x

1−β1

x−β1T1y−

1−β1

y

≤β1T1x−T1y

1−β1x−y ≤β1L1

1−β1x−y.

2.13

Letk∈ {2,3, . . . , N}, then

Ukx−UkyβkTkUk−1x

1−βk

x−βkTkUk−1y−

1−βk

y

≤βkLkUk−1x−Uk−1y

1−βkx−y.

2.14

Hence,

Wx−Wy≤βNLNUN−1x−UN−1y

1−βNx−y

≤βNLNβN−1LN−1UN−2x−UN−2y

βNLN

1−βN−1

1−βNx−y

≤βNLNβN−1LN−1· · ·β2L2U1x−U1y

βNLNβN−1LN−1· · ·β3L3

1−β2

βNLNβN−1LN−1· · ·β4L4

1−β3

· · ·βNLN

1−βN−1

1−βNx−y

≤βNLNβN−1LN−1· · ·β2L2

β1L1

1−β1x−y

βNLNβN−1LN−1· · ·β3L3

1−β2

βNLNβN−1LN−1· · ·β4L4

1−β3

· · ·βNLN

1−βN−1

1−βNx−y βNLNβN−1LN−1· · ·β1L1

βNLNβN−1LN−1· · ·β2L2

1−β1

βNLNβN−1LN−1· · ·β3L3

1−β2

βNLNβN−1LN−1· · ·β4L4

1−β3

· · ·βNLN

1−βN−1

1−βNx−y.

≤LNLN−1· · ·L1LNLN−1· · ·L2LNLN−1· · ·L3

LNLN−1· · ·L4· · ·LNLN−1LN1x−y.

2.15

SinceLi>0 for alli1,2, . . . , N, we get thatWis a Lipschitz mapping.

iiSinceF ⊂ FWis trivial, it suffices to show thatFW ⊂ F. To end this, letp ∈ FWandx∗∈F. Then, we have

p−x∗Wp−x∗βN

TNUN−1p−x∗

1−βN

p−x∗

≤βNUN−1p−x∗

1−βNp−x∗ βNβN−1

TN−1UN−2p−x∗

1−βN−1

p−x∗1−βNp−x∗

≤βNβN−1UN−2p−x∗

1−βNβN−1p−x∗

βNβN−1βN−2

TN−2UN−3p−x∗

1−βN−2

p−x∗1−βNβN−1p−x∗ ≤βNβN−1βN−2UN−3p−x∗

1−βNβN−1βN−2p−x∗

.. .

βNβN−1· · ·β3β2

T2U1p−x∗

1−β2

≤βNβN−1· · ·β2T2U1p−x∗

1−βNβN−1· · ·β2p−x∗ ≤βNβN−1· · ·β2U1p−x∗

1−βNβN−1· · ·β2p−x∗

βNβN−1· · ·β2β1

T1p−x∗

1−β1

p−x∗1−βNβN−1· · ·β2p−x∗ ≤βNβN−1· · ·β2β1T1p−x∗

1−βNβN−1· · ·β2β1p−x∗ ≤βNβN−1· · ·β2β1p−x∗

1−βNβN−1· · ·β2β1p−x∗p−x∗.

2.16

This shows that

_p−x∗_β

NβN−1· · ·β2β1

T1p−x∗

1−β1

p−x∗1−βNβN−1· · ·β2p−x∗,

2.17

and hence

p−x∗β1

T1p−x∗

1−β1

p−x∗. 2.18

Again by2.16, we see thatp−x∗T1p−x∗. Hence

p−x∗T1p−x∗β1

T1p−x∗

1−β1

p−x∗. 2.19

ApplyingLemma 2.5to2.19, we get thatT1ppand henceU1pp.

Again by2.16, we have

p−x∗βNβN−1· · ·β3β2

T2U1p−x∗

1−β2

p−x∗1−βNβN−1· · ·β3p−x∗,

2.20

and hence

p−x∗β2

T2U1p−x∗

1−β2

p−x∗. 2.21

From2.16, we know thatU1p−x∗T2U1p−x∗. SinceU1pp, we have

_p−x∗_T

2p−x∗β2

T2p−x∗

1−β2

p−x∗. 2.22

ApplyingLemma 2.5to2.22, we get thatT2ppand henceU2pp.

By proving in the same manner, we can conclude that Tip pand Uip p for all i1,2, . . . , N−1. Finally, we also have

p−TNp≤p−WpWp−TNpp−Wp

1−βNp−TNp, 2.23

Lemma 2.7. LetCbe a nonempty closed convex subset of a Banach spaceX. Let{Ti}Ni1 be a finite

family of quasi-nonexpansive andLi-Lipschitz mappings ofCinto itself and {βn,i}Ni1 sequences in

0,1such thatβn,i → βiasn → ∞. Moreover, for everyn∈N, letWandWnbe theW-mappings generated byT1, T2, . . . , TNandβ1, β2, . . . , βNandT1, T2, . . . , TNandβn,1, βn,2, . . . , βn,N, respectively. Then

lim

n→ ∞Wnx−Wx0, ∀x∈C. 2.24

Proof. Let x ∈ C and Uk and Un,k be generated by T1, T2, . . . , Tk and β1, β2, . . . , βk and T1, T2, . . . , Tkandβn,1, βn,2, . . . , βn,k, respectively. Then

Un,1x−U1xβn,1−β1

T1x−x≤βn,1−β1T1x−x. 2.25

Letk∈ {2,3, . . . , N}andMmax{TkUk−1xx:k2,3, . . . , N}. Then

Un,kx−Ukxβn,kTkUn,k−1x

1−βn,k

x−βkTkUk−1−

1−βk

x

βn,kTkUn,k−1x−βn,kx−βkTkUk−1βkx

≤βn,kTkUn,k−1x−TkUk−1xβn,k−βkTkUk−1xβn,k−βkx

≤LkUn,k−1x−Uk−1xβn,k−βkM.

2.26

It follows that

Wnx−WxUn,Nx−UNx

≤LNUn,N−1x−UN−1xβn,N−βNM

≤LN

LN−1Un,N−2x−UN−2xβn,N−1−βN−1M

βn,N−βNM LNLN−1Un,N−2x−UN−2xLNβn,N−1−βN−1Mβn,N−βNM

.. .

≤LNLN−1· · ·L3

L2Un,1x−U1xβn,2−β2M

LNLN−1· · ·L4βn,3−β3M· · ·LNβn,N−1−βN−1Mβn,N−βNM

≤LNLN−1· · ·L2βn,1−β1T1x−xLNLN−1· · ·L3βn,2−β2M

LNLN−1· · ·L4βn,3−β3M· · ·LNβn,N−1−βN−1Mβn,N−βNM. 2.27

3. Strong Convergence Theorems

In this section, we prove a strong convergence theorem which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings.

Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, letF:C×C → R be a bifunction satisfying (A1)–(A5), letϕ : C → R∪ {∞}be a proper lower semicontinuous and convex function, letA:H → Hbe anα-inverse strongly monotone mapping, letM:H → 2Hbe a maximal monotone mapping, and let{Ti}Ni1be a finite family of quasi-nonexpansive andLi-Lipschitz mappings ofCinto itself. Assume thatΩ :N_i1FTi∩GEPF, ϕ∩IA, M/∅and either (B1) or (B2) holds. LetWn be theW-mapping generated by T1, T2, . . . , TN and βn,1, βn,2, . . . , βn,N. For an initial pointx0 ∈ HwithC1 Candx1 PC1x0, let{xn},{yn},{zn}, and{un}be sequences generated by

Fun, y

ϕy−ϕun 1 rn

y−un, un−xn ≥0, ∀y∈C,

ynαnxn 1−αnWnun, znJM,λn

yn−λnAyn

,

Cn1

z∈Cn:zn−z ≤yn−z≤ xn−z

,

xn1PCn1x0, ∀n∈N,

3.1

where{αn} ⊂ 0, afor somea ∈0,1,{rn} ⊂ b,∞for someb ∈0,∞and{λn} ⊂ c, dfor somec, d∈0,2α.

Then,{xn},{yn},{zn}, and{un}converge strongly toz0PΩx0.

Proof. Since 0< c≤λn≤d <2αfor alln∈N, we get thatJM,λnI−λnAis nonexpansive for all n∈N. Hence,∞_n1FJM,λnI−λnA IA, Mis closed and convex. ByLemma 2.25, we

know that GEPF, ϕis closed and convex. ByLemma 2.4, we also know thatF:N_i1FTiis

closed and convex. Hence,Ω:N_i1FTi∩GEPF, ϕ∩IA, Mis a nonempty closed convex

set; consequently,PΩx0is well defined for everyx0∈H.

Next, we divide the proof into seven steps.

Step 1. Show thatΩ⊂Cnfor alln∈N.

ByLemma 2.1, we see thatCnis closed and convex for alln∈N. HencePCn1x0is well

defined for everyx0 ∈H,n∈N. Letp∈Ω. FromunSrnxnandpJM,λnp−λnApfor all n∈N, we have

zn−pJM,λn

yn−λnAyn

−JM,λn

p−λnAp

≤yn−p

≤αnxn−p 1−αnWnun−p

≤αnxn−p 1−αnun−p αnxn−p 1−αnSrnxn−Srnp

≤xn−p.

3.2

Step 2. Show that limn→ ∞xn−x0exists.

SinceΩis a nonempty closed convex subset ofC, there exists a unique elementz0

PΩx0∈Ω⊂Cn. FromxnPCnx0, we obtain

xn−x0 ≤ z0−x0. 3.3

Hence{xn−x0}is bounded; so are{yn},{zn}, and{un}.

Sincexn1 PCn1x0∈Cn1⊂Cn, we also have

xn−x0 ≤ xn1−x0. 3.4

From3.3and3.4, we get that limn→ ∞xn−x0exists.

Step 3. Show that{xn}is a Cauchy sequence.

By the construction of the setCn, we know thatxmPCmx0∈Cm⊂Cnform > n. From 2.1, it follows that

xm−xn2≤ xm−x02− xn−x02−→0, 3.5

asm, n → ∞. Hence{xn}is a Cauchy sequence. By the completeness ofHand the closeness

ofC, we can assume thatxn → q∈C.

Step 4. Show thatq∈F.

From3.5, we get

xn1−xn −→0, 3.6

asn → ∞. Sincexn1∈Cn1⊂Cn, we have

zn−xn ≤ zn−xn1xn1−xn ≤2xn1−xn −→0, 3.7

asn → ∞. Hence,zn → qasn → ∞. By the nonexpansiveness ofJM,λn and the inverse

strongly monotonicity ofA, we obtain that

zn−p2≤yn−λnAyn−p−λnAp2

≤yn−p2λnλn−2αAyn−Ap2

≤xn−p2cd−2αAyn−Ap2.

This implies that

c2α−dAyn−Ap2 ≤xn−p2−zn−p2

≤ xn−znxn−pzn−p.

3.9

It follows from3.7that

lim

n→ ∞Ayn−Ap0. 3.10

SinceJM,λn is 1-inverse strongly monotone, we have

zn−p2 JM,λn

yn−λnAyn

−JM,λn

p−λnAp2

≤yn−λnAyn

−p−λnAp

, zn−p

1

2

yn−λnAyn

−p−λnAp2zn−p2

−yn−λnAyn−p−λnAp−zn−p2

≤ 1

2

yn−p2zn−p2−yn−zn−λnAyn−Ap2

≤ 1

2

xn−p2zn−p2−yn−zn22λn

yn−zn, Ayn−Ap

≤ 1

2

xn−p2zn−p2−yn−zn22λnyn−znAyn−Ap.

3.11

This implies that

_zn−p2_≤

xn−p2−yn−zn22λnyn−znAyn−Ap. 3.12

It follows that

yn−zn2≤ xn−znxn−pzn−p 2dyn−znAyn−Ap.

3.13

From3.7and3.10we get

lim

n→ ∞yn−zn0. 3.14

It follows from3.7and3.14that

Wnun−xn 1

1−αn

asn → ∞. SinceSrn is firmly nonexpansive andunSrnxn, we have

un−p2Srnxn−Srnp2

≤Srnxn−Srnp, xn−p

un−p, xn−p

1

2

un−p2xn−p2− xn−un2

,

3.16

which implies that

un−p2≤xn−p2− xn−un2. 3.17

It follows from3.17that

yn−p2≤αnxn−p2 1−αnWnun−p2

≤αnxn−p2 1−αnun−p2

≤αnxn−p2 1−αn

xn−p2− xn−un2

xn−p2−1−αnxn−un2,

3.18

which yields that

1−axn−un2 ≤xn−p2−yn−p2. 3.19

Hence, from3.7and3.14, we also have

lim

n→ ∞xn−un0. 3.20

It follows from3.15and3.20that

lim

n→ ∞un−Wnun0. 3.21

ByLemma 2.7, we also get that limn→ ∞un−Wun 0. FromLemma 2.6i, we know that

W is Lipschitz. Sinceun → qasn → ∞, it is easy to verify that q ∈ FW. Moreover, by

Step 5. Show thatq∈GEPF, ϕ.

SinceunSrnxn, we have

Fun, y

ϕy 1 rn

y−un, un−xn ≥ϕun, ∀y∈C. 3.22

FromA2, we have

ϕy 1 rn

y−un, un−xn ≥F

y, un

ϕun, ∀y∈C. 3.23

It follows fromA5and the weakly lower semicontinuity ofϕ,xn−un/rn → 0, andun → q

that

Fy, qϕq≤ϕy, ∀y∈C. 3.24

Putytty1−tqfor allt∈0,1andy∈C∩domϕ. Sincey∈C∩domϕandq∈C∩domϕ,

we obtain yt ∈ C∩domϕ, and hence Fyt, q ϕq ≤ ϕyt. So by A1,A4, and the

convexity ofϕ, we have

0Fyt, yt

ϕyt

−ϕyt

≤tFyt, y

1−tFyt, q

tϕy 1−tϕq−ϕyt

≤tFyt, y

ϕy−ϕyt

.

3.25

Hence,

Fyt, y

ϕy−ϕyt

≥0. 3.26

Lettingt → 0, it follows fromA3and the weakly semicontinuity ofϕthat

Fq, yϕy≥ϕq 3.27

for ally∈C∩domϕ. Observe that ify∈C\domϕ, thenFq, y ϕy≥ϕqholds. Hence

q∈GEPF, ϕ.

Step 6. Show thatq∈IA, M.

First observe thatAis an1/α-Lipschitz monotone mapping and DA H. From

Lemma 2.3, we know thatMA is maximal monotone. Let v, g ∈ GMA, that is,

g−Av∈Mv. SinceznJM,λnyn−λnAyn, we getyn−λnAyn∈IλnMzn, that is,

1

λn

yn−zn−λnAyn

By the maximal monotonicity ofMA, we have

v−zn, g−Av−

1

λn

yn−zn−λnAyn

≥0, 3.29

and so

v−zn, g ≥

v−zn, Av 1 λn

yn−zn−λnAyn

v−zn, Av−AznAzn−Ayn

1

λn

yn−zn

≥0v−zn, Azn−Ayn

v−zn, 1 λn

yn−zn

.

3.30

It follows fromyn−zn → 0,Ayn−Azn → 0 andzn → qthat

lim

n→ ∞v−zn, gv−q, g ≥0. 3.31

By the maximal monotonicity ofMA, we haveθ∈MAq; consequently,q∈IA, M.

Step 7. Show thatqz0PΩx0.

SincexnPCnx0andΩ⊂Cn, we obtain

x0−xn, xn−p ≥0 ∀p∈Ω. 3.32

By taking the limit in3.32, we obtain

x0−q, q−p ≥0 ∀p∈Ω. 3.33

This shows thatqP_Ωx0z0.

From Steps1–7, we can conclude that{xn},{yn},{zn}, and{un}converge strongly to z0PΩx0. This completes the proof.

4. Applications

As a direct consequence of Theorem 3.1, we obtain some new and interesting results in a Hilbert space as the following theorems. Recall that VIA, Cis the solution set of the classical variational inequality

Theorem 4.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, letF:C×C → R be a bifunction satisfying (A1)–(A5), letϕ : C → R∪ {∞}be a proper lower semicontinuous and convex function, letA : C → H be anα-inverse strongly monotone mapping, and let{Ti}Ni1 be

a finite family of quasi-nonexpansive andLi-Lipschitz mappings of Cinto itself. Assume thatΩ :

N

i1FTi∩GEPF, ϕ∩VIA, C/∅ and either (B1) or (B2) holds. Let Wn be theW-mapping generated byT1, T2, . . . , TN andβn,1, βn,2, . . . , βn,N. For an initial pointx0 ∈ H withC1 Cand

x1PC1x0, let{xn},{yn},{zn}, and{un}be sequences generated by

Fun, y

ϕy−ϕun

1

rn

y−un, un−xn ≥0, ∀y∈C,

ynαnxn 1−αnWnun,

znPC

yn−λnAyn

,

Cn1

z∈Cn:zn−z ≤yn−z≤ xn−z

,

xn1PCn1x0, ∀n∈N,

4.2

where{αn} ⊂0, afor somea∈ 0,1,{rn} ⊂ b,∞for someb∈ 0,∞,and{λn} ⊂c, dfor somec, d∈0,2α.

Then,{xn},{yn},{zn}, and{un}converge strongly toz0PΩx0.

Proof. InTheorem 3.1, takeM ∂δC : H → 2H, whereδC : H → 0,∞is the indicator

function ofC. It is well known that the subdifferential∂δCis a maximal monotone operator.

Then, problem1.7is equivalent to problem4.1and the resolvent operatorJM,λn PC for

alln∈N. This completes the proof.

Next, we give a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings. In order to do this, let us assume that

B3for eachx ∈Handr >0, there exists a bounded subsetDx ⊆Candyx ∈ Csuch

that for anyz∈C\Dx,

Fz, yx

1

r

yx−z, z−x <0. 4.3

Theorem 4.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, letF:C×C → R be a bifunction satisfying (A1)–(A5), letA:H → Hbe anα-inverse strongly monotone mapping, let M : H → 2H _{be a maximal monotone mapping, and let {T}

i}Ni1 be a finite family of

βn,1, βn,2, . . . , βn,N. For an initial pointx0 ∈HwithC1 Candx1 PC1x0, let{xn},{yn},{zn}, and{un}be sequences generated by

Fun, y

1

rn

y−un, un−xn ≥0, ∀y∈C,

ynαnxn 1−αnWnun,

znJM,λn

yn−λnAyn

,

Cn1

z∈Cn:zn−z ≤yn−z≤ xn−z

,

xn1PCn1x0, ∀n∈N,

4.4

where{αn} ⊂0, afor somea∈ 0,1,{rn} ⊂ b,∞for someb∈ 0,∞, and{λn} ⊂c, dfor somec, d∈0,2α.

Then,{xn},{yn},{zn}, and{un}converge strongly toz0PΩx0.

Proof. InTheorem 3.1, takeϕx δCx, for allx ∈ H. Then problem1.3reduces to the

equilibrium problem1.5.

Remark 4.3. Theorem 3.1 improves and extends the main results in 4, 13 and the corresponding results.

Acknowledgments

The authors would like to thank the referee for the valuable suggestions on the manuscript. The authors were supported by the Commission on Higher Education, the Thailand Research Fund, and the Graduate School of Chiang Mai University.

References

1 W. R. Mann, “Mean value methods in iteration,”Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.

2 A. Genel and J. Lindenstrauss, “An example concerning fixed points,”Israel Journal of Mathematics, vol. 22, no. 1, pp. 81–86, 1975.

3 S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.

4 K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,”Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372– 379, 2003.

5 G. Bigi, M. Castellani, and G. Kassay, “A dual view of equilibrium problems,”Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 17–26, 2008.

6 F. Flores-Baz´an, “Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case,”SIAM Journal on Optimization, vol. 11, no. 3, pp. 675–690, 2000.

7 J.-W. Peng, Y.-C. Liou, and J.-C. Yao, “An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions,”Fixed Point Theory and Applications, vol. 2009, Article ID 794178, 21 pages, 2009.

8 L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,”Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.

10 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.

11 S. D. Fl˚am and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,”

Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.

12 A. N. Iusem and W. Sosa, “Iterative algorithms for equilibrium problems,”Optimization, vol. 52, no. 3, pp. 301–316, 2003.

13 A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,”Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.

14 H. B´rezis, “Operateur maximaux monotones,” in Mathematics Studies, vol. 5, North-Holland, Amsterdam, The Netherlands, 1973.

15 B. Lemaire, “Which fixed point does the iteration method select?” inRecent Advances in Optimization, vol. 452, pp. 154–167, Springer, Berlin, Germany, 1997.

16 R. P. Agarwal, Y. J. Cho, and N.-J. Huang, “Sensitivity analysis for strongly nonlinear quasi-variational inclusions,”Applied Mathematics Letters, vol. 13, no. 6, pp. 19–24, 2000.

17 L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “On relaxed viscosity iterative methods for variational inequalities in Banach spaces,”Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 813–822, 2009.

18 L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Mann-type descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces,”Numerical Functional Analysis and Optimization, vol. 29, no. 9-10, pp. 987–1033, 2008.

19 L. C. Ceng, G. Y. Chen, X. X. Huang, and J.-C. Yao, “Existence theorems for generalized vector variational inequalities with pseudomonotonicity and their applications,” Taiwanese Journal of Mathematics, vol. 12, no. 1, pp. 151–172, 2008.

20 L.-C. Ceng, C. Lee, and J.-C. Yao, “Strong weak convergence theorems of implicit hybrid steepest-descent methods for variational inequalities,”Taiwanese Journal of Mathematics, vol. 12, no. 1, pp. 227– 244, 2008.

21 L.-C. Ceng, A. Petrus¸el, and J.-C. Yao, “Weak convergence theorem by a modified extragradient method for nonexpansive mappings and monotone mappings,”Fixed Point Theory, vol. 9, no. 1, pp. 73–87, 2008.

22 L.-C. Ceng, H.-K. Xu, and J.-C. Yao, “A hybrid steepest-descent method for variational inequalities in Hilbert spaces,”Applicable Analysis, vol. 87, no. 5, pp. 575–589, 2008.

23 L. C. Zeng, S. Schaible, and J.-C. Yao, “Hybrid steepest descent methods for zeros of nonlinear operators with applications to variational inequalities,”Journal of Optimization Theory and Applications, vol. 141, no. 1, pp. 75–91, 2009.

24 L.-C. Ceng and J.-C. Yao, “Relaxed viscosity approximation methods for fixed point problems and variational inequality problems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3299–3309, 2008.

25 S. S. Chang, “Set-valued variational inclusions in Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 248, no. 2, pp. 438–454, 2000.

26 P. Cholamjiak, “A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces,”Fixed Point Theory and Applications, vol. 2009, Article ID 719360, 18 pages, 2009.

27 X. P. Ding, “Perturbed Ishikawa type iterative algorithm for generalized quasivariational inclusions,”

Applied Mathematics and Computation, vol. 141, no. 2-3, pp. 359–373, 2003.

28 Y.-P. Fang and N.-J. Huang, “H-monotone operator and resolvent operator technique for variational inclusions,”Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 795–803, 2003.

29 A. Kangtunyakarn and S. Suantai, “A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4448–4460, 2009.

30 A. Kangtunyakarn and S. Suantai, “Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings,”Nonlinear Analysis: Hybrid Systems, vol. 3, no. 3, pp. 296–309, 2009.

32 W. Nilsrakoo and S. Saejung, “Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications,”Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 154–167, 2009.

33 J.-W. Peng, Y. Wang, D. S. Shyu, and J.-C. Yao, “Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems,”Journal of Inequalities and Applications, vol. 2008, Article ID 720371, 15 pages, 2008.

34 J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,”Taiwanese Journal of Mathemat-ics, vol. 12, no. 6, pp. 1401–1432, 2008.

35 J.-W. Peng and J.-C. Yao, “A modified CQ method for equilibrium problems, fixed points and variational inequality,”Fixed Point Theory, vol. 9, no. 2, pp. 515–531, 2008.

36 A. Petrus¸el and J.-C. Yao, “An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems,”Central European Journal of Mathematics, vol. 7, no. 2, pp. 335–347, 2009.

37 S. Plubtieng and W. Sriprad, “A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces,”Fixed Point Theory and Applications, vol. 2009, Article ID 567147, 20 pages, 2009.

38 S. Schaible, J.-C. Yao, and L.-C. Zeng, “A proximal method for pseudomonotone type variational-like inequalities,”Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 497–513, 2006.

39 R. U. Verma, “A-monotonicity and applications to nonlinear variational inclusion problems,”Journal of Applied Mathematics and Stochastic Analysis, vol. 2004, no. 2, pp. 193–195, 2004.

40 L.-C. Zeng, S. M. Guu, and J.-C. Yao, “Hybrid approximate proximal point algorithms for variational inequalities in Banach spaces,”Journal of Inequalities and Applications, vol. 2009, Article ID 275208, 17 pages, 2009.

41 L. C. Zeng, L. J. Lin, and J.-C. Yao, “Auxiliary problem method for mixed variational-like inequalities,”

Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 515–529, 2006.

42 L. C. Zeng and J.-C. Yao, “A hybrid extragradient method for general variational inequalities,”

Mathematical Methods of Operations Research, vol. 69, no. 1, pp. 141–158, 2009.

43 S.-S. Zhang, J. H. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variational inclusion and fixed point problems,”Applied Mathematics and Mechanics, vol. 29, no. 5, pp. 571–581, 2008.

44 G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.

45 C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,”Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006.

46 W. Takahashi,Nonlinear Functional Analysis, Yokohama, Yokohama, Japan, 2000.

47 S. Atsushiba and W. Takahashi, “Strong convergence theorems for a finite family of nonexpansive mappings and applications,”Indian Journal of Mathematics, vol. 41, no. 3, pp. 435–453, 1999.

48 K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,”Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001.

49 W. Takahashi, “Weak and strong convergence theorems for families of nonexpansive mappings and their applications,”Annales Universitatis Mariae Curie-Skłodowska. Sectio A, vol. 51, no. 2, pp. 277–292, 1997.