New Generalized Mixed Equilibrium Problem with Respect to Relaxed Semi-Monotone Mappings in Banach Spaces

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 ℛ the sets of positive integers and real numbers, respectively. Also, we denote by 𝐽 the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽𝑥=𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²=‖𝑥^∗‖²,∀𝑥∈𝐸,(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^∗∗,𝜂∶𝐶×𝐶→𝐸^∗∗ be a mapping and let 𝜉∶𝐸^∗∗→ℛ a function with 𝜉(𝑡𝑧)=𝑡^𝑝𝜉(𝑧) for all 𝑡>0 and 𝑧∈𝐸^∗∗, where 𝑝>1 is a constant. A mapping 𝐴∶𝐶×𝐶→𝐸^∗ is said to be relaxed 𝜂-𝜉 semimonotone [1] if the following two conditions hold:(i)for each fixed 𝑢∈𝐶,𝐴(𝑢,⋅) is relaxed 𝜂-𝜉 monotone; that is,⟨𝐴(𝑢,𝑣)−𝐴(𝑢,𝑤),𝜂(𝑣,𝑤)⟩≥𝜉(𝑣−𝑤),∀𝑣,𝑤∈𝐶;(1.2)(ii)for each fixed 𝑣∈𝐶, 𝐴(⋅,𝑣) is completely continuous; that is, for any net {𝑢_𝑗} in 𝐶, 𝑢_𝑗→𝑢₀ in weak∗ topology of 𝐸^∗∗, then {𝐴(𝑢_𝑗,𝑣)} has a subsequence {𝐴(𝑢_𝑗𝑘,𝑣)}→𝐴(𝑢₀,𝑣) in norm topology of 𝐸^∗.

In case 𝜂(𝑥,𝑦)=𝑥−𝑦 for all 𝑥,𝑦∈𝐶 and 𝜉≡0, 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(𝑐𝑥(𝑥−−𝑦)𝑦,),𝑥𝑥≥<𝑦𝑦,,(1.3) where 𝑐>0 is a constant. Then, 𝐴 is relaxed 𝜂-𝜉 semi-monotone with 𝜉(𝑧)=−𝑐𝑧²𝑐𝑧₂,𝑧≥0,,𝑧<0.(1.4)
Let 𝑓∶𝐶×𝐶→ℛ be a bifunction, 𝜂∶𝐶×𝐶→𝐸^∗∗ a mapping, and 𝜉∶𝐸^∗∗→ℛ, 𝜑∶𝐶→ℛ two real-valued functions, and let 𝐴∶𝐶×𝐶→𝐸^∗ be a 𝜂-𝜉 semi-monotone mapping. We consider the problem of finding 𝑢∈𝐶 such that𝑓(𝑢,𝑣)+⟨𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)≥𝜑(𝑢),∀𝑣∈𝐶,(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 𝑢∈𝐶 is denoted by GMEP(𝑓,𝐴,𝜂,𝜑), that is, GMEP(𝑓,𝐴,𝜂,𝜑)={𝑢∈𝐶∶𝑓(𝑢,𝑣)+⟨𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)≥𝜑(𝑢),∀𝑣∈𝐶}.(1.6)

Now, let us consider some special cases of the problem (1.5).(a) In the case of 𝑓≡0, (1.5) is deduced to the following variational-like inequality problem:find𝑢∈𝐶suchthat⟨𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)−𝜑(𝑢)≥0,∀𝑣∈𝐶.(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 𝑓≡0,𝜑≡0 and 𝜂(𝑣,𝑢)=𝑣−𝑢 for all 𝑣,𝑢∈𝐶, the problem (1.5) is deduced to the following variational inequality problem:Find𝑢∈𝐶suchthat⟨𝐴(𝑢,𝑢),𝑣−𝑢⟩≥0,∀𝑣∈𝐶.(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 𝑗∶𝐸→𝐸^∗∗ is the duality mapping defined by ⟨𝑗𝑥,𝑓⟩=⟨𝑓,𝑥⟩, for all 𝑥∈𝐸,𝑓∈𝐸^∗, 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^∗∗), 𝑓≡0 and 𝜂(𝑣,𝑢)=𝑣−𝑢 for all 𝑣,𝑢∈𝐶, the problem (1.5) is deduced to the following variational inequality problem:find𝑢∈𝐶suchthat⟨𝐴(𝑢,𝑢),𝑣−𝑢⟩+𝜑(𝑣)−𝜑(𝑢)≥0,∀𝑣∈𝐶.(1.9) The problem (1.9) was studied by Chen [2].(d) If 𝐸 is reflexive (i.e., 𝐸=𝐸^∗∗) and 𝐴≡0, (1.5) is deduced to the mixed equilibrium problem:find𝑢∈𝐶suchthat𝑓(𝑢,𝑣)+𝜑(𝑣)≥𝜑(𝑢),∀𝑣∈𝐶.(1.10) The problem (1.10) was considered and studied by Ceng and Yao [3]; Cholamjiak and Suantai [4].(e)In the case of 𝐴≡0 and 𝜑≡0, (1.5) is deduced to the following classical equilibrium problem:find𝑢∈𝐶suchthat𝑓(𝑢,𝑣)≥0,∀𝑣∈𝐶.(1.11) The set of all solution of (1.11) is denoted by EP(𝑓), that is,EP(𝑓)={𝑢∈𝐶∶𝑓(𝑢,𝑣)≥0,∀𝑣∈𝐶}.(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, [5–11]. 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 𝑆∶𝐶→𝐸 is called nonexpansive if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. Also a mapping 𝑆∶𝐶→𝐶 is called asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that ‖𝑆^𝑛𝑥−𝑆^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 and for each 𝑛≥1. 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)={𝑥∈𝐶∶𝑆𝑥=𝑥}. There are several methods for approximating fixed points of a nonexpansive mapping; see, for instance, [16–21]. 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, [22–25]. In 1953, Mann [16] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping 𝑆 in a Hilbert space 𝐻:𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑥_𝑛,∀𝑛∈𝒩,(1.13) where the initial point 𝑥₀ is taken in 𝐶 arbitrarily and {𝛼_𝑛} 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 [26–28]. In 2003, Nakajo and Takahashi [29] introduced the following iterative algorithm for the nonexpansive mapping 𝑆 in the framework of Hilbert spaces:𝑦𝑥₀=𝑥∈𝐶,_𝑛𝐶_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶‖‖𝑧−𝑦_𝑛‖‖≤‖‖𝑧−𝑥_𝑛‖‖,=𝑥𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,𝑛≥0,(1.14) where {𝛼_𝑛}⊂[0,𝛼],𝛼∈[0,1], and 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection from a Hilbert space 𝐻 onto 𝐶_𝑛∩𝑄_𝑛. They proved that {𝑥_𝑛} 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:𝐶𝑥₀=𝑥∈𝐶,_𝑛𝐷=_𝑛co𝑧∈𝐶∶‖𝑧−𝑆𝑧‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆𝑥_𝑛‖‖,=𝑧∈𝑥𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝑥_𝑛≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥,𝑛≥0,(1.15) where co𝐷 denoted the convex closure of the set 𝐷,{𝑡_𝑛} is a sequence in (0,1) with 𝑡_𝑛→0, and 𝑃_{𝐶𝑛∩𝐷𝑛} is the metric projection from 𝐸 onto 𝐶_𝑛∩𝐷_𝑛.

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:𝐶𝑥₀=𝑥∈𝐶,𝐶₀=𝐷₀=𝐶,_𝑛𝐷=_𝑛co𝑧∈𝐶_𝑛−1∶‖𝑧−𝑆^𝑛𝑧‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆^𝑛𝑥_𝑛‖‖,=𝑧∈𝑥𝐷_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥−𝑥_𝑛≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥,𝑛≥0,(1.16) where co𝐷 denotes the convex closure of the set 𝐷,𝐽 is the normalized duality mapping, {𝑡_𝑛} is a sequence in (0,1) with 𝑡_𝑛→0, and 𝑃_{𝐶𝑛∩𝐷𝑛} is the metric projection from 𝐸 onto 𝐶_𝑛∩𝐷_𝑛. The strong convergence theorem of the iterative sequence {𝑥_𝑛} 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={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. A Banach space 𝐸 is said to be strictly convex if for any 𝑥,𝑦∈𝑈,𝑥≠𝑦implies‖𝑥+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,𝑦∈𝑈,‖𝑥−𝑦‖≥𝜀implies‖𝑥+𝑦‖<2(1−𝛿).(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)=inf1−‖‖‖𝑥+𝑦2‖‖‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀.(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 limitlim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.4) exists for all 𝑥,𝑦∈𝑈. Let 𝐶 be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space 𝐸. Then for any 𝑥∈𝐸, there exists a unique point 𝑥₀∈𝐶 such that‖‖𝑥₀−𝑥‖‖≤min_𝑦∈𝐶‖𝑦−𝑥‖.(2.5)

The mapping 𝑃_𝐶∶𝐸→𝐶 defined by 𝑃_𝐶𝑥=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 𝑥∈𝐸, and 𝑦∈𝐶. Then the following are equivalent:(a)𝑦 is a best approximation to 𝑥∶𝑦=𝑃_𝐶𝑥.(b)𝑦 is a solution of the variational inequality:⟨𝑦−𝑧,𝐽(𝑥−𝑦)⟩≥0,∀𝑧∈𝐶,(2.6) where 𝐽 is a duality mapping and 𝑃_𝐶 is the metric projection from 𝐸 onto 𝐶.

It is wellknown that if 𝑃_𝐶 is a metric projection from a real Hilbert space 𝐻 onto a nonempty, closed, and convex subset 𝐶, then 𝑃_𝐶 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 {𝛼_𝑛} be a sequence of real numbers such that 0<𝑏≤𝛼_𝑛≤𝑐<1 for all 𝑛≥1, and let {𝑥_𝑛} and {𝑦_𝑛} be sequences in 𝐸 such that limsup_𝑛→∞‖𝑥_𝑛‖≤𝑑,limsup_𝑛→∞‖𝑦_𝑛‖≤𝑑, and lim_𝑛→∞‖𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑦_𝑛‖=𝑑. Then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=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 𝛾‖‖‖‖𝑆^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖−^𝑛_𝑖=1𝜆_𝑖𝑆𝑥_𝑖‖‖‖‖≤max_{1≤𝑗≤𝑘≤𝑛}‖‖𝑥_𝑗−𝑥_𝑘‖‖−‖‖𝑆𝑥_𝑗−𝑆𝑥_𝑘‖‖,(2.7) for all 𝑛∈𝒩, {𝑥₁,𝑥₂,…,𝑥_𝑛}⊂𝐶, {𝜆₁,𝜆₂,…,𝜆_𝑛}⊂[0,1] with ∑^𝑛𝑖=1𝜆_𝑖=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 𝛾1𝑘_𝑚‖‖‖‖𝑆_𝑚^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖−^𝑛_𝑖=1𝜆_𝑖𝑆^𝑚𝑥_𝑖‖‖‖‖≤max_{1≤𝑗≤𝑘≤𝑛}‖‖𝑥^𝑗−𝑥_𝑘‖‖−1𝑘_𝑚‖‖𝑆^𝑚𝑥_𝑗−𝑆^𝑚𝑥_𝑘‖‖,(2.8) for all 𝑛∈𝒩, {𝑥₁,𝑥₂,…,𝑥_𝑛}⊂𝐶; {𝜆₁,𝜆₂,…,𝜆_𝑛}⊂[0,1] with ∑^𝑛𝑖=1𝜆_𝑖=1 and an asymptotically nonexpansive mapping 𝑆 of 𝐶 into 𝐸 with the sequence {𝑘_𝑚}.

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 𝐺∶𝐵→2^𝑋 is called a KKM mapping if co{𝑥₁,𝑥₂,…,𝑥_𝑚}⊂⋃^𝑚𝑖=1𝐺(𝑥_𝑖) for 𝑥_𝑖∈𝐵 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 𝐺∶𝐵→2^𝑋 be a KKM mapping. If 𝐺(𝑥) is closed for all 𝑥∈𝐵 and is compact for at least one 𝑥∈𝐵, then ⋂_𝑥∈𝐵𝐺(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 𝑇∶𝐶→2^𝐶 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 𝑇∶𝐶→𝐸^∗ and let 𝜂∶𝐶×𝐶→ℛ be two mappings. 𝑇 is said to be 𝜂-hemicontinuous if, for any fixed 𝑥,𝑦∈𝐶, the mapping 𝑓∶[0,1]→(−∞,∞) defined by 𝑓(𝑡)=⟨𝑇(𝑥+𝑡(𝑦−𝑥)),𝜂(𝑦,𝑥)⟩ is continuous at 0⁺.
For solving the mixed equilibrium problem, let us assume the following conditions for a bifunction 𝑓∶𝐶×𝐶→ℛ:
(A1)𝑓(𝑥,𝑥)=0 for all 𝑥∈𝐶;(A2)𝑓 is monotone, that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶;(A3)for all 𝑦∈𝐶, 𝑓(⋅,𝑦) is weakly upper semicontinuous;(A4)for all 𝑥∈𝐶, 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 𝑇∶𝐶→𝐸^∗ be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℛ satisfying (A1) and (A4), and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℛ. Let 𝑟>0 and 𝑧∈𝐶. Assume that(i)𝜂(𝑥,𝑥)=0,forall𝑥∈𝐶;(ii) for any fixed 𝑢,𝑣∈𝐶, the mapping 𝑥↦⟨𝑇𝑣,𝜂(𝑥,𝑢)⟩ is convex. Then the following problems (2.9) and (2.10) are equivalent. Find 𝑥∈𝐶 such that: 𝑓(𝑥,𝑦)+𝜑(𝑦)+⟨𝑇𝑥,𝜂(𝑦,𝑥)⟩+1𝑟⟨𝑦−𝑥,𝐽(𝑥−𝑧)⟩≥𝜑(𝑥),∀𝑦∈𝐶.(2.9) Find 𝑥∈𝐶 such that 𝑓(𝑥,𝑦)+⟨𝑇𝑦,𝜂(𝑦,𝑥)⟩+𝜑(𝑦)+1𝑟⟨𝑦−𝑥,𝐽(𝑥−𝑧)⟩≥𝜑(𝑥)+𝜉(𝑦−𝑥),∀𝑦∈𝐶.(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 𝑇∶𝐶→𝐸^∗ be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℛ satisfying (A1), (A3), and (A4), and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℛ. Let 𝑟>0 and 𝑧∈𝐶. Assume that(i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐶;(ii)for any fixed 𝑢,𝑣∈𝐶, the mapping 𝑥↦⟨𝑇𝑣,𝜂(𝑥,𝑢)⟩ is convex and lower semicontinuous;(iii)𝜉∶𝐸→ℛ is weakly lower semicontinuous; that is,  for any net {𝑥_𝛽},{𝑥_𝛽} converges to 𝑥 in 𝜎(𝐸,𝐸^∗) implies that 𝜉(𝑥)≤liminf𝜉(𝑥_𝛽). Then, the solution set of the problem (2.9) is nonempty, that is, there exists 𝑥₀∈𝐶 such that 𝑓𝑥₀,𝑦+𝑇𝑥₀,𝜂𝑦,𝑥₀+𝜑(𝑦)+1𝑟𝑦−𝑥⁰,𝐽𝑥₀−𝑧≥𝜑𝑥₀,∀𝑦∈𝐶.(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 𝐴∶𝐶×𝐶→𝐸^∗ be a relaxed 𝜂-𝜉 semi-monotone mapping. Let 𝑓∶𝐶×𝐶→ℛ be a bifunction satisfying (A1), (A3), and (A4), and let 𝜑∶𝐶→ℛ∪{+∞} be a proper lower semicontinuous and convex function. Let 𝑟>0 and 𝑧∈𝐶. Assume that(i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐶;(ii)for any fixed 𝑢,𝑣,𝑤∈𝐶, the mapping 𝑥↦⟨𝐴(𝑣,𝑤),𝜂(𝑥,𝑢)⟩ is convex and lower semicontinuous;(iii)for each 𝑥∈𝐶,𝐴(𝑥,⋅)∶𝐶→𝐸^∗ is finite-dimensional continuous: that is, for any finite-dimensional subspace 𝐹⊂𝐸^∗∗,𝐴(𝑥,⋅)∶𝐶∩𝐹→𝐸^∗ is continuous;(iv)𝜉∶𝐸^∗∗→ℛ is convex lower semicontinuous. Then there exists 𝑢₀∈𝐶 such that 𝑓𝑢₀,𝑣+𝐴𝑢₀,𝑢₀,𝜂𝑣,𝑢₀+𝜑(𝑣)+1𝑟𝑣−𝑢⁰,𝐽𝑢₀−𝑧≥𝜑𝑢₀,∀𝑣∈𝐶.(3.1)

Proof. Let 𝐹⊆𝐸^∗∗ be a finite-dimensional subspace with 𝐶_𝐹∶=𝐹∩𝐶≠∅. For each 𝑤∈𝐶, consider the following problem: find 𝑢₀∈𝐶_𝐹 such that 𝑓𝑢₀,𝑣+𝐴𝑤,𝑢₀,𝜂𝑣,𝑢₀+𝜑(𝑣)+1𝑟𝑣−𝑢⁰,𝐽𝑢₀−𝑧−𝜑𝑢₀≥0,∀𝑣∈𝐶_𝐹.(3.2)
Since 𝐶_𝐹⊆𝐹 is bounded closed and convex, 𝐴(𝑤,⋅) is continuous on 𝐶_𝐹 and relaxed 𝜂-𝜉 monotone for each fixed 𝑤∈𝐶, from Lemma 2.10, we know that problem (3.2) has a solution 𝑢₀∈𝐶_𝐹.
Now, define a set-valued mapping 𝐺∶𝐶_𝐹→2^𝐶𝐹 as follows: Gw=𝑢∈𝐶_𝐹∶𝑓(𝑢,𝑣)+⟨𝐴(𝑤,𝑢),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)+1𝑟⟨𝑣−𝑢,𝐽(𝑢−𝑧)⟩−𝜑(𝑢)≥0,∀𝑣∈𝐶^𝐹.(3.3) It follows from Lemma 2.9 that, for each fixed 𝑤∈𝐶_𝐹: 𝑢∈=𝐶^𝐹𝑢∶∈𝑓𝐶(_𝐹𝑢,𝑣)+⟨𝐴(𝑤,𝑢),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)+1𝑟⟨𝑣−𝑢,𝐽(𝑢−𝑧)⟩−𝜑(𝑢)≥0,∀𝑣∈𝐾^𝐹∶𝑓(𝑢,𝑣)+⟨𝐴(𝑤,𝑣),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)+1𝑟⟨𝑣−𝑢,𝐽(𝑢−𝑧)⟩−𝜑(𝑢)≥𝜉(𝑣−𝑢),∀𝑣∈𝐾(^𝐹3.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 𝐺∶𝐶_𝐹→2^𝐶𝐹 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 𝑤^∗∈𝐶_𝐹, that is, 𝑓𝑤^∗,𝑣+⟨𝐴𝑤^∗,𝑤^∗,𝜂𝑣,𝑤^∗+𝜑(𝑣)+1𝑟𝑣−𝑤^∗,𝐽𝑤^∗−𝑧−𝜑𝑤^∗≥0,∀𝑣∈𝐶_𝐹.(3.5) Let ℱ=𝐹⊂𝐸^∗∗∶𝐹isfinitedimensionalwith𝐹∩𝐶≠∅,(3.6) and let 𝑊_𝐹=+𝑢1∈𝐶∶𝑓(𝑢,𝑣)+⟨𝐴(𝑢,𝑣),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)r⟨𝑣−𝑢,𝐽(𝑢−𝑧)⟩−𝜑(𝑢)≥𝜉(𝑣−𝑢),∀𝑣∈𝐶^𝐹,∀𝐹∈ℱ.(3.7) By (3.5) and Lemma 2.9, we know that 𝑊_𝐹 is nonempty and bounded. Denote by 𝑊^∗𝐹 the weak^∗-closure of 𝑊_𝐹 in 𝐸^∗∗. Then, 𝑊^∗𝐹 is weak^∗ compact in 𝐸^∗∗.
For any 𝐹_𝑖∈ℱ, 𝑖=1,2,…,𝑁, we know that 𝑊_⋂^𝑁𝑖=1_𝐹𝑖⊂⋂^𝑁𝑖=1𝑊_𝐹𝑖, so {𝑊^∗𝐹∶𝐹∈ℱ} has the finite intersection property. Therefore, it follows that_𝐹_∈ℱ𝑊^∗𝐹≠∅.(3.8) Let 𝑢₀∈⋂_𝐹∈ℱ𝑊^∗𝐹. We claim that 𝑓𝑢₀,𝑣+𝐴𝑢₀,𝑢₀,𝜂𝑣,𝑢₀+𝜑(𝑣)+1𝑟𝑣−𝑢⁰,𝐽𝑢₀−𝑧−𝜑𝑢₀≥0,∀𝑣∈𝐶.(3.9) Indeed, for each 𝑣∈𝐶, let 𝐹∈ℱ be such that 𝑣∈𝐶_𝐹 and 𝑢₀∈𝐶_𝐹. Then, there exists 𝑢_𝑗∈𝑊_𝐹 such that 𝑢_𝑗⇀𝑢₀. The definition of 𝑊_𝐹 implies that 𝑓𝑢_𝑗,𝑣+𝐴𝑢_𝑗,𝑣,𝜂𝑣,𝑢_𝑗+𝜑(𝑣)+1𝑟𝑣−𝑢^𝑗,𝐽𝑢_𝑗−𝑧−𝜑𝑢_𝑗≥𝜉𝑣−𝑢_𝑗,(3.10) that is 𝑓𝑢_𝑗,𝑣+𝐴𝑢_𝑗,𝑣,𝜂𝑣,𝑢_𝑗+𝜑(𝑣)+1𝑟𝑣−𝑧,𝐽𝑢^𝑗−𝑧−1𝑟‖‖𝑧−𝑢^𝑗‖‖²−𝜑𝑢_𝑗≥𝜉𝑣−𝑢_𝑗(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 𝑓𝑢₀,𝑣+⟨𝐴𝑢₀,𝑣,𝜂𝑣,𝑢₀⟩+𝜑(𝑣)+1𝑟⟨𝑣−𝑢⁰,𝐽𝑢₀−𝑧⟩−𝜑𝑢₀≥𝜉𝑣−𝑢₀,∀𝑣∈𝐶.(3.12) From Lemma 2.9, we have 𝑓𝑢₀,𝑣+𝐴𝑢₀,𝑢₀,𝜂𝑣,𝑢₀+𝜑(𝑣)+1𝑟𝑣−𝑢⁰,𝐽𝑢₀−𝑧−𝜑𝑢₀≥0,∀𝑣∈𝐶.(3.13) Hence, we complete the proof.

Setting 𝐴≡0 and 𝜑≡0 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 𝑓∶𝐶×𝐶→ℛ be a bifunction satisfying (A1), (A3), and (A4). Let 𝑟>0 and 𝑧∈𝐶. Then there exists 𝑢₀∈𝐶 such that 𝑓𝑢₀,𝑣+1𝑟𝑣−𝑢⁰,𝐽𝑢₀−𝑧≥0,∀𝑣∈𝐶.(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 𝐴∶𝐶×𝐶→𝐸^∗ be a relaxed 𝜂-𝜉 semi-monotone mapping. Let 𝑓∶𝐶×𝐶→ℛ be a bifunction satisfying (A1), (A3), and (A4), and let 𝜑∶𝐶→ℛ∪{+∞} be a proper lower semicontinuous and convex function. Let 𝑟>0 and 𝑧∈𝐶. Assume that(i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐶;(ii)for any fixed 𝑢,𝑣,𝑤∈𝐶, the mapping 𝑥↦⟨𝐴(𝑣,𝑤),𝜂(𝑥,𝑢)⟩ is convex and lower semicontinuous;(iii)for each 𝑥∈𝐶,𝐴(𝑥,⋅)∶𝐶→𝐸^∗ is finite-dimensional continuous.(iv)𝜉∶𝐸→ℛ is convex lower semicontinuous.Then, there exists 𝑢₀∈𝐶 such that 𝑓𝑢₀,𝑣+𝐴𝑢₀,𝑢₀,𝜂𝑣,𝑢₀+𝜑(𝑣)+1𝑟𝑣−𝑢⁰,𝐽𝑢₀−𝑧≥𝜑𝑢₀,∀𝑣∈𝐶.(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 𝐴∶𝐶×𝐶→𝐸^∗ be a semi-monotone mapping. Let 𝑓∶𝐶×𝐶→ℛ be a bifunction satisfying (A1), (A3), and (A4), and let 𝜑∶𝐶→ℛ∪{+∞} be a proper lower semicontinuous and convex function. Assume that, for any 𝑟>0 and 𝑧∈𝐶,(i)for any fixed 𝑢,𝑣,𝑤∈𝐶, the mapping 𝑥↦⟨𝐴(𝑣,𝑤),𝑥−𝑢)⟩ is convex and lower semicontinuous;(ii)for each 𝑥∈𝐶,𝐴(𝑥,⋅)∶𝐶→𝐸^∗ is finite-dimensional continuous.Then, there exists 𝑢₀∈𝐶 such that 𝑓𝑢₀,𝑣+𝐴𝑢₀,𝑢₀,𝑣−𝑢₀+𝜑(𝑣)+1𝑟𝑣−𝑢⁰,𝐽𝑢₀−𝑧≥𝜑𝑢₀,∀𝑣∈𝐶.(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 𝐴∶𝐶×𝐶→𝐸^∗ be a relaxed 𝜂-𝜉 semi-monotone mapping. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℛ satisfying (A1)–(A4) and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℛ. For any 𝑟>0, define a mapping Φ_𝑟∶𝐸^∗∗→𝐶 as follows: Φ_𝑟(𝑥)=𝑢∈𝐶∶𝑓(𝑢,𝑣)+⟨𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)+1𝑟⟨𝑣−𝑢,𝐽(𝑢−𝑥)⟩≥𝜑(𝑢),∀𝑣∈𝐶,(3.17) for all 𝑥∈𝐸. Assume that(i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐶;(ii)for any fixed 𝑢,𝑣,𝑤∈𝐶, the mapping 𝑥↦⟨𝐴(𝑣,𝑤),𝜂(𝑥,𝑢)⟩ is convex and lower semicontinuous;(iii)for each 𝑥∈𝐶,𝐴(𝑥,⋅)∶𝐶→𝐸^∗ is finite-dimensional continuous: that is, for any finite-dimensional subspace 𝐹⊂𝐸^∗∗,𝐴(𝑥,⋅)∶𝐶∩𝐹→𝐸^∗ is continuous;(iv)𝜉∶𝐸^∗∗→ℛ is convex lower semicontinuous;(v) for any 𝑥,𝑦∈𝐶, 𝜉(𝑥−𝑦)+𝜉(𝑦−𝑥)≥0;(vi)for any 𝑥,𝑦∈𝐶, 𝐴(𝑥,𝑦)=𝐴(𝑦,𝑥). Then, the following holds:(1)Φ_𝑟 is single-valued;(2)⟨Φ_𝑟𝑥−Φ_𝑟𝑦,𝐽(Φ_𝑟𝑥−𝑥)⟩≤⟨Φ_𝑟𝑥−Φ_𝑟𝑦,𝐽(Φ_𝑟𝑦−𝑦)⟩ for all 𝑥,𝑦∈𝐸;(3)𝐹(Φ_𝑟)=GMEP(𝑓,𝐴,𝜂,𝜑);(4)GMEP(𝑓,𝐴,𝜂,𝜑) is nonempty, closed, and convex.

Proof. For each 𝑥∈𝐸^∗∗, by Lemma 2.10, we conclude that Φ_𝑟(𝑥) is nonempty.(1)We prove that Φ_𝑟 is single-valued. Indeed, for 𝑥∈𝐸^∗∗ and 𝑟>0, let 𝑧₁,𝑧₂∈Φ_𝑟(𝑥). Then,𝑓𝑧₁,𝑣+𝐴𝑧₁,𝑧₁,𝜂𝑣,𝑧₁+𝜑(𝑣)+1𝑟𝑣−𝑧¹𝑓𝑧,𝐽𝑧₁−𝑥≥𝜑𝑧₁,∀𝑣∈𝐶,₂,𝑣+𝐴𝑧₂,𝑧₂,𝜂𝑣,𝑧₂+𝜑(𝑣)+1𝑟𝑣−𝑧²,𝐽𝑧₂−𝑥≥𝜑𝑧₂,∀𝑣∈𝐶.(3.18) Hence, 𝑓𝑧₁,𝑧₂+𝐴𝑧₁,𝑧₁,𝜂𝑧₂,𝑧₁+𝜑𝑧₂+1𝑟𝑧²𝑓𝑧₂−𝑧₁,𝐽𝑧₁−𝑥≥𝜑𝑧₁,,𝑧₁+𝐴𝑧₂,𝑧₂,𝜂𝑧₁,𝑧₂+𝜑𝑧₁+1𝑟𝑧¹−𝑧₂,𝐽𝑧₂−𝑥≥𝜑𝑧₂.(3.19) Adding the two inequalities, from (i) we have 𝑓𝑧₂,𝑧₁+𝑓𝑧₁,𝑧₂+𝐴𝑧₁,𝑧₁−𝐴𝑧₂,𝑧₂,𝜂𝑧₂,𝑧₁+1𝑟𝑧²−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥(≥3.02.0) From (A2), we have 𝐴𝑧₁,𝑧₁−𝐴𝑧₂,𝑧₂,𝜂𝑧₂,𝑧₁+1𝑟𝑧²−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥≥0.(3.21) That is, 1𝑟𝑧²−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥≥𝐴𝑧₂,𝑧₂−𝐴𝑧₁,𝑧₁,𝜂𝑧₂,𝑧₁.(3.22) Calculating the right-hand side of (3.22), we have 𝐴=𝑧₂,𝑧𝐴₂𝑧−𝐴𝑧₁,𝑧₁,𝜂𝑧₂,𝑧₁₂=𝐴𝑧,𝑧₂−𝐴𝑧₂,𝑧₁+𝐴𝑧₂,𝑧₁−𝐴𝑧₁,𝑧₂+𝐴𝑧₁,𝑧₂−𝐴𝑧₁,𝑧₁,𝜂𝑧₂,𝑧₁₂+𝐴𝑧,𝑧₂−𝐴𝑧₂,𝑧₁,𝜂𝑧₂,𝑧₁+𝐴𝑧₂,𝑧₁−𝐴𝑧₁,𝑧₂,𝜂𝑧₂,𝑧₁₁≥2𝜉𝑧₂,𝑧₂−𝐴𝑧₁,𝑧₁,𝜂𝑧₂,𝑧₁−𝑧₁+𝐴𝑧₂,𝑧₁−𝐴𝑧₁,𝑧₂,𝜂𝑧₂,𝑧₁,(3.23) and so, 1𝑟𝑧²−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥≥2𝜉𝑧₂−𝑧₁+𝐴𝑧₂,𝑧₁−𝐴𝑧₁,𝑧₂,𝜂𝑧₂,𝑧₁.(3.24) In (3.24) exchanging the position of 𝑧₁ and 𝑧₂, we get 1𝑟𝑧¹−𝑧₂,𝐽𝑧₂−𝑥−𝐽𝑧₁−𝑥≥2𝜉𝑧₁−𝑧₂+𝐴𝑧₁,𝑧₂−𝐴𝑧₂,𝑧₁,𝜂𝑧₁,𝑧₂.(3.25) Adding the inequalities (3.24) and (3.25) and using (v) and (vi), we have 𝑧₂−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥≥𝑟𝜉𝑧₂−𝑧₁+𝜉𝑧₁−𝑧₂≥0.(3.26) Hence, 0≤𝑧₂−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥=𝑧₂−𝑥−𝑧₁−𝑥,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥.(3.27) Since 𝐽 is monotone and 𝐸^∗∗ is strictly convex, we obtain that 𝑧₁−𝑥=𝑧₂−𝑥 and hence 𝑧₁=𝑧₂. Therefore, Φ_𝑟 is single-valued.(2) For 𝑥,𝑦∈𝐶, we have𝑓Φ_𝑟𝑥,Φ_𝑟𝑦+𝐴Φ_𝑟𝑥,Φ_𝑟𝑥,𝜂Φ_𝑟𝑦,Φ_𝑟𝑥+𝜑Φ_𝑟𝑦−𝜑Φ_𝑟𝑥+1𝑟Φ^𝑟𝑓Φ𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥≥0,_𝑟𝑦,Φ_𝑟𝑥+𝐴Φ_𝑟𝑦,Φ_𝑟𝑦,𝜂Φ_𝑟𝑥,Φ_𝑟𝑦+𝜑Φ_𝑟𝑥−𝜑Φ_𝑟𝑦+1𝑟Φ^𝑟𝑥−Φ_𝑟𝑦,𝐽Φ_𝑟𝑦−𝑦≥(3.02.8) Adding the above two inequalities and by (i) and (A2), we get 𝐴Φ_𝑟𝑥,Φ_𝑟𝑥−𝐴Φ_𝑟𝑦,Φ_𝑟𝑦,𝜂Φ_𝑟𝑦,Φ_𝑟𝑥+1𝑟Φ^𝑟𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥−𝐽Φ_𝑟𝑦−𝑦≥0,(3.29) that is 1𝑟Φ^𝑟𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥−𝐽Φ_𝑟𝑦−𝑦≥𝐴Φ_𝑟𝑦,Φ_𝑟𝑦−𝐴Φ_𝑟𝑥,Φ_𝑟𝑥,𝜂Φ_𝑟𝑦,Φ_𝑟𝑥.(3.30) After calculating (3.30), we have 1𝑟Φ^𝑟𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥−𝐽Φ_𝑟𝑦−𝑦≥+2𝜉𝐴ΦΦ_𝑟𝑦_𝑟,Φ_𝑟𝑥𝑦,Φ_𝑟𝑥−𝐴Φ_𝑟𝑥,Φ_𝑟𝑦,𝜂Φ_𝑟𝑦,Φ_𝑟𝑥.(3.31) In (3.30), exchanging the position of Φ_𝑟𝑥 and Φ_𝑟𝑦, we get 1𝑟⟨Φ^𝑟𝑥−Φ_𝑟𝑦,𝐽Φ_𝑟𝑦−𝑦−𝐽Φ_𝑟𝑥−𝑥⟩≥+2𝜉𝐴ΦΦ_𝑟𝑥_𝑟,Φ_𝑟𝑦𝑥,Φ_𝑟𝑦−𝐴Φ_𝑟𝑦,Φ_𝑟𝑥,𝜂Φ_𝑟𝑥,Φ_𝑟𝑦.(3.32) Adding the inequalities (3.31) and (3.32), use (i) and (vi), we have Φ_𝑟𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥−𝐽Φ_𝑟𝑦−𝑦≥𝑟𝜉Φ_𝑟𝑥,Φ_𝑟𝑦+𝜉Φ_𝑟𝑦,Φ_𝑟𝑥.(3.33) It follows from (iv) that Φ_𝑟𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥−𝐽Φ_𝑟𝑦−𝑦≥0.(3.34) Hence, ⟨Φ_𝑟𝑥−Φ_𝑟𝑦,𝐽Φ_𝑟𝑥−𝑥⟩≤⟨Φ_𝑟𝑥−Φ_𝑟𝑦,𝐽Φ_𝑟𝑦−𝑦⟩.(3.35)(3) Next, we show that 𝐹(Φ_𝑟)=GMEP(𝑓,𝐴,𝜂,𝜑). Indeed, we have the following: 𝑢∈𝐹Φ_𝑟⟺⟺𝑢𝑓=(𝑢,Φ𝑣_𝑟)𝑢+⟨𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)+1⟺⟺𝑓𝑢∈(𝑢,G𝑣)M+E⟨P𝐴((𝑓𝑢,,𝐴𝑢),𝜂,𝜂,𝜑(𝑣),.𝑢)⟩+𝜑(𝑣)≥𝜑𝑟⟨(𝑣𝑢)−,𝑢,∀𝐽𝑣(∈𝑢−𝐶𝑢)⟩≥𝜑(𝑢),∀𝑣∈𝐶(3.36)Hence, 𝐹(Φ_𝑟)=GMEP(𝑓,𝐴,𝜂,𝜑).(4) Finally, we prove that GMEP(𝑓,𝐴,𝜂,𝜑) is nonempty, closed, and convex. For each 𝑣∈𝐶, we define the multivalued mapping 𝐺∶𝐶→2^𝐸∗∗ by𝐺(𝑣)={𝑢∈𝐶∶𝑓(𝑢,𝑣)+⟨𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)≥𝜑(𝑢)}.(3.37) Since 𝑣∈𝐺(𝑣), we have 𝐺(𝑣)≠∅. We prove that 𝐺 is a KKM mapping on 𝐶. Suppose that there exists a finite subset {𝑧₁,𝑧₂,…,𝑧_𝑚} of 𝐶, and 𝛼_𝑖>0 with ∑^𝑚𝑖=1𝛼_𝑖=1 such that 𝑧̂=∑^𝑚𝑖=1𝛼_𝑖𝑧_𝑖∉𝐺(𝑧_𝑖) for all 𝑖=1,2,…,𝑚. Then 𝑓𝑧,̂𝑧_𝑖+𝐴(𝑧,̂𝑧)̂,𝜂𝑧_𝑖,𝑧̂+𝜑𝑧_𝑖−𝜑(𝑧)̂<0,𝑖=1,2,…,𝑚.(3.38) From (A1), (A4), (ii), and the convexity of 𝜑, we have 0==𝑓𝑓(𝑧𝑧,̂𝑧,̂)̂_𝑚+⟨𝐴(𝑧,̂𝑧)̂,𝜂(𝑧,̂𝑧)̂⟩+𝜑(𝑧)̂−𝜑(𝑧)̂_𝑖=1𝛼_𝑖𝑧_𝑖+𝐴(𝑧,̂𝑧)̂,𝜂^𝑚_𝑖=1𝛼_𝑖𝑧_𝑖,𝑧̂+𝜑^𝑚_𝑖=1𝛼_𝑖𝑧_𝑖≤−𝜑(𝑧)_𝑚̂_𝑖=1<0,𝛼_𝑖𝑓𝑧,̂𝑧_𝑖+𝐴(𝑧,̂𝑧)̂,𝜂𝑧_𝑖,𝑧̂+𝜑𝑧_𝑖−𝜑(𝑧)̂(3.39) which is a contradiction. Thus, 𝐺 is a KKM mapping on 𝐶.
Next, we prove that 𝐺(𝑦) is closed for each 𝑦∈𝐶. For any 𝑦∈𝐶, let {𝑥_𝑛} be any sequence in 𝐺(𝑦) such that 𝑥_𝑛→𝑥₀. We claim that 𝑥₀∈𝐺(𝑦). Then, for each 𝑦∈𝐶, we have𝑓𝑥_𝑛,𝑦+𝐴𝑥_𝑛,𝑥_𝑛,𝜂𝑦,𝑥_𝑛+𝜑(𝑦)≥𝜑𝑥_𝑛.(3.40) By monotonicity of 𝐴, we obtain that 𝑓𝑥_𝑛,𝑦+𝐴𝑥_𝑛,𝑦,𝜂𝑦,𝑥_𝑛+𝜑(𝑦)≥𝜑𝑥_𝑛+𝜉𝑦−𝑥_𝑛.(3.41) By (A3), (i), (ii), (iv), lower semicontinuity of 𝜑, and the complete continuity of 𝐴, we obtain the following 𝜑𝑥₀+𝐴𝑥₀,𝑦,𝜂𝑥₀,𝑦≤≤lliimmiinnf_𝑛→∞f_𝑛→∞𝜑𝑥_𝑛+liminf_𝑛→∞𝐴𝑥_𝑛,𝑦,𝜂𝑥_𝑛,𝑦=liminf_𝑛→∞𝜑𝑥_𝑛+𝐴𝑥_𝑛,𝑦,𝜂𝑥_𝑛,𝑦≤limsup_𝑛→∞𝜑𝑥_𝑛−𝐴𝑥_𝑛,𝑦,𝜂𝑦,𝑥_𝑛≤limsup_𝑛→∞𝜑𝑥_𝑛−𝐴𝑥_𝑛,𝑦,𝜂𝑦,𝑥_𝑛≤𝑓𝑥₀𝑓𝑥_𝑛,𝑦+𝜑(𝑦)−𝜉𝑦−𝑥_𝑛,𝑦+𝜑(𝑦)−𝜉𝑦−𝑥₀.(3.42) Hence, 𝑓𝑥₀,𝑦+𝐴𝑥₀,𝑦,𝜂𝑦,𝑥₀+𝜑(𝑦)≥𝜑𝑥₀+𝜉𝑦−𝑥₀,∀𝑦∈𝐶.(3.43) From Lemma 2.9, we have 𝑓𝑥₀,𝑦+𝐴𝑥₀,𝑥₀,𝜂𝑦,𝑥₀+𝜑(𝑦)≥𝜑𝑥₀,∀𝑦∈𝐶.(3.44) This shows that 𝑥₀∈𝐺(𝑦), and hence 𝐺(𝑦) is closed for each 𝑦∈𝐶. Thus, GMEP(𝑓,𝐴,𝜂,𝜑)=⋂_𝑦∈𝐶𝐺(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 ⋂_𝑦∈𝐶𝐺(𝑦)=GMEP(𝑓,𝐴,𝜂,𝜑)≠∅.
Finally, we prove that GMEP(𝑓,𝐴,𝜂,𝜑) is convex. In fact, let 𝑢,𝑣∈𝐹(Φ_𝑟), and 𝑧_𝑡=𝑡𝑢+(1−𝑡)𝑣 for 𝑡∈(0,1). From (2), we know that Φ_𝑟𝑢−Φ_𝑟𝑧_𝑡,𝐽Φ_𝑟𝑧_𝑡−𝑧_𝑡−𝐽Φ_𝑟𝑢−𝑢≥0.(3.45) This yields that 𝑢−Φ_𝑟𝑧_𝑡,𝐽Φ_𝑟𝑧_𝑡−𝑧_𝑡≥0.(3.46) Similarly, we also have 𝑣−Φ_𝑟𝑧_𝑡,𝐽Φ_𝑟𝑧_𝑡−𝑧_𝑡≥0.(3.47) It follows from (3.46) and (3.47) that ‖‖𝑧_𝑡−Φ_𝑟𝑧_𝑡‖‖²==𝑡𝑧𝑢_𝑡−−ΦΦ_𝑟𝑟𝑧_𝑡,𝐽𝑧_𝑡−Φ_𝑟𝑧_𝑡≤0.𝑧_𝑡,𝐽𝑧_𝑡−Φ_𝑟𝑧_𝑡+(1−𝑡)𝑣−Φ_𝑟𝑧_𝑡,𝐽𝑧_𝑡−Φ_𝑟𝑧_𝑡(3.48) Hence, 𝑧_𝑡∈𝐹(Φ_𝑟)=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 𝐴∶𝐶×𝐶→𝐸^∗ be a relaxed 𝜂-𝜉 semi-monotone mapping. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℛ satisfying (A1)–(A4) and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℛ. Let 𝑟>0 and 𝑧∈𝐶 and define a mapping Φ_𝑟∶𝐸→𝐶 as follows: Φ_𝑟(𝑥)=𝑢∈𝐶∶𝑓(𝑢,𝑣)+⟨𝐴(𝑢,𝑢),𝜂(𝑣,𝑢)⟩+𝜑(𝑣)+1𝑟⟨𝑣−𝑢,𝐽(𝑢−𝑥)⟩≥𝜑(𝑢),∀𝑣∈𝐶,(3.49)for all 𝑥∈𝐸. Assume that(i)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐶;(ii)for any fixed 𝑢,𝑣,𝑤∈𝐶, the mapping 𝑥↦⟨𝐴(𝑣,𝑤),𝜂(𝑥,𝑢)⟩ is convex and lower semicontinuous; (iii)for each 𝑥∈𝐶,𝐴(𝑥,⋅)∶𝐶→𝐸^∗ is finite-dimensional continuous;(iv)𝜉∶𝐸→ℛ is convex lower semicontinuous;(v)for any 𝑥,𝑦∈𝐶, 𝜉(𝑥−𝑦)+𝜉(𝑦−𝑥)≥0;(vi)for any 𝑥,𝑦∈𝐶, 𝐴(𝑥,𝑦)=𝐴(𝑦,𝑥). Then, the following holds:(1)Φ_𝑟 is single-valued;(2)⟨Φ_𝑟𝑥−Φ_𝑟𝑦,𝐽(Φ_𝑟𝑥−𝑥)⟩≤⟨Φ_𝑟𝑥−Φ_𝑟𝑦,𝐽(Φ_𝑟𝑦−𝑦)⟩ for all 𝑥,𝑦∈𝐸;(3)𝐹(Φ_𝑟)=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 ℛ satisfying (A1)–(A4), and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℛ. Let 𝐴∶𝐶×𝐶→𝐸^∗ be a relaxed 𝜂-𝜉 semi-monotone and let 𝑆∶𝐶→𝐶 be an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞) such that 𝑘_𝑛→1 as 𝑛→∞. Assume that Ω∶=𝐹(𝑆)∩GMEP(𝑓,𝐴,𝜂,𝜑)≠∅. Let {𝑥_𝑛} be a sequence in 𝐶 generated by 𝐶𝑥₀∈𝐶,𝐷₀=𝐶₀=𝐶,_𝑛𝑢=co𝑧∈𝐶_𝑛−1∶‖𝑧−𝑆^𝑛𝑧‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆^𝑛𝑥_𝑛‖‖,𝑛≥1,_𝑛𝑓∈𝑢𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡_𝑛,𝑦+𝜑(𝑦)+𝐴𝑢_𝑛,𝑢_𝑛,𝜂𝑦,𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛𝐷,𝐽𝑢_𝑛−𝑥_𝑛≥𝜑𝑢_𝑛,∀𝑦∈𝐶,𝑛≥0,_𝑛𝑥=𝑧∈𝐷_𝑛−1∶𝑢_𝑛−𝑧,𝐽𝑥_𝑛−𝑢_𝑛≥0,𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥₀,𝑛≥0,(4.1) where {𝑡_𝑛} and {𝑟_𝑛} are real sequences in (0,1) such that lim_𝑛→∞𝑡_𝑛=0, and liminf_𝑛→∞𝑟_𝑛>0. Then {𝑥_𝑛} converges strongly, as 𝑛→∞, to 𝑃_Ω𝑥₀.

Proof. Firstly, we rewrite the (4.1) as follows: 𝑥𝐶₀∈𝐶,𝐷₀=𝐶₀=𝐶,_𝑛𝐷=co𝑧∈𝐶_𝑛−1∶‖𝑧−𝑆^𝑛𝑧‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆^𝑛𝑥_𝑛‖‖,𝑛≥0,_𝑛𝑥=𝑧∈𝐷_𝑛−1∶⟨Φ_𝑟𝑛𝑥_𝑛−𝑧,𝐽𝑥_𝑛−Φ_𝑟𝑛𝑥_𝑛⟩≥0,𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥₀,𝑛≥0,(4.2) where Φ_𝑟 is the mapping defined by Φ_𝑟(𝑥)=𝑧∈𝐶∶𝑓(𝑧,𝑦)+⟨𝐴(𝑧,𝑧),𝜂(𝑦,𝑧)⟩+𝜑(𝑦)+1𝑟⟨𝑦−𝑧,𝐽(𝑧−𝑥)⟩≥𝜑(𝑧),∀𝑦∈𝐶.(4.3) We first show that the sequence {𝑥_𝑛} is well defined. It is easy to verify that 𝐶_𝑛∩𝐷_𝑛 is closed and convex and Ω⊂𝐶_𝑛 for all 𝑛≥0. Next, we prove that Ω⊂𝐶_𝑛∩𝐷_𝑛. Since 𝐷₀=𝐶, we also have Ω⊂𝐶₀∩𝐷₀. Suppose that Ω⊂𝐶_𝑘−1∩𝐷_𝑘−1 for 𝑘≥2. It follows from Theorem 3.5 (2) that Φ_𝑟𝑘𝑥_𝑘−Φ_𝑟𝑘𝑢,𝐽Φ_𝑟𝑘𝑢−𝑢−𝐽Φ_𝑟𝑘𝑥_𝑘−𝑥_𝑘≥0,(4.4) for all 𝑢∈Ω. This implies that Φ_𝑟𝑘𝑥_𝑘−𝑢,𝐽𝑥_𝑘−Φ_𝑟𝑘𝑥_𝑘≥0,(4.5) for all 𝑢∈Ω. Hence Ω⊂𝐷_𝑘. By the mathematical induction, we get that Ω⊂𝐶_𝑛∩𝐷_𝑛 for each 𝑛≥0, and hence {𝑥_𝑛} is welldefined. Put 𝑤=𝑃_Ω𝑥₀. Since Ω⊂𝐶_𝑛∩𝐷_𝑛 and 𝑥_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}, we have ‖‖𝑥_𝑛+1−𝑥₀‖‖≤‖‖𝑤−𝑥₀‖‖,𝑛≥0.(4.6) Since 𝑥_𝑛+2∈𝐷_𝑛+1⊂𝐷_𝑛 and 𝑥_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥₀, we have ‖‖𝑥_𝑛+1−𝑥₀‖‖≤‖‖𝑥_𝑛+2−𝑥₀‖‖.(4.7) Since {‖𝑥_𝑛−𝑥₀‖} is bounded, we have lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖=𝑑 for some a constant 𝑑. Moreover, by the convexity of 𝐷_𝑛, we also have (1/2)(𝑥_𝑛+1+𝑥_𝑛+2)∈𝐷_𝑛 and hence ‖‖𝑥₀−𝑥_𝑛+1‖‖≤‖‖‖𝑥₀−𝑥^𝑛+1+𝑥_𝑛+22‖‖‖≤21‖‖𝑥⁰−𝑥_𝑛+1‖‖+‖‖𝑥₀−𝑥_𝑛+2‖‖.(4.8) This implies that lim_𝑛→∞‖‖‖12𝑥⁰−𝑥_𝑛+1+12𝑥⁰−𝑥_𝑛+2‖‖‖=lim_𝑛→∞‖‖‖𝑥₀−𝑥^𝑛+1+𝑥_𝑛+22‖‖‖=𝑑.(4.9) By Lemma 2.2, we have lim_𝑛→∞‖‖𝑥_𝑛−𝑥_𝑛+1‖‖=0.(4.10) Next, we show that lim_𝑛→∞‖‖𝑥_𝑛−𝑆𝑥_𝑛‖‖=0.(4.11) To obtain (4.11), we need to show that lim_𝑛→∞‖𝑥_𝑛−𝑆^𝑛−𝑘𝑥_𝑛‖=0,forall𝑘∈𝒩.
Fix 𝑘∈𝒩 and put 𝑚=𝑛−𝑘. Since 𝑥_𝑛=𝑃_{𝐶𝑛−1∩𝐷𝑛−1}𝑥, we have 𝑥_𝑛∈𝐶_𝑛−1⊆⋯⊆𝐶_𝑚. Since 𝑡_𝑚>0, there exist 𝑦₁,…,𝑦_𝑁∈𝐶 and nonnegative numbers 𝜆₁,…,𝜆_𝑁 with 𝜆₁+⋯+𝜆_𝑁=1 such that ‖‖‖‖𝑥_𝑛−^𝑁_𝑖=1𝜆_𝑖𝑦_𝑖‖‖‖‖<𝑡_𝑚,(4.12) and ‖𝑦_𝑖−𝑆^𝑚𝑦_𝑖‖≤𝑡_𝑚‖𝑥_𝑚−𝑆^𝑚𝑥_𝑚‖ for all 𝑖∈{1,…,𝑁}. Put 𝑀=sup_𝑥∈𝐶‖𝑥‖,𝑢=𝑃_𝐹(𝑆)𝑥 and 𝑟₀=sup_𝑛≥1(1+𝑘_𝑛)‖𝑥_𝑛−𝑢‖. Since 𝐶 and {𝑘_𝑚} are bounded, (4.12) implies ‖‖‖‖𝑥_𝑛−1𝑘_{𝑚𝑁𝑖=1}𝜆_𝑖𝑦_𝑖‖‖‖‖≤1−1𝑘_𝑚‖𝑥‖+𝑘1_𝑚‖‖‖‖𝑥_𝑛−^𝑁_𝑖=1𝜆_𝑖𝑦_𝑖‖‖‖‖≤1−1𝑘_𝑚𝑀+𝑡^𝑚,(4.13) and ‖𝑦_𝑖−𝑆^𝑚𝑦_𝑖‖≤𝑡_𝑚‖𝑥_𝑚−𝑆^𝑚𝑥_𝑚‖≤𝑡_𝑚(1+𝑘_𝑚)‖𝑥_𝑚−𝑢‖≤𝑟₀𝑡_𝑚 for all 𝑖∈{1,…,𝑁}. Therefore, ‖‖‖𝑦_𝑖−1𝑘_𝑚𝑆^𝑚𝑦_𝑖‖‖‖≤1−1𝑘_𝑚𝑀+𝑟⁰𝑡_𝑚,(4.14) for all 𝑖∈{1,…,𝑁}. Moreover, asymptotically nonexpansiveness of 𝑆 and (4.6) give that ‖‖‖‖1𝑘_𝑚𝑆^𝑚^𝑁_𝑖=1𝜆_𝑖𝑦_𝑖−𝑆^𝑚𝑥_𝑛‖‖‖‖≤1−1𝑘_𝑚𝑀+𝑡^𝑚.(4.15) It follows from Theorem 2.4, (4.13)–(4.15) that ‖‖𝑥_𝑛−𝑆^𝑚𝑥_𝑛‖‖≤‖‖‖‖𝑥_𝑛−1𝑘_{𝑚𝑁𝑖=1}𝜆_𝑖𝑦_𝑖‖‖‖‖+1𝑘_𝑚‖‖‖‖^𝑁_𝑖=1+1𝜆_𝑖𝑦_𝑖−𝑆^𝑚𝑦_𝑖‖‖‖‖𝑘_𝑚‖‖‖‖^𝑁_𝑖=1𝜆_𝑖𝑆^𝑚𝑦_𝑖−𝑆^𝑚^𝑁_𝑖=1𝜆_𝑖𝑦_𝑖‖‖‖‖+‖‖‖‖𝑘1_𝑚𝑆^𝑚^𝑁_𝑖=1𝜆_𝑖𝑦_𝑖−𝑆^𝑚𝑥_𝑛≤21−1‖‖‖‖𝑘_𝑚𝑀+2𝑡^𝑚+𝑟⁰𝑡_𝑚𝑘_𝑚+𝛾⁻¹max_{1≤𝑖≤𝑗≤𝑁}‖‖‖𝑦^𝑖−𝑦_𝑗‖−1𝑘_𝑚‖𝑆^𝑚𝑦_𝑖−𝑆^𝑚𝑦_𝑗≤21−1‖‖‖𝑘_𝑚𝑀+2𝑡^𝑚+𝑟⁰𝑡_𝑚𝑘_𝑚+𝛾⁻¹max_{1≤𝑖≤𝑗≤𝑁}‖‖‖𝑦^𝑖−1𝑘_𝑚𝑆^𝑚𝑦_𝑖‖‖‖+‖‖‖𝑦_𝑗−1𝑘_𝑚𝑆^𝑚𝑦_𝑗≤21−1‖‖‖𝑘_𝑚𝑀+2𝑡^𝑚+𝑟⁰𝑡_𝑚𝑘_𝑚+𝛾⁻¹21−𝑘1_𝑚𝑀+2𝑟⁰𝑡_𝑚.(4.16) Since lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝑡_𝑛=0, it follows from the last inequality that lim_𝑛→∞‖𝑥_𝑛−𝑆^𝑚𝑥_𝑛‖=0. We have that ‖‖𝑥_𝑛−𝑆𝑥_𝑛‖‖≤=‖‖‖‖𝑥𝑥_𝑛𝑛−𝑆^𝑛−1𝑥_𝑛‖‖+‖‖𝑆^𝑛−1𝑥_𝑛−𝑆𝑥_𝑛‖‖−𝑆^𝑛−1𝑥_𝑛‖‖+𝑘₁‖‖𝑆^𝑛−2𝑥_𝑛−𝑥_𝑛‖‖⟶0as𝑛⟶∞.(4.17) Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑥̃∈𝐶. Therefore, we obtain 𝑥̃∈𝐹(𝑆). Next, we show that 𝑥̃∈GMEP(𝑓,𝐴,𝜂,𝜑). By the construction of 𝐷_𝑛, we see from Theorem 2.1 that Φ_𝑟𝑛𝑥_𝑛=𝑃_𝐷𝑛𝑥_𝑛. Since 𝑥_𝑛+1∈𝐷_𝑛, we get ‖‖𝑥_𝑛−Φ_𝑟𝑛𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖⟶0.(4.18) From (C2), we also have 1𝑟_𝑛‖‖𝐽𝑥_𝑛−Φ_𝑟𝑛𝑥_𝑛‖‖=1𝑟_𝑛‖‖𝑥_𝑛−Φ_𝑟𝑛𝑥_𝑛‖‖⟶0,(4.19) as 𝑛→∞. By (4.19), we also have Φ_𝑟𝑛𝑖𝑥_𝑛𝑖⇀𝑥̃. By the definition of Φ_𝑟𝑛𝑖, for each 𝑦∈𝐶, we obtain 𝑓Φ+^𝑟_𝑛𝑖𝑥1^𝑛_𝑖,𝑦+𝐴Φ_𝑟𝑛𝑖𝑥_𝑛𝑖,Φ_𝑟𝑛𝑖𝑥_𝑛𝑖,𝜂𝑦,Φ^𝑟_𝑛𝑖𝑥_𝑛𝑖+𝜑(𝑦)𝑟_𝑛𝑖𝑦−Φ^𝑟_𝑛𝑖𝑥_𝑛𝑖,𝐽Φ_𝑟𝑛𝑖𝑥_𝑛𝑖−𝑥_𝑛𝑖≥𝜑Φ^𝑟_𝑛𝑖𝑥_𝑛𝑖.(4.20) By (A3), (4.19), (ii), the weakly lower semicontinuity of 𝜑 and complete continuity of 𝐴 we have 𝜑(𝑥̃)≤≤lliimmiinnf_𝑖→∞f_𝑖→∞𝜑Φ_𝑟𝑛𝑖𝑥_𝑛𝑖+𝜑(𝑦)+𝑓limΦin^𝑟_𝑛𝑖𝑥f^𝑛_{𝑖𝑖→∞},𝑦+liminf_𝑖→∞𝐴Φ^𝑟_𝑛𝑖𝑥_𝑛𝑖,Φ_𝑟𝑛𝑖𝑥_𝑛𝑖,𝜂𝑦,Φ^𝑟_𝑛𝑖𝑥_𝑛𝑖1𝑟_𝑛𝑖𝑦−Φ^𝑟_𝑛𝑖≤𝑓(𝑥̃,𝑦)+𝜑(𝑦)+⟨𝐴(𝑥̃,𝑥̃),𝜂𝑥(𝑦,^𝑛_𝑖,𝑥𝐽̃)⟩.Φ_𝑟𝑛𝑖𝑥_𝑛𝑖−𝑥_𝑛𝑖(4.21) Hence, 𝑓(𝑥̃,𝑦)+𝜑(𝑦)+⟨𝐴(𝑥̃,𝑥̃),𝜂(𝑦,𝑥̃)⟩≥𝜑(𝑥̃).(4.22) This shows that 𝑥̃∈GMEP(𝑓,𝐴,𝜂,𝜑), and hence 𝑥̃∈Ω∶=𝐹(𝑆)∩GMEP(𝑓,𝐴,𝜂,𝜑).
Finally, we show that 𝑥_𝑛→𝑤 as 𝑛→∞, where 𝑤∶=𝑃_Ω𝑥₀. By the weakly lower semicontinuity of the norm, it follows from (4.6) that ‖‖𝑥₀−𝑤‖‖≤‖‖𝑥₀−𝑥̃‖‖≤liminf_𝑖→∞‖‖𝑥₀−𝑥_𝑛𝑖‖‖≤limsup_𝑖→∞‖‖𝑥₀−𝑥_𝑛𝑖‖‖≤‖‖𝑥₀−𝑤‖‖.(4.23) This shows that lim_𝑖→∞‖‖𝑥₀−𝑥_𝑛𝑖‖‖=‖‖𝑥₀−𝑤‖‖=‖‖𝑥₀−𝑥̃‖‖,(4.24) and 𝑥̃=𝑤. Since 𝐸^∗∗ is uniformly convex, we obtain that 𝑥₀−𝑥_𝑛𝑖→𝑥₀−𝑤. It follows that 𝑥_𝑛𝑖→𝑤. So, we have 𝑥_𝑛→𝑤 as 𝑛→∞. This completes the proof.

If 𝑆 is a nonexpansive mapping in Theorem 4.1, then we obtain the following result concerning the problem of finding a common element of GMEP(𝑓,𝐴,𝜂,𝜑) and the fixed point set of a nonexpansive mapping in a Banach space setting.

Theorem 4.2. 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 ℛ satisfying (A1)–(A4) and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℛ. Let 𝐴∶𝐶×𝐶→𝐸^∗ be a relaxed 𝜂-𝜉 semi-monotone and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Ω∶=𝐹(𝑆)∩GMEP(𝑓,𝐴,𝜂,𝜑)≠∅. Let {𝑥_𝑛} be a sequence in 𝐶 generated by 𝑥𝐶₀∈𝐶,𝐷₀=𝐶₀=𝐶,_𝑛𝑢=co𝑧∈𝐶_𝑛−1∶‖𝑧−𝑆𝑧‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆𝑥_𝑛‖‖,𝑛≥1,_𝑛𝑓∈𝑢𝐶suchthat_𝑛,𝑦+𝜑(𝑦)+𝐴𝑢_𝑛,𝑢_𝑛,𝜂𝑦,𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛𝐷,𝐽𝑢_𝑛−𝑥_𝑛≥𝜑𝑢_𝑛,∀𝑦∈𝐶,𝑛≥0,_𝑛𝑥=𝑧∈𝐷_𝑛−1∶𝑢_𝑛−𝑧,𝐽𝑥_𝑛−𝑢_𝑛≥0,𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥₀,𝑛≥0,(4.25) where {𝑡_𝑛} and {𝑟_𝑛} are real sequences in (0,1) such that lim_𝑛→∞𝑡_𝑛=0, and liminf_𝑛→∞𝑟_𝑛>0. Then, {𝑥_𝑛} converges strongly, as 𝑛→∞, to 𝑃_Ω𝑥₀.

Putting 𝐴≡0 and 𝜑≡0 in Theorem 4.1, then we have the following result in a Banach space.

Theorem 4.3. Let 𝐸 be a real Banach space with the smooth and uniformly convex second dual space 𝐸^∗∗ and let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸^∗∗. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℛ satisfying (A1)–(A4). Let 𝑆∶𝐶→𝐶 be an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞) such that Ω∶=𝐹(𝑆)∩EP(𝑓)≠∅. Let {𝑥_𝑛} be a sequence in 𝐶 generated by 𝑥𝐶₀∈𝐶,𝐷₀=𝐶₀=𝐶,_𝑛𝑢=co𝑧∈𝐶_𝑛−1∶‖𝑧−𝑆^𝑛𝑧‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆^𝑛𝑥_𝑛‖‖,𝑛≥1,_𝑛𝑓∈𝑢𝐶suchthat_𝑛,𝑦+1𝑟_𝑛𝑦−𝑢_𝑛𝐷,𝐽𝑢_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶,𝑛≥0,_𝑛𝑥=𝑧∈𝐷_𝑛−1∶𝑢_𝑛−𝑧,𝐽𝑥_𝑛−𝑢_𝑛≥0,𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥₀,𝑛≥0,(4.26) where {𝑡_𝑛} and {𝑟_𝑛} are real sequences in (0,1) such that lim_𝑛→∞𝑡_𝑛=0, and liminf_𝑛→∞𝑟_𝑛>0. Then, {𝑥_𝑛} converges strongly, as 𝑛→∞, to 𝑃_Ω𝑥₀.

Putting 𝑓≡0,𝐴≡0,𝜑≡0, and 𝑟_𝑛≡1 in Theorem 4.1 and applying Theorem 2.1, we get 𝑥_𝑛=𝑢_𝑛. Then, we have the following new approximation method concerning the problem of finding a fixed of an asymptotically nonexpansive mapping in a Banach space.

Theorem 4.4. 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 𝑆∶𝐶→𝐶 be an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞) such that 𝐹(𝑆)≠∅. Let {𝑥_𝑛} be a sequence in 𝐶 generated by 𝐶𝑥₀∈𝐶,𝐶₀=𝐶,_𝑛=co𝑧∈𝐶_𝑛−1∶𝑥‖𝑧−𝑆^𝑛𝑧‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆^𝑛𝑥_𝑛‖‖,𝑛≥1,_𝑛+1=𝑃_𝐶𝑛𝑥₀,𝑛≥0,(4.27) where {𝑡_𝑛} and {𝑟_𝑛} is a real sequence in (0,1) such that lim_𝑛→∞𝑡_𝑛=0. Then {𝑥_𝑛} converges strongly, as 𝑛→∞, to 𝑃_𝐹(𝑆)𝑥₀.

If 𝐸 is reflexive (i.e., 𝐸=𝐸^∗∗) smooth and uniformly convex, then the following results can be derived as a corollary of Theorem 4.4.

Corollary 4.5. Let 𝐸 be a reflexive smooth and uniformly convex real Banach space, let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸. Let 𝑆∶𝐶→𝐶 be an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞] such that 𝐹(𝑆)≠∅. Let {𝑥_𝑛} be a sequence in 𝐶 generated by 𝐶𝑥₀∈𝐶,𝐶₀=𝐶,_𝑛=co𝑧∈𝐶_𝑛−1∶𝑥‖𝑧−𝑆^𝑛𝑧‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆^𝑛𝑥_𝑛‖‖,𝑛≥1,_𝑛+1=𝑃_𝐶𝑛𝑥₀,𝑛≥0,(4.28) where {𝑡_𝑛} and {𝑟_𝑛} is a real sequence in (0,1) such that lim_𝑛→∞𝑡_𝑛=0. Then, {𝑥_𝑛} converges strongly, as 𝑛→∞, to 𝑃_𝐹(𝑆)𝑥₀.

Acknowledgment

The first author is supported by the “Centre of Excellence in Mathematics” under the Commission on Higher Education, Ministry of Education, Thailand.