Theorems for Generalized Mixed Equilibrium Problems of a Finite Family of Asymptotically Nonexpansive Mappings in Banach Spaces

Volume 2012, Article ID 859492,21pages doi:10.1155/2012/859492

Research Article

Existence and Strong Convergence

Theorems for Generalized Mixed Equilibrium Problems of a Finite Family of Asymptotically Nonexpansive Mappings in Banach Spaces

Rabian Wangkeeree,

Hossein Dehghan,

and Pakkapon Preechasilp

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

3Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Gava Zang, Zanjan 45137-66731, Iran

Correspondence should be addressed to Rabian Wangkeeree,rabianw@nu.ac.th Received 5 April 2012; Accepted 10 May 2012

Academic Editor: Giuseppe Marino

Copyrightq2012 Rabian Wangkeeree 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.

We first prove the existence of solutions for a generalized mixed equilibrium problem under the new conditions imposed on the given bifunction and introduce the algorithm for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of finite family of asymptotically nonexpansive mappings. Next, the strong convergence theorems are obtained, under some appropriate conditions, in uniformly convex and smooth Banach spaces. The main results extend various results existing in the current literature.

1. Introduction

LetEbe a real Banach space with the dualE^∗andCbe a nonempty closed convex subset of E. We denote byNandRthe sets of positive integers and real numbers, respectively. Also, we denote byJthe normalized duality mapping fromEto 2^E∗defined by

Jx

x^∗∈E^∗:x, x^∗x²x^∗2

, ∀x∈E, 1.1

where·,·denotes the generalized duality pairing. Recall that ifEis smooth, thenJis single valued and if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets ofE. We will still denote byJthe single-valued duality mapping.

A mappingS:C → Eis called nonexpansive ifSx−Sy ≤ x−yfor allx, y∈C. Also a mappingS : C → Cis called asymptotically nonexpansive if there exists a sequence{kn} ⊂ 1,∞with kn → 1 asn → ∞such thatSⁿx−Sⁿy ≤ knx−yfor allx, y ∈ Cand for eachn≥1. Denote byFSthe set of fixed points ofS, that is,FS {x∈C:Sxx}. The following example shows that the class of asymptotically nonexpansive mappings which was first introduced by Goebel and Kirk1is wider than the class of nonexpansive mappings.

Example 1.1see2. LetBHbe the closed unit ball in the Hilbert spaceHl2andS:BH → BHa mapping defined by

Sx1, x2, x3, . . .

0, x₁², a2x2, a3x3, . . .

, 1.2

where{an}is a sequence of real numbers such that 0< ai<1 and_∞

i2ai1/2. Then Sx−Sy≤2x−y, ∀x, y∈BH. 1.3

That is,Sis Lipschitzian but not nonexpansive. Observe that Sⁿx−Sⁿy≤2

n i2

aix−y, ∀x, y∈BH, n≥2. 1.4 Herekn2_n

i2ai → 1 asn → ∞. Therefore,Sis asymptotically nonexpansive but not non- expansive.

A mappingT : C → E^∗is said to be relaxedη-ξmonotone if there exist a mapping η : C×C → Eand a function ξ : E → Rpositively homogeneous of degreep, that is, ξtz t^pξzfor allt >0 andz∈Esuch that

Tx−Ty, η x, y

≥ξ x−y

, ∀x, y∈C, 1.5

wherep >1 is a constant; see3. In the case ofηx, y x−yfor allx,y ∈C,T is said to be relaxedξ-monotone. In the case ofηx, y x−yfor allx, y∈Candξz kz^p, where p >1 andk >0,Tis said to bep-monotone; see4–6. In fact, in this case, ifp2, thenT is a k-strongly monotone mapping. Moreover, every monotone mapping is relaxedη-ξmonotone withηx, y x−yfor allx, y ∈Candξ0. The following is an example ofη-ξmonotone mapping which can be found in3. LetC −∞,∞,Tx−x, and

η x, y

−c

x−y

, x≥y, c

x−y

, x < y, 1.6

wherec >0 is a constant. Then,Tis relaxedη-ξmonotone with

ξz

cz², z≥0,

−cz², z <0. 1.7

A mappingT :C → E^∗is said to beη-hemicontinuous if, for each fixedx, y∈C, the mapping f : 0,1 → −∞,∞defined byft Txty−x, ηy, xis continuous at 0. For a real Banach spaceEwith the dualE^∗and forCa nonempty closed convex subset ofE, let f:C×C → Rbe a bifunction,ϕ:C → Ra real-valued function andT :C → E^∗be a relaxed η-ξmonotone mapping. Recently, Kamraksa and Wangkeeree7introduced the following generalized mixed equilibrium problemGMEP.

Findx∈Csuch thatf x, y

Tx, η y, x

ϕ y

≥ϕx, ∀y∈C. 1.8

The set of suchx∈Cis denoted by GMEPf, T, that is, GMEP

f, T

x∈C:f x, y

Tx, η y, x

ϕ y

≥ϕx,∀y∈C

. 1.9

Special Cases

1IfTis monotone that isTis relaxedη-ξmonotone withηx, y x−yfor allx, y∈Cand ξ0,1.8is reduced to the following generalized equilibrium problemGEP.

Findx∈Csuch thatf x, y

Tx, y−x ϕ

y

≥ϕx, ∀y∈C. 1.10

The solution set of1.10is denoted by GEPf, that is, GEP

f

x∈C:f x, y

Tx, y−x ϕ

y

≥ϕx,∀y∈C

. 1.11

2In the case ofT ≡0 andϕ≡0,1.8is reduced to the following classical equilibrium problem

Find x∈Csuch thatf x, y

≥0, ∀y∈C. 1.12

The set of all solution of1.12is denoted by EPf, that is, EP

f

x∈C:f x, y

≥0,∀y∈C

. 1.13

3In the case off ≡ 0,1.8is reduced to the following variational-like inequality problem3.

Findx∈Csuch that Tx, η y, x

ϕ y

−ϕx≥0, ∀y∈C. 1.14 The generalized mixed equilibrium problemGMEP 1.8is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems. Using the KKM technique introduced by Kanster et al.8and η-ξ monotonicity of the mappingϕ, Kamraksa and Wangkeeree7obtained the existence of solutions of generalized mixed equilibrium problem1.8in a real reflexive Banach space.

Some methods have been proposed to solve the equilibrium problem in a Hilbert space; see, for instance, Blum and Oettli 9, Combettes and Hirstoaga 10, and Moudafi 11. On the other hand, there are several methods for approximation fixed points of a nonexpansive mapping; see, for instance,12–17. Recently, Tada and Takahashi13,16and S. Takahashi and W. Takahashi 17 obtained weak and strong convergence theorems for finding a common elements in the solution set of an equilibrium problem and the set of fixed point of a nonexpansive mapping in a Hilbert space. In particular, Tada and Takahashi16 established a strong convergence theorem for finding a common element of two sets by using the hybrid method introduced in Nakajo and Takahashi18. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.

On the other hand, in 1953, Mann12introduced the following iterative procedure to approximate a fixed point of a nonexpansive mappingSin a Hilbert spaceH:

x_n1αnxn 1−αnSxn, ∀n∈N, 1.15

where the initial pointx0 is taken inCarbitrarily and{αn}is a sequence in0,1. However, we note that Manns iteration process 1.15 has only weak convergence, in general; for instance, see19–21. In 2003, Nakajo and Takahashi18proposed the following sequence for a nonexpansive mappingSin a Hilbert space:

x0x∈C, ynαnxn 1−αnSxn, Cn

z∈C:yn−z≤ xn−z , Qn{z∈C:xn−z, x0−xn ≥0},

x_n1PCn∩Qnx0,

1.16

where 0≤αn≤ a <1 for alln∈N, andPCn∩Dn is the metric projection fromEontoCn∩Dn. Then, they proved that{xn}converges strongly toP_FTx0. Recently, motivated by Nakajo and Takahashi18and Xu22, Matsushita and Takahashi14introduced the iterative algorithm for finding fixed points of nonexpansive mappings in a uniformly convex and smooth Banach space:x0x∈Cand

Cnco{z∈C:z−Sz ≤tnxn−Sxn}, Dn{z∈C:xn−z, Jx−xn ≥0},

xn1PCn∩Dnx, n≥0,

1.17

where coD denotes the convex closure of the set D, {tn} is a sequence in 0, 1 with tn → 0. They proved that {xn} generated by 1.17 converges strongly to a fixed point of S. Very recently, Dehghan23 investigated iterative schemes for finding fixed point of an asymptotically nonexpansive mapping and proved strong convergence theorems in a

uniformly convex and smooth Banach space. More precisely, he proposed the following algorithm:x1 x∈C,C0D0Cand

Cnco{z∈Cn−1:z−Sⁿz ≤tnxn−Sⁿxn}, Dn {z∈D_n−1:xn−z, Jx−xn ≥0},

xn1PCn∩Dnx, n≥0,

1.18

where{tn}is a sequence in0,1withtn → 0 asn → ∞andSis an asymptotically nonex- pansive mapping. It is proved in23that{xn}converges strongly to a fixed point ofS.

On the other hand, recently, Kamraksa and Wangkeeree7studied the hybrid pro- jection algorithm for finding a common element in the solution set of the GMEP and the common fixed point set of a countable family of nonexpansive mappings in a uniformly convex and smooth Banach space.

Motivated by the above mentioned results and the on-going research, we first prove the existence results of solutions for GMEP under the new conditions imposed on the bifunction f. Next, we introduce the following iterative algorithm for finding a common element in the solution set of the GMEP and the common fixed point set of a finite family of asymptotically nonexpansive mappings{S1, S2, . . . , SN}in a uniformly convex and smooth Banach space:x0∈C,D0C0C, and

x1PC0∩D0x0PCx0,

C1co{z∈C:z−S1z ≤t1x1−S1x1}, u1∈Csuch thatf

u1, y ϕ

y

Tu1, η y, u1

1

r1 y−u1, Ju1−x1

, ∀y∈C, D1{z∈C:u1−z, Jx1−u1 ≥0},

x2PC1∩D1x0, ...

CNco{z∈CN−1:z−SNz ≤t1xN−SNxN}, uN∈Csuch thatf

uN, y ϕ

y

TuN, η y, uN

1

rN y−uN, JuN−xN

, ∀y∈C, DN {z∈D_N−1:uN−z, JxN−uN ≥0},

xN1PCN∩DNx0, C_N1co

z∈CN :z−S²₁z≤t1xN1−S²₁x_N1 , u_N1∈Csuch thatf

u_N1, y ϕ

y

Tu_N1, η

y, u_N1 1

rN1 y−uN1, JuN1−xN1

, ∀y∈C,

DN1{z∈DN:uN1−z, JxN1−uN1 ≥0}, x_N2PCN1∩DN1x0,

... C2Nco

z∈C_2N−1:z−S²_Nz≤t1x2N−S²_Nx2N , u2N∈Csuch that f

u2N, y ϕ

y

Tu2N, η y, u2N

1

r2N y−u2N, Ju2N−x2N

, ∀y∈C, D2N{z∈D2N−1:u2N−z, Jx2N−u2N ≥0},

x_2N1 PC2N∩D2Nx0, C2N1co

z∈C2N:z−S³₁z≤t1x2N1−S³₁x2N1 , u_2N1∈Csuch thatf

u2N1, y ϕ

y

Tu2N1, η

y, u2N1 1

r_2N1 y−u2N1, Ju2N1−x2N1

, ∀y∈C, D_2N1{z∈D2N:u_2N1−z, Jx_2N1−u_2N1 ≥0},

x2N2PC_2N1∩D2N1x0, ...

1.19 The above algorithm is called the hybrid iterative algorithm for a finite family of asymptot- ically nonexpansive mappings fromCinto itself. Since, for eachn ≥ 1, it can be written as n h−1Ni, whereiin∈ {1,2, . . . , N},hhn≥1 is a positive integer andhn → ∞ asn → ∞. Hence the above table can be written in the following form:

x0∈C, D0C0C, Cnco

z∈Cn−1:z−S^hn_inz≤tnxn−S^hn_inxn

, n≥1, un∈Csuch thatf

un, y ϕ

y

Tun, η y, un

1

rn y−un, Jun−xn

, ∀y∈C, n≥1, Dn{z∈Dn−1:un−z, Jxn−un ≥0}, n≥1,

xn1PCn∩Dnx0, n≥0.

1.20

Strong convergence theorems are obtained in a uniformly convex and smooth Banach space.

The results presented in this paper extend and improve the corresponding Kimura and Nakajo24, Kamraksa and Wangkeeree7, Dehghan23, and many others.

2. Preliminaries

LetEbe a real Banach space and letU{x∈E:x1}be the unit sphere ofE. A Banach spaceEis said to be strictly convex if for anyx, y∈U,

x /yimpliesxy<2. 2.1

It is also said to be uniformly convex if for eachε∈0,2, there existsδ >0 such that for any x, y∈U,

x−y≥εimpliesxy<21−δ. 2.2 It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a functionδ:0,2 → 0,1called the modulus of convexity ofEas follows:

δε inf

1− xy

2

:x, y∈E,xy1,x−y≥ε

. 2.3

ThenEis uniformly convex if and only ifδε>0 for allε∈0,2. A Banach spaceEis said to be smooth if the limit

limt→0

xty− x

t 2.4

exists for allx, y ∈U. LetCbe a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach spaceE. Then for anyx∈E, there exists a unique pointx0 ∈C such that

x0−x ≤min

y∈Cy−x. 2.5

The mappingPC:E → Cdefined byPCxx0is called the metric projection fromEontoC.

Letx∈Eandu∈C. The following theorem is well known.

Theorem 2.1. LetCbe a nonempty convex subset of a smooth Banach spaceEand letx ∈ Eand y∈C. Then the following are equivalent:

ay is a best approximation tox:yPCx, byis a solution of the variational inequality:

y−z, J x−y

≥0 ∀z∈C, 2.6

whereJis a duality mapping andPCis the metric projection fromEontoC.

It is well known that ifPC is a metric projection from a real Hilbert spaceHonto a nonempty, closed, and convex subsetC, thenPC is nonexpansive. But, in a general Banach space, this fact is not true.

In the sequel one will need the following lemmas.

Lemma 2.2see25. LetE be a uniformly convex Banach space, let{αn}be a sequence of real numbers such that 0 < b ≤ αn ≤ c < 1 for alln ≥ 1, and let {xn} and {yn}be sequences in E such that lim sup_{n→ ∞}xn ≤d, lim sup_{n→ ∞}yn ≤dand lim_{n→ ∞}αnxn 1−αnynd. Then limn→ ∞xn−yn0.

Dehghan23obtained the following useful result.

Theorem 2.3see23. LetCbe a bounded, closed, and convex subset of a uniformly convex Banach spaceE. Then there exists a strictly increasing, convex, and continuous functionγ:0,∞ → 0,∞ such thatγ0 0 and

γ 1

km

S^m

_n

i1

λixi

−ⁿ

i1

λiS^mxi

≤ max

1≤j≤k≤n

xj−xk− 1 km

S^mxj−S^mxk

2.7

for any asymptotically nonexpansive mappingSofCintoCwith{kn}, any elementsx1, x2, . . . , xn∈ C, any numbersλ1, λ2, . . . , λn≥0 with_n

i1λi1 and eachm≥1.

Lemma 2.4see26, Lemma 1.6. LetEbe a uniformly convex Banach space, Cbe a nonempty closed convex subset ofEandS:C → Cbe an asymptotically nonexpansive mapping. ThenI−S is demiclosed at 0, that is, ifxn xandI−Sxn → 0, thenx∈FS.

The following lemma can be found in7.

Lemma 2.5see7, Lemma 3.2. LetCbe a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach spaceE, letT :C → E^∗be anη-hemicontinuous and relaxedη−ξmonotone mapping. Letfbe a bifunction fromC×CtoRsatisfying (A1), (A3), and (A4) and letϕbe a lower semicontinuous and convex function fromCtoR. Letr >0 andz∈C. Assume that

iηx, y ηy, x 0 for allx, y∈C;

iifor any fixedu, v∈C, the mappingx→ Tv, ηx, uis convex and lower semicontinu- ous;

iiiξ:E → Ris weakly lower semicontinuous, that is, for any net{xβ}, xβconverges toxin σE, E^∗which implies thatξx≤lim infξxβ.

Then there existsx0∈Csuch that f

x0, y

Tx0, η y, x0

ϕ y

1

r y−x0, Jx0−z

≥ϕx0, ∀y∈C. 2.8 Lemma 2.6see7, Lemma 3.3. LetCbe a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach spaceE, letT :C → E^∗be anη-hemicontinuous and relaxed η-ξ monotone mapping. Let f be a bifunction from C×CtoR satisfying (A1)–(A4) and letϕbe a lower semicontinuous and convex function fromCtoR. Letr > 0 and define a mapping Φr :E → Cas follows:

Φrx

z∈C:f z, y

Tz, η y, z

ϕ y

1

r y−z, Jz−x

≥ϕz,∀y∈C

2.9

for allx∈E. Assume that

iηx, y ηy, x 0, for allx, y∈C;

iifor any fixedu, v∈C, the mappingx→ Tv, ηx, uis convex and lower semicontinuous and the mappingx→ Tu, ηv, xis lower semicontinuous;

iiiξ:E → Ris weakly lower semicontinuous;

ivfor anyx, y∈C,ξx−y ξy−x≥0.

Then, the following holds:

1 Φr is single valued;

2Φrx−Φry, JΦrx−x ≤ Φrx−Φry, JΦry−yfor allx, y∈E;

3FΦr EPf, T;

4EPf, Tis nonempty closed and convex.

3. Existence of Solutions for GMEP

In this section, we prove the existence results of solutions for GMEP under the new conditions imposed on the bifunctionf.

Theorem 3.1. LetCbe a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach spaceE, letT : C → E^∗be anη-hemicontinuous and relaxedη-ξ monotone mapping. Letfbe a bifunction fromC×CtoRsatisfying the following conditions (A1)–(A4):

A1fx, x 0 for allx∈C;

A2fx, y fy, x≤min{ξx−y, ξy−x}for allx, y∈C;

A3for ally∈C,f·, yis weakly upper semicontinuous;

A4for allx∈C,fx,·is convex.

For anyr >0 andx∈E, define a mappingΦr :E → Cas follows:

Φrx

z∈C:f z, y

Tz, η y, z

ϕ y

1

r y−z, Jz−x

≥ϕz,∀y∈C

, 3.1

whereϕis a lower semicontinuous and convex function fromCtoR. Assume that iηx, y ηy, x 0, for allx, y∈C;

iifor any fixedu, v∈C, the mappingx→ Tv, ηx, uis convex and lower semicontinuous and the mappingx→ Tu, ηv, xis lower semicontinuous;

iiiξ:E → Ris weakly lower semicontinuous.

Then, the following holds:

1 Φr is single valued;

2Φrx−Φry, JΦrx−x ≤ Φrx−Φry, JΦry−yfor allx, y∈E;

3FΦr GMEPf, T;

4GMEPf, Tis nonempty closed and convex.

Proof. For eachx∈E. It follows from Lemma2.5thatΦrxis nonempty.

1We prove thatΦr is single valued. Indeed, forx ∈ Eandr > 0, letz1, z2 ∈Φrx.

Then

fz1, z2 Tz2, ηz2, z1 ϕz2

1

rz1−z2, Jz1−x ≥ϕz1, fz2, z1 Tz1, ηz1, z2

ϕz1

1

rz2−z1, Jz2−x ≥ϕz2.

3.2

Adding the two inequalities, fromiwe have

fz2, z1 fz1, z2 Tz1−Tz2, ηz2, z11

rz2−z1, Jz1−x−Jz2−x ≥0. 3.3 SettingΔ:min{ξz1−z2, ξz2−z1}and usingA2, we have

Δ Tz1−Tz2, ηz2, z1 1

rz2−z1, Jz1−x−Jz2−x ≥0, 3.4 that is,

1

rz2−z1, Jz1−x−Jz2−x ≥ Tz2−Tz1, ηz2, z1

−Δ. 3.5

SinceT is relaxedη-ξmonotone andr >0, one has

z2−z1, Jz1−x−Jz2−x ≥rξz2−z1−Δ≥0. 3.6

In3.5exchanging the position ofz1andz2, we get 1

rz1−z2, Jz2−x−Jz1−x ≥ Tz1−Tz2, ηz1, z2

−Δ, 3.7

that is,

z1−z2, Jz2−x−Jz1−x ≥rξz1−z2−Δ≥0. 3.8

Now, adding the inequalities3.6and3.8, we have

2z2−z1, Jz1−x−Jz2−x ≥0. 3.9

Hence,

0≤ z2−z1, Jz1−x−Jz2−xz2−x−z1−x, Jz1−x−Jz2−x. 3.10 SinceJis monotone andEis strictly convex, we obtain thatz1−xz2−xand hencez1z2. ThereforeSris single valued.

2Forx, y∈C, we have

f

Φrx,Φry

TΦrx, η

Φry,Φrx ϕ

Φry

−ϕΦrx 1

r Φry−Φrx, JΦrx−x

≥0, f

Φry,Φrx

TΦry, η

Φrx,Φry

ϕΦrx−ϕ Φry

1

r Φrx−Φry, J

Φry−y

≥0.

3.11

SettingΛx,y:min{ξΦrx−Φry, ξΦry−Φrx}and applyingA2, we get TΦrx−TΦry, η

Φry,Φrx 1

r Φry−Φrx, JΦrx−x−J

Φry−y

≥ −Λx,y, 3.12

that is, 1

r Φry−Φrx, JΦrx−x−J

Φry−y

≥ TΦry−TΦrx, η

Φry,Φrx

−Λx,y

≥ξ

Φry−Φrx

−Λx,y ≥0.

3.13

In3.13exchanging the position ofΦrxandΦry, we get 1

r Φrx−Φry, J

Φry−y

−JΦrx−x

≥0. 3.14

Adding the inequalities3.13and3.14, we have 2

r Φry−Φrx, JΦrx−x−J

Φry−y

≥0. 3.15

It follows that

Φry−Φrx, JΦrx−x−J

Φry−y

≥0. 3.16

Hence

Φrx−Φry, JΦrx−x

≤ Φrx−Φry, J

Φry−y

. 3.17

The conclusions3,4follow from Lemma2.6.

Example 3.2. Defineξ:R → Randf:R×R → Rby

f x, y

x−y₂

2 , ξx x² ∀x, y∈R. 3.18 It is easy to see that f satisfiesA1,A3, A4, andA2:fx, y fy, x ≤ min{ξx− y, ξx−y},for allx, y∈R×R.

Remark 3.3. Theorem3.1generalizes and improves7, Lemma 3.3in the following manners.

1The condition fx, y fy, x ≤ 0 has been weakened byA2that is fx, y fy, x≤min{ξx−y, ξy−x}for allx, y∈C.

2The control conditionξx−yξy−x≥0 imposed on the mappingξin7, Lemma 3.3can be removed.

IfTis monotone that isT is relaxedη-ξmonotone withηx, y x−yfor allx, y∈C andξ0, we have the following results.

Corollary 3.4. LetCbe a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach spaceE. LetT :C → E^∗be a monotone mapping andf be a bifunction from C×CtoRsatisfying the following conditions (i)–(iv):

ifx, x 0 for allx∈C;

iifx, y fy, x≤0 for allx, y∈C;

iiifor ally∈C,f·, yis weakly upper semicontinuous;

ivfor allx∈C,fx,·is convex.

For anyr >0 andx∈E, define a mappingΦr :E → Cas follows:

Φrx

z∈C:f z, y

Tz, y−z ϕ

y 1

r y−z, Jz−x

≥ϕz,∀y∈C

, 3.19

whereϕis a lower semicontinuous and convex function fromCtoR. Then, the following holds:

1 Φr is single valued;

2Φrx−Φry, JΦrx−x ≤ Φrx−Φry, JΦry−yfor allx, y∈E;

3FΦr GEPf;

4GEPfis nonempty closed and convex.

4. Strong Convergence Theorems

In this section, we prove the strong convergence theorem of the sequence {xn} defined by 1.20for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of a finite family of asymptotically nonexpansive mappings.

Theorem 4.1. Let E be a uniformly convex and smooth Banach space and let C be a nonempty, bounded, closed, and convex subset ofE. Let f be a bifunction fromC×CtoR satisfying (A1)–

(A4). LetT : C → E^∗be anη-hemicontinuous and relaxedη-ξ monotone mapping andϕa lower semicontinuous and convex function fromC toR. Let, for each 1 ≤ i ≤ N,Si : C → C be an asymptotically nonexpansive mapping with a sequence{kn,i}^∞_n1, respectively, such that kn,i → 1 as n → ∞. Assume thatΩ : _N

i1FSi∩GMEPf, T is nonempty. Let {xn} be a sequence generated by1.20, where{tn}and{rn}are real sequences in0,1satisfying limn→ ∞tn 0 and lim inf_{n→ ∞}rn > 0. Then{xn} converges strongly, as n → ∞, to PΩx0, wherePΩ is the metric projection ofEontoΩ.

Proof. First, define the sequence{kn}bykn:max{kn,i: 1≤i≤N}and sokn → 1 asn → ∞ and

S^hn_inx−S^hn_iny≤knx−y ∀x, y∈C, 4.1 wherehn j1 ifjN < n≤j1N,j 1,2. . . , NandnjNin;in∈ {1,2, . . . , N}.

Next, we rewrite the algorithm1.20as the following relation:

x0∈C, D0C0C, Cnco

z∈C_n−1:z−S^hn_inz≤tnxn−S^hn_inxn

, n≥0, Dn{z∈Dn−1:Φrnxn−z, Jxn−Φrnxn ≥0}, n≥1,

xn1PCn∩Dnx0, n≥0,

4.2

whereΦr is the mapping defined by3.19. We show that the sequence{xn}is well defined.

It is easy to verify that Cn ∩Dn is closed and convex andΩ ⊂ Cn for alln ≥ 0. Next, we prove thatΩ ⊂ Cn ∩ Dn. Indeed, since D0 C, we also haveΩ ⊂ C0 ∩ D0. Assume that Ω⊂C_k−1 ∩ D_k−1fork≥2. Utilizing Theorem3.12, we obtain

Φrkxk−Φrku, JΦrku−u−JΦrkxk−xk ≥0, ∀u∈Ω, 4.3

which gives that

Φrkxk−u, Jxk−Φrkxk ≥0, ∀u∈Ω, 4.4 henceΩ⊂ Dk. By the mathematical induction, we get thatΩ ⊂Cn∩Dn for eachn ≥0 and hence{xn}is well defined. Now, we show that

nlim→ ∞xn−xnj0, ∀j1,2, . . . , N. 4.5 PutwP_Ωx0, sinceΩ⊂Cn∩Dnandx_n1PCn∩Dn, we have

xn1−x0 ≤ w−x0, ∀n≥0. 4.6

Sincexn2∈Dn1 ⊂Dnandxn1PCn∩Dnx0, we have

x_n1−x0 ≤ x_n2−x0. 4.7 Hence the sequence{xn−x0}is bounded and monotone increasing and hence there exists a constantdsuch that

nlim→ ∞xn−x0d. 4.8

Moreover, by the convexity ofDn, we also have 1/2xn1xn2∈Dnand hence x0−xn1 ≤x0−xn1xn2

2

≤ 1

2x0−xn1x0−xn2. 4.9 This implies that

nlim→ ∞

1

2x0−x_n1 1

2x0−x_n2 lim

n→ ∞

x0− xn1xn2

2

d. 4.10

By Lemma2.2, we have

nlim→ ∞xn−xn10. 4.11

Furthermore, we can easily see that

nlim→ ∞xn−xnj0, ∀j1,2, . . . , N. 4.12

Next, we show that

nlim→ ∞

xn−S^hn−κ_in−κxn0, for anyκ∈ {1,2, . . . , N}. 4.13 Fixκ∈ {1,2, . . . , N}and putmn−κ. Sincexn PCn−1∩Dn−1x, we havexn∈C_n−1 ⊆ · · · ⊆Cm. Sincetm>0, there existsy1, . . . , yP ∈Cand a nonnegative numberλ1, . . . , λPwithλ1· · ·λP 1 such that

xn−^P

i1

λiyi

< tm, 4.14

yi−S^hm_imyi≤tmxm−S^hm_imxm, ∀i∈ {1, . . . , P}. 4.15

By the boundedness ofCand{kn}, we can put the following:

Msup

x∈Cx, uP^N_i1FSix0, r0sup

n≥11knxn−u. 4.16

This together with4.14implies that

xn− 1 km

P i1

λiyi

≤

1− 1

km

x 1 km

xn−^P

i1

λiyi

≤

1− 1

km

Mtm, yi−S^hm_imyi≤tmxm−S^hm_imxm

≤tmxm−S^hm_imutmS^hm_imu−S^hm_imxm

≤tmxm−utmkmu−xm

≤tm1kmxm−u

≤tmr0,

4.17

for alli∈ {1, . . . , N}. Therefore, for eachi∈ {1, . . . , P}, we get yi− 1

kmS^hm_imyi

≤yi−S^hm_imyi

S^hm_imyi− 1

kmS^hm_imyi

≤r0tm

1− 1 km

M.

4.18

Moreover, since eachSi,i∈ {1,2, . . . , N}, is asymptotically nonexpansive, we can obtain that

1 kmS^hm_im

_P

i1

λiyi

−S^hm_imxn

≤

1 kmS^hm_im

_P

i1

λiyi

− 1

kmS^hm_imxn

1

kmS^hm_imxn−S^hm_imxn

≤

P i1

λiyi−xn

1− 1

km

M

tm

1− 1 km

M.

4.19

It follows from Theorem2.3and the inequalities4.17–4.19that xn−S^hm_imxn≤

xn− 1 km

P i1

λiyi

1

km

P i1

λi

yi−S^hm_imyi 1

km

P i1

λiS^hm_imyi−S^hm_{im P}

i1

λiyi

1 kmS^hm_im

_P

i1

λiyi

−S^hm_imxn

≤2

1− 1 km

Mtm

r0tm

km

γ⁻¹

1≤i≤j≤Nmax

yi−yj− 1 km

S^hm_imyi−S^hm_imyj

2

1− 1 km

M2tm r0tm

km

γ⁻¹

1≤i≤j≤Nmax

yi−yj− 1 km

S^hm_imyi−S^hm_imyj

≤2

1− 1 km

M2tm r0tm

km

γ⁻¹

1≤i≤j≤Nmax

yi− 1

kmS^hm_imyi

yj− 1

kmS^hm_imyj

≤2

1− 1 km

M2tm r0tm

km γ⁻¹

2

1− 1 km

M2r0tm

.

4.20

Since lim_{n→ ∞}kn1 and lim_{n→ ∞}tn0, it follows from the above inequality that

nlim→ ∞

xn−S^hm_imxn0. 4.21

Hence4.13is proved. Next, we show that

nlim→ ∞xn−Slxn0; ∀ l1,2, . . . , N. 4.22 From the construction ofCn, one can easily see that

xn1−S^hn_inxn1≤tnxn−S^hn_inxn. 4.23

The boundedness ofCand limn→ ∞tn0 implies that

nlim→ ∞

xn1−S^hn_inxn10. 4.24 On the other hand, since for any positive integern > N,n n−NmodNandn hn−

1Nin, we have

n−N hn−1Nin hn−N−1Nin−N 4.25 that is

hn−N hn−1, in−N in. 4.26

Thus,

xn−Snxn ≤ xn−xn1xn1−S^hn_inxn1S^hn_inxn1−S_nxn

≤ xn−xn1xn1−S^hn_inxn1S^hn_inxn1−Snxn1Snxn1−Snxn

≤1k1xn−xn1xn1−S^hn_inxn1k1S^hn−1_in xn1−xn1

≤1k1xn−xn1xn1−S^hn_inxn1

k1S^hn−1_in xn1−S^hn−1_in xnS^hn−1_in xn−xnxn−xn1

≤12k1xn−xn1xn1−S^hn_inxn1

k1S^hn−N_in−Nxn1−S^hn−N_in−Nxnk1S^hn−N_in−Nxn−xn

≤12k1xn−x_n1xn1−S^hn_inx_n1 k1k_n−Nx_n1−xnk1S^hn−N_in−Nxn−xn

≤12k1k1k_n−Nxn−x_n1xn1−S^hn_inx_n1k1S^hn−N_in−Nxn−xn. 4.27

Applying the facts4.11,4.13, and4.24to the above inequality, we obtain

nlim→ ∞xn−Snxn0. 4.28

Therefore, for anyj 1,2, . . . , N, we have

xn−S_njxn≤xn−x_njx_nj−S_njx_njS_njx_nj−S_njxn

≤xn−xnjxnj−Snjxnjk1xnj−xn

1k1xn−x_njx_nj−S_njx_nj−→0 as n−→ ∞,

4.29

which gives that

nlim→ ∞xn−Slxn0; ∀l1,2, . . . , N, 4.30 as required. Since{xn}is bounded, there exists a subsequence{xni}of{xn}such thatxni

x∈C. It follows from Lemma2.4thatx∈FSlfor alll1,2, . . . , N. That isx∈_N

i1FSi. Next, we show that x ∈ GMEPf, T. By the construction of Dn, we see from Theorem2.1thatΦrnxnPDnxn. Sincex_n1∈Dn, we get

xn−Φrnxn ≤ xn−xn1 −→0. 4.31

Furthermore, since lim infn→ ∞rn>0, we have 1

rnJxn−Φrnxn 1

rnxn−Φrnxn −→0, 4.32 asn → ∞. By4.32, we also haveΦr_nixni x. By the definition of Φr_ni, for eachy ∈C, we obtain

f

Φr_nixni, y

TΦr_nixni, η

y,Φr_nixni

ϕ y

1 rni

y−Φr_nixni, J

Φr_nixni−xni

≥ϕ Φr_nixni

.

4.33 By A3,4.32,ii, the weakly lower semicontinuity of ϕand η-hemicontinuity ofT, we have

ϕx ≤lim inf

i→ ∞ ϕ Φr_nixni

≤lim inf

i→ ∞ f

Φr_nixni, y

lim inf

i→ ∞

TΦr_nixni, η

y,Φr_nixni

ϕ

y

lim inf

i→ ∞

1 rni

y−Φr_nixni, J

Φr_nixni−xni

≤f x, y

ϕ y

Tx, η y,x

.

4.34

Hence,

f x, y

ϕ y

Tx, η y,x

≥ϕx. 4.35

This shows thatx∈EPf, Tand hencex∈Ω:_N

i1FSi∩GMEPf, T.

Finally, we show thatxn → w asn → ∞, wherew : P_Ωx0. By the weakly lower semicontinuity of the norm, it follows from4.6that

x0−w ≤ x0−x ≤ lim inf

i→ ∞ x0−xni ≤lim sup

i→ ∞ x0−xni ≤ x0−w. 4.36 This shows that

ilim→ ∞x0−xnix0−wx0−x 4.37

andx w. SinceEis uniformly convex, we obtain thatx0−xni → x0−w. It follows that xni → w. So we havexn → wasn → ∞. This completes the proof.