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Journal of Applied Mathematics
Volume 2012, Article ID 395760, 24 pages
http://dx.doi.org/10.1155/2012/395760
Research Article

The Modified Block Iterative Algorithms for Asymptotically Relatively Nonexpansive Mappings and the System of Generalized Mixed Equilibrium Problems

1Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand
2Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

Received 3 March 2012; Accepted 17 June 2012

Academic Editor: Hong-Kun Xu

Copyright © 2012 Kriengsak Wattanawitoon and Poom Kumam. 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

The propose of this paper is to present a modified block iterative algorithm for finding a common element between the set of solutions of the fixed points of two countable families of asymptotically relatively nonexpansive mappings and the set of solution of the system of generalized mixed equilibrium problems in a uniformly smooth and uniformly convex Banach space. Our results extend many known recent results in the literature.

1. Introduction

The equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, networks, elasticity, and optimization, and it has been extended and generalized in many directions.

In the theory of equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. This theory combines theoretical and algorithmic advances with novel domain of applications. Analysis of these problems requires a blend of techniques from convex analysis, functional analysis, and numerical analysis.

Let 𝐸 be a Banach space with norm , 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝐸 denote the dual of 𝐸. Let 𝑓𝑖𝐶×𝐶 be a bifunction, 𝜓𝑖𝐶 be a real-valued function, where is denoted by the set of real numbers, and 𝐴𝑖𝐶𝐸 be a nonlinear mapping. The goal of the system of generalized mixed equilibrium problem is to find 𝑢𝐶 such that 𝑓1(𝑢,𝑦)+𝐴1𝑢,𝑦𝑢+𝜓1(𝑦)𝜓1𝑓(𝑢)0,𝑦𝐶,2(𝑢,𝑦)+𝐴2𝑢,𝑦𝑢+𝜓2(𝑦)𝜓2𝑓(𝑢)0,𝑦𝐶,𝑁(𝑢,𝑦)+𝐴𝑁𝑢,𝑦𝑢+𝜓𝑁(𝑦)𝜓𝑁(𝑢)0,𝑦𝐶.(1.1) If 𝑓𝑖=𝑓, 𝐴𝑖=𝐴, and 𝜓𝑖=𝜓, the problem (1.1) is reduced to the generalized mixed equilibrium problem, denoted by GEMP(𝑓,𝐴,𝜓), to find 𝑢𝐶 such that 𝑓(𝑢,𝑦)+𝐴𝑢,𝑦𝑢+𝜓(𝑦)𝜓(𝑢)0,𝑦𝐶.(1.2) The set of solutions to (1.2) is denoted by Ω, that is, Ω={𝑥𝐶𝑓(𝑢,𝑦)+𝐴𝑢,𝑦𝑢+𝜑(𝑦)𝜑(𝑢)0,𝑦𝐶}.(1.3) If 𝐴=0, the problem (1.2) is reduced to the mixed equilibrium problem for 𝑓, denoted by MEP(𝑓,𝜓), to find 𝑢𝐶 such that 𝑓(𝑢,𝑦)+𝜓(𝑦)𝜓(𝑢)0,𝑦𝐶.(1.4) If 𝑓0, the problem (1.2) is reduced to the mixed variational inequality of Browder type, denoted by VI(𝐶,𝐴,𝜓), is to find 𝑢𝐶 such that 𝐴𝑢,𝑦𝑢+𝜓(𝑦)𝜓(𝑢)0,𝑦𝐶.(1.5) If 𝐴=0 and 𝜓=0, the problem (1.2) is reduced to the equilibrium problem for 𝑓, denoted by EP(𝑓), to find 𝑢𝐶 such that 𝑓(𝑢,𝑦)0,𝑦𝐶.(1.6)

The above formulation (1.6) was shown in [1] to cover monotone inclusion problems, saddle-point problems, variational inequality problems, minimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed-point problem, and optimization problem, which can also be written in the form of an EP(𝑓). In other words, the EP(𝑓) is a unifying model for several problems arising in physics, engineering, science, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions to EP(𝑓); see, for example [14] and references therein. Some solution methods have been proposed to solve the EP(𝑓); see, for example, [2, 415] and references therein. In 2005, Combettes and Hirstoaga [5] introduced an iterative scheme of finding the best approximation to the initial data when EP(𝑓) is nonempty, and they also proved a strong convergence theorem.

A Banach space 𝐸 is said to be strictly convex if (𝑥+𝑦)/2<1 for all 𝑥,𝑦𝐸 with 𝑥=𝑦=1 and 𝑥𝑦. Let 𝑈={𝑥𝐸𝑥=1} be the unit sphere of 𝐸. Then the Banach space 𝐸 is said to be smooth, provided lim𝑡0𝑥+𝑡𝑦𝑥𝑡(1.7) exists for each 𝑥,𝑦𝑈. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦𝐸. The modulus of convexity of 𝐸 is the function 𝛿[0,2][0,1] defined by 𝛿(𝜀)=inf1𝑥+𝑦2.𝑥,𝑦𝐸,𝑥=𝑦=1,𝑥𝑦𝜀(1.8) A Banach space 𝐸 is uniformly convex, if and only if 𝛿(𝜀)>0 for all 𝜀(0,2].

Let 𝐸 be a Banach space, 𝐶 be a closed convex subset of 𝐸, a mapping 𝑇𝐶𝐶 is said to be nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦(1.9) for all 𝑥,𝑦𝐶. We denote by 𝐹(𝑇) the set of fixed points of 𝑇. If 𝐶 is a bounded closed convex set and 𝑇 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑇) is nonempty (see [16]). A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [17] if 𝐶 contains a sequence {𝑥𝑛} which converges weakly to 𝑝 such that lim𝑛𝑥𝑛𝑇𝑥𝑛=0. The set of asymptotic fixed points of 𝑇 will be denoted by  𝐹(𝑇). A point 𝑝𝐶 is said to be a strong asymptotic fixed point of 𝑇, if there exists a sequence {𝑥𝑛}𝐶 such that 𝑥𝑛𝑝 and 𝑥𝑛𝑇𝑥𝑛0. The set of strong asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 from 𝐶 into itself is said to be relatively nonexpansive [1820] if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥) for all 𝑥𝐶 and 𝑝𝐹(𝑇). The asymptotic behavior of a relatively nonexpansive mapping was studied in [21, 22]. 𝑇 is said to be 𝜙-nonexpansive, if 𝜙(𝑇𝑥,𝑇𝑦)𝜙(𝑥,𝑦) for 𝑥,𝑦𝐶. 𝑇 is said to be quas-ϕ-nonexpansive if 𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥) for 𝑥𝐶 and 𝑝𝐹(𝑇). A mapping 𝑇 is said to be asymptotically relatively nonexpansive, if 𝐹(𝑇), and there exists a real sequence {𝑘𝑛}[1,) with 𝑘𝑛1 such that 𝜙(𝑝,𝑇𝑛𝑥)𝑘𝑛𝜙(𝑝,𝑥), forall𝑛1,𝑥𝐶, and 𝑝𝐹(𝑇). {𝑇𝑛}𝑛=0 is said to be a countable family of weak relatively nonexpansive mappings [23] if the following conditions are satisfied: (i)𝐹({𝑇𝑛}𝑛=0); (ii)𝜙(𝑢,𝑇𝑛𝑥)𝜙(𝑢,𝑥),forall𝑢𝐹(𝑇𝑛),𝑥𝐶,𝑛0; (iii)𝐹({𝑇𝑛}𝑛=0)=𝑛=0𝐹(𝑇𝑛). A mapping 𝑇𝐶𝐶 is said to be uniformly L-Lipschitz continuous, if there exists a constant 𝐿>0 such that 𝑇𝑛𝑥𝑇𝑛𝑦𝐿𝑥𝑦,𝑥,𝑦𝐶,𝑛1.(1.10) A mapping 𝑇𝐶𝐶 is said to be closed if for any sequence {𝑥𝑛}𝐶 with 𝑥𝑛𝑥 and 𝑇𝑥𝑛𝑦, then 𝑇𝑥=𝑦. Let {𝑇𝑖}𝑖=1𝐶𝐶 be a sequence of mappings. {𝑇𝑖}𝑖=1 is said to be a countable family of uniformly asymptotically relatively nonexpansive mappings, if 𝑛=1𝐹(𝑇𝑛), and there exists a sequence {𝑘𝑛}[1,) with 𝑘𝑛1 such that for each 𝑖>1𝜙𝑝,𝑇𝑛𝑖𝑥𝑘𝑛𝜙(𝑝,𝑥),𝑝𝑛=1𝐹𝑇𝑛,𝑥𝐶,𝑛1.(1.11)

In 2009, Petrot et al. [24], introduced a hybrid projection method for approximating a common element of the set of solutions of fixed points of hemirelatively nonexpansive (or quasi-𝜙-nonexpansive) mappings in a uniformly convex and uniformly smooth Banach space:𝑥0𝐶,𝐶0𝑦=𝐶,𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑛𝑧𝑛,𝑧𝑛=𝐽1𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑇𝑛𝑥𝑛,𝐶𝑛+1=𝑣𝐶𝑛𝜙𝑣,𝑦𝑛𝜙𝑣,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0.(1.12) They proved that the sequence {𝑥𝑛} converges strongly to 𝑝𝐹(𝑇), where 𝑝Π𝐹(𝑇)𝑥 and Π𝐶 is the generalized projection from 𝐸 onto 𝐹(𝑇). Kumam and Wattanawitoon [25], introduced a hybrid iterative scheme for finding a common element of the set of common fixed points of two quasi-𝜙-nonexpansive mappings and the set of solutions of an equilibrium problem in Banach spaces, by the following manner: 𝑥0𝐶,𝐶0𝑦=𝐶𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑆𝑧𝑛,𝑧𝑛=𝐽1𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑇𝑥𝑛,𝑢𝑛𝑢𝐶suchthat𝑓𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝐶0,𝑦𝐶,𝑛+1=𝑧𝐶𝑛𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0.(1.13) They proved that the sequence {𝑥𝑛} converges strongly to 𝑝𝐹(𝑇)𝐹(𝑆)EP(𝑓), where 𝑝Π𝐹(𝑇)𝐹(𝑆)EP(𝑓)𝑥 under the assumptions (C1) limsup𝑛𝛼𝑛<1, (C2) lim𝑛𝛽𝑛<1, and (C3) liminf𝑛(1𝛼𝑛)𝛽𝑛(1𝛽𝑛)>0.

Recently, Chang et al. [26], introduced the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi-𝜙-asymptotically nonexpansive mappings,𝑥0𝐶chosenarbitrary,𝐶0𝑦=𝐶,𝑛=𝐽1𝛼𝑛,0𝐽𝑥𝑛+𝑖=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖𝑥𝑛,𝐶𝑛+1=𝑣𝐶𝑛𝜙𝑣,𝑦𝑛𝜙𝑣,𝑥𝑛+𝜉𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0,𝑛0,(1.14)where 𝜉𝑛=sup𝑢𝐹(𝑘𝑛1)𝜙(𝑢,𝑥𝑛). Then, they proved that under appropriate control conditions the sequence {𝑥𝑛} converges strongly to Π𝑛=1𝐹(𝑆𝑖)𝑥0.

Very recently, Tan and Chang [27], introduced a new hybrid iterative scheme for finding a common element between set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi-𝜙-asymptotically nonexpansive mappings (which is more general than quasi-𝜙-nonexpansive mappings), and null spaces of finite family of 𝛾-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space.

In this paper, motivated and inspired by Petrot et al. [24], Kumam and Wattanawitoon [25], Chang et al. [26], and Tan and Chang [27], we introduce the new hybrid block algorithm for two countable families of closed and uniformly Lipschitz continuous and uniformly asymptotically relatively nonexpansive mappings in a Banach space. Let {𝑥𝑛} be a sequence defined by 𝑥0𝐶, 𝐶0=𝐶 and 𝑦𝑛=𝐽1𝛽𝑛,0𝐽𝑥𝑛+𝑖=1𝛽𝑛,𝑖𝐽𝑇𝑛𝑖𝑥𝑛,𝑧𝑛=𝐽1𝛼𝑛,0𝐽𝑥𝑛+𝑖=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖𝑦𝑛,𝑢𝑛(𝑖)=𝐾𝑓𝑖,𝑟𝑖𝐾𝑓𝑖1,𝑟𝑖1𝐾𝑓1,𝑟1𝑧𝑛𝐶,𝑖=1,2,,𝑁,𝑛+1=𝑧𝐶𝑛max𝑖=1,2,,𝑁𝜙𝑧,𝑢𝑛(𝑖)𝜙𝑧,𝑥𝑛+𝜃𝑛,𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛+𝜉𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0,𝑛0.(1.15) Under appropriate conditions, we will prove that the sequence {𝑥𝑛} generated by algorithms (1.15) converges strongly to the point Π(𝑁𝑖=1Ω𝑖)(𝑖=1𝐹(𝑇𝑖))(𝑖=1𝐹(𝑆𝑖))𝑥0. Our results extend many known recent results in the literature.

2. Preliminaries

Let 𝐸 be a real Banach space with norm , and let 𝐽 be the normalized duality mapping from 𝐸 into 2𝐸 given by 𝑥𝐽𝑥=𝐸𝑥,𝑥=𝑥𝑥,𝑥=𝑥(2.1) for all 𝑥𝐸, where 𝐸 denotes the dual space of 𝐸 and , the generalized duality pairing between 𝐸 and 𝐸. It is also known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.

We know the following (see [28, 29]):(i)if 𝐸 is smooth, then 𝐽 is single valued;(ii)if 𝐸 is strictly convex, then 𝐽 is one-to-one and 𝑥𝑦,𝑥𝑦>0 holds for all (𝑥,𝑥),(𝑦,𝑦)𝐽 with 𝑥𝑦;(iii)if 𝐸 is reflexive, then 𝐽 is surjective;(iv)if 𝐸 is uniformly convex, then it is reflexive;(v)if 𝐸 is a reflexive and strictly convex, then 𝐽1 is norm-weak-continuous;(vi)𝐸 is uniformly smooth if and only if 𝐸 is uniformly convex;(vii)if 𝐸 is uniformly convex, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸;(viii)each uniformly convex Banach space 𝐸 has the Kadec-Klee property, that is, for any sequence {𝑥𝑛}𝐸, if 𝑥𝑛𝑥𝐸 and 𝑥𝑛𝑥, then 𝑥𝑛𝑥.

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Throughout this paper, we denote by 𝜙 the function defined by 𝜙(𝑥,𝑦)=𝑥22𝑥,𝐽𝑦+𝑦2,for𝑥,𝑦𝐸.(2.2) Following Alber [30], the generalized projection Π𝐶𝐸𝐶 is a map that assigns to an arbitrary point 𝑥𝐸 the minimum point of the function 𝜙(𝑥,𝑦), that is, Π𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem 𝜙𝑥,𝑥=inf𝑦𝐶𝜙(𝑦,𝑥).(2.3) Existence and uniqueness of the operator Π𝐶 follows from the properties of the functional 𝜙(𝑥,𝑦) and strict monotonicity of the mapping 𝐽. It is obvious from the definition of function 𝜙 that (see [30]) (𝑦𝑥)2)𝜙(𝑦,𝑥)(𝑦+𝑥2,𝑥,𝑦𝐸.(2.4) If 𝐸 is a Hilbert space, then 𝜙(𝑥,𝑦)=𝑥𝑦2.

If 𝐸 is a reflexive, strictly convex, and smooth Banach space, then for 𝑥,𝑦𝐸, 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦. It is sufficient to show that if 𝜙(𝑥,𝑦)=0, then 𝑥=𝑦. From (2.4), we have 𝑥=𝑦. This implies that 𝑥,𝐽𝑦=𝑥2=𝐽𝑦2. From the definition of 𝐽, one has 𝐽𝑥=𝐽𝑦. Therefore, we have 𝑥=𝑦; see [28, 29] for more details.

We also need the following lemmas for the proof of our main results.

Lemma 2.1 (see Kamimura and Takahashi [31]). Let 𝐸 be a uniformly convex and smooth real Banach space, and let {𝑥𝑛},{𝑦𝑛} be two sequences of 𝐸. If 𝜙(𝑥𝑛,𝑦𝑛)0 and either {𝑥𝑛} or {𝑦𝑛} is bounded, then 𝑥𝑛𝑦𝑛0.

Lemma 2.2 (see Alber [30]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑥𝐸. Then, 𝑥0=Π𝐶𝑥 if and only if 𝑥0𝑦,𝐽𝑥𝐽𝑥00,𝑦𝐶.(2.5)

Lemma 2.3 (see Alber [30]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑥𝐸. Then 𝜙𝑦,Π𝐶𝑥Π+𝜙𝐶𝑥,𝑥𝜙(𝑦,𝑥),𝑦𝐶.(2.6)

Lemma 2.4 (see Chang et al. [26]). Let 𝐸 be a uniformly convex Banach space, 𝑟>0 a positive number, and 𝐵𝑟(0) a closed ball of 𝐸. Then, for any given sequence {𝑥𝑖}𝑖=1𝐵𝑟(0) and for any given sequence {𝜆𝑖}𝑖=1 of positive number with 𝑛=1𝜆𝑛=1, there exists a continuous, strictly increasing, and convex function 𝑔[0,2𝑟)[0,) with 𝑔(0)=0 such that for any positive integers 𝑖,𝑗 with 𝑖<𝑗, 𝑛=1𝜆𝑛𝑥𝑛2𝑛=1𝜆𝑛𝑥𝑛2𝜆𝑖𝜆𝑗𝑔𝑥𝑖𝑥𝑗.(2.7)

Lemma 2.5 (see Chang et al. [26]). Let 𝐸 be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇𝐶𝐶 be a closed and asymptotically relatively nonexpansive mapping with a sequence {𝑘𝑛}[1,),𝑘𝑛1. Then 𝐹(𝑇) is closed and convex subset of 𝐶.

For solving the generalized mixed equilibrium problem (or a system of generalized mixed equilibrium problem), let us assume that the bifunction 𝑓𝐶×𝐶 and 𝜓𝐶 is convex and lower semicontinuous satisfies the following conditions: (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 each 𝑥,𝑦,𝑧𝐶, limsup𝑡0𝑓(𝑡𝑧+(1𝑡)𝑥,𝑦)𝑓(𝑥,𝑦);(2.8)(A4) for each 𝑥𝐶, 𝑦𝑓(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.6 (see Chang et al. [26]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝐴𝐶𝐸 be a continuous and monotone mapping, 𝜓𝐶 is convex and lower semicontinuous and 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4). For 𝑟>0 and 𝑥𝐸, then there exists 𝑢𝐶 such that 1𝑓(𝑢,𝑦)+𝐴𝑢,𝑦𝑢+𝜓(𝑦)𝜓(𝑢)+𝑟𝑦𝑢,𝐽𝑢𝐽𝑥0,𝑦𝐶.(2.9) Define a mapping 𝐾𝑓,𝑟𝐶𝐶 as follows: 𝐾𝑓,𝑟1(𝑥)=𝑢𝐶𝑓(𝑢,𝑦)+𝐴𝑢,𝑦𝑢+𝜓(𝑦)𝜓(𝑢)+𝑟𝑦𝑢,𝐽𝑢𝐽𝑥0,𝑦𝐶(2.10) for all 𝑥𝐸. Then, the following hold: (i)𝐾𝑓,𝑟 is singlevalued; (ii)𝐾𝑓,𝑟 is firmly nonexpansive, that is, for all 𝑥,𝑦𝐸, 𝐾𝑓,𝑟𝑥𝐾𝑓,𝑟𝑦,𝐽𝐾𝑓,𝑟𝑥𝐽𝐾𝑓,𝑟𝑦𝐾𝑓,𝑟𝑥𝐾𝑓,𝑟𝑦,𝐽𝑥𝐽𝑦;(iii)𝐹(𝐾𝑓,𝑟)=𝐹(𝐾𝑓,𝑟); (iv)𝑢𝐶 is a solution of variational equation (2.9) if and only if 𝑢𝐶 is a fixed point of 𝐾𝑓,𝑟; (v)𝐹(𝐾𝑓,𝑟)=Ω; (vi)Ω is closed and convex; (vii)𝜙(𝑝,𝐾𝑓,𝑟𝑧)+𝜙(𝐾𝑓,𝑟𝑧,𝑧)𝜙(𝑝,𝑧), forall𝑝𝐹(𝐾𝑓,𝑟), 𝑧𝐸.

3. Main Results

Theorem 3.1. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝐴𝑖𝐶𝐸 be a continuous and monotone mapping, 𝜓𝑖𝐶 be a lower semi-continuous and convex function, 𝑓𝑖 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4), 𝐾𝑓𝑖,𝑟𝑖 is the mapping defined by (2.10) where 𝑟𝑖𝑟>0, and let {𝑇𝑖}𝑖=1, {𝑆𝑖}𝑖=1 be countable families of closed and uniformly 𝐿𝑖, 𝜇𝑖-Lipschitz continuous and asymptotically relatively nonexpansive mapping with sequence {𝑘𝑛},{𝜁𝑛}[1,);  𝑘𝑛1,𝜁𝑛1 such that =(𝑁𝑖=1Ω𝑖)(𝑖=1𝐹(𝑇𝑖))(𝑖=1𝐹(𝑆𝑖)). Let {𝑥𝑛} be a sequence generated by 𝑥0𝐶 and 𝐶0=𝐶, 𝑦𝑛=𝐽1𝛽𝑛,0𝐽𝑥𝑛+𝑖=1𝛽𝑛,𝑖𝐽𝑇𝑛𝑖𝑥𝑛,𝑧𝑛=𝐽1𝛼𝑛,0𝐽𝑥𝑛+𝑖=1𝛼𝑛,𝑖𝐽𝑆𝑛𝑖𝑦𝑛,𝑢𝑛(𝑖)=𝐾𝑓𝑖,𝑟𝑖𝐾𝑓𝑖1,𝑟𝑖1𝐾𝑓1,𝑟1𝑧𝑛𝐶,𝑖=1,2,,𝑁,𝑛+1=𝑧𝐶𝑛max𝑖=1,2,,𝑁𝜙𝑧,𝑢𝑛(𝑖)𝜙𝑧,𝑥𝑛+𝜃𝑛,𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛+𝜉𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0,𝑛0,(3.1) where 𝜉𝑛=sup𝑝(𝑘𝑛1)𝜙(𝑝,𝑥𝑛), 𝜃𝑛=𝛿𝑛+𝜉𝑛𝜁𝑛, and 𝛿𝑛=sup𝑝(𝜁𝑛1)𝜙(𝑝,𝑥𝑛). The coefficient sequences {𝛼𝑛,𝑖} and {𝛽𝑛,𝑖}[0,1] satisfy the following: (i)𝑖=0𝛼𝑛,𝑖=1; (ii)𝑖=0𝛽𝑛,𝑖=1; (iii)liminf𝑛𝛼𝑛,0𝛼𝑛,𝑖>0, forall𝑖1; (iv)liminf𝑛𝛽𝑛,0𝛽𝑛,𝑖>0, forall𝑖1, Ω𝑖,𝑖=1,2,,𝑁 is the set of solutions to the following generalized mixed equilibrium problem: 𝑓𝑖(𝑧,𝑦)+𝐴𝑖𝑧,𝑦𝑧+𝜓𝑖(𝑦)𝜓𝑖(𝑧)0,𝑦𝐶,𝑖=1,2,,𝑁.(3.2) Then the sequence {𝑥𝑛} converges strongly to Π𝑥0.

Proof. We first show that 𝐶𝑛, forall𝑛0 is closed and convex. Clearly 𝐶0=𝐶 is closed and convex. Suppose that 𝐶𝑘 is closed and convex for some 𝑘>1. For each 𝑧𝐶𝑘, we see that 𝜙(𝑧,𝑢𝑘(𝑖))𝜙(𝑧,𝑥𝑘) is equivalent to 2𝑧,𝑥𝑘𝑧,𝑢𝑘(𝑖)𝑥𝑘2𝑢𝑘(𝑖)2.(3.3) By the set of 𝐶𝑘+1, we have 𝐶𝑛+1=𝑧𝐶𝑛max𝑖=1,2,,𝑁𝜙𝑧,𝑢𝑛(𝑖)𝜙𝑧,𝑥𝑛+𝜃𝑛=𝑁𝑖=1𝑧𝐶𝜙𝑧,𝑢𝑛(𝑖)𝜙𝑧,𝑥𝑛+𝜃𝑛.(3.4) Hence, 𝐶𝑛+1 is also closed and convex.
By taking Θ𝑗𝑛=𝐾𝑟𝑗,𝑓𝑖𝐾𝑟𝑗1,𝑓𝑗1𝐾𝑟1,𝑓1 for any 𝑗{1,2,,𝑖} and Θ0𝑛=𝐼 for all 𝑛1. We note that 𝑢𝑛(𝑖)=Θ𝑖𝑛𝑧𝑛.
Next, we show that 𝐶𝑛,forall𝑛1. For 𝑛1, we have 𝐶=𝐶1. For any given 𝑝=(𝑁𝑖=1Ω𝑖)(𝑖=1𝐹(𝑇𝑖))(𝑖=1𝐹(𝑆𝑖)). By (3.1) and Lemma 2.4, we have 𝜙𝑝,𝑦𝑛=𝜙𝑝,𝐽1𝑖=0𝛽𝑛,𝑖𝐽𝑇𝑛𝑖𝑥𝑛=𝑝2𝑖=0𝛽𝑛,𝑖2𝑝,𝐽𝑇𝑛𝑖𝑥𝑛+𝑖=0𝛽𝑛,𝑖𝐽𝑇𝑛𝑖𝑥𝑛2𝑝2𝑖=0𝛽𝑛,𝑖2𝑝,𝐽𝑇𝑛𝑖𝑥𝑛+𝑖=0𝛽𝑛,𝑖𝐽𝑇𝑛𝑖𝑥𝑛2𝛽𝑛,0𝛽𝑛,𝑖𝑔𝐽𝑇𝑛0𝑥𝑛𝐽𝑇𝑛𝑖𝑥𝑛=𝑝2𝑖=0𝛽𝑛,𝑖2𝑝,𝐽𝑇𝑛𝑖𝑥𝑛+𝑖=0𝛽𝑛,𝑖𝑇𝑛𝑖𝑥𝑛2𝛽𝑛,0𝛽𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑖𝑥𝑛=𝑖=0𝛽𝑛,𝑖𝜙𝑝,𝑇𝑛𝑖𝑥𝑛𝛽𝑛,0𝛽𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑖𝑥𝑛𝑘𝑛𝜙𝑝,𝑥𝑛𝛽𝑛,0𝛽𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑖𝑥𝑛𝜙𝑝,𝑥𝑛+sup𝑝𝐹𝑘𝑛𝜙1𝑝,𝑥𝑛𝛽𝑛,0𝛽𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑖𝑥𝑛𝜙𝑝,𝑥𝑛+𝜉𝑛𝛽𝑛,0𝛽𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑖𝑥𝑛𝜙𝑝,𝑥𝑛+𝜉𝑛,(3.5) where 𝜉𝑛=sup𝑝(𝑘𝑛1)𝜙(𝑝,𝑥𝑛).
By (3.1) and (3.5), we note that 𝜙𝑝,𝑢𝑛(𝑖)=𝜙𝑝,Θ𝑖𝑛𝑧𝑛𝜙𝑝,𝑧𝑛𝜙𝑝,𝐽1𝛼𝑛,0𝐽𝑥𝑛+𝑖=1𝐽𝑆𝑛𝑖𝑦𝑛=𝑝22𝑝,𝛼𝑛,0𝐽𝑥𝑛+𝑖=1𝐽𝑆𝑛𝑖𝑦𝑛+𝛼𝑛,0𝐽𝑥𝑛+𝑖=1𝐽𝑆𝑛𝑖𝑦𝑛2𝑝22𝛼𝑛,0𝑝,𝐽𝑥𝑛2𝑖=1𝛼𝑛,𝑖𝑝,𝐽𝑆𝑛𝑖𝑦𝑛+𝛼𝑛,0𝑥𝑛2+𝑖=1𝑆𝑛𝑖𝑦𝑛2𝛼𝑛,0𝛼𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑖𝑦𝑛𝛼𝑛,0𝜙𝑝,𝑥𝑛+𝑖=1𝛼𝑛,𝑖𝜙𝑝,𝑆𝑛𝑖𝑦𝑛𝛼𝑛,0𝛼𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑖𝑦𝑛𝛼𝑛,0𝜙𝑝,𝑥𝑛+𝜁𝑛𝑖=1𝛼𝑛,𝑖𝜙𝑝,𝑦𝑛𝛼𝑛,0𝛼𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑖𝑦𝑛𝛼𝑛,0𝜙𝑝,𝑥𝑛+𝜁𝑛𝑖=1𝛼𝑛,𝑖𝜙𝑝,𝑥𝑛+𝜉𝑛𝛼𝑛,0𝛼𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑖𝑦𝑛𝛼𝑛,0𝜙𝑝,𝑥𝑛+𝜁𝑛𝑖=1𝛼𝑛,𝑖𝜙𝑝,𝑥𝑛+𝜉𝑛𝜁𝑛𝑖=1𝛼𝑛,𝑖𝛼𝑛,0𝛼𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑖𝑦𝑛𝜁𝑛𝜙𝑝,𝑥𝑛+𝜉𝑛𝜁𝑛𝑖=1𝛼𝑛,𝑖𝛼𝑛,0𝛼𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑖𝑦𝑛𝜙𝑝,𝑥𝑛+sup𝑝𝐹𝜁𝑛𝜙1𝑝,𝑥𝑛+𝜉𝑛𝜁𝑛𝑖=1𝛼𝑛,𝑖𝛼𝑛,0𝛼𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑖𝑦𝑛𝜙𝑝,𝑥𝑛+𝛿𝑛+𝜉𝑛𝜁𝑛𝛼𝑛,0𝛼𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑖𝑦𝑛𝜙𝑝,𝑥𝑛+𝜃𝑛,(3.6) where 𝛿𝑛=sup𝑝(𝜁𝑛1)𝜙(𝑝,𝑥𝑛), 𝜃𝑛=𝛿𝑛+𝜉𝑛𝜁𝑛. By assumptions on {𝑘𝑛} and {𝜁𝑛}, we have 𝜉𝑛=sup𝑝𝑘𝑛𝜙1𝑝,𝑥𝑛sup𝑝𝑘𝑛()1𝑝+𝑀2𝛿0as𝑛,(3.7)𝑛=sup𝑝𝜁𝑛𝜙1𝑝,𝑥𝑛sup𝑝𝜁𝑛1(𝑝+𝑀)20as𝑛,(3.8) where 𝑀=sup𝑛0𝑥𝑛.
So, we have 𝑝𝐶𝑛+1. This implies that 𝐶𝑛,forall𝑛0 and also {𝑥𝑛} is well defined.
From Lemma 2.2 and 𝑥𝑛=Π𝐶𝑛𝑥0, we have 𝑥𝑛𝑧,𝐽𝑥0𝐽𝑥𝑛0,𝑧𝐶𝑛,𝑥𝑛𝑝,𝐽𝑥0𝐽𝑥𝑛0,𝑝𝐶𝑛.(3.9) From Lemma 2.3, one has 𝜙𝑥𝑛,𝑥0Π=𝜙𝐶𝑛𝑥0,𝑥0𝜙𝑝,𝑥0𝜙𝑝,𝑥𝑛𝜙𝑝,𝑥0(3.10) for all 𝑝𝐶𝑛 and 𝑛1. Then, the sequence {𝜙(𝑥𝑛,𝑥0)} is also bounded. Thus {𝑥𝑛} is bounded. Since 𝑥𝑛=Π𝐶𝑛𝑥0 and 𝑥𝑛+1=Π𝐶𝑛+1𝑥0𝐶𝑛+1𝐶𝑛, we have 𝜙𝑥𝑛,𝑥0𝑥𝜙𝑛+1,𝑥0,𝑛.(3.11) Therefore, {𝜙(𝑥𝑛,𝑥0)} is nondecreasing. Hence, the limit of {𝜙(𝑥𝑛,𝑥0)} exists. By the construction of 𝐶𝑛, one has that 𝐶𝑚𝐶𝑛 and 𝑥𝑚=Π𝐶𝑚𝑥0𝐶𝑛 for any positive integer 𝑚𝑛. It follows that 𝜙𝑥𝑚,𝑥𝑛𝑥=𝜙𝑚,Π𝐶𝑛𝑥0𝑥𝜙𝑚,𝑥0Π𝜙𝐶𝑛𝑥0,𝑥0𝑥=𝜙𝑚,𝑥0𝑥𝜙𝑛,𝑥0.(3.12) Letting 𝑚,𝑛0 in (3.12), we get 𝜙(𝑥𝑚,𝑥𝑛)0. It follows from Lemma 2.1, that 𝑥𝑚𝑥𝑛0 as 𝑚,𝑛. That is, {𝑥𝑛} is a Cauchy sequence.
Since {𝑥𝑛} is bounded and 𝐸 is reflexive, there exists a subsequence {𝑥𝑛𝑖}{𝑥𝑛} such that 𝑥𝑛𝑖𝑢. Since 𝐶𝑛 is closed and convex and 𝐶𝑛+1𝐶𝑛, this implies that 𝐶𝑛 is weakly closed and 𝑢𝐶𝑛 for each 𝑛0. since 𝑥𝑛=Π𝐶𝑛𝑥0, we have 𝜙𝑥𝑛𝑖,𝑥0𝜙𝑢,𝑥0,𝑛𝑖0.(3.13) Since liminf𝑛𝑖𝜙𝑥𝑛𝑖,𝑥0=liminf𝑛𝑖𝑥𝑛𝑖2𝑥2𝑛𝑖,𝐽𝑥0+𝑥02𝑢22𝑢,𝐽𝑥0𝑥+02=𝜙𝑢,𝑥0.(3.14) We have 𝜙𝑢,𝑥0liminf𝑛𝑖𝜙𝑥𝑛𝑖,𝑥0limsup𝑛𝑖𝜙𝑥𝑛𝑖,𝑥0𝜙𝑢,𝑥0.(3.15) This implies that lim𝑛𝑖𝜙(𝑥𝑛𝑖,𝑥0)=𝜙(𝑢,𝑥0). That is, 𝑥𝑛𝑖𝑢. Since 𝑥𝑛𝑖𝑢, by the Kadec-klee property of 𝐸, we obtain that lim𝑛𝑥𝑛𝑖=𝑢.(3.16) If there exists some subsequence {𝑥𝑛𝑗}{𝑥𝑛} such that 𝑥𝑛𝑗𝑞, then we have 𝜙(𝑢,𝑞)=lim𝑛𝑖,𝑛𝑗𝜙𝑥𝑛𝑖,𝑥𝑛𝑗lim𝑛𝑖,𝑛𝑗𝜙𝑥𝑛𝑖,𝑥0Π𝜙𝐶𝑛𝑗𝑥0,𝑥0=lim𝑛𝑖,𝑛𝑗𝜙𝑥𝑛𝑖,𝑥0𝑥𝜙𝑛𝑗𝑥0,𝑥0=0.(3.17) Therefore, we have 𝑢=𝑞. This implies that lim𝑛𝑥𝑛=𝑢.(3.18) Since 𝜙𝑥𝑛+1,𝑥𝑛𝑥=𝜙𝑛+1,Π𝐶𝑛𝑥0𝑥𝜙𝑛+1,𝑥0Π𝜙𝐶𝑛𝑥0,𝑥0𝑥=𝜙𝑛+1,𝑥0𝑥𝜙𝑛,𝑥0(3.19) for all 𝑛, we also have lim𝑛𝜙𝑥𝑛+1,𝑥𝑛=0.(3.20) Since 𝑥𝑛+1=Π𝐶𝑛+1𝑥0𝐶𝑛+1 and by the definition of 𝐶𝑛+1, for 𝑖=1,2,,𝑁, we have 𝜙𝑥𝑛+1,𝑢𝑖𝑛𝑥𝜙𝑛+1,𝑥𝑛+𝜃𝑛.(3.21) Noticing that lim𝑛𝜙(𝑥𝑛+1,𝑥𝑛)=0, we obtain lim𝑛𝜙𝑥𝑛+1,𝑢𝑖𝑛=0,for𝑖=1,2,,𝑁.(3.22) It then yields that lim𝑛(𝑥𝑛+1𝑢𝑖𝑛)=0,forall𝑖=1,2,,𝑁. Since lim𝑛𝑥𝑛+1=𝑢, we have lim𝑛𝑢𝑖𝑛=𝑢,𝑖=1,2,,𝑁.(3.23) Hence, lim𝑛𝐽𝑢𝑖𝑛=𝐽𝑢,𝑖=1,2,,𝑁.(3.24) From Lemma 2.1 and (3.22), we have lim𝑛𝑥𝑛+1𝑥𝑛=lim𝑛𝑥𝑛+1𝑢𝑖𝑛=0,𝑖=1,2,,𝑁.(3.25) By the triangle inequality, we get lim𝑛𝑥𝑛𝑢𝑖𝑛=0,𝑖=1,2,,𝑁.(3.26) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim𝑛𝐽𝑥𝑛𝐽𝑢𝑖𝑛=lim𝑛𝐽𝑥𝑛+1𝐽𝑢𝑖𝑛=0,𝑖=1,2,,𝑁.(3.27)
Now, we prove that 𝑢(𝑖=1𝐹(𝑇𝑖))(𝑖=1𝐹(𝑆𝑖)). From the construction of 𝐶𝑛, we obtain that 𝜙𝑥𝑛+1,𝑦𝑛𝑥𝜙𝑛+1,𝑥𝑛+𝜉𝑛.(3.28) From (3.7) and (3.20), we have lim𝑛𝜙𝑥𝑛+1,𝑦𝑛=0.(3.29) By Lemma 2.1, we also have lim𝑛𝑥𝑛+1𝑦𝑛=0.(3.30) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim𝑛𝐽𝑥𝑛+1𝐽𝑦𝑛=0.(3.31) From (2.4) and (3.29), we have (𝑥𝑛+1𝑦𝑛)20. Since 𝑥𝑛+1𝑢, it yields that 𝑦𝑛𝑢as𝑛.(3.32) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, it follows that J𝑦𝑛𝐽𝑢as𝑛.(3.33) This implies that {𝐽𝑦𝑛} is bounded in 𝐸. Since 𝐸 is reflexive, there exists a subsequence {𝐽𝑦𝑛𝑖}{𝐽𝑦𝑛} such that 𝐽𝑦𝑛𝑖𝑟𝐸. Since 𝐸 is reflexive, we see that 𝐽(𝐸)=𝐸. Hence, there exists 𝑥𝐸 such that 𝐽𝑥=𝑟. We note that 𝜙𝑥𝑛𝑖+1,𝑦𝑛𝑖=𝑥𝑛𝑖+12𝑥2𝑛𝑖+1,𝐽𝑦𝑛𝑖+𝑦𝑛𝑖2=𝑥𝑛𝑖+12𝑥2𝑛𝑖+1,𝐽𝑦𝑛𝑖+𝐽𝑦𝑛𝑖2.(3.34) Taking the limit interior of both side and in view of weak lower semicontinuity of norm , we have 0𝑢22𝑢,𝑟+𝑟2=𝑢22𝑢,𝐽𝑥+𝐽𝑥2=𝑢22𝑢,𝐽𝑥+𝑥2=𝜙(𝑢,𝑥),(3.35) that is, 𝑢=𝑥. This implies that 𝑟=𝐽𝑢 and so 𝐽𝑦𝑛𝐽𝑝. It follows from lim𝑛𝐽𝑦𝑛=𝐽𝑢, as 𝑛 and the Kadec-Klee property of 𝐸 that 𝐽𝑦𝑛𝑖𝐽𝑢 as 𝑛. Note that 𝐽1𝐸𝐸 is hemicontinuous, it yields that 𝑦𝑛𝑖𝑢. It follows from lim𝑛𝑢𝑛=𝑢, as 𝑛 and the Kadec-Klee property of 𝐸 that lim𝑛𝑖𝑦𝑛𝑖=𝑢.
By similar, we can prove that lim𝑛𝑦𝑛=𝑢.(3.36) By (3.20) and (3.30), we obtain lim𝑛𝑥𝑛𝑦𝑛=0.(3.37) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim𝑛𝐽𝑥𝑛𝐽𝑦𝑛=0.(3.38) So, from (3.27) and (3.31), by the triangle inequality, we get lim𝑛𝐽𝑦𝑛𝐽𝑢𝑖𝑛=0,for𝑖=1,2,,𝑁.(3.39) Since 𝐽1 is uniformly norm-to-norm continuous on bounded sets, we note that lim𝑛𝑦𝑛𝑢𝑖𝑛=0,for𝑖=1,2,,𝑁.(3.40) Since 𝜙𝑝,𝑥𝑛𝜙𝑝,𝑦𝑛=𝑥𝑛2𝑦𝑛22𝑝,𝐽𝑥𝑛𝐽𝑦𝑛𝑥𝑛2𝑦𝑛2+2𝑝𝐽𝑥𝑛𝐽𝑦𝑛𝑥𝑛𝑦𝑛𝑥𝑛+𝑦𝑛+2𝑝𝐽𝑥𝑛𝐽𝑦𝑛.(3.41) From (3.37) and (3.38), we obtain 𝜙𝑝,𝑥𝑛𝜙𝑝,𝑦𝑛0,𝑛.(3.42) On the other hand, we observe that, for 𝑖=1,2,,𝑁. 𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑖𝑛=𝑥𝑛2𝑢𝑖𝑛22𝑝,𝐽𝑥𝑛𝐽𝑢𝑖𝑛𝑥𝑛2𝑢𝑖𝑛2+2𝑝𝐽𝑥𝑛𝐽𝑢𝑖𝑛𝑥𝑛𝑢𝑖𝑛𝑥𝑛+𝑢𝑖𝑛+2𝑝𝐽𝑥𝑛𝐽𝑢𝑖𝑛.(3.43) From (3.22) and (3.27), we have 𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑖𝑛0,𝑛,𝑖=1,2,,𝑁.(3.44) For any 𝑝𝑁𝑖=1Ω𝑖(𝑖=1F(𝑇𝑖))(𝑖=1𝐹(𝑆𝑖)), it follows from (3.5) that 𝛽𝑛,0𝛽𝑛,𝑖𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑖𝑥𝑛𝜙𝑝,𝑥𝑛+𝜉𝑛𝜙𝑝,𝑦𝑛.(3.45)From condition, liminf𝑛𝛽𝑛,0𝛽𝑛,𝑖>0, property of 𝑔, (3.7), and (3.42), we have that 𝐽𝑥𝑛𝐽𝑇𝑛𝑖𝑥𝑛0,𝑛,𝑖=1,2,,𝑁.(3.46) Since 𝑥𝑛𝑢 and 𝐽 is uniformly norm-to-norm continuous. It yields 𝐽𝑥𝑛𝐽𝑝. Hence from (3.46), we have 𝑥𝑛𝑇𝑛𝑖𝑥𝑛0,𝑛,𝑖=1,2,,𝑁.(3.47) Since 𝑥𝑛𝑢, this implies that lim𝑛𝐽𝑇𝑛𝑖𝑥𝑛𝐽𝑢 as 𝑛. Since 𝐽1𝐸𝐸 is hemicontinuous, it follows that 𝑇𝑛𝑖𝑥𝑛𝑢,foreach𝑖1.(3.48) On the other hand, for each 𝑖1, we have 𝑇𝑛𝑖𝑥𝑛||𝑇𝑢=𝑛𝑖𝑥𝑛||𝑇𝑢𝑛𝑖𝑥𝑛𝑢0,𝑛.(3.49) from this, together with (3.48) and the Kadec-Klee property of 𝐸, we obtain 𝑇𝑛𝑖𝑥𝑛𝑢,foreach𝑖1.(3.50) On the other hand, by the assumption that 𝑇𝑖 is uniformly 𝐿𝑖-Lipschitz continuous, we have 𝑇𝑖𝑛+1𝑥𝑛𝑇𝑛𝑖𝑥𝑛𝑇𝑖𝑛+1𝑥𝑛𝑇𝑖𝑛+1𝑥𝑛+1+𝑇𝑖𝑛+1𝑥𝑛+1𝑥𝑛+1+𝑥𝑛+1𝑥𝑛+𝑥𝑛𝑇𝑛𝑖𝑥𝑛𝐿𝑖𝑥+1𝑛+1𝑥𝑛+𝑇𝑖𝑛+1𝑥𝑛+1𝑥𝑛+1+𝑥𝑛𝑇𝑛𝑖𝑥𝑛.(3.51) By (3.18) and (3.50), we obtain lim𝑛𝑇𝑖𝑛+1𝑥𝑛𝑇𝑛𝑖𝑥𝑛=0,𝑖1,(3.52) and lim𝑛𝑇𝑖𝑛+1𝑥𝑛=𝑢, that is, 𝑇𝑖𝑇𝑛𝑥𝑛𝑢, forall𝑖1. By the closeness of 𝑇𝑖, we have 𝑇𝑖𝑢=u, forall𝑖1. This implies that 𝑢𝑖=1𝐹(𝑇𝑖).
By the similar way, we can prove that for each 𝑖1𝐽𝑥𝑛𝐽𝑆𝑛𝑖𝑦𝑛0,𝑛.(3.53) Since 𝑥𝑛𝑢 and 𝐽 is uniformly norm-to-norm continuous. it yields 𝐽𝑥𝑛𝐽𝑝. Hence from (3.53), we have 𝑥𝑛𝑆𝑛𝑖𝑦𝑛0,𝑛.(3.54) Since 𝑥𝑛𝑢, this implies that lim𝑛𝐽𝑆𝑛𝑖𝑦𝑛𝐽𝑢 as 𝑛. Since 𝐽1𝐸𝐸 is hemicontinuous, it follows that 𝑆𝑛𝑖𝑦𝑛𝑢,foreach𝑖1.(3.55) On the other hand, for each 𝑖1, we have 𝑆𝑛𝑖𝑦𝑛||𝑆𝑢=𝑛𝑖𝑦𝑛||𝑆𝑢𝑛𝑖𝑦𝑛𝑢0,𝑛.(3.56) From this, together with (3.54) and the Kadec-Klee property of 𝐸, we obtain 𝑆𝑛𝑖𝑦𝑛𝑢,foreach𝑖1.(3.57) On the other hand, by the assumption that 𝑆𝑖 is uniformly 𝜇𝑖-Lipschitz continuous, we have 𝑆𝑖𝑛+1𝑦𝑛𝑆𝑛𝑖𝑦𝑛𝑆𝑖𝑛+1𝑦𝑛𝑆𝑖𝑛+1𝑦𝑛+1+𝑆𝑖𝑛+1𝑦𝑛+1𝑦𝑛+1+𝑦𝑛+1𝑦𝑛+𝑦𝑛𝑆𝑛𝑖𝑦𝑛𝜇𝑖𝑦+1𝑛+1𝑦𝑛+𝑆𝑖𝑛+1𝑦𝑛+1𝑦𝑛+1+𝑦𝑛𝑆𝑛𝑖𝑦𝑛.(3.58) By (3.36) and (3.57), we obtain lim𝑛𝑆𝑖𝑛+1𝑦𝑛𝑆𝑛𝑖𝑦𝑛=0(3.59) and lim𝑛𝑆𝑖𝑛+1𝑦𝑛=𝑢, that is, 𝑆𝑖𝑇𝑛𝑦𝑛𝑢. By the closeness of 𝑆𝑖, we have 𝑆𝑖𝑢=𝑢, forall𝑖1. This implies that 𝑢𝑖=1𝐹(𝑆𝑖). Hence 𝑢(𝑖=1𝐹(𝑇𝑖))(𝑖=1𝐹(𝑆𝑖)).
Next, we prove that 𝑢𝑁𝑖=1Ω𝑖. For any 𝑝, for each 𝑖=1,2,,𝑁, we have 𝜙𝑢𝑖𝑛,𝑧𝑛Θ=𝜙𝑖𝑛𝑧𝑛,𝑧𝑛𝜙𝑝,𝑧𝑛𝜙𝑝,Θ𝑖𝑛𝑧𝑛=𝜙𝑝,𝑧𝑛𝜙𝑝,𝑢𝑖𝑛𝜙𝑝,𝑥𝑛+𝜃𝑛𝜙𝑝,𝑢𝑖𝑛0,as𝑛.(3.60) It then yields that lim𝑛(𝑢𝑖𝑛𝑧𝑛)=0. Since lim𝑛𝑢𝑖𝑛=𝑢, forall𝑖1, we have lim𝑛𝑧𝑛=𝑢.(3.61) Hence, lim𝑛𝐽𝑧𝑛=𝐽𝑢.(3.62) This together with lim𝑛𝑢𝑖𝑛=𝑢 show that for each 𝑖=1,2,,𝑁, lim𝑛𝑢𝑖𝑛𝑢𝑛𝑖1=lim𝑛𝐽𝑢𝑖𝑛𝐽𝑢𝑛𝑖1=0,(3.63) where 𝑢0𝑛=𝑧𝑛. On the other hand, we have 𝑢𝑖𝑛=𝐾𝑓𝑖,𝑟𝑖𝑢𝑛𝑖1,foreach𝑖=2,3,,𝑁,(3.64) and 𝑢𝑖𝑛 is a solution of the following variational equation 𝑓𝑖𝑢𝑖𝑛+𝐴,𝑦𝑖𝑢𝑖𝑛,𝑦𝑢𝑖𝑛+𝜓𝑖(𝑦)𝜓𝑖𝑢𝑖𝑛+1𝑟𝑖𝑦𝑢𝑖𝑛,𝐽𝑢𝑖𝑛𝐽𝑢𝑛𝑖10,𝑦𝐶.(3.65) By condition (A2), we note that 𝐴𝑖𝑢𝑖𝑛,𝑦𝑢𝑖𝑛+𝜓𝑖(𝑦)𝜓𝑖𝑢𝑖𝑛+1𝑟𝑖𝑦𝑢𝑖𝑛,𝐽𝑢𝑖𝑛𝐽𝑢𝑛𝑖1𝑓𝑖𝑢𝑖𝑛,𝑦𝑓𝑖𝑦,𝑢𝑖𝑛,𝑦𝐶.(3.66) By (A4), (3.63), and 𝑢𝑖𝑛𝑢 for each 𝑖=2,3,,𝑁, we have 𝐴𝑖𝑢,𝑦𝑢+𝜓𝑖(𝑦)𝜓𝑖(𝑢)𝑓𝑖(𝑦,𝑢),𝑦𝐶.(3.67) For 0<𝑡<1 and 𝑦𝐶, define 𝑦𝑡=𝑡𝑦+(1𝑡)𝑢. Noticing that 𝑦,𝑢𝐶, we obtain 𝑦𝑡𝐶, which yields that 𝐴𝑖𝑢,𝑦𝑡𝑢+𝜓𝑖𝑦𝑡𝜓𝑖(𝑢)𝑓𝑖𝑦𝑡,𝑢.(3.68) In view of the convexity of 𝜙 it yields 𝑡𝐴𝑖𝜓𝑢,𝑦𝑢+𝑡𝑖(𝑦)𝜓𝑖(𝑢)𝑓𝑖𝑦𝑡,𝑢.(3.69) It follows from (A1) and (A4) that 0=𝑓𝑖𝑦𝑡,𝑦𝑡𝑡𝑓𝑖𝑦𝑡+,𝑦(1𝑡)𝑓𝑖𝑦𝑡,𝑢𝑡𝑓𝑖𝑦𝑡,𝑦+(1𝑡)𝑡𝐴𝑖𝜓𝑢,𝑦𝑢+𝑖(𝑦)𝜓𝑖(.𝑢)(3.70) Let 𝑡0, from (A3), we obtain the following: 𝑓i(𝑢,𝑦)+𝐴𝑖𝑢,𝑦𝑢+𝜓𝑖(𝑦)𝜓𝑖(𝑢)0,𝑦𝐶,𝑖=1,2,,𝑁.(3.71) This implies that 𝑢 is a solution of the system of generalized mixed equilibrium problem (3.2), that is, 𝑢𝑁𝑖=1Ω𝑖. Hence, 𝑢=(𝑁𝑖=1Ω𝑖)(𝑖=1𝐹(𝑇𝑖))(𝑖=1𝐹(𝑆𝑖)).
Finally, we show that 𝑥𝑛𝑢=Π𝐹𝑥0. Indeed from 𝑤𝐹𝐶𝑛 and 𝑥𝑛=Π𝐶𝑛𝑥0, we have the following: 𝜙𝑥𝑛,𝑥0𝜙𝑤,𝑥0,𝑛0.(3.72) This implies that 𝜙𝑢,𝑥0=lim𝑛𝜙𝑥𝑛,𝑥0𝜙𝑤,𝑥0.(3.73) From the definition of Π𝐹𝑥0 and (3.73), we see that 𝑢=𝑤. This completes the proof.

Since every asymptotically relatively nonexpansive mappings is quasi-𝜙-nonexpansive mappings, hence we obtain the following corollary.

Corollary 3.2. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝐴𝑖𝐶𝐸 be a continuous and monotone mapping, 𝜓𝑖𝐶 be a lower semi-continuous and convex function, 𝑓𝑖 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4), 𝐾𝑓𝑖,𝑟𝑖 is the mapping defined by (2.10) where 𝑟𝑖𝑟>0, and let {𝑇𝑖}𝑖=1, {𝑆𝑖}𝑖=1 be countable families of closed and quasi-𝜙-nonexpansive mapping such that =(𝑁𝑖=1Ω𝑖)(𝑖=1𝐹(𝑇𝑖))(𝑖=1𝐹(𝑆𝑖)). Let {𝑥𝑛} be a sequence generated by 𝑥0𝐶 and 𝐶0=𝐶, such that 𝑦𝑛=𝐽1𝛽𝑛,0𝐽𝑥𝑛+𝑖=1𝛽𝑛,𝑖𝐽𝑇𝑖𝑥𝑛,𝑧𝑛=𝐽1𝛼𝑛,0𝐽𝑥𝑛+𝑖=1𝛼𝑛,𝑖𝐽𝑆𝑖𝑦𝑛,𝑢𝑛(𝑖)=𝐾𝑓𝑖,𝑟𝑖𝐾𝑓𝑖1,𝑟𝑖1𝐾𝑓1,𝑟1𝑧𝑛𝐶,𝑖=1,2,,𝑁,𝑛+1=𝑧𝐶𝑛max𝑖=1,2,,𝑁𝜙𝑧,𝑢𝑛(𝑖)𝜙𝑧,𝑥𝑛,𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0,𝑛0,(3.74) where Π𝐶 is the generalized projection from 𝐸 onto 𝐶, 𝐽 is the duality mapping on 𝐸. The coefficient sequences {𝛼𝑛,𝑖} and {𝛽𝑛,𝑖}[0,1], satisfying: (i)𝑖=0𝛼𝑛,i=1; (ii)𝑖=0𝛽𝑛,𝑖=1; (iii)liminf𝑛𝛼𝑛,0𝛼𝑛,𝑖>0, forall𝑖1; (iv)liminf𝑛𝛽𝑛,0𝛽𝑛,𝑖>0, forall𝑖1. Ω𝑖,𝑖=1,2,,𝑁 is the set of solutions to the following generalized mixed equilibrium problem: 𝑓𝑖(𝑧,𝑦)+𝐴𝑖𝑧,𝑦𝑧+𝜓𝑖(𝑦)𝜓𝑖(𝑧)0,𝑦𝐶,𝑖=1,2,,𝑁.(3.75) Then the sequence {𝑥𝑛} converges strongly to Π𝑥0.

If 𝐴𝑖=𝐴,𝜓𝑖=𝜓, and 𝑓𝑖=𝑓 for all 𝑖1 in Theorem 3.1, we obtain the following corollary.

Corollary 3.3. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝐴𝐶𝐸 be a continuous and monotone mapping, 𝜓𝐶 be a lower semicontinuous and convex function, 𝑓 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4), 𝐾𝑓,𝑟 be the mapping define by (2.10) where 𝑟>0, and let {𝑇𝑖}𝑖=1, {𝑆𝑖}𝑖=1 be countable families of closed and uniformly 𝐿𝑖, 𝜇𝑖-Lipschitz continuous, and asymptotically relatively nonexpansive mappings with sequence {𝑘𝑛},{𝜁𝑛}[1,);  𝑘𝑛1,𝜁𝑛1 such that =Ω(