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

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 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 will still denote by 𝐽 the single-valued duality mapping.

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. Denote by 𝐹(𝑆) the set of fixed points of 𝑆, that is, 𝐹(𝑆)={𝑥∈𝐶∶𝑆𝑥=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 𝐵_𝐻 be the closed unit ball in the Hilbert space 𝐻=𝑙₂ and 𝑆∶𝐵_𝐻→𝐵_𝐻 a mapping defined by 𝑆𝑥₁,𝑥₂,𝑥₃,…=0,𝑥²¹,𝑎₂𝑥₂,𝑎₃𝑥₃,…,(1.2) where {𝑎_𝑛} is a sequence of real numbers such that 0<𝑎_𝑖<1 and ∏^∞𝑖=2𝑎_𝑖=1/2. Then ‖𝑆𝑥−𝑆𝑦‖≤2‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐵_𝐻.(1.3) That is, 𝑆 is Lipschitzian but not nonexpansive. Observe that ‖𝑆^𝑛𝑥−𝑆^𝑛𝑦‖≤2^𝑛_𝑖=2𝑎_𝑖‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐵_𝐻,𝑛≥2.(1.4) Here 𝑘_𝑛=2∏^𝑛𝑖=2𝑎_𝑖→1 as 𝑛→∞. Therefore, 𝑆 is asymptotically nonexpansive but not nonexpansive.

A mapping 𝑇∶𝐶→𝐸^∗ is said to be relaxed 𝜂-𝜉 monotone if there exist a mapping 𝜂∶𝐶×𝐶→𝐸 and a function 𝜉∶𝐸→ℝ positively homogeneous of degree 𝑝, that is, 𝜉(𝑡𝑧)=𝑡^𝑝𝜉(𝑧) for all 𝑡>0 and 𝑧∈𝐸 such that ⟨𝑇𝑥−𝑇𝑦,𝜂(𝑥,𝑦)⟩≥𝜉(𝑥−𝑦),∀𝑥,𝑦∈𝐶,(1.5) where 𝑝>1 is a constant; see [3]. In the case of 𝜂(𝑥,𝑦)=𝑥−𝑦 for all 𝑥, 𝑦∈𝐶, 𝑇 is said to be relaxed 𝜉-monotone. In the case of 𝜂(𝑥,𝑦)=𝑥−𝑦 for all 𝑥,𝑦∈𝐶 and 𝜉(𝑧)=𝑘‖𝑧‖^𝑝, where 𝑝>1 and 𝑘>0, 𝑇 is said to be 𝑝-monotone; see [4–6]. 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)=𝑥−𝑦 for all 𝑥,𝑦∈𝐶 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(𝑐𝑥(𝑥−−𝑦)𝑦,),𝑥𝑥≥<𝑦𝑦,,(1.6) where 𝑐>0 is a constant. Then, 𝑇 is relaxed 𝜂-𝜉 monotone with 𝜉(𝑧)=𝑐𝑧²−𝑐𝑧,₂𝑧≥0,,𝑧<0.(1.7) A mapping 𝑇∶𝐶→𝐸^∗ is said to be 𝜂-hemicontinuous if, for each 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 a real Banach space 𝐸 with the dual 𝐸^∗ and for 𝐶 a nonempty closed convex subset of 𝐸, let 𝑓∶𝐶×𝐶→ℝ be a bifunction, 𝜑∶𝐶→ℝ a real-valued function and 𝑇∶𝐶→𝐸^∗ be a relaxed 𝜂-𝜉 monotone mapping. Recently, Kamraksa and Wangkeeree [7] introduced the following generalized mixed equilibrium problem (GMEP). Find𝑥∈𝐶suchthat𝑓(𝑥,𝑦)+⟨𝑇𝑥,𝜂(𝑦,𝑥)⟩+𝜑(𝑦)≥𝜑(𝑥),∀𝑦∈𝐶.(1.8) The set of such 𝑥∈𝐶 is denoted by GMEP(𝑓,𝑇), that is, GMEP(𝑓,𝑇)={𝑥∈𝐶∶𝑓(𝑥,𝑦)+⟨𝑇𝑥,𝜂(𝑦,𝑥)⟩+𝜑(𝑦)≥𝜑(𝑥),∀𝑦∈𝐶}.(1.9)

Special Cases
(1) If 𝑇 is monotone that is 𝑇 is relaxed 𝜂-𝜉 monotone with 𝜂(𝑥,𝑦)=𝑥−𝑦 for all 𝑥,𝑦∈𝐶 and 𝜉=0, (1.8) is reduced to the following generalized equilibrium problem (GEP). Find𝑥∈𝐶suchthat𝑓(𝑥,𝑦)+⟨𝑇𝑥,𝑦−𝑥⟩+𝜑(𝑦)≥𝜑(𝑥),∀𝑦∈𝐶.(1.10) The solution set of (1.10) is denoted by GEP(𝑓), that is, GEP(𝑓)={𝑥∈𝐶∶𝑓(𝑥,𝑦)+⟨𝑇𝑥,𝑦−𝑥⟩+𝜑(𝑦)≥𝜑(𝑥),∀𝑦∈𝐶}.(1.11)
(2) In the case of 𝑇≡0 and 𝜑≡0, (1.8) is reduced to the following classical equilibrium problem Find𝑥∈𝐶suchthat𝑓(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.12) The set of all solution of (1.12) is denoted by EP(𝑓), that is, EP(𝑓)={𝑥∈𝐶∶𝑓(𝑥,𝑦)≥0,∀𝑦∈𝐶}.(1.13)
(3) In the case of 𝑓≡0, (1.8) is reduced to the following variational-like inequality problem [3]. Find𝑥∈𝐶suchthat⟨𝑇𝑥,𝜂(𝑦,𝑥)⟩+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶.(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, [12–17]. 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: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑥_𝑛,∀𝑛∈ℕ,(1.15) where the initial point 𝑥₀ is taken in 𝐶 arbitrarily and {𝛼_𝑛} is a sequence in [0,1]. However, we note that Manns iteration process (1.15) has only weak convergence, in general; for instance, see [19–21]. In 2003, Nakajo and Takahashi [18] proposed the following sequence for a nonexpansive mapping 𝑆 in a Hilbert space: 𝑦𝑥₀=𝑥∈𝐶,_𝑛𝐶_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑥_𝑛,𝑄=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖,=𝑧∈𝑥𝐶∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,(1.16) where 0≤𝛼_𝑛≤𝑎<1 for all 𝑛∈ℕ, and 𝑃_{𝐶𝑛∩𝐷𝑛} is the metric projection from 𝐸 onto 𝐶_𝑛∩𝐷_𝑛. Then, they proved that {𝑥_𝑛} converges strongly to 𝑃_𝐹(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=𝑥∈𝐶 and 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,(1.17) where co𝐷 denotes the convex closure of the set 𝐷, {𝑡_𝑛} is a sequence in (0, 1) with 𝑡_𝑛→0. They proved that {𝑥_𝑛} 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=𝑥∈𝐶, 𝐶₀=𝐷₀=𝐶 and 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,(1.18) where {𝑡_𝑛} is a sequence in (0,1) with 𝑡_𝑛→0 as 𝑛→∞ and 𝑆 is an asymptotically nonexpansive mapping. It is proved in [23] that {𝑥_𝑛} 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,…,𝑆_𝑁} in a uniformly convex and smooth Banach space: 𝑥₀∈𝐶, 𝐷₀=𝐶₀=𝐶, and 𝐶₁𝑥₁=𝑃_{𝐶0∩𝐷0}𝑥₀=𝑃_𝐶𝑥₀,𝑢₁=co𝑧∈𝐶∶‖‖𝑧−𝑆₁𝑧‖‖≤𝑡₁‖‖𝑥₁−𝑆₁𝑥₁‖‖,∈𝐶suchthat𝑓𝑢₁,𝑦+𝜑(𝑦)+𝑇𝑢₁,𝜂𝑦,𝑢₁+1𝑟₁𝑦−𝑢₁𝐷,𝐽𝑢₁−𝑥₁,∀𝑦∈𝐶,₁=𝑧∈𝐶∶𝑥𝑢₁−𝑧,𝐽𝑥₁−𝑢₁≥0₂,𝐶_𝑁=𝑃⋮_{𝐶1∩𝐷1}𝑥₀,𝑢_𝑁=co𝑧∈𝐶_𝑁−1∶‖‖𝑧−𝑆_𝑁𝑧‖‖≤𝑡₁‖‖𝑥_𝑁−𝑆_𝑁𝑥_𝑁‖‖,∈𝐶suchthat𝑓𝑢_𝑁,𝑦+𝜑(𝑦)+𝑇𝑢_𝑁,𝜂𝑦,𝑢_𝑁+1𝑟_𝑁𝑦−𝑢_𝑁𝐷,𝐽𝑢_𝑁−𝑥_𝑁,∀𝑦∈𝐶,_𝑁=𝑧∈𝐷𝑥_𝑁−1∶𝑢_𝑁−𝑧,𝐽𝑥_𝑁−𝑢_𝑁≥0_𝑁+1,𝐶_𝑁+1=𝑃_{𝐶𝑁∩𝐷𝑁}𝑥₀,𝑢_𝑁+1=co𝑧∈𝐶_𝑁∶‖‖𝑧−𝑆²¹𝑧‖‖≤𝑡₁‖‖𝑥_𝑁+1−𝑆²¹𝑥_𝑁+1‖‖,∈𝐶suchthat𝑓+𝑢1_𝑁+1,𝑦+𝜑(𝑦)+𝑇𝑢_𝑁+1,𝜂𝑦,𝑢_𝑁+1𝑟_𝑁+1𝑦−𝑢_𝑁+1𝐷,𝐽𝑢_𝑁+1−𝑥_𝑁+1,∀𝑦∈𝐶,_𝑁+1=𝑧∈𝐷_𝑁𝑥∶𝑢_𝑁+1−𝑧,𝐽𝑥_𝑁+1−𝑢_𝑁+1≥0_𝑁+2,𝐶_2𝑁=𝑃⋮_{𝐶𝑁+1∩𝐷𝑁+1}𝑥₀,𝑢_2𝑁=co𝑧∈𝐶_2𝑁−1∶‖‖𝑧−𝑆^2𝑁𝑧‖‖≤𝑡₁‖‖𝑥_2𝑁−𝑆^2𝑁𝑥_2𝑁‖‖,∈𝐶suchthat𝑓+𝑢1_2𝑁,𝑦+𝜑(𝑦)+𝑇𝑢_2𝑁,𝜂𝑦,𝑢_2𝑁𝑟_2𝑁𝑦−𝑢_2𝑁𝐷,𝐽𝑢_2𝑁−𝑥_2𝑁,∀𝑦∈𝐶,_2𝑁=𝑧∈𝐷𝑥_2𝑁−1∶𝑢_2𝑁−𝑧,𝐽𝑥_2𝑁−𝑢_2𝑁≥0_2𝑁+1,𝐶_2𝑁+1=𝑃_{𝐶2𝑁∩𝐷2𝑁}𝑥₀,𝑢_2𝑁+1=co𝑧∈𝐶_2𝑁∶‖‖𝑧−S³¹𝑧‖‖≤𝑡₁‖‖𝑥_2𝑁+1−𝑆³¹𝑥_2𝑁+1‖‖,∈𝐶suchthat𝑓+𝑢1_2𝑁+1,𝑦+𝜑(𝑦)+𝑇𝑢_2𝑁+1,𝜂𝑦,𝑢_2𝑁+1𝑟_2𝑁+1𝑦−𝑢_2𝑁+1𝐷,𝐽𝑢_2𝑁+1−𝑥_2𝑁+1,∀𝑦∈𝐶,_2𝑁+1=𝑧∈𝐷_2𝑁𝑥∶𝑢_2𝑁+1−𝑧,𝐽𝑥_2𝑁+1−𝑢_2𝑁+1≥0_2𝑁+2,=𝑃⋮_{𝐶2𝑁+1∩𝐷2𝑁+1}𝑥₀,(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 𝑛≥1, it can be written as 𝑛=(ℎ−1)𝑁+𝑖, where 𝑖=𝑖(𝑛)∈{1,2,…,𝑁}, ℎ=ℎ(𝑛)≥1 is a positive integer and ℎ(𝑛)→∞ as 𝑛→∞. Hence the above table can be written in the following form: 𝐶𝑥₀∈𝐶,𝐷₀=𝐶₀=𝐶,_𝑛=𝑢co𝑧∈𝐶_𝑛−1∶‖‖𝑧−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑧‖‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛‖‖,𝑛≥1,_𝑛∈𝐶suchtha+t𝑓1𝑢_𝑛,𝑦+𝜑(𝑦)+𝑇𝑢_𝑛,𝜂𝑦,𝑢_𝑛𝑟_𝑛𝑦−𝑢_𝑛𝐷,𝐽𝑢_𝑛−𝑥_𝑛,∀𝑦∈𝐶,𝑛≥1,_𝑛=𝑧∈𝐷𝑥_𝑛−1∶𝑢_𝑛−𝑧,𝐽𝑥_𝑛−𝑢_𝑛≥0,𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥₀,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 Nakajo [24], Kamraksa and Wangkeeree [7], Dehghan [23], and many others.

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 limit lim_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. Let 𝑥∈𝐸 and 𝑢∈𝐶. The following theorem is well known.

Theorem 2.1. Let 𝐶 be a nonempty convex subset of a smooth Banach space 𝐸 and let 𝑥∈𝐸 and 𝑦∈𝐶. Then the following are equivalent: (a)𝑦is 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 well known 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 one will need the following lemmas.

Lemma 2.2 (see [25]). 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.

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 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‖‖(2.7) for any asymptotically nonexpansive mapping 𝑆 of 𝐶 into 𝐶 with {𝑘_𝑛}, any elements 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝐶, any numbers 𝜆₁,𝜆₂,…,𝜆_𝑛≥0 with ∑^𝑛𝑖=1𝜆_𝑖=1 and each 𝑚≥1.

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 𝑆∶𝐶→𝐶 be an asymptotically nonexpansive mapping. Then (𝐼−𝑆) is demiclosed at 0, that is, if 𝑥_𝑛⇀𝑥 and (𝐼−𝑆)𝑥_𝑛→0, then 𝑥∈𝐹(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 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^∗) which implies that 𝜉(𝑥)≤liminf𝜉(𝑥_𝛽). Then 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.(2.8)

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 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)–(A4) and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℛ. Let 𝑟>0 and define a mapping Φ_𝑟∶𝐸→𝐶 as follows: Φ_𝑟(𝑥)=𝑧∈𝐶∶𝑓(𝑧,𝑦)+⟨𝑇𝑧,𝜂(𝑦,𝑧)⟩+𝜑(𝑦)+1𝑟⟨𝑦−𝑧,𝐽(𝑧−𝑥)⟩≥𝜑(𝑧),∀𝑦∈C(2.9) 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∈𝐶, the mapping 𝑥↦⟨𝑇𝑣,𝜂(𝑥,𝑢)⟩ is convex and lower semicontinuous and the mapping 𝑥↦⟨𝑇𝑢,𝜂(𝑣,𝑥)⟩ is lower semicontinuous; (iii)𝜉∶𝐸→ℛ is weakly lower semicontinuous; (iv)for any 𝑥,𝑦∈𝐶, 𝜉(𝑥−𝑦)+𝜉(𝑦−𝑥)≥0. 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)=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 𝑇∶𝐶→𝐸^∗ be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying the following conditions (A1)–(A4): (A1)𝑓(𝑥,𝑥)=0 for all 𝑥∈𝐶; (A2)𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤min{𝜉(𝑥−𝑦),𝜉(𝑦−𝑥)} 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. For any 𝑟>0 and 𝑥∈𝐸, define a mapping Φ_𝑟∶𝐸→𝐶 as follows: Φ_𝑟(𝑥)=𝑧∈𝐶∶𝑓(𝑧,𝑦)+⟨𝑇𝑧,𝜂(𝑦,𝑧)⟩+𝜑(𝑦)+1𝑟⟨𝑦−𝑧,𝐽(𝑧−𝑥)⟩≥𝜑(𝑧),∀𝑦∈𝐶,(3.1) where 𝜑 is a lower semicontinuous and convex function from 𝐶 to ℝ. 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 and the mapping 𝑥↦⟨𝑇𝑢,𝜂(𝑣,𝑥)⟩ is lower semicontinuous; (iii)𝜉∶𝐸→ℝ is weakly lower semicontinuous. 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,𝑇); (4)GMEP(𝑓,𝑇) is nonempty closed and convex.

Proof. For each 𝑥∈𝐸. It follows from Lemma 2.5 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_𝑟𝑥. Then 𝑓𝑧₁,𝑧₂+𝑇𝑧₂,𝜂𝑧₂,𝑧₁+𝜑𝑧₂+1𝑟𝑧¹𝑓𝑧₂−𝑧₂,𝐽𝑧₁−𝑥≥𝜑𝑧₁,,𝑧₁+𝑇𝑧₁,𝜂𝑧₁,𝑧₂+𝜑𝑧₁+1𝑟𝑧²−𝑧₁,𝐽𝑧₂−𝑥≥𝜑𝑧₂.(3.2) Adding the two inequalities, from (i) we have 𝑓𝑧₂,𝑧₁+𝑓𝑧₁,𝑧₂+⟨𝑇𝑧₁−𝑇𝑧₂,𝜂𝑧₂,𝑧₁⟩+1𝑟𝑧²−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥≥0.(3.3) Setting Δ∶=min{𝜉(𝑧₁−𝑧₂),𝜉(𝑧₂−𝑧₁)} and using (A2), we have Δ+𝑇𝑧₁−𝑇𝑧₂,𝜂𝑧₂,𝑧₁+1𝑟𝑧²−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥≥0,(3.4) that is, 1𝑟𝑧²−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥≥𝑇𝑧₂−𝑇𝑧₁,𝜂𝑧₂,𝑧₁−Δ.(3.5) Since 𝑇 is relaxed 𝜂-𝜉 monotone and 𝑟>0, one has 𝑧₂−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥≥𝑟𝜉𝑧₂−𝑧₁−Δ≥0.(3.6) In (3.5) exchanging the position of 𝑧₁ and 𝑧₂, we get 1𝑟𝑧¹−𝑧₂,𝐽𝑧₂−𝑥−𝐽𝑧₁−𝑥≥𝑇𝑧₁−𝑇𝑧₂,𝜂𝑧₁,𝑧₂−Δ,(3.7) that is, 𝑧₁−𝑧₂,𝐽𝑧₂−𝑥−𝐽𝑧₁−𝑥≥𝑟𝜉𝑧₁−𝑧₂−Δ≥0.(3.8) Now, adding the inequalities (3.6) and (3.8), we have 2𝑧₂−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥≥0.(3.9) Hence, 0≤𝑧₂−𝑧₁,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥=𝑧₂−𝑥−𝑧₁−𝑥,𝐽𝑧₁−𝑥−𝐽𝑧₂−𝑥.(3.10) Since 𝐽 is monotone and 𝐸 is strictly convex, we obtain that 𝑧₁−𝑥=𝑧₂−𝑥 and hence 𝑧₁=𝑧₂. Therefore 𝑆_𝑟 is single valued.
(2) For 𝑥,𝑦∈𝐶, we have 𝑓Φ_𝑟𝑥,Φ_𝑟𝑦+𝑇Φ_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+𝑇Φ_𝑟𝑦,𝜂Φ_𝑟𝑥,Φ_𝑟𝑦+𝜑Φ_𝑟𝑥−𝜑Φ_𝑟𝑦+1𝑟Φ^𝑟𝑥−Φ_𝑟𝑦,𝐽Φ_𝑟𝑦−𝑦≥0.(3.11) Setting Λ_𝑥,𝑦∶=min{𝜉(Φ_𝑟𝑥−Φ_𝑟𝑦),𝜉(Φ_𝑟𝑦−Φ_𝑟𝑥)} and applying (A2), we get 𝑇Φ_𝑟𝑥−𝑇Φ_𝑟𝑦,𝜂Φ_𝑟𝑦,Φ_𝑟𝑥+1𝑟Φ^𝑟𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥−𝐽Φ_𝑟𝑦−𝑦≥−Λ_𝑥,𝑦,(3.12) that is, 1𝑟Φ^𝑟𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥−𝐽Φ_𝑟𝑦−𝑦≥𝑇Φ_𝑟𝑦−𝑇Φ_𝑟𝑥,𝜂Φ_𝑟𝑦,Φ_𝑟𝑥−Λ_𝑥,𝑦≥𝜉Φ_𝑟𝑦−Φ_𝑟𝑥−Λ_𝑥,𝑦≥0.(3.13) In (3.13) exchanging the position of Φ_𝑟𝑥 and Φ_𝑟𝑦, we get 1𝑟Φ^𝑟𝑥−Φ_𝑟𝑦,𝐽Φ_𝑟𝑦−𝑦−𝐽Φ_𝑟𝑥−𝑥≥0.(3.14) Adding the inequalities (3.13) and (3.14), we have 2𝑟Φ^𝑟𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥−𝐽Φ_𝑟𝑦−𝑦≥0.(3.15) It follows that Φ_𝑟𝑦−Φ_𝑟𝑥,𝐽Φ_𝑟𝑥−𝑥−𝐽Φ_𝑟𝑦−𝑦≥0.(3.16) Hence Φ_𝑟𝑥−Φ_𝑟𝑦,𝐽Φ_𝑟𝑥−𝑥≤Φ_𝑟𝑥−Φ_𝑟𝑦,𝐽Φ_𝑟𝑦−𝑦.(3.17) The conclusions (3), (4) follow from Lemma 2.6.

Example 3.2. Define 𝜉∶ℝ→ℝ and 𝑓∶ℝ×ℝ→ℝ by 𝑓(𝑥,𝑦)=(𝑥−𝑦)²2,𝜉(𝑥)=𝑥²∀𝑥,𝑦∈ℝ.(3.18) It is easy to see that 𝑓 satisfies (A1), (A3), (A4), and (A2): 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤min{𝜉(𝑥−𝑦),𝜉(𝑥−𝑦)}, for  all (𝑥,𝑦)∈ℝ×ℝ.

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(𝑥−𝑦),𝜉(𝑦−𝑥)} for all 𝑥,𝑦∈𝐶.(2)The control condition 𝜉(𝑥−𝑦)+𝜉(𝑦−𝑥)≥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)=𝑥−𝑦 for all 𝑥,𝑦∈𝐶 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 𝑇∶𝐶→𝐸^∗ be a monotone mapping and 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying the following conditions (i)–(iv): (i)𝑓(𝑥,𝑥)=0 for all 𝑥∈𝐶; (ii)𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶; (iii)for all 𝑦∈𝐶, 𝑓(⋅,𝑦) is weakly upper semicontinuous; (iv)for all 𝑥∈𝐶, 𝑓(𝑥,⋅) is convex. For any 𝑟>0 and 𝑥∈𝐸, define a mapping Φ_𝑟∶𝐸→𝐶 as follows: Φ_𝑟(𝑥)=𝑧∈𝐶∶𝑓(𝑧,𝑦)+⟨𝑇𝑧,𝑦−𝑧⟩+𝜑(𝑦)+1𝑟⟨𝑦−𝑧,𝐽(𝑧−𝑥)⟩≥𝜑(𝑧),∀𝑦∈𝐶,(3.19) where 𝜑 is a lower semicontinuous and convex function from 𝐶 to ℝ. 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)=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 {𝑥_𝑛} 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 ℝ satisfying (A1)–(A4). Let 𝑇∶𝐶→𝐸^∗ be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping and 𝜑 a lower semicontinuous and convex function from 𝐶 to ℝ. Let, for each 1≤𝑖≤𝑁, 𝑆_𝑖∶𝐶→𝐶 be an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛,𝑖}^∞𝑛=1, respectively, such that 𝑘_𝑛,𝑖→1 as 𝑛→∞. Assume that Ω∶=⋂^𝑁𝑖=1𝐹(𝑆_𝑖)∩GMEP(𝑓,𝑇) is nonempty. Let {𝑥_𝑛} be a sequence generated by (1.20), where {𝑡_𝑛} and {𝑟_𝑛} are real sequences in (0,1) satisfying lim_𝑛→∞𝑡_𝑛=0 and liminf_𝑛→∞𝑟_𝑛>0. Then {𝑥_𝑛} converges strongly, as 𝑛→∞, to 𝑃_Ω𝑥₀, where 𝑃_Ω is the metric projection of 𝐸 onto Ω.

Proof. First, define the sequence {𝑘_𝑛} by 𝑘_𝑛∶=max{𝑘_𝑛,𝑖∶1≤𝑖≤𝑁} and so 𝑘_𝑛→1 as 𝑛→∞ and ‖‖𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑦‖‖≤𝑘_𝑛‖𝑥−𝑦‖∀𝑥,𝑦∈𝐶,(4.1) where ℎ(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: 𝐶𝑥₀∈𝐶,𝐷₀=𝐶₀=𝐶,_𝑛𝐷=co𝑧∈𝐶_𝑛−1∶‖‖𝑧−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑧‖‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛‖‖,𝑛≥0,_𝑛=𝑧∈𝐷_𝑛−1∶𝑥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 (3.19). We 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. Indeed, since 𝐷₀=𝐶, we also have Ω⊂𝐶₀∩𝐷₀. Assume that Ω⊂𝐶_𝑘−1∩𝐷_𝑘−1 for 𝑘≥2. Utilizing Theorem 3.1 (2), we obtain Φ_𝑟𝑘𝑥_𝑘−Φ_𝑟𝑘𝑢,𝐽Φ_𝑟𝑘𝑢−𝑢−𝐽Φ_𝑟𝑘𝑥_𝑘−𝑥_𝑘≥0,∀𝑢∈Ω,(4.3) which gives that Φ_𝑟𝑘𝑥_𝑘−𝑢,𝐽𝑥_𝑘−Φ_𝑟𝑘𝑥_𝑘≥0,∀𝑢∈Ω,(4.4) 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 well defined. Now, we show that lim_𝑛→∞‖‖𝑥_𝑛−𝑥_𝑛+𝑗‖‖=0,∀𝑗=1,2,…,𝑁.(4.5) 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) Hence the sequence {‖𝑥_𝑛−𝑥₀‖} is bounded and monotone increasing and hence there exists a constant 𝑑 such that lim_𝑛→∞‖‖𝑥_𝑛−𝑥₀‖‖=𝑑.(4.8) 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‖‖.(4.9) 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.(4.10) By Lemma 2.2, we have lim_𝑛→∞‖‖𝑥_𝑛−𝑥_𝑛+1‖‖=0.(4.11) Furthermore, we can easily see that lim_𝑛→∞‖‖𝑥_𝑛−𝑥_𝑛+𝑗‖‖=0,∀𝑗=1,2,…,𝑁.(4.12) Next, we show that lim_𝑛→∞‖‖𝑥_𝑛−𝑆^{ℎ(𝑛−𝜅)𝑖(𝑛−𝜅)}𝑥_𝑛‖‖=0,forany𝜅∈{1,2,…,𝑁}.(4.13) Fix 𝜅∈{1,2,…,𝑁} and put 𝑚=𝑛−𝜅. Since 𝑥_𝑛=𝑃_{𝐶𝑛−1∩𝐷𝑛−1}𝑥, we have 𝑥_𝑛∈𝐶_𝑛−1⊆⋯⊆𝐶_𝑚. Since 𝑡_𝑚>0, there exists 𝑦₁,…,𝑦_𝑃∈𝐶 and a nonnegative number 𝜆₁,…,𝜆_𝑃 with 𝜆₁+⋯+𝜆_𝑃=1 such that ‖‖‖‖𝑥_𝑛−^𝑃_𝑖=1𝜆_𝑖𝑦_𝑖‖‖‖‖<𝑡_𝑚‖‖𝑦,(4.14)_𝑖−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖‖‖≤𝑡_𝑚‖‖𝑥_𝑚−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑚‖‖,∀𝑖∈{1,…,𝑃}.(4.15) By the boundedness of 𝐶 and {𝑘_𝑛}, we can put the following: 𝑀=sup_𝑥∈𝐶‖𝑥‖,𝑢=𝑃_⋂^𝑁𝑖=1_{𝐹(𝑆𝑖)}𝑥₀,𝑟₀=sup_𝑛≥11+𝑘_𝑛‖‖𝑥_𝑛−𝑢‖‖.(4.16) This together with (4.14) implies that ‖‖‖‖𝑥_𝑛−1𝑘_{𝑚𝑃𝑖=1}𝜆_𝑖𝑦_𝑖‖‖‖‖≤1−1𝑘_𝑚‖𝑥‖+𝑘1_𝑚‖‖‖‖𝑥_𝑛−^𝑃_𝑖=1𝜆_𝑖𝑦_𝑖‖‖‖‖≤1−1𝑘_𝑚𝑀+𝑡^𝑚‖‖𝑦_𝑖,−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖‖‖≤≤𝑡𝑡_𝑚𝑚‖‖𝑥_𝑚−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑚‖‖≤𝑡_𝑚‖‖𝑥_𝑚−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑢‖‖+𝑡_𝑚‖‖𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑢−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑚‖‖≤𝑡_𝑚‖‖𝑥_𝑚−𝑢‖‖+𝑡_𝑚𝑘_𝑚‖‖𝑢−𝑥_𝑚‖‖≤𝑡_𝑚1+𝑘_𝑚‖‖𝑥_𝑚−𝑢‖‖𝑟₀,(4.17) for all 𝑖∈{1,…,𝑁}. Therefore, for each 𝑖∈{1,…,𝑃}, we get ‖‖‖𝑦_𝑖−1𝑘_𝑚𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖‖‖‖≤‖‖𝑦_𝑖−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖‖‖+‖‖‖𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖−1𝑘_𝑚𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖≤𝑟₀‖‖‖𝑡_𝑚+1−1𝑘_𝑚𝑀.(4.18) Moreover, since each 𝑆_𝑖, 𝑖∈{1,2,…,𝑁}, is asymptotically nonexpansive, we can obtain that ‖‖‖‖1𝑘_𝑚𝑆^{ℎ(𝑚)𝑖(𝑚)}^𝑃_𝑖=1𝜆_𝑖𝑦_𝑖−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑛‖‖‖‖≤‖‖‖‖1𝑘_𝑚𝑆^{ℎ(𝑚)𝑖(𝑚)}^𝑃_𝑖=1𝜆_𝑖𝑦_𝑖−𝑘1_𝑚𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑛+‖‖‖1‖‖‖‖𝑘_𝑚𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑛−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑛≤‖‖‖‖_𝑃‖‖‖_𝑖=1𝜆_𝑖𝑦_𝑖−𝑥_𝑛‖‖‖‖+1−1=𝑡_𝑚𝑘_𝑚𝑀+1−1𝑘_𝑚𝑀.(4.19) It follows from Theorem 2.3 and the inequalities (4.17)–(4.19) that ‖‖𝑥_𝑛−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑛‖‖≤‖‖‖‖𝑥_𝑛−1𝑘_{𝑚𝑃𝑖=1}𝜆_𝑖𝑦_𝑖‖‖‖‖+1𝑘_𝑚‖‖‖‖^𝑃_𝑖=1+1𝜆_𝑖𝑦^𝑖−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖‖‖‖‖𝑘_𝑚‖‖‖‖^𝑃_𝑖=1𝜆_𝑖𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖−𝑆^{ℎ(𝑚)𝑖(𝑚)}^𝑃_𝑖=1𝜆_𝑖𝑦_𝑖‖‖‖‖+‖‖‖‖𝑘1_𝑚𝑆^{ℎ(𝑚)𝑖(𝑚)}^𝑃_𝑖=1𝜆_𝑖𝑦_𝑖−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑛≤21−1‖‖‖‖𝑘_𝑚𝑀+𝑡^𝑚+𝑟⁰𝑡_𝑚𝑘_𝑚+𝛾⁻¹max_{1≤𝑖≤𝑗≤𝑁}‖‖𝑦^𝑖−𝑦_𝑗‖‖−1𝑘_𝑚=21−1‖‖𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑗‖‖𝑘_𝑚𝑀+2𝑡^𝑚+𝑟⁰𝑡_𝑚𝑘_𝑚+𝛾⁻¹max_{1≤𝑖≤𝑗≤𝑁}‖‖𝑦^𝑖−𝑦_𝑗‖‖−1𝑘_𝑚≤21−1‖‖𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑗‖‖𝑘_𝑚𝑀+2𝑡^𝑚+𝑟⁰𝑡_𝑚𝑘_𝑚+𝛾⁻¹max_{1≤𝑖≤𝑗≤𝑁}‖‖‖𝑦^𝑖−1𝑘_𝑚𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑖‖‖‖+‖‖‖𝑦_𝑗−1𝑘_𝑚𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑦_𝑗≤21−1‖‖‖𝑘_𝑚𝑀+2𝑡^𝑚+𝑟⁰𝑡_𝑚𝑘_𝑚+𝛾⁻¹21−𝑘1_𝑚𝑀+2𝑟⁰𝑡_𝑚.(4.20) Since lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝑡_𝑛=0, it follows from the above inequality that lim_𝑛→∞‖‖𝑥_𝑛−𝑆^{ℎ(𝑚)𝑖(𝑚)}𝑥_𝑛‖‖=0.(4.21) Hence (4.13) is proved. Next, we show that lim_𝑛→∞‖‖𝑥_𝑛−𝑆_𝑙𝑥_𝑛‖‖=0;∀𝑙=1,2,…,𝑁.(4.22) From the construction of 𝐶_𝑛, one can easily see that ‖‖𝑥_𝑛+1−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1‖‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛‖‖.(4.23) The boundedness of 𝐶 and lim_𝑛→∞𝑡_𝑛=0 implies that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1‖‖=0.(4.24) On the other hand, since for any positive integer 𝑛>𝑁, 𝑛=(𝑛−𝑁)(mod𝑁) and 𝑛=(ℎ(𝑛)−1)𝑁+𝑖(𝑛), we have 𝑛−𝑁=(ℎ(𝑛)−1)𝑁+𝑖(𝑛)=(ℎ(𝑛−𝑁)−1)𝑁+𝑖(𝑛−𝑁)(4.25) that is ℎ(𝑛−𝑁)=ℎ(𝑛)−1,𝑖(𝑛−𝑁)=𝑖(𝑛).(4.26) Thus, ‖‖𝑥_𝑛−𝑆_𝑛𝑥_𝑛‖‖≤≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1‖‖+‖‖𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1−S_𝑛𝑥_𝑛‖‖≤1+−𝑘𝑥_𝑛+11‖‖+‖‖𝑥_𝑛+1−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1‖‖+‖‖𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1−𝑆_𝑛𝑥_𝑛+1‖‖+‖‖𝑆_𝑛𝑥_𝑛+1−𝑆_𝑛𝑥_𝑛‖‖≤1+𝑘₁‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1‖‖+𝑘₁‖‖𝑆^{ℎ(𝑛)−1𝑖(𝑛)}𝑥_𝑛+1−𝑥_𝑛+1‖‖+𝑘₁‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1‖‖‖‖𝑆^{ℎ(𝑛)−1𝑖(𝑛)}𝑥_𝑛+1−𝑆^{ℎ(𝑛)−1𝑖(𝑛)}𝑥_𝑛≤1+2𝑘₁‖‖+‖‖𝑆^{ℎ(𝑛)−1𝑖(𝑛)}𝑥_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+𝑘₁‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1‖‖≤1+‖‖2𝑆𝑘^{ℎ(𝑛−𝑁)𝑖(𝑛−𝑁)}₁𝑥_𝑛+1−𝑆^{ℎ(𝑛−𝑁)𝑖(𝑛−𝑁)}𝑥_𝑛‖‖+𝑘₁‖‖𝑆^{ℎ(𝑛−𝑁)𝑖(𝑛−𝑁)}𝑥_𝑛−𝑥_𝑛‖‖+𝑘₁‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1‖‖≤1+𝑘2𝑘_𝑛−𝑁1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝑘₁‖‖𝑆^{ℎ(𝑛−𝑁)𝑖(𝑛−𝑁)}𝑥_𝑛−𝑥_𝑛‖‖+𝑘₁𝑘_𝑛−𝑁‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑆^{ℎ(𝑛)𝑖(𝑛)}𝑥_𝑛+1‖‖+𝑘₁‖‖𝑆^{ℎ(𝑛−𝑁)𝑖(𝑛−𝑁)}𝑥_𝑛−𝑥_𝑛‖‖.(4.27) Applying the facts (4.11), (4.13), and (4.24) to the above inequality, we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑆_𝑛𝑥_𝑛‖‖=0.(4.28) Therefore, for any 𝑗=1,2,…,𝑁, we have ‖‖𝑥_𝑛−𝑆_𝑛+𝑗𝑥_𝑛‖‖≤≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+𝑗‖‖+‖‖𝑥_𝑛+𝑗−𝑆_𝑛+𝑗𝑥_𝑛+𝑗‖‖+‖‖𝑆_𝑛+𝑗𝑥_𝑛+𝑗−𝑆_𝑛+𝑗𝑥_𝑛‖‖=1+−𝑘₁𝑥_𝑛+𝑗‖‖+‖‖𝑥_𝑛+𝑗−𝑆_𝑛+𝑗𝑥_𝑛+𝑗‖‖+𝑘₁‖‖𝑥_𝑛+𝑗−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑥_𝑛+𝑗‖‖+‖‖𝑥_𝑛+𝑗−𝑆_𝑛+𝑗𝑥_𝑛+𝑗‖‖⟶0as𝑛⟶∞,(4.29) which gives that lim_𝑛→∞‖‖𝑥_𝑛−𝑆_𝑙𝑥_𝑛‖‖=0;∀𝑙=1,2,…,𝑁,(4.30) as required. Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑥̃∈𝐶. It follows from Lemma 2.4 that 𝑥̃∈𝐹(𝑆_𝑙) for all 𝑙=1,2,…,𝑁. That is 𝑥∈⋂^𝑁𝑖=1𝐹(𝑆_𝑖).
Next, we show that 𝑥̃∈GMEP(𝑓,𝑇). 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.(4.31) Furthermore, since liminf_𝑛→∞𝑟_𝑛>0, we have 1𝑟_𝑛‖‖𝐽𝑥_𝑛−Φ_𝑟𝑛𝑥_𝑛‖‖=1𝑟_𝑛‖‖𝑥_𝑛−Φ_𝑟𝑛𝑥_𝑛‖‖⟶0,(4.32) as 𝑛→∞. By (4.32), we also have Φ_𝑟𝑛𝑖𝑥_𝑛𝑖⇀𝑥̃. By the definition of Φ_𝑟𝑛𝑖, for each 𝑦∈𝐶, we obtain 𝑓Φ_𝑟𝑛𝑖𝑥_𝑛𝑖,𝑦+𝑇Φ_𝑟𝑛𝑖𝑥_𝑛𝑖,𝜂𝑦,Φ_𝑟𝑛𝑖𝑥_𝑛𝑖+𝜑(𝑦)+1𝑟_𝑛𝑖𝑦−Φ^𝑟_𝑛𝑖≥𝜑Φ_𝑟𝑛𝑖𝑥_𝑛𝑖,𝐽Φ_𝑟𝑛𝑖𝑥_𝑛𝑖−𝑥_𝑛𝑖𝑥_𝑛𝑖.(4.33) By (A3), (4.32), (ii), the weakly lower semicontinuity of 𝜑 and 𝜂-hemicontinuity of 𝑇, we have 𝜑(𝑥̃)≤≤lliimmiinnf_𝑖→∞f_𝑖→∞𝜑Φ_𝑟𝑛𝑖𝑥_𝑛𝑖+𝜑(𝑦)+𝑓limΦin^𝑟_𝑛𝑖𝑥f^𝑛_{𝑖𝑖→∞},𝑦+liminf_𝑖→∞𝑇Φ^𝑟_𝑛𝑖𝑥_𝑛𝑖,𝜂𝑦,Φ_𝑟𝑛𝑖𝑥_𝑛𝑖1𝑟_𝑛𝑖𝑦−Φ^𝑟_𝑛𝑖≤𝑓(𝑥̃,𝑦)+𝜑(𝑦)+⟨𝑇𝑥̃,𝜂(𝑦,𝑥̃)⟩𝑥.^𝑛_𝑖,𝐽Φ_𝑟𝑛𝑖𝑥_𝑛𝑖−𝑥_𝑛𝑖(4.34) Hence, 𝑓(𝑥̃,𝑦)+𝜑(𝑦)+⟨𝑇𝑥̃,𝜂(𝑦,𝑥̃)⟩≥𝜑(𝑥̃).(4.35) This shows that 𝑥̃∈EP(𝑓,𝑇) and hence 𝑥̃∈Ω∶=⋂^𝑁𝑖=1𝐹(𝑆_𝑖)∩GMEP(𝑓,𝑇).
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‖‖.(4.36) This shows that lim_𝑖→∞‖‖𝑥₀−𝑥_𝑛𝑖‖‖=‖‖𝑥₀−𝑤‖‖=‖‖𝑥₀−𝑥̃‖‖(4.37) 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.

5. Corollaries

Setting 𝑆_𝑖≡𝑆, 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 ℝ satisfying (A1)–(A4). Let 𝑇∶𝐶→𝐸^∗ be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping and 𝜑 a lower semicontinuous and convex function from 𝐶 to ℝ. Let 𝑆 be an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}, such that 𝑘_𝑛→1 as 𝑛→∞. Assume that Ω∶=𝐹(𝑆)∩GMEP(𝑓,𝑇) is nonempty. Let {𝑥_𝑛} be a sequence 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,_𝑛∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑓𝑢_𝑛,𝑦+𝜑(𝑦)+𝑇𝑢_𝑛,𝜂𝑦,𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛𝐷,𝐽𝑢_𝑛−𝑥_𝑛,∀𝑦∈𝐶,𝑛≥1,_𝑛=𝑧∈𝐷𝑥_𝑛−1∶𝑢_𝑛−𝑧,𝐽𝑥_𝑛−𝑢_𝑛≥0,𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥₀,𝑛≥0,(5.1) where {𝑡_𝑛} and {𝑟_𝑛} are real sequences in (0,1) satisfying lim_𝑛→∞𝑡_𝑛=0 and liminf_𝑛→∞𝑟_𝑛>0. Then {𝑥_𝑛} converges strongly, as 𝑛→∞, to 𝑃_Ω𝑥₀, where 𝑃_Ω 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 ℝ satisfying (A1)–(A4). Let 𝑇∶𝐶→𝐸^∗ be an 𝜂-hemicontinuous and relaxed 𝜂-𝜉 monotone mapping and 𝜑 a lower semicontinuous and convex function from 𝐶 to ℝ. Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that Ω∶=𝐹(𝑆)∩GMEP(𝑓,𝑇)≠∅. Let {𝑥_𝑛} be the sequence in 𝐶 generated by 𝐶𝑥₀∈𝐶,𝐷₀=𝐶₀=𝐶,_𝑛𝑢=co𝑧∈𝐶_𝑛−1∶‖𝑧−𝑆𝑧‖≤𝑡_𝑛‖‖𝑥_𝑛−𝑆𝑥_𝑛‖‖,𝑛≥1,_𝑛∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑓𝑢_𝑛,𝑦+𝜑(𝑦)+𝑇𝑢_𝑛,𝜂𝑦,𝑢_𝑛+1𝑟𝑦−𝑢^𝑛𝐷∀𝑦∈𝐶,,𝑛𝐽≥𝑢0_𝑛,_𝑛−𝑥_𝑛≥𝜑𝑢_𝑛,=𝑧∈𝐷𝑥_𝑛−1∶𝑢_𝑛−𝑧,𝐽𝑥_𝑛−𝑢_𝑛≥0,𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥₀,𝑛≥0,(5.2) where {𝑡_𝑛} and {𝑟_𝑛} are real sequences in (0,1) satisfying lim_𝑛→∞𝑡_𝑛=0 and liminf_𝑛→∞𝑟_𝑛>0. Then, the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑥₀.
If one takes 𝑇≡0 and 𝜑≡0 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 ℝ satisfying (A1)–(A4) and let 𝑆 be an asymptotically nonexpansive mapping of C into itself such that Ω∶=⋂^∞𝑛=0𝐹(𝑆_𝑛)∩EP(𝑓)≠∅. Let {𝑥_𝑛} be the 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,_𝑛∈𝐶suchthat𝑓𝑢_𝑛,𝑦+1𝑟_𝑛𝑦−𝑢_𝑛𝐷,𝐽𝑢_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶,𝑛≥0,_𝑛=𝑧∈𝐷𝑥_𝑛−1∶𝑢_𝑛−𝑧,𝐽𝑥_𝑛−𝑢_𝑛≥0,𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝐷𝑛}𝑥₀,𝑛≥0,(5.3) where {𝑡_𝑛} and {𝑟_𝑛} are real sequences in (0,1) satisfying lim_𝑛→∞𝑡_𝑛=0 and liminf_𝑛→∞𝑟_𝑛>0. Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑥₀.

If one takes 𝑓≡0 and 𝑇≡0 and 𝜑≡0 in Theorem 4.1, then one obtains the following result.

Theorem 5.4. Let 𝐸 be a uniformly convex and smooth Banach space, 𝐶 a nonempty, bounded, closed, and convex subset of 𝐸 and 𝑆 an asymptotically nonexpansive mapping of 𝐶 into itself such that Ω∶=⋂^∞𝑛=0𝐹(𝑆_𝑛)≠∅. Let {𝑥_𝑛} be the 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.(5.4) If {𝑡_𝑛}⊂(0,1) and lim_𝑛→∞𝑡_𝑛=0, then {𝑥_𝑛} converges strongly to 𝑃_Ω𝑥₀.

Acknowledgments

The first author is supported by the “Centre of Excellence in Mathematics” under the Commission on Higher Education, Ministry of Education, Thailand, and the second author is supported by grant under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand. Finally, The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.