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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 790592, 25 pages
http://dx.doi.org/10.1155/2012/790592
Research Article

New Generalized Mixed Equilibrium Problem with Respect to Relaxed Semi-Monotone 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

Received 24 November 2011; Accepted 12 January 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Rabian Wangkeeree and Pakkapon Preechasilp. 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 introduce the new generalized mixed equilibrium problem with respect to relaxed semimonotone mappings. Using the KKM technique, we obtain the existence of solutions for the generalized mixed equilibrium problem in Banach spaces. Furthermore, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the fixed point set of an asymptotically nonexpansive mapping. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main results extend various results existing in the current literature.

1. Introduction

Let 𝐸 be a Banach space with the dual 𝐸 and let 𝐸 denote the dual space of 𝐸. If 𝐸=𝐸, then 𝐸 is called reflexive. We denote by 𝒩 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 shall still denote by 𝐽 the single-valued duality mapping.

Let 𝐶 be a nonempty subset of 𝐸,𝜂𝐶×𝐶𝐸 be a mapping and let 𝜉𝐸 a function with 𝜉(𝑡𝑧)=𝑡𝑝𝜉(𝑧) for all 𝑡>0 and 𝑧𝐸, where 𝑝>1 is a constant. A mapping 𝐴𝐶×𝐶𝐸 is said to be relaxed 𝜂-𝜉 semimonotone [1] if the following two conditions hold:(i)for each fixed 𝑢𝐶,𝐴(𝑢,) is relaxed 𝜂-𝜉 monotone; that is,𝐴(𝑢,𝑣)𝐴(𝑢,𝑤),𝜂(𝑣,𝑤)𝜉(𝑣𝑤),𝑣,𝑤𝐶;(1.2)(ii)for each fixed 𝑣𝐶, 𝐴(,𝑣) is completely continuous; that is, for any net {𝑢𝑗} in 𝐶, 𝑢𝑗𝑢0 in weak topology of 𝐸, then {𝐴(𝑢𝑗,𝑣)} has a subsequence {𝐴(𝑢𝑗𝑘,𝑣)}𝐴(𝑢0,𝑣) in norm topology of 𝐸.

In case 𝜂(𝑥,𝑦)=𝑥𝑦 for all 𝑥,𝑦𝐶 and 𝜉0, 𝐴 is called semi-monotone [2]. The following is an example of 𝜂-𝜉 semi-monotone mapping.

Example 1.1. Let 𝐶=(,),𝐴(𝑥,𝑦)=𝑥+𝑦, and 𝜂(𝑥,𝑦)=𝑐(𝑥𝑦),𝑥𝑦,𝑐(𝑥𝑦),𝑥<𝑦,(1.3) where 𝑐>0 is a constant. Then, 𝐴 is relaxed 𝜂-𝜉 semi-monotone with 𝜉(𝑧)=𝑐𝑧2,𝑧0,𝑐𝑧2,𝑧<0.(1.4)
Let 𝑓𝐶×𝐶 be a bifunction, 𝜂𝐶×𝐶𝐸 a mapping, and 𝜉𝐸, 𝜑𝐶 two real-valued functions, and let 𝐴𝐶×𝐶𝐸 be a 𝜂-𝜉 semi-monotone mapping. We consider the problem of finding 𝑢𝐶 such that𝑓(𝑢,𝑣)+𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)𝜑(𝑢),𝑣𝐶,(1.5) which is called the generalized mixed equilibrium problem with respect to relaxed 𝜂-𝜉 semi-monotone mapping (GMEP(𝑓,𝐴,𝜂,𝜑)). The set of such 𝑢𝐶 is denoted by GMEP(𝑓,𝐴,𝜂,𝜑), that is, GMEP(𝑓,𝐴,𝜂,𝜑)={𝑢𝐶𝑓(𝑢,𝑣)+𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)𝜑(𝑢),𝑣𝐶}.(1.6)

Now, let us consider some special cases of the problem (1.5).(a) In the case of 𝑓0, (1.5) is deduced to the following variational-like inequality problem:nd𝑢𝐶suchthat𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)𝜑(𝑢)0,𝑣𝐶.(1.7) The problem (1.7) was studied by Fang and Huang [1]. Using the KKM technique and 𝜂-𝜉 monotonicity of the mapping 𝜑, they [1] obtained the existence of solutions of the variational-like inequality problem (1.7) in a real Banach space.(b) In the case of 𝑓0,𝜑0 and 𝜂(𝑣,𝑢)=𝑣𝑢 for all 𝑣,𝑢𝐶, the problem (1.5) is deduced to the following variational inequality problem:Find𝑢𝐶suchthat𝐴(𝑢,𝑢),𝑣𝑢0,𝑣𝐶.(1.8) The problem (1.8) was studied by Chen [2]. They obtained the existence results of solutions in a real Banach space.

When 𝐸 is a reflexive Banach space, we know 𝐸=𝑗(𝐸), where 𝑗𝐸𝐸 is the duality mapping defined by 𝑗𝑥,𝑓=𝑓,𝑥, for all 𝑥𝐸,𝑓𝐸, which is an isometric mapping, so we may regard 𝐸=𝐸 under an isometry. The following problems can be derived as special cases of the problem (1.5).(c) In case 𝐸 is reflexive (i.e., 𝐸=𝐸), 𝑓0 and 𝜂(𝑣,𝑢)=𝑣𝑢 for all 𝑣,𝑢𝐶, the problem (1.5) is deduced to the following variational inequality problem:nd𝑢𝐶suchthat𝐴(𝑢,𝑢),𝑣𝑢+𝜑(𝑣)𝜑(𝑢)0,𝑣𝐶.(1.9) The problem (1.9) was studied by Chen [2].(d) If 𝐸 is reflexive (i.e., 𝐸=𝐸) and 𝐴0, (1.5) is deduced to the mixed equilibrium problem:nd𝑢𝐶suchthat𝑓(𝑢,𝑣)+𝜑(𝑣)𝜑(𝑢),𝑣𝐶.(1.10) The problem (1.10) was considered and studied by Ceng and Yao [3]; Cholamjiak and Suantai [4].(e)In the case of 𝐴0 and 𝜑0, (1.5) is deduced to the following classical equilibrium problem:nd𝑢𝐶suchthat𝑓(𝑢,𝑣)0,𝑣𝐶.(1.11) The set of all solution of (1.11) is denoted by EP(𝑓), that is,EP(𝑓)={𝑢𝐶𝑓(𝑢,𝑣)0,𝑣𝐶}.(1.12) Numerous problems in physics, optimization, and economics can be reduced to find a solution of the equilibrium problem, variational inequality problem, and related optimization problems; see, for instance, [511]. Some methods have been proposed to solve the equilibrium problem in a Hilbert space; see, for instance, Blum and Oettli [12]; Combettes and Hirstoaga [13]; Moudafi [14].

Let 𝐶 be a nonempty, closed convex subset of 𝐸. 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. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [15] as an important generalization of nonexpansive mappings. Denote by 𝐹(𝑆) the set of fixed points of 𝑆, that is, 𝐹(𝑆)={𝑥𝐶𝑆𝑥=𝑥}. There are several methods for approximating fixed points of a nonexpansive mapping; see, for instance, [1621]. Furthermore, since 1972, a host of authors have studied weak and strong convergence problems of the iterative processes for the class of asymptotically nonexpansive mappings; see, for instance, [2225]. In 1953, Mann [16] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping 𝑆 in a Hilbert space 𝐻:𝑥𝑛+1=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑆𝑥𝑛,𝑛𝒩,(1.13) where the initial point 𝑥0 is taken in 𝐶 arbitrarily and {𝛼𝑛} is a sequence in [0,1]. However, we note that Mann's iteration process (1.13) has only weak convergence, in general; for instance, see [2628]. In 2003, Nakajo and Takahashi [29] introduced the following iterative algorithm for the nonexpansive mapping 𝑆 in the framework of Hilbert spaces:𝑥0𝑦=𝑥𝐶,𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑆𝑥𝑛,𝐶𝑛=𝑧𝐶𝑧𝑦𝑛𝑧𝑥𝑛,𝑄𝑛=𝑧𝐶𝑥𝑛𝑧,𝑥𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝑄𝑛𝑥,𝑛0,(1.14) where {𝛼𝑛}[0,𝛼],𝛼[0,1], and 𝑃𝐶𝑛𝑄𝑛 is the metric projection from a Hilbert space 𝐻 onto 𝐶𝑛𝑄𝑛. They proved that {𝑥𝑛} generated by (1.14) converges strongly to a fixed point of 𝑆. Xu [30] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection.

Matsushita and Takahashi [17] introduced the following iterative algorithm in the framework of Banach spaces:𝑥0𝐶=𝑥𝐶,𝑛=co𝑧𝐶𝑧𝑆𝑧𝑡𝑛𝑥𝑛𝑆𝑥𝑛,𝐷𝑛=𝑥𝑧𝐶𝑛𝑧,𝐽𝑥𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥,𝑛0,(1.15) where co𝐷 denoted the convex closure of the set 𝐷,{𝑡𝑛} is a sequence in (0,1) with 𝑡𝑛0, and 𝑃𝐶𝑛𝐷𝑛 is the metric projection from 𝐸 onto 𝐶𝑛𝐷𝑛.

Very recently, Dehghan [24] introduced the following iterative algorithm for finding fixed points of an asymptotically nonexpansive mapping 𝑆 in a uniformly convex and smooth Banach space:𝑥0=𝑥𝐶,𝐶0=𝐷0𝐶=𝐶,𝑛=co𝑧𝐶𝑛1𝑧𝑆𝑛𝑧𝑡𝑛𝑥𝑛𝑆𝑛𝑥𝑛,𝐷𝑛=𝑧𝐷𝑛1𝑥𝑛𝑧,𝐽𝑥𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥,𝑛0,(1.16) where co𝐷 denotes the convex closure of the set 𝐷,𝐽 is the normalized duality mapping, {𝑡𝑛} is a sequence in (0,1) with 𝑡𝑛0, and 𝑃𝐶𝑛𝐷𝑛 is the metric projection from 𝐸 onto 𝐶𝑛𝐷𝑛. The strong convergence theorem of the iterative sequence {𝑥𝑛} defined by (1.16) is obtained in a uniformly convex and smooth Banach space.

In this paper, motivated and inspired by the above results, we first suggest and analyze the new generalized mixed equilibrium problem with respect to relaxed 𝜂-𝜉 semi-monotone mapping. Using the KKM technique, we obtain the existence of solutions for such problem in a Banach space. Next, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the fixed point set of an asymptotically nonexpansive mapping. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main results extend various results existing in the current literature.

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+𝑦2𝑥,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 limitlim𝑡0𝑥+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 that𝑥0𝑥min𝑦𝐶𝑦𝑥.(2.5)

The mapping 𝑃𝐶𝐸𝐶 defined by 𝑃𝐶𝑥=𝑥0 is called the metric projection from 𝐸 onto 𝐶. The following theorem is wellknown.

Theorem 2.1 (see [31]). Let 𝐶 be a nonempty, closed convex subset of a smooth Banach space 𝐸 and let 𝑥𝐸, and 𝑦𝐶. Then the following are equivalent:(a)𝑦 is a best approximation to 𝑥𝑦=𝑃𝐶𝑥.(b)𝑦 is a solution of the variational inequality:𝑦𝑧,𝐽(𝑥𝑦)0,𝑧𝐶,(2.6) where 𝐽 is a duality mapping and 𝑃𝐶 is the metric projection from 𝐸 onto 𝐶.

It is wellknown 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, we will need the following lemmas.

Lemma 2.2 (see [32]). 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.

Theorem 2.3 (see [33]). 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 𝛾𝑆𝑛𝑖=1𝜆𝑖𝑥𝑖𝑛𝑖=1𝜆𝑖𝑆𝑥𝑖max1𝑗𝑘𝑛𝑥𝑗𝑥𝑘𝑆𝑥𝑗𝑆𝑥𝑘,(2.7) for all 𝑛𝒩, {𝑥1,𝑥2,,𝑥𝑛}𝐶, {𝜆1,𝜆2,,𝜆𝑛}[0,1] with 𝑛𝑖=1𝜆𝑖=1 and nonexpansive mapping 𝑆 of 𝐶 into 𝐸.

Theorem 2.4 (see [24]). 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,(2.8) for all 𝑛𝒩, {𝑥1,𝑥2,,𝑥𝑛}𝐶; {𝜆1,𝜆2,,𝜆𝑛}[0,1] with 𝑛𝑖=1𝜆𝑖=1 and an asymptotically nonexpansive mapping 𝑆 of 𝐶 into 𝐸 with the sequence {𝑘𝑚}.

Now, let us recall the following well-known concepts and results.

Definition 2.5. Let 𝐵 be a subset of topological vector space 𝑋. A mapping 𝐺𝐵2𝑋 is called a KKM mapping if co{𝑥1,𝑥2,,𝑥𝑚}𝑚𝑖=1𝐺(𝑥𝑖) for 𝑥𝑖𝐵 and 𝑖=1,2,,𝑚, where co𝐴 denotes the convex hull of the set 𝐴.

Lemma 2.6 (see [34]). Let 𝐵 be a nonempty subset of a Hausdorff topological vector space 𝑋, and let 𝐺𝐵2𝑋 be a KKM mapping. If 𝐺(𝑥) is closed for all 𝑥𝐵 and is compact for at least one 𝑥𝐵, then 𝑥𝐵𝐺(𝑥).

Theorem 2.7 (see [35] (Kakutani-Fan-Glicksberg Fixed Point Theorem)). Let E be a locally convex Hausdorff topological vector space and 𝐶 a nonempty, convex, and compact subset of 𝐸. Suppose 𝑇𝐶2𝐶 is a upper semi-continuous mapping with nonempty, closed, and convex values. Then 𝑇 has a fixed point in 𝐶.

Definition 2.8 (see [36]). Let 𝐶 be a nonempty, closed convex of a Banach space 𝐸. Let 𝑇𝐶𝐸 and let 𝜂𝐶×𝐶 be two mappings. 𝑇 is said to be 𝜂-hemicontinuous if, for any fixed 𝑥,𝑦𝐶, the mapping 𝑓[0,1](,) defined by 𝑓(𝑡)=𝑇(𝑥+𝑡(𝑦𝑥)),𝜂(𝑦,𝑥) is continuous at 0+.
For solving the mixed equilibrium problem, let us assume the following conditions for a bifunction 𝑓𝐶×𝐶:
(A1)𝑓(𝑥,𝑥)=0 for all 𝑥𝐶;(A2)𝑓 is monotone, that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)0 for all 𝑥,𝑦𝐶;(A3)for all 𝑦𝐶, 𝑓(,𝑦) is weakly upper semicontinuous;(A4)for all 𝑥𝐶, 𝑓(𝑥,) is convex. The following lemmas can be found in [37].

Lemma 2.9 (see [37]). 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) 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,𝑥)=0,forall𝑥𝐶;(ii) for any fixed 𝑢,𝑣𝐶, the mapping 𝑥𝑇𝑣,𝜂(𝑥,𝑢) is convex. Then the following problems (2.9) and (2.10) are equivalent. Find 𝑥𝐶 such that: 1𝑓(𝑥,𝑦)+𝜑(𝑦)+𝑇𝑥,𝜂(𝑦,𝑥)+𝑟𝑦𝑥,𝐽(𝑥𝑧)𝜑(𝑥),𝑦𝐶.(2.9) Find 𝑥𝐶 such that 1𝑓(𝑥,𝑦)+𝑇𝑦,𝜂(𝑦,𝑥)+𝜑(𝑦)+𝑟𝑦𝑥,𝐽(𝑥𝑧)𝜑(𝑥)+𝜉(𝑦𝑥),𝑦𝐶.(2.10)

Lemma 2.10 (see [37]). 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), (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) implies that 𝜉(𝑥)liminf𝜉(𝑥𝛽). Then, the solution set of the problem (2.9) is nonempty, that is, there exists 𝑥0𝐶 such that 𝑓𝑥0+,𝑦𝑇𝑥0,𝜂𝑦,𝑥01+𝜑(𝑦)+𝑟𝑦𝑥0𝑥,𝐽0𝑥𝑧𝜑0,𝑦𝐶.(2.11)

3. Existence Results of Generalized Mixed Equilibrium Problem

In this section, we prove the following crucial lemma concerning the generalized mixed equilibrium problem with respect to relaxed 𝜂-𝜉 semi-monotone mapping (GMEP(𝑓,𝐴,𝜂,𝜑)) in a real Banach space with the smooth and strictly convex second dual space.

Lemma 3.1. Let 𝐸 be a real Banach space with the smooth and strictly convex second dual space 𝐸, let 𝐶 be a nonempty bounded closed convex subset of 𝐸, let 𝐴𝐶×𝐶𝐸 be a relaxed 𝜂-𝜉 semi-monotone mapping. Let 𝑓𝐶×𝐶 be a bifunction satisfying (A1), (A3), and (A4), and let 𝜑𝐶{+} be a proper lower semicontinuous and convex function. Let 𝑟>0 and 𝑧𝐶. Assume that(i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦𝐶;(ii)for any fixed 𝑢,𝑣,𝑤𝐶, the mapping 𝑥𝐴(𝑣,𝑤),𝜂(𝑥,𝑢) is convex and lower semicontinuous;(iii)for each 𝑥𝐶,𝐴(𝑥,)𝐶𝐸 is finite-dimensional continuous: that is, for any finite-dimensional subspace 𝐹𝐸,𝐴(𝑥,)𝐶𝐹𝐸 is continuous;(iv)𝜉𝐸 is convex lower semicontinuous. Then there exists 𝑢0𝐶 such that 𝑓𝑢0+𝐴𝑢,𝑣0,𝑢0,𝜂𝑣,𝑢01+𝜑(𝑣)+𝑟𝑣𝑢0𝑢,𝐽0𝑢𝑧𝜑0,𝑣𝐶.(3.1)

Proof. Let 𝐹𝐸 be a finite-dimensional subspace with 𝐶𝐹=𝐹𝐶. For each 𝑤𝐶, consider the following problem: find 𝑢0𝐶𝐹 such that 𝑓𝑢0+𝐴,𝑣𝑤,𝑢0,𝜂𝑣,𝑢01+𝜑(𝑣)+𝑟𝑣𝑢0𝑢,𝐽0𝑢𝑧𝜑00,𝑣𝐶𝐹.(3.2)
Since 𝐶𝐹𝐹 is bounded closed and convex, 𝐴(𝑤,) is continuous on 𝐶𝐹 and relaxed 𝜂-𝜉 monotone for each fixed 𝑤𝐶, from Lemma 2.10, we know that problem (3.2) has a solution 𝑢0𝐶𝐹.
Now, define a set-valued mapping 𝐺𝐶𝐹2𝐶𝐹 as follows: Gw=𝑢𝐶𝐹1𝑓(𝑢,𝑣)+𝐴(𝑤,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)+𝑟𝑣𝑢,𝐽(𝑢𝑧)𝜑(𝑢)0,𝑣𝐶𝐹.(3.3) It follows from Lemma 2.9 that, for each fixed 𝑤𝐶𝐹: 𝑢𝐶𝐹1𝑓(𝑢,𝑣)+𝐴(𝑤,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)+𝑟𝑣𝑢,𝐽(𝑢𝑧)𝜑(𝑢)0,𝑣𝐾𝐹=𝑢𝐶𝐹1𝑓(𝑢,𝑣)+𝐴(𝑤,𝑣),𝜂(𝑣,𝑢)+𝜑(𝑣)+𝑟𝑣𝑢,𝐽(𝑢𝑧)𝜑(𝑢)𝜉(𝑣𝑢),𝑣𝐾𝐹.(3.4) Since every convex lower semicontinuous function in Banach spaces is weakly lower semicontinuous, the proper convex lower semicontinuity of 𝜑 and 𝜉, condition (ii), (A3) and (A4) implies that 𝐺𝐶𝐹2𝐶𝐹 has nonempty bounded closed and convex values. Using (A3) and the complete continuity of 𝐴(,𝑢), we can conclude that 𝐺 is upper semicontinuous. It follows from Theorem 2.7 that 𝐺 has a fixed point 𝑤𝐶𝐹, that is, 𝑓𝑤𝑤,𝑣+𝐴,𝑤,𝜂𝑣,𝑤1+𝜑(𝑣)+𝑟𝑣𝑤𝑤,𝐽𝑤𝑧𝜑0,𝑣𝐶𝐹.(3.5) Let =𝐹𝐸𝐹isnitedimensionalwith𝐹𝐶,(3.6) and let 𝑊𝐹=+1𝑢𝐶𝑓(𝑢,𝑣)+𝐴(𝑢,𝑣),𝜂(𝑣,𝑢)+𝜑(𝑣)r𝑣𝑢,𝐽(𝑢𝑧)𝜑(𝑢)𝜉(𝑣𝑢),𝑣𝐶𝐹,𝐹.(3.7) By (3.5) and Lemma 2.9, we know that 𝑊𝐹 is nonempty and bounded. Denote by 𝑊𝐹 the weak-closure of 𝑊𝐹 in 𝐸. Then, 𝑊𝐹 is weak compact in 𝐸.
For any 𝐹𝑖, 𝑖=1,2,,𝑁, we know that 𝑊𝑁𝑖=1𝐹𝑖𝑁𝑖=1𝑊𝐹𝑖, so {𝑊𝐹𝐹} has the finite intersection property. Therefore, it follows that𝐹𝑊𝐹.(3.8) Let 𝑢0𝐹𝑊𝐹. We claim that 𝑓𝑢0+𝐴𝑢,𝑣0,𝑢0,𝜂𝑣,𝑢01+𝜑(𝑣)+𝑟𝑣𝑢0𝑢,𝐽0𝑢𝑧𝜑00,𝑣𝐶.(3.9) Indeed, for each 𝑣𝐶, let 𝐹 be such that 𝑣𝐶𝐹 and 𝑢0𝐶𝐹. Then, there exists 𝑢𝑗𝑊𝐹 such that 𝑢𝑗𝑢0. The definition of 𝑊𝐹 implies that 𝑓𝑢𝑗+𝐴𝑢,𝑣𝑗,𝑣,𝜂𝑣,𝑢𝑗1+𝜑(𝑣)+𝑟𝑣𝑢𝑗𝑢,𝐽𝑗𝑢𝑧𝜑𝑗𝜉𝑣𝑢𝑗,(3.10) that is 𝑓𝑢𝑗+𝐴𝑢,𝑣𝑗,𝑣,𝜂𝑣,𝑢𝑗1+𝜑(𝑣)+𝑟𝑢𝑣𝑧,𝐽𝑗1𝑧𝑟𝑧𝑢𝑗2𝑢𝜑𝑗𝜉𝑣𝑢𝑗,(3.11) for all 𝑗=1,2,. Using the complete continuity of 𝐴(,𝑢), (A3), (ii), the continuity of 𝐽, the convex lower semicontinuity of 𝜑, 𝜉, and 2, and letting 𝑗, we get 𝑓𝑢0𝑢,𝑣+𝐴0,𝑣,𝜂𝑣,𝑢01+𝜑(𝑣)+𝑟𝑣𝑢0𝑢,𝐽0𝑢𝑧𝜑0𝜉𝑣𝑢0,𝑣𝐶.(3.12) From Lemma 2.9, we have 𝑓𝑢0+𝐴𝑢,𝑣0,𝑢0,𝜂𝑣,𝑢01+𝜑(𝑣)+𝑟𝑣𝑢0𝑢,𝐽0𝑢𝑧𝜑00,𝑣𝐶.(3.13) Hence, we complete the proof.

Setting 𝐴0 and 𝜑0 in Lemma 3.1, we have the following result.

Corollary 3.2. Let 𝐸 be a real Banach space with the smooth and strictly convex second dual space 𝐸, let 𝐶 be a nonempty bounded closed convex subset of 𝐸. Let 𝑓𝐶×𝐶 be a bifunction satisfying (A1), (A3), and (A4). Let 𝑟>0 and 𝑧𝐶. Then there exists 𝑢0𝐶 such that 𝑓𝑢0+1,𝑣𝑟𝑣𝑢0𝑢,𝐽0𝑧0,𝑣𝐶.(3.14)

If 𝐸 is reflexive (i.e., 𝐸=𝐸) smooth and strictly convex real Banach space, then we have the following result.

Corollary 3.3. Let 𝐸 be a reflexive smooth and strictly convex Banach space, let 𝐶 be a nonempty bounded closed convex subset of 𝐸, let 𝐴𝐶×𝐶𝐸 be a relaxed 𝜂-𝜉 semi-monotone mapping. Let 𝑓𝐶×𝐶 be a bifunction satisfying (A1), (A3), and (A4), and let 𝜑𝐶{+} be a proper lower semicontinuous and convex function. Let 𝑟>0 and 𝑧𝐶. Assume that(i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦𝐶;(ii)for any fixed 𝑢,𝑣,𝑤𝐶, the mapping 𝑥𝐴(𝑣,𝑤),𝜂(𝑥,𝑢) is convex and lower semicontinuous;(iii)for each 𝑥𝐶,𝐴(𝑥,)𝐶𝐸 is finite-dimensional continuous.(iv)𝜉𝐸 is convex lower semicontinuous.Then, there exists 𝑢0𝐶 such that 𝑓𝑢0+𝐴𝑢,𝑣0,𝑢0,𝜂𝑣,𝑢01+𝜑(𝑣)+𝑟𝑣𝑢0𝑢,𝐽0𝑢𝑧𝜑0,𝑣𝐶.(3.15)

If 𝐸 is reflexive (i.e., 𝐸=𝐸) smooth and strictly convex, 𝐴 is semi-monotone, then we obtain the following result.

Corollary 3.4. Let 𝐸 be a reflexive smooth and strictly convex Banach space, let 𝐶 be a nonempty bounded closed convex subset of 𝐸, let 𝐴𝐶×𝐶𝐸 be a semi-monotone mapping. Let 𝑓𝐶×𝐶 be a bifunction satisfying (A1), (A3), and (A4), and let 𝜑𝐶{+} be a proper lower semicontinuous and convex function. Assume that, for any 𝑟>0 and 𝑧𝐶,(i)for any fixed 𝑢,𝑣,𝑤𝐶, the mapping 𝑥𝐴(𝑣,𝑤),𝑥𝑢) is convex and lower semicontinuous;(ii)for each 𝑥𝐶,𝐴(𝑥,)𝐶𝐸 is finite-dimensional continuous.Then, there exists 𝑢0𝐶 such that 𝑓𝑢0+𝐴𝑢,𝑣0,𝑢0,𝑣𝑢01+𝜑(𝑣)+𝑟𝑣𝑢0𝑢,𝐽0𝑢𝑧𝜑0,𝑣𝐶.(3.16)

Theorem 3.5. Let 𝐸 be a real Banach space with the smooth and strictly convex second dual space 𝐸, let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸, let 𝐴𝐶×𝐶𝐸 be a relaxed 𝜂-𝜉 semi-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 . For any 𝑟>0, define a mapping Φ𝑟𝐸𝐶 as follows: Φ𝑟1(𝑥)=𝑢𝐶𝑓(𝑢,𝑣)+𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)+𝑟𝑣𝑢,𝐽(𝑢𝑥)𝜑(𝑢),𝑣𝐶,(3.17) for all 𝑥𝐸. Assume that(i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦𝐶;(ii)for any fixed 𝑢,𝑣,𝑤𝐶, the mapping 𝑥𝐴(𝑣,𝑤),𝜂(𝑥,𝑢) is convex and lower semicontinuous;(iii)for each 𝑥𝐶,𝐴(𝑥,)𝐶𝐸 is finite-dimensional continuous: that is, for any finite-dimensional subspace 𝐹𝐸,𝐴(𝑥,)𝐶𝐹𝐸 is continuous;(iv)𝜉𝐸 is convex lower semicontinuous;(v) for any 𝑥,𝑦𝐶, 𝜉(𝑥𝑦)+𝜉(𝑦𝑥)0;(vi)for any 𝑥,𝑦𝐶, 𝐴(𝑥,𝑦)=𝐴(𝑦,𝑥). 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,𝐴,𝜂,𝜑);(4)GMEP(𝑓,𝐴,𝜂,𝜑) is nonempty, closed, and convex.

Proof. For each 𝑥𝐸, by Lemma 2.10, we conclude that Φ𝑟(𝑥) is nonempty.(1)We prove that Φ𝑟 is single-valued. Indeed, for 𝑥𝐸 and 𝑟>0, let 𝑧1,𝑧2Φ𝑟(𝑥). Then,𝑓𝑧1+𝐴𝑧,𝑣1,𝑧1,𝜂𝑣,𝑧11+𝜑(𝑣)+𝑟𝑣𝑧1𝑧,𝐽1𝑧𝑥𝜑1𝑓𝑧,𝑣𝐶,2+𝐴𝑧,𝑣2,𝑧2,𝜂𝑣,𝑧21+𝜑(𝑣)+𝑟𝑣𝑧2𝑧,𝐽2𝑧𝑥𝜑2,𝑣𝐶.(3.18) Hence, 𝑓𝑧1,𝑧2+𝐴𝑧1,𝑧1𝑧,𝜂2,𝑧1𝑧+𝜑2+1𝑟𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝜑1,𝑓𝑧2,𝑧1+𝐴𝑧2,𝑧2𝑧,𝜂1,𝑧2𝑧+𝜑1+1𝑟𝑧1𝑧2𝑧,𝐽2𝑧𝑥𝜑2.(3.19) Adding the two inequalities, from (i) we have 𝑓𝑧2,𝑧1𝑧+𝑓1,𝑧2+𝐴𝑧1,𝑧1𝑧𝐴2,𝑧2𝑧,𝜂2,𝑧1+1𝑟𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝑥0.(3.20) From (A2), we have 𝐴𝑧1,𝑧1𝑧𝐴2,𝑧2𝑧,𝜂2,𝑧1+1𝑟𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝑥0.(3.21) That is, 1𝑟𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝐴𝑧𝑥2,𝑧2𝑧𝐴1,𝑧1𝑧,𝜂2,𝑧1.(3.22) Calculating the right-hand side of (3.22), we have 𝐴𝑧2,𝑧2𝑧𝐴1,𝑧1𝑧,𝜂2,𝑧1=𝐴𝑧2,𝑧2𝑧𝐴2,𝑧1𝑧+𝐴2,𝑧1𝑧𝐴1,𝑧2𝑧+𝐴1,𝑧2𝑧𝐴1,𝑧1𝑧,𝜂2,𝑧1=𝐴𝑧2,𝑧2𝑧𝐴2,𝑧1𝑧,𝜂2,𝑧1+𝐴𝑧2,𝑧1𝑧𝐴1,𝑧2𝑧,𝜂2,𝑧1+𝐴𝑧1,𝑧2𝑧𝐴1,𝑧1𝑧,𝜂2,𝑧1𝑧2𝜉2𝑧1+𝐴𝑧2,𝑧1𝑧𝐴1,𝑧2𝑧,𝜂2,𝑧1,(3.23) and so, 1𝑟𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝑧𝑥2𝜉2𝑧1+𝐴𝑧2,𝑧1𝑧𝐴1,𝑧2𝑧,𝜂2,𝑧1.(3.24) In (3.24) exchanging the position of 𝑧1 and 𝑧2, we get 1𝑟𝑧1𝑧2𝑧,𝐽2𝑧𝑥𝐽1𝑧𝑥2𝜉1𝑧2+𝐴𝑧1,𝑧2𝑧𝐴2,𝑧1𝑧,𝜂1,𝑧2.(3.25) Adding the inequalities (3.24) and (3.25) and using (v) and (vi), we have 𝑧2𝑧1𝑧,𝐽1𝑧𝑥𝐽2𝜉𝑧𝑥𝑟2𝑧1𝑧+𝜉1𝑧20.(3.26) Hence, 𝑧02𝑧1𝑧,𝐽1𝑧𝑥𝐽2=𝑧𝑥2𝑧𝑥1𝑧𝑥,𝐽1𝑧𝑥𝐽2𝑥.(3.27) Since 𝐽 is monotone and 𝐸 is strictly convex, we obtain that 𝑧1𝑥=𝑧2𝑥 and hence 𝑧1=𝑧2. Therefore, Φ𝑟 is single-valued.(2) For 𝑥,𝑦𝐶, we have𝑓Φ𝑟𝑥,Φ𝑟𝑦+𝐴Φ𝑟𝑥,Φ𝑟𝑥Φ,𝜂𝑟𝑦,Φ𝑟𝑥Φ+𝜑𝑟𝑦Φ𝜑𝑟𝑥+1𝑟Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟𝑓Φ𝑥𝑥0,𝑟𝑦,Φ𝑟𝑥+𝐴Φ𝑟𝑦,Φ𝑟𝑦Φ,𝜂𝑟𝑥,Φ𝑟𝑦Φ+𝜑𝑟𝑥Φ𝜑𝑟𝑦+1𝑟Φ𝑟𝑥Φ𝑟Φ𝑦,𝐽𝑟𝑦𝑦0.(3.28) Adding the above two inequalities and by (i) and (A2), we get 𝐴Φ𝑟𝑥,Φ𝑟𝑥Φ𝐴𝑟𝑦,Φ𝑟𝑦Φ,𝜂𝑟𝑦,Φ𝑟𝑥+1𝑟Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟Φ𝑥𝑥𝐽𝑟𝑦𝑦0,(3.29) that is 1𝑟Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟Φ𝑥𝑥𝐽𝑟𝐴Φ𝑦𝑦𝑟𝑦,Φ𝑟𝑦Φ𝐴𝑟𝑥,Φ𝑟𝑥Φ,𝜂𝑟𝑦,Φ𝑟𝑥.(3.30) After calculating (3.30), we have 1𝑟Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟Φ𝑥𝑥𝐽𝑟Φ𝑦𝑦2𝜉𝑟𝑦,Φ𝑟𝑥+𝐴Φ𝑟𝑦,Φ𝑟𝑥Φ𝐴𝑟𝑥,Φ𝑟𝑦Φ,𝜂𝑟𝑦,Φ𝑟𝑥.(3.31) In (3.30), exchanging the position of Φ𝑟𝑥 and Φ𝑟𝑦, we get 1𝑟Φ𝑟𝑥Φ𝑟Φ𝑦,𝐽𝑟Φ𝑦𝑦𝐽𝑟Φ𝑥𝑥2𝜉𝑟𝑥,Φ𝑟𝑦+𝐴Φ𝑟𝑥,Φ𝑟𝑦Φ𝐴𝑟𝑦,Φ𝑟𝑥Φ,𝜂𝑟𝑥,Φ𝑟𝑦.(3.32) Adding the inequalities (3.31) and (3.32), use (i) and (vi), we have Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟Φ𝑥𝑥𝐽𝑟𝜉Φ𝑦𝑦𝑟𝑟𝑥,Φ𝑟𝑦Φ+𝜉𝑟𝑦,Φ𝑟𝑥.(3.33) It follows from (iv) that Φ𝑟𝑦Φ𝑟Φ𝑥,𝐽𝑟Φ𝑥𝑥𝐽𝑟𝑦𝑦0.(3.34) Hence, Φ𝑟𝑥Φ𝑟Φ𝑦,𝐽𝑟𝑥𝑥Φ𝑟𝑥Φ𝑟Φ𝑦,𝐽𝑟𝑦𝑦.(3.35)(3) Next, we show that 𝐹(Φ𝑟)=GMEP(𝑓,𝐴,𝜂,𝜑). Indeed, we have the following: Φ𝑢𝐹𝑟𝑢=Φ𝑟𝑢1𝑓(𝑢,𝑣)+𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)+𝑟𝑣𝑢,𝐽(𝑢𝑢)𝜑(𝑢),𝑣𝐶𝑓(𝑢,𝑣)+𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)𝜑(𝑢),𝑣𝐶𝑢GMEP(𝑓,𝐴,𝜂,𝜑).(3.36)Hence, 𝐹(Φ𝑟)=GMEP(𝑓,𝐴,𝜂,𝜑).(4) Finally, we prove that GMEP(𝑓,𝐴,𝜂,𝜑) is nonempty, closed, and convex. For each 𝑣𝐶, we define the multivalued mapping 𝐺𝐶2𝐸 by𝐺(𝑣)={𝑢𝐶𝑓(𝑢,𝑣)+𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)𝜑(𝑢)}.(3.37) Since 𝑣𝐺(𝑣), we have 𝐺(𝑣). We prove that 𝐺 is a KKM mapping on 𝐶. Suppose that there exists a finite subset {𝑧1,𝑧2,,𝑧𝑚} of 𝐶, and 𝛼𝑖>0 with 𝑚𝑖=1𝛼𝑖=1 such that ̂𝑧=𝑚𝑖=1𝛼𝑖𝑧𝑖𝐺(𝑧𝑖) for all 𝑖=1,2,,𝑚. Then 𝑓̂𝑧,𝑧𝑖+𝐴𝑧(̂𝑧,̂𝑧),𝜂𝑖𝑧,̂𝑧+𝜑𝑖𝜑(̂𝑧)<0,𝑖=1,2,,𝑚.(3.38) From (A1), (A4), (ii), and the convexity of 𝜑, we have 0=𝑓(̂𝑧,̂𝑧)+𝐴(̂𝑧,̂𝑧),𝜂(̂𝑧,̂𝑧)+𝜑(̂𝑧)𝜑(̂𝑧)=𝑓̂𝑧,𝑚𝑖=1𝛼𝑖𝑧𝑖+𝐴(̂𝑧,̂𝑧),𝜂𝑚𝑖=1𝛼𝑖𝑧𝑖,̂𝑧+𝜑𝑚𝑖=1𝛼𝑖𝑧𝑖𝜑(̂𝑧)𝑚𝑖=1𝛼𝑖𝑓̂𝑧,𝑧𝑖+𝑧𝐴(̂𝑧,̂𝑧),𝜂𝑖𝑧,̂𝑧+𝜑𝑖𝜑(̂𝑧)<0,(3.39) which is a contradiction. Thus, 𝐺 is a KKM mapping on 𝐶.
Next, we prove that 𝐺(𝑦) is closed for each 𝑦𝐶. For any 𝑦𝐶, let {𝑥𝑛} be any sequence in 𝐺(𝑦) such that 𝑥𝑛𝑥0. We claim that 𝑥0𝐺(𝑦). Then, for each 𝑦𝐶, we have𝑓𝑥𝑛+𝐴𝑥,𝑦𝑛,𝑥𝑛,𝜂𝑦,𝑥𝑛𝑥+𝜑(𝑦)𝜑𝑛.(3.40) By monotonicity of 𝐴, we obtain that 𝑓𝑥𝑛+𝐴𝑥,𝑦𝑛,𝑦,𝜂𝑦,𝑥𝑛𝑥+𝜑(𝑦)𝜑𝑛+𝜉𝑦𝑥𝑛.(3.41) By (A3), (i), (ii), (iv), lower semicontinuity of 𝜑, and the complete continuity of 𝐴, we obtain the following 𝜑𝑥0+𝐴𝑥0𝑥,𝑦,𝜂0,𝑦liminf𝑛𝜑𝑥𝑛+liminf𝑛𝐴𝑥𝑛𝑥,𝑦,𝜂𝑛,𝑦liminf𝑛𝜑𝑥𝑛+𝐴𝑥𝑛𝑥,𝑦,𝜂𝑛,𝑦=liminf𝑛𝜑𝑥𝑛𝐴𝑥𝑛,𝑦,𝜂𝑦,𝑥𝑛limsup𝑛𝜑𝑥𝑛𝐴𝑥𝑛,𝑦,𝜂𝑦,𝑥𝑛limsup𝑛𝑓𝑥𝑛,𝑦+𝜑(𝑦)𝜉𝑦𝑥𝑛𝑥𝑓0,𝑦+𝜑(𝑦)𝜉𝑦𝑥0.(3.42) Hence, 𝑓𝑥0+𝐴𝑥,𝑦0,𝑦,𝜂𝑦,𝑥0𝑥+𝜑(𝑦)𝜑0+𝜉𝑦𝑥0,𝑦𝐶.(3.43) From Lemma 2.9, we have 𝑓𝑥0+𝐴𝑥,𝑦0,𝑥0,𝜂𝑦,𝑥0𝑥+𝜑(𝑦)𝜑0,𝑦𝐶.(3.44) This shows that 𝑥0𝐺(𝑦), and hence 𝐺(𝑦) is closed for each 𝑦𝐶. Thus, GMEP(𝑓,𝐴,𝜂,𝜑)=𝑦𝐶𝐺(𝑦) is also closed.
Next, we observe that 𝐺(𝑦) is weakly compact. In fact, since 𝐶 is bounded, closed, and convex, we also have 𝐺(𝑦), which is weakly compact in the weak topology. By Lemma 2.6, we can conclude that 𝑦𝐶𝐺(𝑦)=GMEP(𝑓,𝐴,𝜂,𝜑).
Finally, we prove that GMEP(𝑓,𝐴,𝜂,𝜑) is convex. In fact, let 𝑢,𝑣𝐹(Φ𝑟), and 𝑧𝑡=𝑡𝑢+(1𝑡)𝑣 for 𝑡(0,1). From (2), we know that Φ𝑟𝑢Φ𝑟𝑧𝑡Φ,𝐽𝑟𝑧𝑡𝑧𝑡Φ𝐽𝑟𝑢𝑢0.(3.45) This yields that 𝑢Φ𝑟𝑧𝑡Φ,𝐽𝑟𝑧𝑡𝑧𝑡0.(3.46) Similarly, we also have 𝑣Φ𝑟𝑧𝑡Φ,𝐽𝑟𝑧𝑡𝑧𝑡0.(3.47) It follows from (3.46) and (3.47) that 𝑧𝑡Φ𝑟𝑧𝑡2=𝑧𝑡Φ𝑟𝑧𝑡𝑧,𝐽𝑡Φ𝑟𝑧𝑡=𝑡𝑢Φ𝑟𝑧𝑡𝑧,𝐽𝑡Φ𝑟𝑧𝑡+(1𝑡)𝑣Φ𝑟𝑧𝑡𝑧,𝐽𝑡Φ𝑟𝑧𝑡0.(3.48) Hence, 𝑧𝑡𝐹(Φ𝑟)=GMEP(𝑓,𝐴,𝜂,𝜑) and hence GMEP(𝑓,𝐴,𝜂,𝜑) is convex. This completes the proof.

If 𝐸 is reflexive (i.e., 𝐸=𝐸) smooth and strictly convex, then the following result can be derived as a corollary of Theorem 3.5

Corollary 3.6. Let 𝐸 be a reflexive smooth and strictly convex Banach space, let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸, and let 𝐴𝐶×𝐶𝐸 be a relaxed 𝜂-𝜉 semi-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 𝑧𝐶 and define a mapping Φ𝑟𝐸𝐶 as follows: Φ𝑟1(𝑥)=𝑢𝐶𝑓(𝑢,𝑣)+𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)+𝜑(𝑣)+𝑟𝑣𝑢,𝐽(𝑢𝑥)𝜑(𝑢),𝑣𝐶,(3.49)for all 𝑥𝐸. Assume that(i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦𝐶;(ii)for any fixed 𝑢,𝑣,𝑤𝐶, the mapping 𝑥𝐴(𝑣,𝑤),𝜂(𝑥,𝑢) is convex and lower semicontinuous; (iii)for each 𝑥𝐶,𝐴(𝑥,)𝐶𝐸 is finite-dimensional continuous;(iv)𝜉𝐸 is convex lower semicontinuous;(v)for any 𝑥,𝑦𝐶, 𝜉(𝑥𝑦)+𝜉(𝑦𝑥)0;(vi)for any 𝑥,𝑦𝐶, 𝐴(𝑥,𝑦)=𝐴(𝑦,𝑥). 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,𝐴,𝜂,𝜑);(4)GMEP(𝑓,𝐴,𝜂,𝜑) is nonempty, closed, and convex.

4. Strong Convergence Theorems

In this section, we prove a strong convergence theorem by using a hybrid projection algorithm for an asymptotically nonexpansive mapping in a uniformly convex and smooth Banach space.

Theorem 4.1. Let 𝐸 be a real Banach space with the smooth and uniformly convex second dual space 𝐸, 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 𝜑 be a lower semicontinuous and convex function from 𝐶 to . Let 𝐴𝐶×𝐶𝐸 be a relaxed 𝜂-𝜉 semi-monotone and let 𝑆𝐶𝐶 be an asymptotically nonexpansive mapping with a sequence {𝑘𝑛}[1,) such that 𝑘𝑛1 as 𝑛. Assume that Ω=𝐹(𝑆)GMEP(𝑓,𝐴,𝜂,𝜑). Let {𝑥𝑛} be a sequence in 𝐶 generated by 𝑥0𝐶,𝐷0=𝐶0𝐶=𝐶,𝑛=co𝑧𝐶𝑛1𝑧𝑆𝑛𝑧𝑡𝑛𝑥𝑛𝑆𝑛𝑥𝑛𝑢,𝑛1,𝑛𝑓𝑢𝐶𝑠𝑢𝑐𝑡𝑎𝑡𝑛𝐴𝑢,𝑦+𝜑(𝑦)+𝑛,𝑢𝑛,𝜂𝑦,𝑢𝑛+1𝑟𝑛𝑦𝑢𝑛𝑢,𝐽𝑛𝑥𝑛𝑢𝜑𝑛𝐷,𝑦𝐶,𝑛0,𝑛=𝑧𝐷𝑛1𝑢𝑛𝑥𝑧,𝐽𝑛𝑢𝑛𝑥0,𝑛1,𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥0,𝑛0,(4.1) where {𝑡𝑛} and {𝑟𝑛} are real sequences in (0,1) such that lim𝑛𝑡𝑛=0, and liminf𝑛𝑟𝑛>0. Then {𝑥𝑛} converges strongly, as 𝑛, to 𝑃Ω𝑥0.

Proof. Firstly, we rewrite the (4.1) as follows: 𝑥0𝐶,𝐷0=𝐶0𝐶=𝐶,𝑛=co𝑧𝐶𝑛1𝑧𝑆𝑛𝑧𝑡𝑛𝑥𝑛𝑆𝑛𝑥𝑛𝐷,𝑛0,𝑛=𝑧𝐷𝑛1Φ𝑟𝑛𝑥𝑛𝑥𝑧,𝐽𝑛Φ𝑟𝑛𝑥𝑛𝑥0,𝑛1,𝑛+1=𝑃𝐶𝑛𝐷𝑛𝑥0,𝑛0,(4.2) where Φ𝑟 is the mapping defined by Φ𝑟1(𝑥)=𝑧𝐶𝑓(𝑧,𝑦)+𝐴(𝑧,𝑧),𝜂(𝑦,𝑧)+𝜑(𝑦)+𝑟𝑦𝑧,𝐽(𝑧𝑥)𝜑(𝑧),𝑦𝐶.(4.3) We first 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. Since 𝐷0=𝐶, we also have Ω𝐶0𝐷0. Suppose that Ω𝐶𝑘1𝐷𝑘1 for 𝑘2. It follows from Theorem 3.5 (2) that Φ𝑟𝑘𝑥𝑘Φ𝑟𝑘Φ𝑢,𝐽𝑟𝑘Φ𝑢𝑢𝐽𝑟𝑘𝑥𝑘𝑥𝑘0,(4.4) for all 𝑢Ω. This implies that Φ𝑟𝑘𝑥𝑘𝑥𝑢,𝐽𝑘Φ𝑟𝑘𝑥𝑘0,(4.5) for all 𝑢Ω. Hence Ω𝐷𝑘. By the mathematical induction, we get that Ω𝐶𝑛𝐷𝑛 for each 𝑛0, and hence {𝑥𝑛} is welldefined. 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) Since {𝑥𝑛𝑥0} is bounded, we have lim𝑛𝑥𝑛𝑥0=𝑑 for some a constant 𝑑. 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.(4.8) This implies that lim𝑛12𝑥0𝑥𝑛+1+12𝑥0𝑥𝑛+2=lim𝑛𝑥0𝑥𝑛+1+𝑥𝑛+22=𝑑.(4.9) By Lemma 2.2, we have lim𝑛𝑥𝑛𝑥𝑛+1=0.(4.10) Next, we show that lim𝑛𝑥𝑛𝑆𝑥𝑛=0.(4.11) To obtain (4.11), we need to show that lim𝑛𝑥𝑛𝑆𝑛𝑘𝑥𝑛=0,forall𝑘𝒩.
Fix 𝑘𝒩 and put 𝑚=𝑛𝑘. Since 𝑥𝑛=𝑃𝐶𝑛1𝐷𝑛1𝑥, we have 𝑥𝑛𝐶𝑛1𝐶𝑚. Since 𝑡𝑚>0, there exist 𝑦1,,𝑦𝑁𝐶 and nonnegative numbers 𝜆1,,𝜆𝑁 with 𝜆1++𝜆𝑁=1 such that 𝑥𝑛𝑁𝑖=1𝜆𝑖𝑦𝑖<𝑡𝑚,(4.12) and 𝑦𝑖𝑆𝑚𝑦𝑖𝑡𝑚𝑥𝑚𝑆𝑚𝑥𝑚 for all 𝑖{1,,𝑁}. Put 𝑀=sup𝑥𝐶𝑥,𝑢=𝑃𝐹(𝑆)𝑥 and 𝑟0=sup𝑛1(1+𝑘𝑛)𝑥𝑛𝑢. Since 𝐶 and {𝑘𝑚} are bounded, (4.12) implies 𝑥𝑛1𝑘𝑚𝑁𝑖=1𝜆𝑖𝑦𝑖11𝑘𝑚1𝑥+𝑘𝑚𝑥𝑛𝑁𝑖=1𝜆𝑖𝑦𝑖11𝑘𝑚𝑀+𝑡𝑚,(4.13) and 𝑦𝑖𝑆𝑚𝑦𝑖𝑡𝑚𝑥𝑚𝑆𝑚𝑥𝑚𝑡𝑚(1+𝑘𝑚)𝑥𝑚𝑢𝑟0𝑡𝑚 for all 𝑖{1,,𝑁}. Therefore, 𝑦𝑖1𝑘𝑚𝑆𝑚𝑦𝑖11𝑘𝑚𝑀+𝑟0𝑡𝑚,(4.14) for all 𝑖{1,,𝑁}. Moreover, asymptotically nonexpansiveness of 𝑆 and (4.6) give that 1𝑘𝑚𝑆𝑚𝑁𝑖=1𝜆𝑖𝑦𝑖𝑆𝑚𝑥𝑛11𝑘𝑚𝑀+𝑡𝑚.(4.15) It follows from Theorem 2.4, (4.13)–(4.15) that 𝑥𝑛𝑆𝑚𝑥𝑛𝑥𝑛1𝑘𝑚𝑁𝑖=1𝜆𝑖𝑦𝑖+1𝑘𝑚𝑁𝑖=1𝜆𝑖𝑦𝑖𝑆𝑚𝑦𝑖+1𝑘𝑚𝑁𝑖=1𝜆𝑖𝑆𝑚𝑦𝑖𝑆𝑚𝑁𝑖=1𝜆𝑖𝑦𝑖+1𝑘𝑚𝑆𝑚𝑁𝑖=1𝜆𝑖𝑦𝑖𝑆𝑚𝑥𝑛121𝑘𝑚𝑀+2𝑡𝑚+𝑟0𝑡𝑚𝑘𝑚+𝛾1max1𝑖𝑗𝑁𝑦𝑖𝑦𝑗1𝑘𝑚𝑆𝑚𝑦𝑖𝑆𝑚𝑦𝑗121𝑘𝑚𝑀+2𝑡𝑚+𝑟0𝑡𝑚𝑘𝑚+𝛾1max1𝑖𝑗𝑁𝑦𝑖1𝑘𝑚𝑆𝑚𝑦𝑖+𝑦𝑗1𝑘𝑚𝑆𝑚𝑦𝑗121𝑘𝑚𝑀+2𝑡𝑚+𝑟0𝑡𝑚𝑘𝑚+𝛾1211𝑘𝑚𝑀+2𝑟0𝑡𝑚.(4.16) Since lim𝑛𝑘𝑛=1 and lim𝑛𝑡𝑛=0, it follows from the last inequality that lim𝑛𝑥𝑛𝑆𝑚𝑥𝑛=0. We have that 𝑥𝑛𝑆𝑥𝑛=𝑥𝑛𝑆𝑛1𝑥𝑛+𝑆𝑛1𝑥𝑛𝑆𝑥𝑛𝑥𝑛𝑆𝑛1𝑥𝑛+𝑘1𝑆𝑛2𝑥𝑛𝑥𝑛