A New Hybrid Algorithm for Solving a System of Generalized Mixed Equilibrium Problems, Solving a Family of Quasi

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Volume 2011, Article ID 106323,16pages doi:10.1155/2011/106323

Research Article

A New Hybrid Algorithm for Solving a System of

Generalized Mixed Equilibrium Problems, Solving

a Family of Quasi-

φ

-Asymptotically Nonexpansive

Mappings, and Obtaining Common Fixed Points in

Banach Space

J. F. Tan and S. S. Chang

Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China

Correspondence should be addressed to J. F. Tan,[email protected]

and S. S. Chang,[email protected]

Received 14 February 2011; Accepted 14 April 2011

Academic Editor: Vittorio Colao

Copyrightq2011 J. F. Tan and S. S. Chang. 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.

The main purpose of this paper is to introduce a new hybrid iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, set of common

fixed points of a family of quasi-φ-asymptotically nonexpansive mappings, and null spaces of finite

family ofγ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth

real Banach space. The results presented in the paper improve and extend the corresponding results announced by some authors.

1. Introduction

Throughout this paper, we assume that E is a real Banach space with a dual E∗, C is a nonempty closed convex subset ofE, and ·,·is the duality pairing between members of

EandE∗. The mappingJ:E → 2E∗defined by

Jx f∗∈E∗:x, f∗x2;f∗x, x∈E 1.1

is called the normalized duality mapping.

LetF : C×C → Rbe a bifunction, letB : C → E∗ be a nonlinear mapping, and letΦ:C → Rbe a proper extended real-valued function. The “so-called” generalized mixed

equilibrium problem forF,B,Φis to findx∗∈Csuch that

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The set of solutions of1.2is denoted by GMEPF, B,Φ, that is,

GMEPF, B,Φ x∈C:Fx∗, yy−x∗, Bx∗ Φy−Φx∗≥0, ∀y∈C. 1.3

Special Examples

1If Φ ≡ 0, then the problem1.2is reduced to the generalized equilibrium problem

GEP, and the set of its solutions is denoted by

GEPF, B x∈C:Fx∗, yy−x∗, Bx∗≥0, ∀y∈C. 1.4

2IfB≡0, then the problem1.2is reduced to the mixed equilibrium problemMEP, and the set of its solutions is denoted by

MEPF, B x∈C:Fx∗, y Φy−Φx∗≥0, ∀y∈C. 1.5

These show that the problem1.2is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of1.2. Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach space

see, e.g.,1–3.

LetEbe a smooth, strictly convex, and reflexive Banach space, and letCbe a nonempty closed convex subset ofE. Throughout this paper, the Lyapunov functionφ:E×E → Ris defined by

φx, yx2−2x, Jyy2, ∀x, y∈E. 1.6

Following Alber4, the generalized projectionΠC :E → Cis defined by

ΠCx arg min

y∈C

φy, x, ∀x∈E. 1.7

LetCbe a nonempty closed convex subset ofE, letS:C → Cbe a mapping, and let

FSbe the set of fixed points ofS. A pointp ∈Cis said to be an asymptotic fixed point of T

if there exists a sequence{xn} ⊂ Csuch thatxn pand||xn−Sxn|| → 0. We denoted the

set of all asymptotic fixed points ofSbyFS. A pointp∈Cis said to be a strong asymptotic

fixed point ofSif there exists a sequence{xn} ⊂Csuch thatxn → pand||xn−Sxn|| → 0. We

denoted the set of all strongly asymptotic fixed points ofSbyFS. A mappingS:C → Cis said to be nonexpansive if

Sx−Sy≤x−y, ∀x, y∈C. 1.8

A mappingS:C → Cis said to be relatively nonexpansive5ifFS/∅,FS FS

and

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A mappingS:C → Cis said to be weak relatively nonexpansive6ifFS/∅,FS

FSand

φp, Sx≤φp, x, ∀x∈C, p∈FS. 1.10

A mappingS:C → Cis said to be closed if for any sequence{xn} ⊂Cwithxn → x

andSxn → y, thenSxy.

A mappingS:C → Cis said to be quasi-φ-nonexpansive ifFS/∅and

φp, Sx≤φp, x, ∀x∈C, p∈FS. 1.11

A mappingS:C → Cis said to be quasi-φ-asymptotically nonexpansive, ifFS/∅and there exists a real sequence{kn} ⊂1,∞withkn → 1 such that

φp, Snx≤knφ

p, x, ∀n≥1, x∈C, p∈FS. 1.12

From the definition, it is easy to know that each relatively nonexpansive mapping is closed. The class of quasi-φ-asymptotically nonexpansive mappings contains properly the class of quasi-φ-nonexpansive mappings as a subclass. The class of quasi-φ-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, and the class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.

A mappingA: C → E∗ is said to beα-inverse strongly monotone if there existsα >0 such that

x−y, Ax−Ay ≥αAx−Ay2. 1.13

IfAis anα-inverse strongly monotone mapping, then it is 1/α-Lipschitzian.

Iterative approximation of fixed points for relatively nonexpansive mappings in the setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushita and Takahashi5obtained weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Su et al. 6, 7, Zegeye and Shahzad 8, Wattanawitoon and Kumam 9, and Zhang 10 extend the notion from relatively nonexpansive mappings or quasi-φ-nonexpansive mappings to

quasi-φ-asymptotically nonexpansive mappings and also prove some convergence theorems to approximate a common fixed point of quasi-φ-nonexpansive mappings or quasi-φ -asymptotically nonexpansive mappings.

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2. Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will be needed in proving our main results.

A Banach spaceEis said to be strictly convex if||xy||/2 <1 for allx, y ∈U {z ∈ E:||z||1}withx /y. It is said to be uniformly convex if for each∈0,2, there existsδ >0 such that||xy||/2≤1−δfor allx, y∈Uwith||x−y|| ≥. The convexity modulus of Eis the functionδE:0,2 → 0,1defined by

δE inf

1−1 2

xy:x, y∈U,x−y≥

, 2.1

for all ∈ 0,2. It is well known thatδEis a strictly increasing and continuous function withδE0 0, andδE/is nondecreasing for all ∈0,2. Letp >1, thenEis said to be

p-uniformly convex if there exists a constantc >0 such thatδE≥cp_{, for all}_∈₀_,₂_{. The}

spaceEis said to be smooth if the limit

lim

t→0

_x_ty₋_x

t 2.2

exists for allx, y ∈ U. AndEis said to be uniformly smooth if the limit exists uniformly in

x, y∈U.

In the sequel, we will make use of the following lemmas.

Lemma 2.1see11. LetEbe a 2-uniformly convex real Banach space, then for allx, y ∈E, the

inequality||x−y|| ≤2/c2_||_Jx₋_Jy_||_{holds, where 0}_{< c}_≤_{1, and}_c_{is called the 2-uniformly convex}

constant ofE.

Lemma 2.2see12. LetEbe a smooth, strict convex, and reflexive Banach space, and letCbe a

nonempty closed convex subset ofE, then the following conclusions hold:

iφx,ΠCy φΠCy, y≤φx, y, for allx∈C,y∈E,

iiletx∈Eandz∈C, then

z ΠCx⇐⇒ z−y, Jx−Jz ≥0, ∀y∈C. 2.3

Lemma 2.3see12. LetEbe a uniformly convex and smooth Banach space, and let{xn},{yn}be

sequences ofE. Ifφxn, yn → 0as n → ∞and either{xn}or{yn}is bounded, thenxn−yn →

0asn → ∞.

Lemma 2.4see10. LetEbe a uniformly convex Banach space, letr be a positive number, and

letBr0be a closed ball ofE. For any given points{x1, x2, . . . , xn, . . .} ⊂ Br0and for any given

positive numbers{λ1, λ2, . . .}with

_∞

n1λn 1, there exists a continuous, strictly increasing, and

convex functiong:0,2r → 0,∞withg0 0 such that for anyi, j∈ {1,2, . . .},i < j,

∞

n1

λnxn

2

≤∞

n1

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For solving the generalized mixed equilibrium problem, let us assume that the bifunctionF:C×C → Rsatisfies the following conditions:

A1Fx, x 0 for allx∈C,

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

A3lim sup_t_↓0Fxtz−x, y≤Fx, y, for allx, y, z∈C,

A4the functiony→Fx, yis convex and lower semicontinuous.

Lemma 2.5see13. LetEbe a smooth, strict convex, and reflexive Banach space, and letCbe

a nonempty closed convex subset ofE. Let F : C×C → Rbe a bifunction satisfying conditions

(A1)–(A4). Letr >0 andx∈E, then there existsz∈Csuch that

Fz, y1 r

y−z, Jz−Jx≥0, ∀y∈C. 2.5

By the same way as given in the proofs of14, Lemmas 2.8 and 2.9, we can prove that the bifunction

Γx, yFx, y Φy−Φx y−x, Bx, ∀x, y∈C 2.6

satisfies conditionsA1–A4and the following conclusion holds.

Lemma 2.6. LetEbe a smooth, strictly convex, and reflexive Banach space, and letCbe a nonempty

closed convex subset ofE. LetF:C×C → Rbe a bifunction satisfying conditions (A1)–(A4), letB:

C → E∗be aβ-inverse strongly monotone mapping, and letΦ:C → Rbe a lower semicontinuous

and convex function. For givenr >0 andx∈E, define a mappingKΓ_r :E → Cby

KΓ_rx

z∈C:Fz, y Φy−Φz y−z, Bz1 r

y−z, Jz−Jx≥0, ∀y∈C

,

2.7

then the following hold:

iKΓ_r is single valued,

iiKΓ_r is a firmly nonexpansive-type mapping, that is, for allx, y,∈E,

KΓ_rx−KΓ_ry, JK_rΓx−JK_rΓy≤K_rΓx−K_rΓy, Jx−Jy, 2.8

iiiFKΓ_r GMEPF,Φ, B,

ivGMEPF,Φ, Bis closed and convex,

vφp, KΓ_rx φK_rΓx, x≤φp, x, for allp∈FKΓ_r.

In the sequel, we make use of the functionV :E×E∗ → Rdefined by

Vx, x∗ x2−2x, x∗x∗2, 2.9

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Lemma 2.7see4. LetEbe a smooth, strict convex, and reflexive Banach space withE∗ as its dual, then

Vx, x∗ 2J−1x∗−x, y∗≤Vx, x∗y∗, 2.10

for allx∈Eandx∗, y∗∈E∗.

3. Main Results

In this section, we will propose the following new iterative scheme{xn}for finding a common

element of set of solutions for a system of generalized mixed equilibrium problems, the set of common fixed points of a family of quasi-φ-asymptotically nonexpansive mappings, and null spaces of finite family ofγ-inverse strongly monotone mappings in the setting of 2-uniformly convex and uniformly smooth real Banach spaces:

x0∈C0C,

yn ΠCJ−1Jxn−λAn1xn,

znJ−1

αn,0Jxn

∞

i1

αn,iJTinyn

,

unKΓMrM,nK

ΓM−1

rM−1,n· · ·K

Γ2

r2,nK

Γ1

r1,nzn,

Cn1 v∈Cn:φv, un≤φv, xn ξn

,

xn1 ΠCn1x0, n≥0,

3.1

whereKΓkrk,n :C → C,k 1,2, . . . , Mis the mapping defined by2.7,An AnmodN,rk,n ∈

d,∞for somed >0 and 0< λ < c2γ/2, wherecis the 2-uniformly convex constant ofE, for eachn≥1,αn,0

_∞

i1αn,i1 and for eachj≥1, lim infn→ ∞αn,0αn,j >0.

Definition 3.1. A countable family of mappings{Ti : C → C}is said to be uniformly

quasi-ϕ-asymptotically nonexpansive mappings if there exists a sequence{kn} ⊂ 1,∞withkn → 1

such that for eachi≥1

_Tn

ix−Tiny≤knx−y, ∀x, y∈C, and for eachn≥1. 3.2

Theorem 3.2. LetCbe a nonempty closed and convex subset of a 2-uniformly convex and uniformly

smooth real Banach spaceEwith a dualE∗. Let{Ti:C → C}∞i1be a countable family of closed and

uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence{kn} ⊂1,∞such that

kn → 1. Suppose further that for eachi ≥ 1,Ti is uniformlyLi-Lipschitzian. LetAn : C → E∗, n1,2, . . . , Nbe a finite family ofγn-inverse strongly monotone mappings, and letγmin{γn, n

1,2, . . . , N}. Let{Fm : C×C → R, m 1,2, . . . , M} be a finite family of equilibrium functions

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semicontinuous convex function, and let {Bm : C → E∗, m 1,2, . . . , M} be a finite family of

βm-inverse strongly monotone mappings. If

Ω ∞ i1

FTi∩ N

n1

A−1_n 0∩

M

m1

GMEPFm, Bm,Φm 3.3

is a nonempty and bounded subset inCandξn sup_p_∈Ωkn−1φp, xn, then the sequence{xn}

defined by3.1converges strongly to some pointx∗∈Ω.

Proof. We divide the proof ofTheorem 3.2into five steps.

(I) Sequences

{x

n

},

{y

n

}, and

{T

_in

y

n

}

all are bounded.

In fact, sincexn ΠCnx0, for anyp∈Ω, fromLemma 2.2, we have

φxn, x0 φΠCnx0, x0≤φ

p, x0

−φp, xn

≤φp, x0

. 3.4

This implies that the sequence{φxn, x0}is bounded, and so{xn}is bounded.

On the other hand, by Lemmas2.1and2.7, we have that

φp, yn

φp,ΠCJ−1Jxn−λAn1xn

≤φp, J−1Jxn−λAn1xn

Vp, Jxn−λAn1xn

≤Vp,Jxn−λAn1xn λAn1xn

−2J−1Jxn−λAn1xn−p, λAn1xn

Vp, Jxn

−2λJ−1Jxn−λAn1xn−p, An1xn

φp, xn

−2λxn−p, An1xn −2λJ−1Jxn−λAn1xn−xn, An1xn φp, xn

−2λxn−p, An1xn−An1p sinceAnp0, ∀n≥1

−2λJ−1Jxn−λAn1xn−xn, An1xn

≤φp, xn

−2λγAn1xn22λJ−1Jxn−λAn1xn−J−1Jxn× An1xn

≤φp, xn

−2λγAn1xn24λ

2

c2 An1xn

2 _{by Lemma 2}_.₁

≤φp, xn

2λ

2λ c2 −γ

An1xn2.

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Thus, using the fact thatλ≤c2_/₂_γ_{, we have that}

φp, yn

≤φp, xn

. 3.6

Moreover, by the assumption that{Ti :C → C}∞i1is a countable family of uniformly

quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn,i} ⊂ 1,∞ such that kn

sup_i_≥1kn,i → 1n → ∞, hence for any givenp∈Ω, from3.6we have that

ϕp, T_inyn

≤knϕ

p, yn

≤knϕ

p, xn

, ∀n≥1, i≥1. 3.7

Hence, for eachi≥1,{Tn

i yn}is also bounded, denoted by

M sup

n≥0, i≥1 xn,

_y_n_,_Tn

iyn<∞. 3.8

By the way, from the definition of{ξn}, it is easy to see that

ξnsup p∈Ω

kn−1φ

p, xn

≤sup

p∈Ω

kn−1pM

2_−→₀ _n_{−→ ∞}_. _3.9

(II) For each

n

≥

0,

C

n

is a closed and convex subset of

C

and

Ω

⊂

C

n

.

It is obvious thatC0Cis closed and convex. Suppose thatCnis closed and convex for some n≥1. Since the inequalityφv, un≤φv, xn ξnis equivalent to

2v, Jxn−Jun ≤ xn2− un2ξn, 3.10

therefore, we have

Cn1

v∈Cn: 2v, Jxn−Jun ≤ xn2− un2ξn

. 3.11

This implies thatCn1 is closed and convex. Thus, for eachn≥ 0,Cnis a closed and convex

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Next, we prove that Ω ⊂ Cn for alln ≥ 0. Indeed, it is obvious that Ω ⊂ C0 C.

Suppose thatΩ ⊂ Cnfor somen≥ 1. SinceEis uniformly smooth,E∗is uniformly convex.

For any givenp∈Ω⊂Cnand for any positive integerj >0, fromLemma 2.4, we have

φp, un

φp, KrΓMM,nK

ΓM−1

rM−1,n· · ·K

Γ2

r2,nK

Γ1

r1,nzn

≤φp, zn

φ

p, J−1

αn,0Jxn

∞

i1

αn,iJTinyn

p2−2

p, αn,0Jxn

∞

i1

αn,iJTinyn

αn,0Jxn

∞

i1

αn,iJTinyn

2

≤p2−2αn,0p, Jxn −2

∞

i1

αn,i

p, JT_inyn

αn,0xn2

∞

i1

αn,iT_inyn2−αn,0αn,jgJxn−JT_jnyn

αn,0φ

p, xn

∞

i1

αn,iφ

p, T_inyn

−αn,0αn,jgJxn−JT_jnyn

≤αn,0φ

p, xn

∞

i1

αn,iknφ

p, yn

−αn,0αn,jgJxn−JTjnyn

.

3.12

Having this together with3.6, we have

φp, un

≤φp, zn

≤αn,0φ

p, xn

∞

i1

αn,iknφ

p, xn

≤knφ

p, xn

≤φp, xn

sup

z∈Ω

kn−1φz, xn−αn,0αn,jgJxn−JTjnyn

φp, xn

ξn−αn,0αn,jgJxn−JTjnyn

≤φp, xn

ξn.

3.13

Hence,p∈Cn1andΩ⊂Cnfor alln≥0.

(III)

{x

n

}

is a Cauchy sequence.

Sincexn ΠCnx0andxn1 ΠCn1x0∈Cn1⊂Cn, we have that

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which implies that the sequence {φxn, x0} is nondecreasing and bounded, and so

limn→ ∞φxn, x0exists. Hence, for any positive integerm, usingLemma 2.2we have

φxnm, xn φxnm,ΠCnx0≤φxnm, x0−φxn, x0, 3.15

for alln≥0. Since limn→ ∞φxn, x0exists, we obtain that

φxnm, xn−→0 n−→ ∞, ∀m∈Z. 3.16

Thus, byLemma 2.3, we have thatxnm−xn → 0 asn → ∞. This implies that the sequence

{xn}is a Cauchy sequence inC. SinceCis a nonempty closed subset of Banach spaceE, it is

complete. Hence, there exists anx∗inCsuch that

xn−→x∗ n−→ ∞. 3.17

(IV) We show that

x

∗

∈

∞i1

F

T

i

.

Sincexn1∈Cn1by the structure ofCn1, we have that

φxn1, un≤φxn1, xn ξn. 3.18

Again by3.16andLemma 2.3, we get that limn→ ∞||xn1−un||0. But

xn−un ≤ xn−xn1xn1−un. 3.19

Thus,

lim

n→ ∞xn−un0. 3.20

This implies thatun → x∗ asn → ∞. SinceJ is norm-to-norm uniformly continuous on

bounded subsets ofE, we have that

lim

n→ ∞Jxn−Jun0. 3.21

From3.13,3.20, and3.21, we have that

αn,0αn,jgJxn−JTjnyn

≤φp, xn

−φp, un

ξn

xn2− un22

p, Jun−Jxn

ξn

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In view of condition lim infn→ ∞αn,0αn,j >0, we see that

gJxn−JTjnyn

−→0 n−→ ∞. 3.23

It follows from the property ofgthat

Jxn−JT_jnyn−→0 n−→ ∞. 3.24

Sincexn → x∗andJis uniformly continuous, it yieldsJxn → Jx∗. Hence, from3.24, we

have

JTn

jyn−→Jx∗ n−→ ∞. 3.25

SinceE∗is uniformly smooth andJ−1is uniformly continuous, it follows that

T_jnyn−→x∗ n−→ ∞, ∀j≥1. 3.26

Moreover, using inequalities3.12and3.5, we obtain that

φp, un

≤αn,0φ

p, xn

∞

i1

αn,iknφ

p, xn

∞ i1

αn,ikn2λ

2

c2λ−γ

An1xn2

≤φp, xn

ξnkn2λ

2

c2λ−γ

An1xn2, ∀p∈Ω.

3.27

This implies that

kn2λ

γ− 2 c2λ

An1xn2≤φ

p, xn

−φp, un

ξn, 3.28

that is,

lim

n→ ∞An1xn

2 ₀_. _3.29

It follows from3.1and3.29that we have

lim

n→ ∞yn−x

∗ _lim

n→ ∞

ΠCJ−1Jxn−λAn1xn−x∗

≤ lim

n→ ∞

J−1Jxn−λAn1xn−x∗0.

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Furthermore, by the assumption that for eachj ≥1,Tjis uniformlyLi-Lipschitz continuous,

hence, we have

T_jn1yn−T_jnyn≤T_jn1yn−T_jn1yn1Tjn1yn1−yn1yn1−ynyn−Tjnyn

≤Lj1yn1−ynT_jn1yn1−yn1yn−T_jnyn.

3.31

This together with3.26and3.30yields

lim

n→ ∞

T_jn1yn−T_jnyn0. 3.32

Hence, from3.26, we have

lim

n→ ∞T

n1

j ynx∗, 3.33

that is,

lim

n→ ∞TjT

n

jyn x∗. 3.34

In view of3.26and the closeness ofTj, it yields thatTjx∗x∗for allj≥1. This implies that x∗∈∞_j₁FTj.

(IV) Now, we prove that

x

∗

∈

N_n₁

A

−n1

0

.

It follows from3.29that

lim

n→ ∞An1xn0. 3.35

Since limn→ ∞xnx∗, we have that for every subsequence{xnj}j≥1of{xn}n≥0, limj→ ∞xnj x∗

and

lim

j→ ∞Anj1xnj0. 3.36

Let{nq}q≥1 ⊂ Nbe an increasing sequence of natural numbers such thatAnq1 A1, for all

q∈N, then limp→ ∞||xnq−x∗||0 and

0 lim

q→ ∞Anq1xnq_qlim_{→ ∞}A1xnq. 3.37

SinceA1isγ-inverse strongly monotone, it is Lipschitz continuous, and thus

A1x∗A1

lim

q→ ∞xnq

lim

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Hence,

x∗∈A−1₁ 0. 3.39

Continuing this process, we obtain thatx∗∈A−1_i 0, for alli1,2, . . . , N. Hence,

x∗∈ N

n1

A−1_n 0. 3.40

(V) Next, we prove that

x

∗

∈

M_m₁

GMEP

F

m

, B

m

,

Φ

m

.

PuttingSm

n KrΓmm,nK

Γm−1

rm−1,n· · ·K

Γ2

r2,nK

Γ1

r1,n form∈ {1,2, . . . , M}andS 0

n Ifor alln ∈N. For any p∈Ω, we have

φSm_nzn,Smn−1zn

≤φp,Sm_n−1zn

−φp,Sm_nzn

≤φp, zn

−φp,Sm_nzn

≤φp, xn

ξn−φ

p,Sm_nzn by3.13

φp, xn

ξn−φ

p, un

.

3.41

It follows from3.22that limn→ ∞φSmnzn,Snm−1zn 0. SinceEis 2-uniformly convex and

uniformly smooth Banach space and{zn}is bounded, we have that

lim

n→ ∞

Sm

nzn− Smn−1zn0, m1,2, . . . , M. 3.42

Sincexn → x∗ andun → x∗, now we prove that for eachm 1,2, . . . , M,Smnzn → x∗ as n → ∞. In fact, ifmM, then we have

lim

n→ ∞

SM

n zn− SMn −1zn_nlim_{→ ∞}un− SMn −1zn0, 3.43

that is,SM−1

n zn → x∗. By induction, the conclusion can be obtained. SinceJis norm-to-norm

uniformly continuous on bounded subsets ofE, we get

lim

n→ ∞

JS_nmzn−JSnm−1zn0, 3.44

and sincerk,n∈d,∞for somed >0, we have that

lim

n→ ∞ JSm

nzn−JSmn−1zn

rm,n

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Next, sinceΓmSm

nzn, y 1/rm,ny− Smnzn, JSmnzn−JSnm−1zn ≥0, for ally∈C, this implies

that

1

rm,n

y− S_nmzn, JSnmzn−JSnm−1zn

≥ −ΓmSm nzn, y

≥Γmy,S_nmzn

, ∀y∈C. 3.46

This implies that

Γmy,Sm nzn

≤ 1

rm,n

y− Sm

nzn, JSmnzn−JSnm−1zn

≤M1y

_J_Sm

nzn−JSmn−1zn rm,n

,

3.47

for someM1≥0. Sincey→Γmx, yis a convex and lower semicontinuous, we obtain from

3.45and3.47that

Γmy, x∗≤lim inf

n→ ∞ Γm

y,Sm_nzn

≤0, ∀y∈C. 3.48

For anyt∈0,1andy ∈C, thenyt ty 1−tx∗ ∈C. SinceΓm satisfies conditionsA1

andA4, from3.48, we have

0 Γmyt, yt

≤tΓmyt, y

1−tΓmyt, x∗

≤tΓmyt, y

, ∀m1,2, . . . M. 3.49

Deletet, and then lett → 0, by conditionA3, we have

0≤Γmx∗, y, ∀y∈C, ∀m1,2, . . . M, 3.50

that is, for eachm1,2, . . . , M, we have

Fm

x∗, yy−x∗, Bmx∗ Ψm

y−Ψmx∗≥0, ∀y∈C. 3.51

Therefore, we have that

x∗∈ M

m1

GMEPFm, Bm,Φm. 3.52

This completes the proof.

Remark 3.3. 1Theorem 3.2not only improves and extends the main results in3,6–10but

also improves and extends the corresponding results of Chang et al.1,15, Wang et al.16, Su et al.17, and Kang et al.18.

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problems, monotone variational inequality problems, variational inclusion problems, and equilibrium problems in some Banach spaces. For saving space, we will give them in another paper.

Acknowledgment

The authors would like to express their thanks to the referees for their helpful comments and suggestions.

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