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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 859492, 21 pages
http://dx.doi.org/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

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

Received 5 April 2012; Accepted 10 May 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 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.

Abstract

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

Let 𝐸 be a real Banach space with the dual 𝐸 and 𝐶 be a nonempty closed convex subset of 𝐸. We denote by and the sets of positive integers and real numbers, respectively. Also, we denote by 𝐽 the normalized duality mapping from 𝐸 to 2𝐸 defined by 𝑥𝐽𝑥=𝐸𝑥,𝑥=𝑥2=𝑥2,𝑥𝐸,(1.1) where , denotes the generalized duality pairing. Recall that if 𝐸 is smooth, then 𝐽 is single valued and if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸. We will still denote by 𝐽 the single-valued duality mapping.

A mapping 𝑆𝐶𝐸 is called nonexpansive if 𝑆𝑥𝑆𝑦𝑥𝑦 for all 𝑥,𝑦𝐶. Also a mapping 𝑆𝐶𝐶 is called asymptotically nonexpansive if there exists a sequence {𝑘𝑛}[1,) with 𝑘𝑛1 as 𝑛 such that 𝑆𝑛𝑥𝑆𝑛𝑦𝑘𝑛𝑥𝑦 for all 𝑥,𝑦𝐶 and for each 𝑛1. Denote by 𝐹(𝑆) the set of fixed points of 𝑆, that is, 𝐹(𝑆)={𝑥𝐶𝑆𝑥=𝑥}. The following example shows that the class of asymptotically nonexpansive mappings which was first introduced by Goebel and Kirk [1] is wider than the class of nonexpansive mappings.

Example 1.1 (see [2]). Let 𝐵𝐻 be the closed unit ball in the Hilbert space 𝐻=𝑙2 and 𝑆𝐵𝐻𝐵𝐻 a mapping defined by 𝑆𝑥1,𝑥2,𝑥3=,0,𝑥21,𝑎2𝑥2,𝑎3𝑥3,,(1.2) where {𝑎𝑛} is a sequence of real numbers such that 0<𝑎𝑖<1 and 𝑖=2𝑎𝑖=1/2. Then 𝑆𝑥𝑆𝑦2𝑥𝑦,𝑥,𝑦𝐵𝐻.(1.3) That is, 𝑆 is Lipschitzian but not nonexpansive. Observe that 𝑆𝑛𝑥𝑆𝑛𝑦2𝑛𝑖=2𝑎𝑖𝑥𝑦,𝑥,𝑦𝐵𝐻,𝑛2.(1.4) Here 𝑘𝑛=2𝑛𝑖=2𝑎𝑖1 as 𝑛. Therefore, 𝑆 is asymptotically nonexpansive but not nonexpansive.

A mapping 𝑇𝐶𝐸 is said to be relaxed 𝜂-𝜉 monotone if there exist a mapping 𝜂𝐶×𝐶𝐸 and a function 𝜉𝐸 positively homogeneous of degree 𝑝, that is, 𝜉(𝑡𝑧)=𝑡𝑝𝜉(𝑧) for all 𝑡>0 and 𝑧𝐸 such that 𝑇𝑥𝑇𝑦,𝜂(𝑥,𝑦)𝜉(𝑥𝑦),𝑥,𝑦𝐶,(1.5) where 𝑝>1 is a constant; see [3]. In the case of 𝜂(𝑥,𝑦)=𝑥𝑦 for all 𝑥, 𝑦𝐶, 𝑇 is said to be relaxed 𝜉-monotone. In the case of 𝜂(𝑥,𝑦)=𝑥𝑦 for all 𝑥,𝑦𝐶 and 𝜉(𝑧)=𝑘𝑧𝑝, where 𝑝>1 and 𝑘>0, 𝑇 is said to be 𝑝-monotone; see [46]. In fact, in this case, if 𝑝=2, then 𝑇 is a 𝑘-strongly monotone mapping. Moreover, every monotone mapping is relaxed 𝜂-𝜉 monotone with 𝜂(𝑥,𝑦)=𝑥𝑦 for all 𝑥,𝑦𝐶 and 𝜉=0. The following is an example of 𝜂-𝜉 monotone mapping which can be found in [3]. Let 𝐶=(,), 𝑇𝑥=𝑥, and 𝜂(𝑥,𝑦)=𝑐(𝑥𝑦),𝑥𝑦,𝑐(𝑥𝑦),𝑥<𝑦,(1.6) where 𝑐>0 is a constant. Then, 𝑇 is relaxed 𝜂-𝜉 monotone with 𝜉(𝑧)=𝑐𝑧2,𝑧0,𝑐𝑧2,𝑧<0.(1.7) A mapping 𝑇𝐶𝐸 is said to be 𝜂-hemicontinuous if, for each fixed 𝑥,𝑦𝐶, the mapping 𝑓[0,1](,+) defined by 𝑓(𝑡)=𝑇(𝑥+𝑡(𝑦𝑥)),𝜂(𝑦,𝑥) is continuous at 0+. For a real Banach space 𝐸 with the dual 𝐸 and for 𝐶 a nonempty closed convex subset of 𝐸, let 𝑓𝐶×𝐶 be a bifunction, 𝜑𝐶 a real-valued function and 𝑇𝐶𝐸 be a relaxed 𝜂-𝜉 monotone mapping. Recently, Kamraksa and Wangkeeree [7] introduced the following generalized mixed equilibrium problem (GMEP). Find𝑥𝐶suchthat𝑓(𝑥,𝑦)+𝑇𝑥,𝜂(𝑦,𝑥)+𝜑(𝑦)𝜑(𝑥),𝑦𝐶.(1.8) The set of such 𝑥𝐶 is denoted by GMEP(𝑓,𝑇), that is, GMEP(𝑓,𝑇)={𝑥𝐶𝑓(𝑥,𝑦)+𝑇𝑥,𝜂(𝑦,𝑥)+𝜑(𝑦)𝜑(𝑥),𝑦𝐶}.(1.9)

Special Cases
(1) If 𝑇 is monotone that is 𝑇 is relaxed 𝜂-𝜉 monotone with 𝜂(𝑥,𝑦)=𝑥𝑦 for all 𝑥,𝑦𝐶 and 𝜉=0, (1.8) is reduced to the following generalized equilibrium problem (GEP). Find𝑥𝐶suchthat𝑓(𝑥,𝑦)+𝑇𝑥,𝑦𝑥+𝜑(𝑦)𝜑(𝑥),𝑦𝐶.(1.10) The solution set of (1.10) is denoted by GEP(𝑓), that is, GEP(𝑓)={𝑥𝐶𝑓(𝑥,𝑦)+𝑇𝑥,𝑦𝑥+𝜑(𝑦)𝜑(𝑥),𝑦𝐶}.(1.11)
(2) In the case of 𝑇0 and 𝜑0, (1.8) is reduced to the following classical equilibrium problem Find𝑥𝐶suchthat𝑓(𝑥,𝑦)0,𝑦𝐶.(1.12) The set of all solution of (1.12) is denoted by EP(𝑓), that is, EP(𝑓)={𝑥𝐶𝑓(𝑥,𝑦)0,𝑦𝐶}.(1.13)
(3) In the case of 𝑓0, (1.8) is reduced to the following variational-like inequality problem [3]. Find𝑥𝐶suchthat𝑇𝑥,𝜂(𝑦,𝑥)+𝜑(𝑦)𝜑(𝑥)0,𝑦𝐶.(1.14)

The generalized mixed equilibrium problem (GMEP) (1.8) is 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. [8] and 𝜂-𝜉 monotonicity of the mapping 𝜑, Kamraksa and Wangkeeree [7] obtained the existence of solutions of generalized mixed equilibrium problem (1.8) in 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, [1217]. Recently, Tada and Takahashi [13, 16] and 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 Takahashi [16] established a strong convergence theorem for finding a common element of two sets by using the hybrid method introduced in Nakajo and Takahashi [18]. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.

On the other hand, in 1953, Mann [12] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping 𝑆 in a Hilbert space 𝐻: 𝑥𝑛+1=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑆𝑥𝑛,𝑛,(1.15) where the initial point 𝑥0 is taken in 𝐶 arbitrarily and {𝛼𝑛} is a sequence in [0,1]. However, we note that Manns iteration process (1.15) has only weak convergence, in general; for instance, see [1921]. In 2003, Nakajo and Takahashi [18] proposed the following sequence for a nonexpansive mapping 𝑆 in a Hilbert space: 𝑥0𝑦=𝑥𝐶,𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑆𝑥𝑛,𝐶𝑛=𝑦𝑧𝐶𝑛𝑥𝑧𝑛,𝑄𝑧𝑛=𝑧𝐶𝑥𝑛𝑧,𝑥0𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝑄𝑛𝑥0,(1.16) where 0𝛼𝑛𝑎<1 for all 𝑛, and 𝑃𝐶𝑛𝐷𝑛 is the metric projection from 𝐸 onto 𝐶𝑛𝐷𝑛. Then, they proved that {𝑥𝑛} converges strongly to 𝑃𝐹(𝑇)𝑥0. Recently, motivated by Nakajo and Takahashi [18] and Xu [22], Matsushita and Takahashi [14] introduced the iterative algorithm for finding fixed points of nonexpansive mappings in a uniformly convex and smooth Banach space: 𝑥0=𝑥𝐶 and 𝐶𝑛=co𝑧𝐶𝑧𝑆𝑧𝑡𝑛𝑥𝑛𝑆𝑥𝑛,𝐷𝑛=𝑥𝑧𝐶𝑛𝑧,𝐽𝑥𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥,𝑛0,(1.17) where co𝐷 denotes the convex closure of the set 𝐷, {𝑡𝑛} is a sequence in (0, 1) with 𝑡𝑛0. They proved that {𝑥𝑛} generated by (1.17) converges strongly to a fixed point of 𝑆. Very recently, Dehghan [23] 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=𝑥𝐶, 𝐶0=𝐷0=𝐶 and 𝐶𝑛=co𝑧𝐶𝑛1𝑧𝑆𝑛𝑧𝑡𝑛𝑥𝑛𝑆𝑛𝑥𝑛,𝐷𝑛=𝑧𝐷𝑛1𝑥𝑛𝑧,𝐽𝑥𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥,𝑛0,(1.18) where {𝑡𝑛} is a sequence in (0,1) with 𝑡𝑛0 as 𝑛 and 𝑆 is an asymptotically nonexpansive mapping. It is proved in [23] that {𝑥𝑛} converges strongly to a fixed point of 𝑆.

On the other hand, recently, Kamraksa and Wangkeeree [7] studied the hybrid projection 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: 𝑥0𝐶, 𝐷0=𝐶0=𝐶, and 𝑥1=𝑃𝐶0𝐷0𝑥0=𝑃𝐶𝑥0,𝐶1=co𝑧𝐶𝑧𝑆1𝑧𝑡1𝑥1𝑆1𝑥1,𝑢1𝑢𝐶suchthat𝑓1,𝑦+𝜑(𝑦)+𝑇𝑢1,𝜂𝑦,𝑢1+1𝑟1𝑦𝑢1𝑢,𝐽1𝑥1𝐷,𝑦𝐶,1=𝑢𝑧𝐶1𝑥𝑧,𝐽1𝑢1,𝑥02=𝑃𝐶1𝐷1𝑥0,𝐶𝑁=co𝑧𝐶𝑁1𝑧𝑆𝑁𝑧𝑡1𝑥𝑁𝑆𝑁𝑥𝑁,𝑢𝑁𝑢𝐶suchthat𝑓𝑁,𝑦+𝜑(𝑦)+𝑇𝑢𝑁,𝜂𝑦,𝑢𝑁+1𝑟𝑁𝑦𝑢𝑁𝑢,𝐽𝑁𝑥𝑁𝐷,𝑦𝐶,𝑁=𝑧𝐷𝑁1𝑢𝑁𝑥𝑧,𝐽𝑁𝑢𝑁,𝑥0𝑁+1=𝑃𝐶𝑁𝐷𝑁𝑥0,𝐶𝑁+1=co𝑧𝐶𝑁𝑧𝑆21𝑧𝑡1𝑥𝑁+1𝑆21𝑥𝑁+1,𝑢𝑁+1𝑢𝐶suchthat𝑓𝑁+1,𝑦+𝜑(𝑦)+𝑇𝑢𝑁+1,𝜂𝑦,𝑢𝑁+1+1𝑟𝑁+1𝑦𝑢𝑁+1𝑢,𝐽𝑁+1𝑥𝑁+1𝐷,𝑦𝐶,𝑁+1=𝑧𝐷𝑁𝑢𝑁+1𝑥𝑧,𝐽𝑁+1𝑢𝑁+1,𝑥0𝑁+2=𝑃𝐶𝑁+1𝐷𝑁+1𝑥0,𝐶2𝑁=co𝑧𝐶2𝑁1𝑧𝑆2𝑁𝑧𝑡1𝑥2𝑁𝑆2𝑁𝑥2𝑁,𝑢2𝑁𝑢𝐶suchthat𝑓2𝑁,𝑦+𝜑(𝑦)+𝑇𝑢2𝑁,𝜂𝑦,𝑢2𝑁+1𝑟2𝑁𝑦𝑢2𝑁𝑢,𝐽2𝑁𝑥2𝑁𝐷,𝑦𝐶,2𝑁=𝑧𝐷2𝑁1𝑢2𝑁𝑥𝑧,𝐽2𝑁𝑢2𝑁,𝑥02𝑁+1=𝑃𝐶2𝑁𝐷2𝑁𝑥0,𝐶2𝑁+1=co𝑧𝐶2𝑁𝑧S31𝑧𝑡1𝑥2𝑁+1𝑆31𝑥2𝑁+1,𝑢2𝑁+1𝑢𝐶suchthat𝑓2𝑁+1,𝑦+𝜑(𝑦)+𝑇𝑢2𝑁+1,𝜂𝑦,𝑢2𝑁+1+1𝑟2𝑁+1𝑦𝑢2𝑁+1𝑢,𝐽2𝑁+1𝑥2𝑁+1𝐷,𝑦𝐶,2𝑁+1=𝑧𝐷2𝑁𝑢2𝑁+1𝑥𝑧,𝐽2𝑁+1𝑢2𝑁+1,𝑥02𝑁+2=𝑃𝐶2𝑁+1𝐷2𝑁+1𝑥0,(1.19) The above algorithm is called the hybrid iterative algorithm for a finite family of asymptotically nonexpansive mappings from 𝐶 into itself. Since, for each 𝑛1, it can be written as 𝑛=(1)𝑁+𝑖, where 𝑖=𝑖(𝑛){1,2,,𝑁}, =(𝑛)1 is a positive integer and (𝑛) as 𝑛. Hence the above table can be written in the following form: 𝑥0𝐶,𝐷0=𝐶0𝐶=𝐶,𝑛=co𝑧𝐶𝑛1𝑧𝑆(𝑛)𝑖(𝑛)𝑧𝑡𝑛𝑥𝑛𝑆(𝑛)𝑖(𝑛)𝑥𝑛𝑢,𝑛1,𝑛𝑢𝐶suchthat𝑓𝑛,𝑦+𝜑(𝑦)+𝑇𝑢𝑛,𝜂𝑦,𝑢𝑛+1𝑟𝑛𝑦𝑢𝑛𝑢,𝐽𝑛𝑥𝑛𝐷,𝑦𝐶,𝑛1,𝑛=𝑧𝐷𝑛1𝑢𝑛𝑥𝑧,𝐽𝑛𝑢𝑛𝑥0,𝑛1,𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥0,𝑛0.(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 Nakajo [24], Kamraksa and Wangkeeree [7], Dehghan [23], and many others.

2. Preliminaries

Let 𝐸 be a real Banach space and let 𝑈={𝑥𝐸𝑥=1} be the unit sphere of 𝐸. A Banach space 𝐸 is said to be strictly convex if for any 𝑥,𝑦𝑈, 𝑥𝑦implies𝑥+𝑦<2.(2.1) It is also said to be uniformly convex if for each 𝜀(0,2], there exists 𝛿>0 such that for any 𝑥,𝑦𝑈, 𝑥𝑦𝜀implies𝑥+𝑦<2(1𝛿).(2.2) It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function 𝛿[0,2][0,1] called the modulus of convexity of 𝐸 as follows: 𝛿(𝜀)=inf1𝑥+𝑦2.𝑥,𝑦𝐸,𝑥=𝑦=1,𝑥𝑦𝜀(2.3) Then 𝐸 is uniformly convex if and only if 𝛿(𝜀)>0 for all 𝜀(0,2]. A Banach space 𝐸 is said to be smooth if the limit lim𝑡0𝑥+𝑡𝑦𝑥𝑡(2.4) exists for all 𝑥,𝑦𝑈. Let 𝐶 be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space 𝐸. Then for any 𝑥𝐸, there exists a unique point 𝑥0𝐶 such that 𝑥0𝑥min𝑦𝐶𝑦𝑥.(2.5) The mapping 𝑃𝐶𝐸𝐶 defined by 𝑃𝐶𝑥=𝑥0 is called the metric projection from 𝐸 onto 𝐶. Let 𝑥𝐸 and 𝑢𝐶. The following theorem is well known.

Theorem 2.1. Let 𝐶 be a nonempty convex subset of a smooth Banach space 𝐸 and let 𝑥𝐸 and 𝑦𝐶. Then the following are equivalent: (a)𝑦is a best approximation to 𝑥𝑦=𝑃𝐶𝑥,(b)𝑦 is a solution of the variational inequality: 𝑦𝑧,𝐽(𝑥𝑦)0𝑧𝐶,(2.6) where 𝐽 is a duality mapping and 𝑃𝐶 is the metric projection from 𝐸 onto 𝐶.

It is well known that if 𝑃𝐶 is a metric projection from a real Hilbert space 𝐻 onto a nonempty, closed, and convex subset 𝐶, then 𝑃𝐶 is nonexpansive. But, in a general Banach space, this fact is not true.

In the sequel one will need the following lemmas.

Lemma 2.2 (see [25]). Let 𝐸 be a uniformly convex Banach space, let {𝛼𝑛} be a sequence of real numbers such that 0<𝑏𝛼𝑛𝑐<1 for all 𝑛1, and let {𝑥𝑛} and {𝑦𝑛} be sequences in 𝐸 such that limsup𝑛𝑥𝑛𝑑, limsup𝑛𝑦𝑛𝑑 and lim𝑛𝛼𝑛𝑥𝑛+(1𝛼𝑛)𝑦𝑛=𝑑. Then lim𝑛𝑥𝑛𝑦𝑛=0.

Dehghan [23] obtained the following useful result.

Theorem 2.3 (see [23]). Let 𝐶 be a bounded, closed, and convex subset of a uniformly convex Banach space 𝐸. Then there exists a strictly increasing, convex, and continuous function 𝛾[0,)[0,) such that 𝛾(0)=0 and 𝛾1𝑘𝑚𝑆𝑚𝑛𝑖=1𝜆𝑖𝑥𝑖𝑛𝑖=1𝜆𝑖𝑆𝑚𝑥𝑖max1𝑗𝑘𝑛𝑥𝑗𝑥𝑘1𝑘𝑚𝑆𝑚𝑥𝑗𝑆𝑚𝑥𝑘(2.7) for any asymptotically nonexpansive mapping 𝑆 of 𝐶 into 𝐶 with {𝑘𝑛}, any elements 𝑥1,𝑥2,,𝑥𝑛𝐶, any numbers 𝜆1,𝜆2,,𝜆𝑛0 with 𝑛𝑖=1𝜆𝑖=1 and each 𝑚1.

Lemma 2.4 (see [26, Lemma 1.6]). Let 𝐸 be a uniformly convex Banach space, 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑆𝐶𝐶 be an asymptotically nonexpansive mapping. Then (𝐼𝑆) is demiclosed at 0, that is, if 𝑥𝑛𝑥 and (𝐼𝑆)𝑥𝑛0, then 𝑥𝐹(𝑆).

The following lemma can be found in [7].

Lemma 2.5 (see [7, Lemma 3.2]). Let 𝐶 be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑇𝐶𝐸 be an 𝜂-hemicontinuous and relaxed 𝜂𝜉 monotone mapping. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1), (A3), and (A4) and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to . Let 𝑟>0 and 𝑧𝐶. Assume that (i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦𝐶; (ii)for any fixed 𝑢,𝑣𝐶, the mapping 𝑥𝑇𝑣,𝜂(𝑥,𝑢) is convex and lower semicontinuous; (iii)𝜉𝐸 is weakly lower semicontinuous, that is, for any net {𝑥𝛽},𝑥𝛽 converges to 𝑥 in 𝜎(𝐸,𝐸) which implies that 𝜉(𝑥)liminf𝜉(𝑥𝛽). Then there exists 𝑥0𝐶 such that 𝑓𝑥0+,𝑦𝑇𝑥0,𝜂𝑦,𝑥01+𝜑(𝑦)+𝑟𝑦𝑥0𝑥,𝐽0𝑥𝑧𝜑0,𝑦𝐶.(2.8)

Lemma 2.6 (see [7, Lemma 3.3]). Let 𝐶 be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑇𝐶𝐸 be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4) and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to . Let 𝑟>0 and define a mapping Φ𝑟𝐸𝐶 as follows: Φ𝑟1(𝑥)=𝑧𝐶𝑓(𝑧,𝑦)+𝑇𝑧,𝜂(𝑦,𝑧)+𝜑(𝑦)+𝑟𝑦𝑧,𝐽(𝑧𝑥)𝜑(𝑧),𝑦C(2.9) for all 𝑥𝐸. Assume that (i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0, for all 𝑥,𝑦𝐶; (ii)for any fixed 𝑢,𝑣𝐶, the mapping 𝑥𝑇𝑣,𝜂(𝑥,𝑢) is convex and lower semicontinuous and the mapping 𝑥𝑇𝑢,𝜂(𝑣,𝑥) is lower semicontinuous; (iii)𝜉𝐸 is weakly lower semicontinuous; (iv)for any 𝑥,𝑦𝐶, 𝜉(𝑥𝑦)+𝜉(𝑦𝑥)0. Then, the following holds: (1)Φ𝑟 is single valued; (2)Φ𝑟𝑥Φ𝑟𝑦,𝐽(Φ𝑟𝑥𝑥)Φ𝑟𝑥Φ𝑟𝑦,𝐽(Φ𝑟𝑦𝑦) for all 𝑥,𝑦𝐸; (3)𝐹(Φ𝑟)=EP(𝑓,𝑇); (4)EP(𝑓,𝑇) is 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 bifunction 𝑓.

Theorem 3.1. Let 𝐶 be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑇𝐶𝐸 be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying the following conditions (A1)–(A4): (A1)𝑓(𝑥,𝑥)=0 for all 𝑥𝐶; (A2)𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)min{𝜉(𝑥𝑦),𝜉(𝑦𝑥)} for all 𝑥,𝑦𝐶; (A3)for all 𝑦𝐶, 𝑓(,𝑦) is weakly upper semicontinuous; (A4)for all 𝑥𝐶, 𝑓(𝑥,) is convex. For any 𝑟>0 and 𝑥𝐸, define a mapping Φ𝑟𝐸𝐶 as follows: Φ𝑟1(𝑥)=𝑧𝐶𝑓(𝑧,𝑦)+𝑇𝑧,𝜂(𝑦,𝑧)+𝜑(𝑦)+𝑟,𝑦𝑧,𝐽(𝑧𝑥)𝜑(𝑧),𝑦𝐶(3.1) where 𝜑 is a lower semicontinuous and convex function from 𝐶 to . Assume that (i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0, for all 𝑥,𝑦𝐶; (ii)for any fixed 𝑢,𝑣𝐶, the mapping 𝑥𝑇𝑣,𝜂(𝑥,𝑢) is convex and lower semicontinuous and the mapping 𝑥𝑇𝑢,𝜂(𝑣,𝑥) is lower semicontinuous; (iii)𝜉𝐸 is weakly lower semicontinuous. Then, the following holds: (1)Φ𝑟 is single valued; (2)Φ𝑟𝑥Φ𝑟𝑦,𝐽(Φ𝑟𝑥𝑥)Φ𝑟𝑥Φ𝑟𝑦,𝐽(Φ𝑟𝑦𝑦) for all 𝑥,𝑦𝐸; (3)𝐹(Φ𝑟)=GMEP(𝑓,𝑇); (4)GMEP(𝑓,𝑇) is nonempty closed and convex.

Proof. For each 𝑥𝐸. It follows from Lemma 2.5 that Φ𝑟(𝑥) is nonempty.
(1) We prove that Φ𝑟 is single valued. Indeed, for 𝑥𝐸 and 𝑟>0, let 𝑧1,𝑧2Φ𝑟𝑥. Then 𝑓𝑧1,𝑧2+𝑇𝑧2𝑧,𝜂2,𝑧1𝑧+𝜑2+1𝑟𝑧1𝑧2𝑧,𝐽1𝑧𝑥𝜑1,𝑓𝑧2,𝑧1+𝑇𝑧1𝑧,𝜂1,𝑧2𝑧+𝜑1+1𝑟𝑧2𝑧1𝑧,𝐽2𝑧𝑥𝜑2.(3.2) Adding the two inequalities, from (i) we have 𝑓𝑧2,𝑧1𝑧+𝑓1,𝑧2+𝑇𝑧1𝑇𝑧2𝑧,𝜂2,𝑧11+𝑟𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝑥0.(3.3) Setting Δ=min{𝜉(𝑧1𝑧2),𝜉(𝑧2𝑧1)} and using (A2), we have Δ+𝑇𝑧1𝑇𝑧2𝑧,𝜂2,𝑧1+1𝑟𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝑥0,(3.4) that is, 1𝑟𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝑥𝑇𝑧2𝑇𝑧1𝑧,𝜂2,𝑧1Δ.(3.5) Since 𝑇 is relaxed 𝜂-𝜉 monotone and 𝑟>0, one has 𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝜉𝑧𝑥𝑟2𝑧1Δ0.(3.6) In (3.5) exchanging the position of 𝑧1 and 𝑧2, we get 1𝑟𝑧1𝑧2𝑧,𝐽2𝑧𝑥𝐽1𝑥𝑇𝑧1𝑇𝑧2𝑧,𝜂1,𝑧2Δ,(3.7) that is, 𝑧1𝑧2𝑧,𝐽2𝑧𝑥𝐽1𝜉𝑧𝑥𝑟1𝑧2Δ0.(3.8) Now, adding the inequalities (3.6) and (3.8), we have 2𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝑥0.(3.9) Hence, 𝑧02𝑧1𝑧,𝐽1𝑧𝑥𝐽2=𝑧𝑥2𝑧𝑥1𝑧𝑥,𝐽1𝑧𝑥𝐽2𝑥.(3.10) Since 𝐽 is monotone and 𝐸 is strictly convex, we obtain that 𝑧1𝑥=𝑧2𝑥 and hence 𝑧1=𝑧2. Therefore 𝑆𝑟 is single valued.
(2) For 𝑥,𝑦𝐶, we have 𝑓Φ𝑟𝑥,Φ𝑟𝑦+𝑇Φ𝑟Φ𝑥,𝜂𝑟𝑦,Φ𝑟𝑥Φ+𝜑𝑟𝑦Φ𝜑𝑟𝑥+1𝑟Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟𝑓Φ𝑥𝑥0,𝑟𝑦,Φ𝑟𝑥+𝑇Φ𝑟Φ𝑦,𝜂𝑟𝑥,Φ𝑟𝑦Φ+𝜑𝑟𝑥Φ𝜑𝑟𝑦+1𝑟Φ𝑟𝑥Φ𝑟Φ𝑦,𝐽𝑟𝑦𝑦0.(3.11) Setting Λ𝑥,𝑦=min{𝜉(Φ𝑟𝑥Φ𝑟𝑦),𝜉(Φ𝑟𝑦Φ𝑟𝑥)} and applying (A2), we get 𝑇Φ𝑟𝑥𝑇Φ𝑟Φ𝑦,𝜂𝑟𝑦,Φ𝑟𝑥+1𝑟Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟Φ𝑥𝑥𝐽𝑟𝑦𝑦Λ𝑥,𝑦,(3.12) that is, 1𝑟Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟Φ𝑥𝑥𝐽𝑟𝑦𝑦𝑇Φ𝑟𝑦𝑇Φ𝑟Φ𝑥,𝜂𝑟𝑦,Φ𝑟𝑥Λ𝑥,𝑦Φ𝜉𝑟𝑦Φ𝑟𝑥Λ𝑥,𝑦0.(3.13) In (3.13) exchanging the position of Φ𝑟𝑥 and Φ𝑟𝑦, we get 1𝑟Φ𝑟𝑥Φ𝑟Φ𝑦,𝐽𝑟Φ𝑦𝑦𝐽𝑟𝑥𝑥0.(3.14) Adding the inequalities (3.13) and (3.14), we have 2𝑟Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟Φ𝑥𝑥𝐽𝑟𝑦𝑦0.(3.15) It follows that Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟Φ𝑥𝑥𝐽𝑟𝑦𝑦0.(3.16) Hence Φ𝑟𝑥Φ𝑟Φ𝑦,𝐽𝑟Φ𝑥𝑥𝑟𝑥Φ𝑟Φ𝑦,𝐽𝑟𝑦𝑦.(3.17) The conclusions (3), (4) follow from Lemma 2.6.

Example 3.2. Define 𝜉 and 𝑓× by 𝑓(𝑥,𝑦)=(𝑥𝑦)22,𝜉(𝑥)=𝑥2𝑥,𝑦.(3.18) It is easy to see that 𝑓 satisfies (A1), (A3), (A4), and (A2): 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)min{𝜉(𝑥𝑦),𝜉(𝑥𝑦)}, for  all (𝑥,𝑦)×.

Remark 3.3. Theorem 3.1 generalizes and improves [7, Lemma 3.3] in the following manners.(1)The condition 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)0 has been weakened by (A2) that is 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)min{𝜉(𝑥𝑦),𝜉(𝑦𝑥)} for all 𝑥,𝑦𝐶.(2)The control condition 𝜉(𝑥𝑦)+𝜉(𝑦𝑥)0 imposed on the mapping 𝜉 in [7, Lemma 3.3] can be removed.
If 𝑇 is monotone that is 𝑇 is relaxed 𝜂-𝜉 monotone with 𝜂(𝑥,𝑦)=𝑥𝑦 for all 𝑥,𝑦𝐶 and 𝜉=0, we have the following results.

Corollary 3.4. Let 𝐶 be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝑇𝐶𝐸 be a monotone mapping and 𝑓 be a bifunction from 𝐶×𝐶 to satisfying the following conditions (i)–(iv): (i)𝑓(𝑥,𝑥)=0 for all 𝑥𝐶; (ii)𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)0 for all 𝑥,𝑦𝐶; (iii)for all 𝑦𝐶, 𝑓(,𝑦) is weakly upper semicontinuous; (iv)for all 𝑥𝐶, 𝑓(𝑥,) is convex. For any 𝑟>0 and 𝑥𝐸, define a mapping Φ𝑟𝐸𝐶 as follows: Φ𝑟1(𝑥)=𝑧𝐶𝑓(𝑧,𝑦)+𝑇𝑧,𝑦𝑧+𝜑(𝑦)+𝑟,𝑦𝑧,𝐽(𝑧𝑥)𝜑(𝑧),𝑦𝐶(3.19) where 𝜑 is a lower semicontinuous and convex function from 𝐶 to . Then, the following holds: (1)Φ𝑟 is single valued; (2)Φ𝑟𝑥Φ𝑟𝑦,𝐽(Φ𝑟𝑥𝑥)Φ𝑟𝑥Φ𝑟𝑦,𝐽(Φ𝑟𝑦𝑦) for all 𝑥,𝑦𝐸; (3)𝐹(Φ𝑟)=GEP(𝑓);(4)GEP(𝑓) is nonempty closed and convex.

4. Strong Convergence Theorems

In this section, we prove the strong convergence theorem of the sequence {𝑥𝑛} defined by (1.20) for 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 of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4). Let 𝑇𝐶𝐸 be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping and 𝜑 a lower semicontinuous and convex function from 𝐶 to . Let, for each 1𝑖𝑁, 𝑆𝑖𝐶𝐶 be an asymptotically nonexpansive mapping with a sequence {𝑘𝑛,𝑖}𝑛=1, respectively, such that 𝑘𝑛,𝑖1 as 𝑛. Assume that Ω=𝑁𝑖=1𝐹(𝑆𝑖)GMEP(𝑓,𝑇) is nonempty. Let {𝑥𝑛} be a sequence generated by (1.20), where {𝑡𝑛} and {𝑟𝑛} are real sequences in (0,1) satisfying lim𝑛𝑡𝑛=0 and liminf𝑛𝑟𝑛>0. Then {𝑥𝑛} converges strongly, as 𝑛, to 𝑃Ω𝑥0, where 𝑃Ω is the metric projection of 𝐸 onto Ω.

Proof. First, define the sequence {𝑘𝑛} by 𝑘𝑛=max{𝑘𝑛,𝑖1𝑖𝑁} and so 𝑘𝑛1 as 𝑛 and 𝑆(𝑛)𝑖(𝑛)𝑥𝑆(𝑛)𝑖(𝑛)𝑦𝑘𝑛𝑥𝑦𝑥,𝑦𝐶,(4.1) where (𝑛)=𝑗+1 if 𝑗𝑁<𝑛(𝑗+1)𝑁, 𝑗=1,2,𝑁 and 𝑛=𝑗𝑁+𝑖(𝑛); 𝑖(𝑛){1,2,,𝑁}. Next, we rewrite the algorithm (1.20) as the following relation: 𝑥0𝐶,𝐷0=𝐶0𝐶=𝐶,𝑛=co𝑧𝐶𝑛1𝑧𝑆(𝑛)𝑖(𝑛)𝑧𝑡𝑛𝑥𝑛𝑆(𝑛)𝑖(𝑛)𝑥𝑛𝐷,𝑛0,𝑛=𝑧𝐷𝑛1Φ𝑟𝑛𝑥𝑛𝑥𝑧,𝐽𝑛Φ𝑟𝑛𝑥𝑛𝑥0,𝑛1,𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥0,𝑛0,(4.2) where Φ𝑟 is the mapping defined by (3.19). We show that the sequence {𝑥𝑛} is well defined. It is easy to verify that 𝐶𝑛𝐷𝑛 is closed and convex and Ω𝐶𝑛 for all 𝑛0. Next, we prove that Ω𝐶𝑛𝐷𝑛. Indeed, since 𝐷0=𝐶, we also have Ω𝐶0𝐷0. Assume that Ω𝐶𝑘1𝐷𝑘1 for 𝑘2. Utilizing Theorem 3.1 (2), we obtain Φ𝑟𝑘𝑥𝑘Φ𝑟𝑘Φ𝑢,𝐽𝑟𝑘Φ𝑢𝑢𝐽𝑟𝑘𝑥𝑘𝑥𝑘0,𝑢Ω,(4.3) which gives that Φ𝑟𝑘𝑥𝑘𝑥𝑢,𝐽𝑘Φ𝑟𝑘𝑥𝑘0,𝑢Ω,(4.4) hence Ω𝐷𝑘. By the mathematical induction, we get that Ω𝐶𝑛𝐷𝑛 for each 𝑛0 and hence {𝑥𝑛} is well defined. Now, we show that lim𝑛𝑥𝑛𝑥𝑛+𝑗=0,𝑗=1,2,,𝑁.(4.5) Put 𝑤=𝑃Ω𝑥0, since Ω𝐶𝑛𝐷𝑛 and 𝑥𝑛+1=𝑃𝐶𝑛𝐷𝑛, we have 𝑥𝑛+1𝑥0𝑤𝑥0,𝑛0.(4.6) Since 𝑥𝑛+2𝐷𝑛+1𝐷𝑛 and 𝑥𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥0, we have 𝑥𝑛+1𝑥0𝑥𝑛+2𝑥0.(4.7) Hence the sequence {𝑥𝑛𝑥0} is bounded and monotone increasing and hence there exists a constant 𝑑 such that lim𝑛𝑥𝑛𝑥0=𝑑.(4.8) Moreover, by the convexity of 𝐷𝑛, we also have 1/2(𝑥𝑛+1+𝑥𝑛+2)𝐷𝑛 and hence 𝑥0𝑥𝑛+1𝑥0𝑥𝑛+1+𝑥𝑛+2212𝑥0𝑥𝑛+1+𝑥0𝑥𝑛+2.(4.9) This implies that lim𝑛12𝑥0𝑥𝑛+1+12𝑥0𝑥𝑛+2=lim𝑛𝑥0𝑥𝑛+1+𝑥𝑛+22=𝑑.(4.10) By Lemma 2.2, we have lim𝑛𝑥𝑛𝑥𝑛+1=0.(4.11) Furthermore, we can easily see that lim𝑛𝑥𝑛𝑥𝑛+𝑗=0,𝑗=1,2,,𝑁.(4.12) Next, we show that lim𝑛𝑥𝑛𝑆(𝑛𝜅)𝑖(𝑛𝜅)𝑥𝑛=0,forany𝜅{1,2,,𝑁}.(4.13) Fix 𝜅{1,2,,𝑁} and put 𝑚=𝑛𝜅. Since 𝑥𝑛=𝑃𝐶𝑛1𝐷𝑛1𝑥, we have 𝑥𝑛𝐶𝑛1𝐶𝑚. Since 𝑡𝑚>0, there exists 𝑦1,,𝑦𝑃𝐶 and a nonnegative number 𝜆1,,𝜆𝑃 with 𝜆1++𝜆𝑃=1 such that 𝑥𝑛𝑃𝑖=1𝜆𝑖𝑦𝑖<𝑡𝑚,𝑦(4.14)𝑖𝑆(𝑚)𝑖(𝑚)𝑦𝑖𝑡𝑚𝑥𝑚𝑆(𝑚)𝑖(𝑚)𝑥𝑚,𝑖{1,,𝑃}.(4.15) By the boundedness of 𝐶 and {𝑘𝑛}, we can put the following: 𝑀=sup𝑥𝐶𝑥,𝑢=𝑃𝑁𝑖=1𝐹(𝑆𝑖)𝑥0,𝑟0=sup𝑛11+𝑘𝑛𝑥𝑛.𝑢(4.16) This together with (4.14) implies that 𝑥𝑛1𝑘𝑚𝑃𝑖=1𝜆𝑖𝑦𝑖11𝑘𝑚1𝑥+𝑘𝑚𝑥𝑛𝑃𝑖=1𝜆𝑖𝑦𝑖11𝑘𝑚𝑀+𝑡𝑚,𝑦𝑖𝑆(𝑚)𝑖(𝑚)𝑦𝑖𝑡𝑚𝑥𝑚𝑆(𝑚)𝑖(𝑚)𝑥𝑚𝑡𝑚𝑥𝑚𝑆(𝑚)𝑖(𝑚)𝑢+𝑡𝑚𝑆(𝑚)𝑖(𝑚)𝑢𝑆(𝑚)𝑖(𝑚)𝑥𝑚𝑡𝑚𝑥𝑚𝑢+𝑡𝑚𝑘𝑚𝑢𝑥𝑚𝑡𝑚1+𝑘𝑚𝑥𝑚𝑢𝑡𝑚𝑟0,(4.17) for all 𝑖{1,,𝑁}. Therefore, for each 𝑖{1,,𝑃}, we get 𝑦𝑖1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑦𝑖𝑦𝑖𝑆(𝑚)𝑖(𝑚)𝑦𝑖+𝑆(𝑚)𝑖(𝑚)𝑦𝑖1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑦𝑖𝑟0𝑡𝑚+11𝑘𝑚𝑀.(4.18) Moreover, since each 𝑆𝑖, 𝑖{1,2,,𝑁}, is asymptotically nonexpansive, we can obtain that 1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑃𝑖=1𝜆𝑖𝑦𝑖𝑆(𝑚)𝑖(𝑚)𝑥𝑛1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑃𝑖=1𝜆𝑖𝑦𝑖1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑥𝑛+1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑥𝑛𝑆(𝑚)𝑖(𝑚)𝑥𝑛𝑃𝑖=1𝜆𝑖𝑦𝑖𝑥𝑛+11𝑘𝑚𝑀=𝑡𝑚+11𝑘𝑚𝑀.(4.19) It follows from Theorem 2.3 and the inequalities (4.17)–(4.19) that 𝑥𝑛𝑆(𝑚)𝑖(𝑚)𝑥𝑛𝑥𝑛1𝑘𝑚𝑃𝑖=1𝜆𝑖𝑦𝑖+1𝑘𝑚𝑃𝑖=1𝜆𝑖𝑦𝑖𝑆(𝑚)𝑖(𝑚)𝑦𝑖+1𝑘𝑚𝑃𝑖=1𝜆𝑖𝑆(𝑚)𝑖(𝑚)𝑦𝑖𝑆(𝑚)𝑖(𝑚)𝑃𝑖=1𝜆𝑖𝑦𝑖+1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑃𝑖=1𝜆𝑖𝑦𝑖𝑆(𝑚)𝑖(𝑚)𝑥𝑛121𝑘𝑚𝑀+𝑡𝑚+𝑟0𝑡𝑚𝑘𝑚+𝛾1max1𝑖𝑗𝑁𝑦𝑖𝑦𝑗1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑦𝑖𝑆(𝑚)𝑖(𝑚)𝑦𝑗1=21𝑘𝑚𝑀+2𝑡𝑚+𝑟0𝑡𝑚𝑘𝑚+𝛾1max1𝑖𝑗𝑁𝑦𝑖𝑦𝑗1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑦𝑖𝑆(𝑚)𝑖(𝑚)𝑦𝑗121𝑘𝑚𝑀+2𝑡𝑚+𝑟0𝑡𝑚𝑘𝑚+𝛾1max1𝑖𝑗𝑁𝑦𝑖1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑦𝑖+𝑦𝑗1𝑘𝑚𝑆(𝑚)𝑖(𝑚)𝑦𝑗121𝑘𝑚𝑀+2𝑡𝑚+𝑟0𝑡𝑚𝑘𝑚+𝛾1211𝑘𝑚𝑀+2𝑟0𝑡𝑚.(4.20) Since lim𝑛𝑘𝑛=1 and lim𝑛𝑡𝑛=0, it follows from the above inequality that lim𝑛𝑥𝑛𝑆(𝑚)𝑖(𝑚)𝑥𝑛=0.(4.21) Hence (4.13) is proved. Next, we show that lim𝑛𝑥𝑛𝑆𝑙𝑥𝑛=0;𝑙=1,2,,𝑁.(4.22) From the construction of 𝐶𝑛, one can easily see that 𝑥𝑛+1𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1𝑡𝑛𝑥𝑛𝑆(𝑛)𝑖(𝑛)𝑥𝑛.(4.23) The boundedness of 𝐶 and lim𝑛𝑡𝑛=0 implies that lim𝑛𝑥𝑛+1𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1=0.(4.24) On the other hand, since for any positive integer 𝑛>𝑁, 𝑛=(𝑛𝑁)(mod𝑁) and 𝑛=((𝑛)1)𝑁+𝑖(𝑛), we have 𝑛𝑁=((𝑛)1)𝑁+𝑖(𝑛)=((𝑛𝑁)1)𝑁+𝑖(𝑛𝑁)(4.25) that is (𝑛𝑁)=(𝑛)1,𝑖(𝑛𝑁)=𝑖(𝑛).(4.26) Thus, 𝑥𝑛𝑆𝑛𝑥𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1+𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1S𝑛𝑥𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1+𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1𝑆𝑛𝑥𝑛+1+𝑆𝑛𝑥𝑛+1𝑆𝑛𝑥𝑛1+𝑘1𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1+𝑘1𝑆(𝑛)1𝑖(𝑛)𝑥𝑛+1𝑥𝑛+11+𝑘1𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1+𝑘1𝑆(𝑛)1𝑖(𝑛)𝑥𝑛+1𝑆(𝑛)1𝑖(𝑛)𝑥𝑛+𝑆(𝑛)1𝑖(𝑛)𝑥𝑛𝑥𝑛+𝑥𝑛𝑥𝑛+11+2𝑘1𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1+𝑘1𝑆(𝑛𝑁)𝑖(𝑛𝑁)𝑥𝑛+1𝑆(𝑛𝑁)𝑖(𝑛𝑁)𝑥𝑛+𝑘1𝑆(𝑛𝑁)𝑖(𝑛𝑁)𝑥𝑛𝑥𝑛1+2𝑘1𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1+𝑘1𝑘𝑛𝑁𝑥𝑛+1𝑥𝑛+𝑘1𝑆(𝑛𝑁)𝑖(𝑛𝑁)𝑥𝑛𝑥𝑛1+2𝑘1+𝑘1𝑘𝑛𝑁𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑆(𝑛)𝑖(𝑛)𝑥𝑛+1+𝑘1𝑆(𝑛𝑁)𝑖(𝑛𝑁)𝑥𝑛𝑥𝑛.(4.27) Applying the facts (4.11), (4.13), and (4.24) to the above inequality, we obtain lim𝑛𝑥𝑛𝑆𝑛𝑥𝑛=0.(4.28) Therefore, for any 𝑗=1,2,,𝑁, we have 𝑥𝑛𝑆𝑛+𝑗𝑥𝑛𝑥𝑛𝑥𝑛+𝑗+𝑥𝑛+𝑗𝑆𝑛+𝑗𝑥𝑛+𝑗+𝑆𝑛+𝑗𝑥𝑛+𝑗𝑆𝑛+𝑗𝑥𝑛𝑥𝑛𝑥𝑛+𝑗+𝑥𝑛+𝑗𝑆𝑛+𝑗𝑥𝑛+𝑗+𝑘1𝑥𝑛+𝑗𝑥𝑛=1+𝑘1𝑥𝑛𝑥𝑛+𝑗+𝑥𝑛+𝑗𝑆𝑛+𝑗𝑥𝑛+𝑗0as𝑛,(4.29) which gives that lim𝑛𝑥𝑛𝑆𝑙𝑥𝑛=0;𝑙=1,2,,𝑁,(4.30) as required. Since {𝑥𝑛} is bounded, there exists a subsequence {𝑥𝑛𝑖} of {𝑥𝑛} such that 𝑥𝑛𝑖̃𝑥𝐶. It follows from Lemma 2.4 that ̃𝑥𝐹(𝑆𝑙) for all 𝑙=1,2,,𝑁. That is 𝑥𝑁𝑖=1𝐹(𝑆𝑖).
Next, we show that ̃𝑥GMEP(𝑓,𝑇). By the construction of 𝐷𝑛, we see from Theorem 2.1 that Φ𝑟𝑛𝑥𝑛=𝑃𝐷𝑛𝑥𝑛. Since 𝑥𝑛+1𝐷𝑛, we get 𝑥𝑛Φ𝑟𝑛𝑥𝑛𝑥𝑛𝑥𝑛+10.(4.31) Furthermore, since liminf𝑛𝑟𝑛>0, we have 1𝑟𝑛𝐽𝑥𝑛Φ𝑟𝑛𝑥𝑛=1𝑟𝑛𝑥𝑛Φ𝑟𝑛𝑥𝑛0,(4.32) as 𝑛. By (4.32), we also have Φ𝑟𝑛𝑖𝑥𝑛𝑖̃𝑥. By the definition of Φ𝑟𝑛𝑖, for each 𝑦𝐶, we obtain 𝑓Φ𝑟𝑛𝑖𝑥𝑛𝑖+,𝑦𝑇Φ𝑟𝑛𝑖𝑥𝑛𝑖,𝜂𝑦,Φ𝑟𝑛𝑖𝑥𝑛𝑖1+𝜑(𝑦)+𝑟𝑛𝑖𝑦Φ𝑟𝑛𝑖𝑥𝑛𝑖Φ,𝐽𝑟𝑛𝑖𝑥𝑛𝑖𝑥𝑛𝑖Φ𝜑𝑟𝑛𝑖𝑥𝑛𝑖.(4.33) By (A3), (4.32), (ii), the weakly lower semicontinuity of 𝜑 and 𝜂-hemicontinuity of 𝑇, we have 𝜑(̃𝑥)liminf𝑖𝜑Φ𝑟𝑛𝑖𝑥𝑛𝑖liminf𝑖𝑓Φ𝑟𝑛𝑖𝑥𝑛𝑖,𝑦+liminf𝑖𝑇Φ𝑟𝑛𝑖𝑥𝑛𝑖,𝜂𝑦,Φ𝑟𝑛𝑖𝑥𝑛𝑖+𝜑(𝑦)+liminf𝑖1𝑟𝑛𝑖𝑦Φ𝑟𝑛𝑖𝑥𝑛𝑖Φ,𝐽𝑟𝑛𝑖𝑥𝑛𝑖𝑥𝑛𝑖𝑓(̃𝑥,𝑦)+𝜑(𝑦)+𝑇̃𝑥,𝜂(𝑦,̃𝑥).(4.34) Hence, 𝑓(̃𝑥,𝑦)+𝜑(𝑦)+𝑇̃𝑥,𝜂(𝑦,̃𝑥)𝜑(̃𝑥).(4.35) This shows that ̃𝑥EP(𝑓,𝑇) and hence ̃𝑥Ω=𝑁𝑖=1𝐹(𝑆𝑖)GMEP(𝑓,𝑇).
Finally, we show that 𝑥𝑛𝑤 as 𝑛, where 𝑤=𝑃Ω𝑥0. By the weakly lower semicontinuity of the norm, it follows from (4.6) that 𝑥0𝑥𝑤0̃𝑥liminf𝑖𝑥0𝑥𝑛𝑖limsup𝑖𝑥0𝑥𝑛𝑖𝑥0.𝑤(4.36) This shows that lim𝑖𝑥0𝑥𝑛𝑖=𝑥0=𝑥𝑤0̃𝑥(4.37) and ̃𝑥=𝑤. Since 𝐸 is uniformly convex, we obtain that 𝑥0𝑥𝑛𝑖𝑥0𝑤. It follows that 𝑥𝑛𝑖𝑤. So we have 𝑥𝑛𝑤 as 𝑛. This completes the proof.

5. Corollaries

Setting 𝑆𝑖𝑆, an asymptotically nonexpansive mapping, in Theorem 4.1 then we have the following result.

Theorem 5.1. Let 𝐸 be a uniformly convex and smooth Banach space and let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4). Let 𝑇𝐶𝐸 be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping and 𝜑 a lower semicontinuous and convex function from 𝐶 to . Let 𝑆 be an asymptotically nonexpansive mapping with a sequence {𝑘𝑛}, such that 𝑘𝑛1 as 𝑛. Assume that Ω=𝐹(𝑆)GMEP(𝑓,𝑇) is nonempty. Let {𝑥𝑛} be a sequence generated by 𝑥0𝐶,𝐷0=𝐶0𝐶=𝐶,𝑛=co𝑧𝐶𝑛1𝑧S𝑛𝑧𝑡𝑛𝑥𝑛𝑆𝑛𝑥𝑛𝑢,𝑛1,𝑛𝑢𝐶𝑠𝑢𝑐𝑡𝑎𝑡𝑓𝑛,𝑦+𝜑(𝑦)+𝑇𝑢𝑛,𝜂𝑦,𝑢𝑛+1𝑟𝑛𝑦𝑢𝑛𝑢,𝐽𝑛𝑥𝑛𝐷,𝑦𝐶,𝑛1,𝑛=𝑧𝐷𝑛1𝑢𝑛𝑥𝑧,𝐽𝑛𝑢𝑛𝑥0,𝑛1,𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥0,𝑛0,(5.1) where {𝑡𝑛} and {𝑟𝑛} are real sequences in (0,1) satisfying lim𝑛𝑡𝑛=0 and liminf𝑛𝑟𝑛>0. Then {𝑥𝑛} converges strongly, as 𝑛, to 𝑃Ω𝑥0, where 𝑃Ω is the metric projection of 𝐸 onto Ω.

It's well known that each nonexpansive mapping is an asymptotically nonexpansive mapping, then Theorem 4.1 works for nonexpansive mapping.

Theorem 5.2. Let 𝐸 be a uniformly convex and smooth Banach space and let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4). Let 𝑇𝐶𝐸 be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping and 𝜑 a lower semicontinuous and convex function from 𝐶 to . Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that Ω=𝐹(𝑆)GMEP(𝑓,𝑇). Let {𝑥𝑛} be the sequence in 𝐶 generated by 𝑥0𝐶,𝐷0=𝐶0𝐶=𝐶,𝑛=co𝑧𝐶𝑛1𝑧𝑆𝑧𝑡𝑛𝑥𝑛𝑆𝑥𝑛𝑢,𝑛1,𝑛𝑢𝐶𝑠𝑢𝑐𝑡𝑎𝑡𝑓𝑛,𝑦+𝜑(𝑦)+𝑇𝑢𝑛,𝜂𝑦,𝑢𝑛+1𝑟𝑦𝑢𝑛𝑢,𝐽𝑛𝑥𝑛𝑢𝜑𝑛,𝐷𝑦𝐶,𝑛0,𝑛=𝑧𝐷𝑛1𝑢𝑛𝑥𝑧,𝐽𝑛𝑢𝑛𝑥0,𝑛1,𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥0,𝑛0,(5.2) where {𝑡𝑛} and {𝑟𝑛} are real sequences in (0,1) satisfying lim𝑛𝑡𝑛=0 and liminf𝑛𝑟𝑛>0. Then, the sequence {𝑥𝑛} converges strongly to 𝑃Ω𝑥0.
If one takes 𝑇0 and 𝜑0 in Theorem 4.1, then one obtains the following result concerning an equilibrium problem in a Banach space setting.

Theorem 5.3. Let 𝐸 be a uniformly convex and smooth Banach space and let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4) and let 𝑆 be an asymptotically nonexpansive mapping of C into itself such that Ω=𝑛=0𝐹(𝑆𝑛)EP(𝑓). Let {𝑥𝑛} be the sequence in 𝐶 generated by 𝑥0𝐶,𝐷0=𝐶0𝐶=𝐶,