Generalized Mixed Equilibrium Problems and Fixed Point Problem for a Countable Family of Total Quasi-𝜙-Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

The purpose of this paper is first to introduce the concept of total quasi-𝜙-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a countable family of total quasi-𝜙-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in an uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

1. Introduction

Throughout this paper, we denote by ℝ and ℝ⁺ the set of all real numbers and all nonnegative real numbers, respectively. We also assume that 𝐸 is a real Banach space, 𝐸^∗ is the dual space of 𝐸, 𝐶 is a nonempty closed convex subset of 𝐸, and ⟨⋅,⋅⟩ is the pairing between 𝐸 and 𝐸^∗. In the sequel, we denote the strong convergence and weak convergence of a sequence {𝑥_𝑛} by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively, and 𝐽∶𝐸→2^𝐸∗ is the normalized duality mapping defined by 𝐽(𝑥)=𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖=‖𝑥^∗‖,𝑥∈𝐸.(1.1) Let 𝜓∶𝐶→ℝ be a proper real-valued function, 𝐴∶𝐶→𝐸^∗ a nonlinear mapping, and 𝐹∶𝐶×𝐶→ℝ a bifunction. The “so called” generalized mixed equilibrium problem for 𝐹,𝐴,𝜓 is to find 𝑥^∗∈𝐶 such that 𝐹𝑥^∗,𝑦+⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩+𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶.(1.2) We denote the set of solutions of (1.2) by GMEP(𝐹,𝐴,𝜓), that is, GMEP(𝐹,𝐴,𝜓)=𝑥^∗∈𝐶∶𝐹𝑥^∗,𝑦+⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩+𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶.(1.3)

Special Examples
(i)If 𝐴=0, then the problem (1.2) is reduced to the mixed equilibrium problem (MEP), and the set of its solutions is denoted by MEP(𝐹,𝜓)=𝑥^∗∈𝐶∶Θ𝑥^∗,𝑦+𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶.(1.4)(ii) If 𝜓≡0, then the problem (1.2) is reduced to the generalized equilibrium problem (GEP), and the set of its solutions is denoted by GEP(𝐹,𝐴)=𝑥^∗∈𝐶∶𝐹𝑥^∗,𝑦+⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩≥0,∀𝑦∈𝐶.(1.5)(iii)If 𝐴=0,𝜓=0, then the problem (1.2) is reduced to the equilibrium problem (EP), and the set of its solutions is denoted by EP(𝐹)=𝑥^∗∈𝐶∶𝐹𝑥^∗,𝑦≥0,∀𝑦∈𝐶.(1.6)(iv) If 𝐹=0, then the problem (1.2) is reduced to the mixed variational inequality of Browder type (VI), and the set of its solutions is denoted by VI(𝐶,𝐴,𝜓)=𝑥^∗∈𝐶∶⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩+𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶.(1.7)

These show that the problem (1.2) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.2). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach space (see, e.g., [1–5]).

A Banach space 𝐸 is said to be strictly convex if ‖𝑥+𝑦‖/2<1 for all 𝑥,𝑦∈𝑈={𝑧∈𝐸∶‖𝑧‖=1} with 𝑥≠𝑦. 𝐸 is said to be uniformly convex if, for each 𝜖∈(0,2], there exists 𝛿>0 such that ‖𝑥+𝑦‖/2<1−𝛿 for all 𝑥,𝑦∈𝑈 with ‖𝑥−𝑦‖≥𝜖. 𝐸 is said to be smooth if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(1.8) exists for all 𝑥,𝑦∈𝑈. 𝐸 is said to be uniformly smooth if the above limit exists uniformly in 𝑥,𝑦∈𝑈.

Remark 1.1. The following basic properties for Banach space 𝐸 and for the normalized duality mapping 𝐽 can be found in Cioranescu [6].(i)If 𝐸 is an arbitrary Banach space, then 𝐽 is monotone and bounded;(ii)If 𝐸 is a strictly convex Banach space, then 𝐽 is strictly monotone;(iii)If 𝐸 is a a smooth Banach space, then 𝐽 is single-valued, and hemicontinuous; that is, 𝐽 is continuous from the strong topology of 𝐸 to the weak star topology of 𝐸^∗;(iv) If 𝐸 is a uniformly smooth Banach space, then 𝐽 is uniformly continuous on each bounded subset of 𝐸;(v)If 𝐸 is a reflexive and strictly convex Banach space with a strictly convex dual 𝐸^∗ and 𝐽^∗∶𝐸^∗→𝐸 is the normalized duality mapping in 𝐸^∗, then 𝐽⁻¹=𝐽^∗,𝐽𝐽^∗=𝐼_𝐸^∗ and 𝐽^∗𝐽=𝐼_𝐸;(vi) If 𝐸 is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping 𝐽 is single valued, one to one and onto;(vii) A Banach space 𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex. If 𝐸 is uniformly smooth, then it is smooth and reflexive.

Recall that a Banach space 𝐸 has the Kadec-Klee property, if for any sequence {𝑥_𝑛}⊂𝐸 and 𝑥∈𝐸 with 𝑥_𝑛⇀𝑥∈𝐸 and ‖𝑥_𝑛‖→‖𝑥‖, then 𝑥_𝑛→𝑥 (as 𝑛→∞). It is well known that if 𝐸 is a uniformly convex Banach space, then 𝐸 has the Kadec-Klee property.

Next we assume that 𝐸 is a smooth, strictly convex and reflexive Banach space and 𝐶 is a nonempty closed convex subset of 𝐸. In the sequel, we always use 𝜙∶𝐸×𝐸→ℝ⁺ to denote the Lyapunov functional defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸.(1.9)

It is obvious from the definition of 𝜙 that (‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)²,∀𝑥,𝑦∈𝐸.(1.10)

Following Alber [7], the generalized projection Π_𝐶∶𝐸→𝐶 is defined by Π_𝐶(𝑥)=arginf_𝑦∈𝐶𝜙(𝑦,𝑥),∀𝑥∈𝐸.(1.11)

Let 𝑇∶𝐶→𝐶 be a mapping and 𝐹(𝑇) be the set of fixed points of 𝑇.

Recall that a point 𝑝∈𝐶 is said to be an asymptotic fixed point of 𝑇 if there exists a sequence {𝑥_𝑛}⊂𝐶 such that 𝑥_𝑛⇀𝑝 and ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0. We denoted the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇). A point 𝑝∈𝐶 is said to be a strong asymptotic fixed point of 𝑇, if there exists a sequence {𝑥_𝑛}⊂𝐶 such that 𝑥_𝑛→𝑝 and ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0. We denoted the set of all strong asymptotic fixed points of 𝑇 by 𝐹(𝑇).

Definition 1.2. (1) A mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.12)
(2) A mapping 𝑇∶𝐶→𝐶 is said to be relatively nonexpansive [8] if 𝐹(𝑇)≠∅,𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(1.13)
(3) A mapping 𝑇∶𝐶→𝐶 is said to be weak relatively nonexpansive [9] if 𝐹(𝑇)≠∅,𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(1.14)
(4) A mapping 𝑇∶𝐶→𝐶 is said to be closed, if for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, then 𝑇𝑥=𝑦.

Definition 1.3. (1) A mapping 𝑇∶𝐶→𝐶 is said to be quasi-𝜙-nonexpansive [10] if 𝐹(𝑇)≠∅ and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(1.15)
(2) A mapping 𝑇∶𝐶→𝐶 is said to be quasi-𝜙-asymptotically nonexpansive [11], if 𝐹(𝑇)≠∅ and there exists a real sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that 𝜙(𝑝,𝑇^𝑛𝑥)≤𝑘_𝑛𝜙(𝑝,𝑥),∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹(𝑇).(1.16)
(3) A mapping 𝑇∶𝐶→𝐶 is said to be uniformly 𝐿-Lipschitz continuous, if there exists a constant 𝐿>0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶,∀𝑛≥1.(1.17)

Definition 1.4. (1) A mapping 𝑇∶𝐶→𝐶 is said to be total quasi-𝜙-asymptotically nonexpansive if 𝐹(𝑇)≠∅ and there exist nonnegative real sequences {𝜈_𝑛},{𝜇_𝑛} with 𝜈_𝑛→0,𝜇_𝑛→0 (as 𝑛→∞) and a strictly increasing continuous function 𝜁∶ℝ⁺→ℝ⁺ with 𝜁(0)=0 such that for all 𝑥∈𝐶,𝑃∈𝐹(𝑇)𝜙(𝑝,𝑇^𝑛𝑥)≤𝜙(𝑝,𝑥)+𝜈_𝑛𝜁(𝜙(𝑝,𝑥))+𝜇_𝑛,∀𝑛≥1.(1.18)
(2) A countable family of mappings {𝑇_𝑛}∶𝐶→𝐶 is said to be uniformly total quasi-𝜙-asymptotically nonexpansive, if ⋂^∞𝑖=1𝐹(𝑇_𝑖)≠∅ and there exist nonnegative real sequences {𝜈_𝑛},{𝜇_𝑛} with 𝜈_𝑛→0,𝜇_𝑛→0 (as 𝑛→∞) and a strictly increasing continuous function 𝜁∶ℝ⁺→ℝ⁺ with 𝜁(0)=0 such that for all 𝑥∈𝐶,𝑝∈⋂^∞𝑖=1𝐹(𝑇_𝑖)𝜙𝑝,𝑇^𝑛𝑖𝑥≤𝜙(𝑝,𝑥)+𝜈_𝑛𝜁(𝜙(𝑝,𝑥))+𝜇_𝑛,∀𝑛≥1.(1.19)

Remark 1.5. From the definition, it is easy to know that(1)each relatively nonexpansive mapping is closed;(2) taking 𝜁(𝑡)=𝑡,𝑡≥0,𝜈_𝑛=(𝑘_𝑛−1) and 𝜇_𝑛=0, then (1.16) can be rewritten as 𝜙𝑝,𝑇^𝑛𝑖𝑥≤𝜙(𝑝,𝑥)+𝜈_𝑛𝜁(𝜙(𝑝,𝑥))+𝜇_𝑛,∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹(𝑇).(1.20) This implies that each quasi-𝜙-asymptotically nonexpansive mapping must be a total quasi-𝜙-asymptotically nonexpansive mapping, but the converse is not true;(3) the class of quasi-𝜙-asymptotically nonexpansive mappings contains properly the class of quasi-𝜙-nonexpansive mappings as a subclass, but the converse is not true;(4)the class of quasi-𝜙-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse is not true;(5)the class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.

A mapping 𝐴∶𝐶→𝐸^∗ is said to be 𝛼-inverse strongly monotone, if there exists 𝛼>0 such that ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖².(1.21)

Remark 1.6. If 𝐴 is an 𝛼-inverse strongly monotone mapping, then it is 1/𝛼-Lipschitz continuous.
Iterative approximation of fixed points for relatively nonexpansive mappings in the setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushita and Takahashi [12] obtained some weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Ofoedu and Malonza [4], Zhang [5], Su et al. [13], Zegeye and Shahzad [14], Wattanawitoon and Kumam [15], Qin et al. [16], Takahashi and Zembayashi [17], Chang et al. [18, 19], Yao et al. [20, 21], Qin et al. [22], and Cho et al. [23, 24] extend the notions from relatively nonexpansive mappings, weakly relatively nonexpansive mappings or quasi-𝜙-nonexpansive mappings to quasi-𝜙-asymptotically nonexpansive mappings and also prove some strongence theorems to approximate a common fixed point of quasi-𝜙-nonexpansive mappings or quasi-𝜙-asymptotically nonexpansive mappings.

The purpose of this paper is first to introduce the concept of total quasi-𝜙-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases, and then by using a hybrid algorithm to introduce a new iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a countable family of total quasi-𝜙-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results improve and extend the corresponding results in [8, 11–25].

2. Preliminaries

First, we recall some definitions and conclusions.

Lemma 2.1 (see [7, 26]). Let 𝐸 be a smooth, strictly convex and reflexive Banach space and 𝐶 be a nonempty closed convex subset of 𝐸. Then the following conclusions hold:(a)𝜙(𝑥,Π_𝐶𝑦)+𝜙(Π_𝐶𝑦,𝑦)≤𝜙(𝑥,𝑦) for all 𝑥∈𝐶 and 𝑦∈𝐸;(b) if 𝑥∈𝐸 and 𝑧∈𝐶, then 𝑧=Π_𝐶𝑥⟺⟨𝑧−𝑦,𝐽𝑥−𝐽𝑧⟩≥0,∀𝑦∈𝐶;(2.1)(c) for 𝑥,𝑦∈𝐸,𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦.

Remark 2.2. If 𝐸 is a real Hilbert space 𝐻, then 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖² and Π_𝐶 is the metric projection 𝑃_𝐶 of 𝐻 onto 𝐶.

Lemma 2.3 (see [18]). Let 𝐸 be a uniformly convex Banach space, 𝑟>0 a positive number, and 𝐵_𝑟(0) a closed ball of 𝐸. Then, for any given sequence {𝑥_𝑖}^∞𝑖=1⊂𝐵_𝑟(0) and for any given sequence {𝜆_𝑖}^∞𝑖=1 of positive numbers with ∑^∞𝑖=1𝜆_𝑖=1, then 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=1𝜆_𝑛𝑥_𝑛‖‖‖‖²≤^∞_𝑛=1𝜆_𝑛‖‖𝑥_𝑛‖‖²−𝜆_𝑖𝜆_𝑗𝑔‖‖𝑥_𝑖−𝑥_𝑗‖‖.(2.2)

Lemma 2.4. Let 𝐸 be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a closed and total quasi-𝜙-asymptotically nonexpansive mapping with nonnegative real sequences {𝜈_𝑛},{𝜇_𝑛} and a strictly increasing continuous functions 𝜁∶ℝ⁺→ℝ⁺ such that 𝜇₁=0,𝜈_𝑛→0,𝜇_𝑛→0 (as 𝑛→∞) and 𝜁(0)=0. Then 𝐹(𝑇) is a closed convex subset of 𝐶.

Proof. Letting {𝑝_𝑛} be a sequence in 𝐹(𝑇) with 𝑝_𝑛→𝑝 (as 𝑛→∞), we prove that 𝑝∈𝐹(𝑇). In fact, from the definition of 𝑇, we have 𝜙𝑝_𝑛,𝑇𝑝≤𝜙𝑝_𝑛,𝑝+𝜈₁𝜁𝜙𝑝_𝑛,𝑝+𝜇₁⟶0(as𝑛⟶∞).(2.3) Therefore we have lim_𝑛→∞𝜙𝑝_𝑛,𝑇𝑝=lim_𝑛→∞‖‖𝑝^𝑛‖‖²−2⟨𝑝_𝑛,𝐽𝑇𝑝⟩+‖𝑇𝑝‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑇𝑝⟩+‖𝑇𝑝‖²=𝜙(𝑝,𝑇𝑝)=0,(2.4) that is, 𝑝∈𝐹(𝑇).
Next we prove that 𝐹(𝑇) is convex. For any 𝑝,𝑞∈𝐹(𝑇),𝑡∈(0,1), putting 𝑤=𝑡𝑝+(1−𝑡)𝑞, we prove that 𝑤∈𝐹(𝑇). Indeed, in view of the definition of 𝜙(𝑥,𝑦), we have 𝜙(𝑤,𝑇^𝑛𝑤)=‖𝑤‖²−2⟨𝑤,𝐽𝑇^𝑛𝑤⟩+‖𝑇^𝑛𝑤‖²=‖𝑤‖²−2𝑡⟨𝑝,𝐽𝑇^𝑛𝑤⟩−2(1−𝑡)⟨𝑞,𝐽𝑇^𝑛𝑤⟩+‖𝑇^𝑛𝑤‖²=‖𝑤‖²+𝑡𝜙(𝑝,𝑇^𝑛𝑤)+(1−𝑡)𝜙(𝑞,𝑇^𝑛𝑤)−𝑡‖𝑝‖²−(1−𝑡)‖𝑞‖²≤−‖𝑤𝑡‖‖𝑝²‖+₂𝑡𝜙(𝑝,𝑤)+𝜈_𝑛𝜁(𝜙(𝑝,𝑤))+𝜇_𝑛+(1−𝑡)𝜙(𝑞,𝑤)+𝜈_𝑛𝜁(𝜙(𝑞,𝑤))+𝜇_𝑛−(1−𝑡)‖𝑞‖²=×‖𝑤‖‖𝑞²‖+₂𝑡‖𝑝‖²−2⟨𝑝,𝐽𝑤⟩+‖𝑤‖²+𝑡𝜈_𝑛𝜁(𝜙(𝑝,𝑤))+𝜇_𝑛+(1−𝑡)−2⟨𝑞,𝐽𝑤⟩+‖𝑤‖²+(1−𝑡)𝜈_𝑛𝜁(𝜙(𝑞,𝑤))+𝜇_𝑛−𝑡‖𝑝‖²−(1−𝑡)‖𝑞‖²=‖𝑤‖²−2⟨𝑤,𝐽𝑤⟩+‖𝑤‖²+𝑡𝜈_𝑛𝜁(𝜙(𝑝,𝑤))+(1−𝑡)𝜈_𝑛𝜁(𝜙(𝑞,𝑤))+𝜇_𝑛=𝑡𝜈_𝑛𝜁(𝜙(𝑝,𝑤))+(1−𝑡)𝜈_𝑛𝜁(𝜙(𝑞,𝑤))+𝜇_𝑛.(2.5) Since 𝜇_𝑛→0 and 𝜈_𝑛→0, we have 𝜙(𝑤,𝑇^𝑛𝑤)→0 (as 𝑛→∞). From (1.10) we have ‖𝑇^𝑛𝑤‖→‖𝑤‖. Consequently ‖𝐽𝑇^𝑛𝑤‖→‖𝐽𝑤‖. This implies that {𝐽𝑇^𝑛𝑤} is a bounded sequence. Since 𝐸 is reflexive, 𝐸^∗ is also reflexive. So we can assume that 𝐽𝑇^𝑛𝑤⇀𝑓₀∈𝐸^∗.(2.6) Again since 𝐸 is reflexive, we have 𝐽(𝐸)=𝐸^∗. Therefore there exists 𝑥∈𝐸 such that 𝐽𝑥=𝑓₀. By virtue of the weakly lower semicontinuity of norm ‖⋅‖, we have 0=liminf_𝑛→∞𝜙(𝑤,𝑇^𝑛𝑤)==lliimmiinnf_𝑛→∞f_𝑛→∞‖𝑤‖²−2⟨𝑤,𝐽(𝑇^𝑛𝑤)⟩+‖𝑇^𝑛𝑤‖²≥‖𝑤‖₂‖𝑤‖²−2⟨𝑤,𝐽(𝑇^𝑛𝑤)⟩+‖𝐽(𝑇^𝑛𝑤)‖²−2⟨𝑤,𝑓₀⟩+‖‖𝑓₀‖‖²=‖𝑤‖²−2⟨𝑤,𝐽𝑥⟩+‖𝐽𝑥‖²=‖𝑤‖²−2⟨𝑤,𝐽𝑥⟩+‖𝑥‖²=𝜙(𝑤,𝑥),(2.7) that is, 𝑤=𝑥 which implies that 𝑓₀=𝐽𝑤. Hence from (2.6) we have 𝐽𝑇^𝑛𝑤⇀𝐽𝑤∈𝐸^∗. Since ‖𝐽𝑇^𝑛𝑤‖→‖𝑤‖ and 𝐸^∗ has the Kadec-Klee property, we have 𝐽𝑇^𝑛𝑤→𝐽𝑤. Since 𝐸 is uniformly smooth, 𝐸^∗ is uniformly convex, which in turn implies that 𝐸^∗ is smooth. From Remark 1.1(iii) it yields that 𝐽⁻¹∶𝐸^∗→𝐸 is hemi-continuous. Therefore we have 𝑇^𝑛𝑤⇀𝑤. Again since ‖𝑇^𝑛𝑤‖→‖𝑤‖, by using the Kadec-Klee property of 𝐸, we have 𝑇^𝑛𝑤→𝑤. This implies that 𝑇𝑇^𝑛𝑤=𝑇^𝑛+1𝑤→𝑤. Since 𝑇 is closed, we have 𝑤=𝑇𝑤.
This completes the proof of Lemma 2.4.

Lemma 2.5. Let 𝐸 be a smooth, strictly convex and reflexive Banach space and 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑓∶𝐶×𝐶→ℝ be a bifunction satisfying the following conditions:(𝐴1) 𝑓(𝑥,𝑥)=0,forall𝑥∈𝐶,(𝐴2)𝑓 is monotone, that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐶,(𝐴3)limsup_𝑡↓0𝑓(𝑥+𝑡(𝑧−𝑥),𝑦)≤𝑓(𝑥,𝑦)forall𝑥,𝑧,𝑦∈𝐶,(𝐴4) The function 𝑦↦𝑓(𝑥,𝑦) is convex and lower semi-continuous.
Then the following conclusions hold:(1) (Blum and Oettli [27]) for any given 𝑟>0 and 𝑥∈𝐸, there exists a unique 𝑧∈𝐶 such that 𝑓(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶;(2.8)(2) (Takahashi and Zembayashi [28]) for any given 𝑟>0 and 𝑥∈𝐸, define a mapping 𝐾^𝑓𝑟∶𝐸→𝐶 by 𝐾^𝑓𝑟(𝑥)=𝑧∈𝐶∶𝑓(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶,𝑥∈𝐸.(2.9) Then, the following conclusions hold:(a)𝐾^𝑓𝑟 is single-valued;(b)𝐾^𝑓𝑟 is firmly nonexpansive-type mapping, that is, for all𝑧,𝑦∈𝐸, 𝐾^𝑓𝑟𝑧−𝐾^𝑓𝑟𝑦,𝐽𝐾^𝑓𝑟𝑧−𝐽𝐾^𝑓𝑟𝑦≤𝐾^𝑓𝑟𝑧−𝐾^𝑓𝑟𝑦,𝐽𝑧−𝐽𝑦;(2.10)(c)𝐹(𝐾^𝑓𝑟)=EP(𝑓) and 𝐾^𝑓𝑟 is quasi-𝜙-nonexpansive;(d)EP(𝑓) is closed and convex;(e)𝜙(𝑞,𝐾^𝑓𝑟𝑥)+𝜙(𝐾^𝑓𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥),forall𝑞∈𝐹(𝐾^𝑓𝑟).
For solving the generalized mixed equilibrium problem (1.2), let us assume that the following conditions are satisfied:(1)𝐸 is a smooth, strictly convex, and reflexive Banach space and 𝐶 is a nonempty closed convex subset of 𝐸;(2)𝐴∶𝐶→𝐸^∗ is 𝛽-inverse strongly monotone mapping;(3)𝐹∶𝐶×𝐶→ℝ is bifunction satisfying the conditions (A1), (A3), (A4) in Lemma 2.5 and the following condition (𝐴2)^:(𝐴2)^ for some 𝛾≥0 with 𝛾≤𝛽𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤𝛾‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐶;(2.11)(4)𝜓∶𝐶→ℝ is a lower semicontinuous and convex function.
Under the assumptions as above, we have the following results.

Lemma 2.6. Let 𝐸,𝐶,𝐴,𝐹,𝜓 satisfy the above conditions (1)–(4). Denote by Γ(𝑥,𝑦)=𝐹(𝑥,𝑦)+𝜓(𝑦)−𝜓(𝑥)+⟨𝐴𝑥,𝑦−𝑥⟩,∀𝑥,𝑦∈𝐶.(2.12) For any given 𝑟>0 and 𝑥∈𝐸, define a mapping 𝐾^Γ𝑟∶𝐸→𝐶 by 𝐾^Γ𝑟(𝑥)=𝑧∈𝐶∶Γ(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.13) Then, the following hold:(a)𝐾^Γ𝑟 is single-valued;(b)𝐾^Γ𝑟 is a firmly nonexpansive-type mapping, that is, for all 𝑧,𝑦∈𝐸, 𝐾^Γ𝑟(𝑧)−𝐾^Γ𝑟(𝑦),𝐽𝐾^Γ𝑟(𝑧)−𝐽𝐾^Γ_r(𝑦)≤𝐾^Γ𝑟(𝑧)−𝐾^Γ𝑟(𝑦),𝐽𝑧−𝐽𝑦;(2.14)(c)𝐹(𝐾^Γ𝑟)=EP(Γ)=GMEP(𝐹,𝐴,𝜓);(d)GMEP(𝐹,𝐴,𝜓) is closed and convex;(e)  𝜙𝑞,𝐾^Γ𝑟𝑥+𝜙𝐾^Γ𝑟𝑥,𝑥≤𝜙(𝑞,𝑥),∀𝑞∈𝐹𝐾^Γ𝑟.(2.15)

Proof. It follows from Lemma 2.5 that in order to prove the conclusions of Lemma 2.6 it is sufficient to prove that the function Γ∶𝐶×𝐶→ℝ satisfies the conditions (A1)–(A4) in Lemma 2.5.
In fact, by the similar method as given in the proof of Lemma 2.4 in [1], we can prove that the function Γ satisfies the conditions (A1), (A3), and (A4). Now we prove that Γ also satisfies the conditions (A2).
Indeed, for any 𝑥,𝑦∈𝐶, by condition (A2)′, we have Γ(𝑥,𝑦)+Γ(𝑦,𝑥)=≤=+𝐹𝐹(𝛾𝐹((−𝑥𝑥(,,𝑦𝛽𝑦𝑦,)))𝑥‖++)𝐴+𝐹𝜓𝑥𝜓((−𝑦𝑦(),𝐴𝑥𝑥−)𝑦)−‖𝜓−₂𝜓(⟨𝑥𝐴()𝑦𝑥+)−+⟨𝐴𝐴⟨𝐴𝑥𝑦,,𝑦𝑦𝑥,𝑥−−−𝑥𝑦⟩⟩𝑦⟩≤0.(2.16) This implies that the function Γ satisfies the conditions (A2). Therefore the conclusions of Lemma 2.6 can be obtained from Lemma 2.3 immediately.

Remark 2.7. It follows from Lemma 2.5 that the mapping 𝐾^Γ𝑟 is a relatively nonexpansive mapping. Thus, it is quasi-𝜙-nonexpansive.

3. Main Results

In this section, we shall use the hybrid method to prove some strong convergence theorems for finding a common element of the set of solutions for a system of the generalized mixed equilibrium problems (1.2) and the set of common fixed points of a countable family of total quasi-𝜙-asymptotically nonexpansive mappings in Banach spaces.

In the sequel, we assume that 𝐸,𝐶,{𝑆_𝑖}^∞𝑖=1,{𝐴_𝑖}^𝑀𝑖=1,{𝐹_𝑖}^𝑀𝑖=1,{𝜓_𝑖}^𝑀𝑖=1 satisfy the following conditions.(1)Let 𝐸 be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and 𝐶 a nonempty closed convex subset of 𝐸.(2) Let 𝑆_𝑖∶𝐶→𝐶 be a countable family of closed and uniformly total quasi-𝜙-asymptotically nonexpansive mappings with nonnegative real sequences {𝜈_𝑛},{𝜇_𝑛} and a strictly increasing continuous functions 𝜁∶ℝ⁺→ℝ⁺ such that 𝜈_𝑛→0,𝜇_𝑛→0 (as 𝑛→∞) and 𝜇₁=0,𝜁(0)=0. Suppose further that for each 𝑖≥1,𝑆_𝑖 is a uniformly 𝐿_𝑖-Lipschitz mapping, that is, there exists a constant 𝐿_𝑖>0 such that ‖‖𝑆^𝑛𝑖𝑥−𝑆^𝑛𝑖𝑦‖‖≤𝐿_𝑖‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶,∀𝑛≥1.(3.1)(3) Let 𝐴_𝑖∶𝐶→𝐸^∗(𝑖=1,2,…,𝑀) be a finite family of 𝛽_𝑖-inverse strongly monotone mappings.(4) Let 𝐹_𝑖∶𝐶→ℝ(𝑖=1,2,…,𝑀) be a finite family of bifunction satisfying the conditions (A1), (A3), (A4), and the following condition (A2)′:(𝐴2)^ For each 𝑖=1,2,…,𝑀 there exists 𝛾_𝑖≥0 with 𝛾_𝑖≤𝛽_𝑖 such that 𝐹_𝑖(𝑥,𝑦)+𝐹_𝑖(𝑦,𝑥)≤𝛾_𝑖‖‖𝐴_𝑖𝑥−𝐴_𝑖𝑦‖‖²,∀𝑥,𝑦∈𝐶;(3.2)(5) Let 𝜓_𝑖∶𝐶→ℝ(𝑖=1,2,…,𝑀) be a finite family of lower semicontinuous and convex functions.

Theorem 3.1. Let 𝐸,𝐶,{𝑆_𝑖}^∞𝑖=1,{𝐴_𝑖}^𝑀𝑖=1,{𝐹_𝑖}^𝑀𝑖=1,{𝜓_𝑖}^𝑀𝑖=1 be the same as above. Suppose that ℱ∶=^∞_𝑖=1𝐹𝑇_𝑖^𝑀_𝑗=1GMEP𝐹_𝑗,𝐴_𝑗,𝜓_𝑗(3.3) is a nonempty and bounded subset of 𝐶. For any given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑥𝑧₀∈𝐶₀=𝐶,_𝑛=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛𝑦_𝑛,𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛,𝐶=_𝑛+1𝐾^Γ𝑀_{𝑟𝑀,𝑛}𝐾^Γ𝑀−1_{𝑟M−1,𝑛}⋯𝐾^Γ2_𝑟2,𝑛𝐾^Γ1_𝑟1,𝑛𝑦_𝑛,𝑥_𝑛+1=𝑣∈𝐶_𝑛∶𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+𝜂_𝑛,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.4) where 𝜂_𝑛=𝜈_𝑛sup_𝑢∈ℱ𝜁𝜙𝑢,𝑥_𝑛+𝜇_𝑛,∀𝑛≥1,(3.5)𝐾^Γ𝑖_{𝑟𝑖,𝑛}∶𝐸→𝐶,𝑖=1,2,…,𝑀 is the mapping defined by (2.13) with Γ=Γ_𝑖,𝑟=𝑟_𝑖,𝑛, and Γ_𝑖(𝑥,𝑦)=𝐹_𝑖(𝑥,𝑦)+⟨𝐴_𝑖𝑥,𝑦−𝑥⟩+𝜓_𝑖(𝑦)−𝜓_𝑖(𝑥),∀𝑥,𝑦∈𝐶.(3.6)𝑟_𝑘,𝑛∈[𝑑,∞),𝑘=1,2,…,𝑀,𝑛≥1 for some 𝑑>0,Π_𝐶𝑛+1 is the generalized projection of 𝐸 onto the set 𝐶_𝑛+1,and {𝛼_𝑛,𝑖},{𝛼_𝑛} are sequences in [0,1] satisfying the following conditions:(a)∑^∞𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0;(b)liminf_𝑛→∞𝛼_𝑛,0⋅𝛼_𝑛,𝑖>0 for all 𝑖≥1;(c)0<𝛼≤𝛼_𝑛<1 for some 𝛼∈(0,1).Then {𝑥_𝑛} converges strongly to Π_ℱ𝑥₀, where Π_ℱ is the generalized projection from 𝐸 onto ℱ.

Proof. We divide the proof of Theorem 3.1 into five steps.
(i) We first prove that ℱ and 𝐶_𝑛 both are closed and convex subset of 𝐶 for all 𝑛≥0.
In fact, it follows from Lemmas 2.4 and 2.6 that 𝐹(𝑆_𝑖),𝑖≥1 and GMEP(𝐹_𝑗,𝐴_𝑗,𝜓_𝑗)(𝑗=1,2,…,𝑀) both are closed and convex. Therefore ℱ is a closed and convex subset in 𝐶. Furthermore, it is obvious that 𝐶₀=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for some 𝑛≥1. Since the inequality 𝜙(𝑣,𝑢_𝑛)≤𝜙(𝑣,𝑥_𝑛)+𝜂_𝑛 is equivalent to 2⟨𝑣,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖‖𝑥_𝑛‖‖²−‖‖𝑢_𝑛‖‖²+𝜂_𝑛,(3.7) therefore, we have𝐶_𝑛+1=𝑣∈𝐶_𝑛∶2⟨𝑣,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖‖𝑥_𝑛‖‖²−‖‖𝑢_𝑛‖‖²+𝜂_𝑛.(3.8)
This implies that 𝐶_𝑛+1 is closed and convex. The desired conclusions are proved. These in turn show that Π_ℱ𝑥₀ and Π_𝐶𝑛𝑥₀ are well defined.
(ii) We prove that {𝑥_𝑛} and {𝑆^𝑛𝑖𝑥_𝑛}^∞𝑛=0 for all 𝑖≥1 are both bounded sequences in 𝐶.
By the definition of 𝐶_𝑛, we have 𝑥_𝑛=Π_𝐶𝑛𝑥₀ for all 𝑛≥0. It follows from Lemma 2.1 (a) that 𝜙𝑥_𝑛,𝑥₀=≤𝜙𝜙𝑢Π,𝑥_𝐶𝑛0𝑥₀,𝑥₀≤𝜙𝑢,𝑥₀−𝜙𝑢,Π_𝐶𝑛𝑥₀,∀𝑛≥0,𝑢∈ℱ.(3.9) This implies that {𝜙(𝑥_𝑛,𝑥₀)} is bounded. By virtue of (1.10), {𝑥_𝑛} is bounded. Since 𝜙(𝑢,𝑆^𝑛𝑖𝑥_𝑛)≤𝜙(𝑢,𝑥_𝑛)+𝜈_𝑛𝜁(𝜙(𝑢,𝑥_𝑛))+𝜇_𝑛 for all 𝑢∈ℱ and 𝑖≥1,{𝑆^𝑛𝑖𝑥_𝑛} is bounded for all 𝑖≥1, and so {𝑧_𝑛}is bounded in 𝐸. Denote 𝑀 by 𝑀=sup_{𝑛≥0,𝑖≥1}‖‖𝑥_𝑛‖‖,‖‖𝑆^𝑛𝑖𝑥_𝑛‖‖,‖‖𝑧_𝑛‖‖<∞.(3.10)
In view of the structure of {𝐶_𝑛}, we have 𝐶_𝑛+1⊂𝐶_𝑛,𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀. This implies that 𝑥_𝑛+1∈𝐶_𝑛 and 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀,∀𝑛≥1.(3.11) Therefore {𝜙(𝑥_𝑛,𝑥₀)} is convergent. Without loss of generality, we can assume that lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀=𝑟≥0.(3.12)(iii) Next, we prove that ℱ∶=⋂^∞𝑖=1𝐹(𝑆_𝑖)⋂⋂^𝑀𝑖=1GMEP(𝐹_𝑖,𝐴_𝑖,𝜓_𝑖)⊂𝐶_𝑛 for all 𝑛≥0.
Indeed, it is obvious that ℱ⊂𝐶₀=𝐶. Suppose that ℱ⊂𝐶_𝑛 for some 𝑛≥0. Since 𝑢_𝑛=𝐾^Γ𝑀_{𝑟𝑀,𝑛}𝐾^Γ𝑀−1_{𝑟𝑀−1,𝑛}⋯𝐾^Γ2_𝑟2,𝑛𝐾^Γ1_𝑟1,𝑛𝑦_𝑛, by Lemma 2.6 and Remark 2.7, 𝐾^Γ𝑖_{𝑟𝑖,𝑛} is quasi-𝜙-nonexpansive. Again since 𝐸 is uniformly smooth, 𝐸^∗ is uniformly convex. Hence, For any given 𝑢∈ℱ⊂𝐶_𝑛 and for any positive integer 𝑗>0, from Lemma 2.3 we have 𝜙𝑢,𝑢_𝑛==𝜙𝜙𝑢𝑢,,𝐽𝐾^Γ𝑀_{𝑟𝑀,𝑛−1}𝐾^Γ𝑀−1_{𝑟𝑀−1,𝑛}⋯𝐾^Γ2_𝑟2,𝑛𝐾^Γ1_𝑟1,𝑛𝑦_𝑛≤𝜙𝑢,𝑦^𝑛≤‖𝑢‖₂𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛−2𝑢,𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛+‖‖𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛‖‖²≤‖𝑢‖²−2𝑢,𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛+𝛼_𝑛‖‖𝐽𝑧_𝑛‖‖²+1−𝛼_𝑛‖‖𝐽𝑥_𝑛‖‖²=‖𝑢‖²−2𝑢,𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛+𝛼_𝑛‖‖𝑧_𝑛‖‖²+1−𝛼_𝑛‖‖𝑥_𝑛‖‖²==𝛼𝛼_𝑛𝑛𝜙𝑢,𝑧_𝑛+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛𝜙𝑢,𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛+1−𝛼^𝑛=𝛼_𝑛𝜙𝑢,𝑥_𝑛⎛⎜⎜⎝‖𝑢‖₂−2𝛼_𝑛,0⟨𝑢,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖𝑢,𝐽𝑆^𝑛𝑖𝑥_𝑛+‖‖‖‖𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛+1−𝛼_𝑛‖‖‖‖²⎞⎟⎟⎠≤𝛼_𝑛𝜙𝑢,𝑥_𝑛‖𝑢‖²−2𝛼_𝑛,0⟨𝑢,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖𝑢,𝐽𝑆^𝑛𝑖𝑥_𝑛+𝛼_𝑛,0‖‖𝐽𝑥_𝑛‖‖²+^∞_𝑖=1𝛼_𝑛,𝑖‖‖𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖²≤𝛼_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛‖𝑢‖²−2𝛼_𝑛,0⟨𝑢,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖𝑢,𝐽𝑆^𝑛𝑖𝑥_𝑛+𝛼_𝑛,0‖‖𝑥_𝑛‖‖²+^∞_𝑖=1𝛼_𝑛,𝑖‖‖𝑆^𝑛𝑖𝑥_𝑛‖‖²=𝛼_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛𝛼^𝑛,0𝜙𝑢,𝑥_𝑛+^∞_𝑖=1+1−𝛼_𝑛𝛼_𝑛,𝑖𝜙𝑢,𝑆^𝑛𝑖𝑥_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖≤𝛼_𝑛𝜙𝑢,𝑥_𝑛𝛼^𝑛,0𝜙𝑢,𝑥_𝑛+^∞_𝑖=1−𝛼_𝑛,0𝛼_𝑛,𝑖𝜙𝑢,𝑥_𝑛+𝜈_𝑛𝜁𝜙𝑢,𝑥_𝑛+𝜇_𝑛≤𝛼_𝑛𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛𝜙𝑢,𝑥^𝑛+1−𝛼_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝜈_𝑛𝜁𝜙𝑢,𝑥_𝑛+𝜇_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖≤𝜙𝑢,𝑥_𝑛𝜙𝑢,𝑥_𝑛=𝜙𝑢,𝑥_𝑛+𝛼_𝑛𝜈^𝑛sup_𝑢∈ℱ𝜁𝜙𝑢,𝑥_𝑛+𝜇_𝑛−𝛼^𝑛𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖≤𝜙𝑢,𝑥_𝑛+𝛼_𝑛𝜂_𝑛−𝛼_𝑛𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖+𝜂_𝑛.(3.13) Hence 𝑢∈𝐶_𝑛+1 and so ℱ⊂𝐶_𝑛 for all 𝑛≥0. By the way, from the definition of {𝜂_𝑛} and 𝜁 and (3.10), it is easy to see that 𝜂_𝑛=𝜈_𝑛sup_𝑢∈ℱ𝜁𝜙𝑢,𝑥_𝑛+𝜇_𝑛≤𝜈_𝑛sup_𝑢∈ℱ𝜁(‖𝑢‖+𝑀)²+𝜇_𝑛⟶0(as𝑛⟶∞).(3.14)(IV) Now, we prove that {𝑥_𝑛} converges strongly to some point 𝑝∈ℱ∶=^∞_𝑖=1𝐹𝑆_𝑖^∞_𝑗=1GMEP𝐹_𝑗,𝐴_𝑗,𝜓_𝑗.(3.15)
First, we prove that {𝑥_𝑛} converges strongly to some point 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖).
In fact, since {𝑥_𝑛} is bounded in 𝐶 and 𝐸 is reflexive, there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑝. Again since 𝐶_𝑛 is closed and convex for each 𝑛≥1, it is weakly closed, and so 𝑝∈𝐶_𝑛 for each 𝑛≥0. Since 𝑥_𝑛=Π_𝐶𝑛𝑥₀, from the definition of Π_𝐶𝑛, we have 𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝,𝑥₀,𝑛≥0.(3.16) Since liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀=≥‖li𝑝m‖i₂nf_{𝑛𝑖→∞}‖‖𝑥^𝑛_𝑖‖‖²−2𝑥_𝑛𝑖,𝐽𝑥₀+‖‖𝑥₀‖‖²−2⟨𝑝,𝐽𝑥₀⟩+‖‖𝑥₀‖‖²=𝜙𝑝,𝑥₀,(3.17) we have 𝜙𝑝,𝑥₀≤liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤limsup_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝,𝑥₀.(3.18) This implies that lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛𝑖,𝑥₀)=𝜙(𝑝,𝑥₀), that is, ‖𝑥_𝑛𝑖‖→‖𝑝‖. In view of the Kadec-Klee property of 𝐸, we obtain that lim_𝑛→∞𝑥_𝑛𝑖=𝑝.
Now we prove that 𝑥_𝑛→𝑝(𝑛→∞). In fact, if there exists a subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑗→𝑞, then we have 𝜙(𝑝,𝑞)==lliimm_{𝑛𝑖→∞,𝑛𝑗→∞𝑛𝑖→∞,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥_𝑛𝑗≤lim_{𝑛𝑖→∞,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖𝑥₀−𝜙Π_𝐶𝑛𝑗𝑥₀,𝑥₀𝜙𝑥_𝑛𝑖,𝑥₀−𝜙𝑥_𝑛𝑗,𝑥₀=0(by(3.12)).(3.19) Therefore we have 𝑝=𝑞. This implies that lim_𝑛→∞𝑥_𝑛=𝑝.(3.20)
Now we prove that 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖). In fact, by the construction of 𝐶_𝑛, we have that 𝐶_𝑛+1⊂𝐶_𝑛 and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀. Therefore by Lemma 2.1(a) we have 𝜙𝑥_𝑛+1,𝑥_𝑛=≤𝜙𝜙𝑥𝑥_𝑛+1𝑛+1,Π_𝐶𝑛𝑥₀=𝜙𝑥_𝑛+1,𝑥₀−𝜙Π_𝐶𝑛𝑥₀,𝑥₀,𝑥₀−𝜙𝑥_𝑛,𝑥₀⟶0(as𝑛⟶∞).(3.21) In view of 𝑥_𝑛+1∈𝐶_𝑛 and noting the construction of 𝐶_𝑛+1 we obtain that 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝜂_𝑛⟶0(as𝑛⟶∞).(3.22) From (1.10) it yields (‖𝑥_𝑛+1‖−‖𝑢_𝑛‖)²→0. Since‖𝑥_𝑛+1‖→‖𝑝‖, we have ‖‖𝑢_𝑛‖‖⟶‖𝑝‖(as𝑛⟶∞).(3.23) Hence we have ‖‖𝐽𝑢_𝑛‖‖⟶‖𝐽𝑝‖(as𝑛⟶∞).(3.24)
This implies that {𝐽𝑢_𝑛} is bounded in 𝐸^∗. Since 𝐸 is reflexive, and so 𝐸^∗ is reflexive, there exists a subsequence {𝐽𝑢_𝑛𝑖}⊂{𝐽𝑢_𝑛} such that 𝐽𝑢_𝑛𝑖⇀𝑝₀∈𝐸^∗. In view of the reflexive of 𝐸, we see that 𝐽(𝐸)=𝐸^∗. Hence there exists 𝑥∈𝐸 such that 𝐽𝑥=𝑝₀. Since 𝜙𝑥_𝑛𝑖+1,𝑢_𝑛𝑖=‖‖𝑥_𝑛𝑖+1‖‖²−2𝑥_𝑛𝑖+1,𝐽𝑢_𝑛𝑖+‖‖𝑢_𝑛𝑖‖‖²=‖‖𝑥_𝑛𝑖+1‖‖²−2𝑥_𝑛𝑖+1,𝐽𝑢_𝑛𝑖+‖‖𝐽𝑢_𝑛𝑖‖‖²(3.25) taking liminf_𝑛→∞ on the both sides of above equality and in view of the weak lower semicontinuity of norm ‖⋅‖, then it yields that 0≥‖𝑝‖²−2⟨𝑝,𝑝₀⟩+‖‖𝑝₀‖‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝐽𝑥‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝑥‖²=𝜙(𝑝,𝑥).(3.26) That is 𝑝=𝑥. This implies that 𝑝₀=𝐽𝑝, and so 𝐽𝑢_𝑛⇀𝐽𝑝. It follows from (3.24) and the Kadec-Klee property of 𝐸^∗ that 𝐽𝑢_𝑛𝑖→𝐽𝑝 (as 𝑛→∞). Note that 𝐽⁻¹∶𝐸^∗→𝐸 is hemi-continuous, it yields that 𝑢_𝑛𝑖⇀𝑝. It follows from (3.23) and the Kadec-Klee property of 𝐸 that lim_{𝑛𝑖→∞}𝑢_𝑛𝑖=𝑝.
By the similar way as given in the proof of (3.20), we can also prove that lim_𝑛→∞𝑢_𝑛=𝑝.(3.27)
From (3.20) and (3.27) we have that ‖‖𝑥_𝑛−𝑢_𝑛‖‖⟶0(as𝑛⟶∞).(3.28) Since 𝐽 is uniformly continuous on any bounded subset of 𝐸, we have ‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖‖⟶0(as𝑛⟶∞).(3.29) For any 𝑗≥1 and any 𝑢∈ℱ, it follows from (3.13), (3.20), and (3.27) that 𝛼_𝑛𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢_𝑛+𝛼_𝑛𝜂_𝑛.(3.30) Since 𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢_𝑛=≤‖‖||‖‖𝑥𝑥_𝑛𝑛‖‖²−‖‖𝑢_𝑛‖‖²−2⟨𝑢,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖‖𝑥_𝑛‖‖²−‖‖𝑢_𝑛‖‖²||+2‖𝑢‖⋅‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖‖−𝑢_𝑛‖‖‖‖𝑥_𝑛‖‖+‖‖𝑢_𝑛‖‖+2‖𝑢‖⋅‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖‖,(3.31) from (3.28) and (3.29), it follows that 𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢_𝑛⟶0(𝑛⟶∞).(3.32) In view of condition (b) and condition (c), we have that 𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖⟶0(as𝑛⟶∞).(3.33) It follows from the property of 𝑔 that ‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖⟶0,(as𝑛⟶∞).(3.34) Since 𝑥_𝑛→𝑝 and 𝐽 is uniformly continuous, it yields 𝐽𝑥_𝑛→𝐽𝑝. Hence from (3.34) we have 𝐽𝑆^𝑛𝑗𝑥_𝑛⟶𝐽𝑝(as𝑛⟶∞).(3.35) Since 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous, it follows that 𝑆^𝑛𝑗𝑥_𝑛⇀𝑝(∀𝑗≥1).(3.36) On the other hand, for each 𝑗≥1 we have ||‖‖𝑆^𝑛𝑗𝑥_𝑛‖‖−‖𝑝‖||=||‖‖𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖−‖𝐽𝑝‖||≤‖‖𝐽𝑆^𝑛𝑗𝑥_𝑛−𝐽𝑝‖‖⟶0(as𝑛⟶∞).(3.37) This together with (3.36) shows that 𝑆^𝑛𝑗𝑥_𝑛⟶𝑝(foreach𝑗≥1).(3.38)
Furthermore, by the assumption that for each 𝑗≥1,𝑆_𝑗 is uniformly 𝐿_𝑖-Lipschitz continuous, hence we have ‖‖𝑆^𝑗𝑛+1𝑥_𝑛−𝑆^𝑛𝑗𝑥_𝑛‖‖≤+‖‖‖‖𝑆𝑥^𝑗𝑛+1_𝑛+1𝑥_𝑛−𝑆^𝑗𝑛+1𝑥_𝑛+1‖‖+‖‖𝑆^𝑗𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖≤𝐿_𝑗−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑆^𝑛𝑗𝑥_𝑛‖‖+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑆^𝑗𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛−𝑆^𝑛𝑗𝑥_𝑛‖‖.(3.39) This together with (3.20) and (3.38), yields ‖𝑆^𝑗𝑛+1𝑥_𝑛−𝑆^𝑛𝑗𝑥_𝑛‖→0 (as 𝑛→∞). Hence from (3.36) we have 𝑆^𝑗𝑛+1𝑥_𝑛→𝑝, that is, 𝑆_𝑗𝑆^𝑛𝑗𝑥_𝑛→𝑝. In view of (3.38) and the closeness of 𝑆_𝑗, it yields that 𝑆_𝑗𝑝=𝑝,forall𝑗≥1. This implies that 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖).
Next, we prove that 𝑝∈⋂^𝑀𝑖=1GMEP(𝐹_𝑖,𝐴_𝑖,𝜓_𝑖). Denote that 𝑢^𝑛(𝑚)=𝐾^Γ𝑚_{𝑟𝑚,𝑛}𝐾^Γ𝑚−1_{𝑟𝑚−1,𝑛}⋯𝐾^Γ2_𝑟2,𝑛𝐾^Γ1_𝑟1,𝑛𝑦_𝑛,𝑚=1,2,…,𝑀−1,𝑢^𝑛(𝑀)=𝑢_𝑛.(3.40) By the similar method as in the proof of (3.13), we can prove that 𝜙𝑢,𝑢^𝑛(𝑚)≤𝜙𝑢,𝑥^𝑛+𝜂_𝑛,𝑚=1,2,…,𝑀,𝑢∈ℱ,∀𝑛≥1.(3.41) It follows from Lemma 2.6, (2.15), (3.32) that for any 𝑢∈ℱ, 𝜙𝑢^𝑛(𝑀),𝑢^{𝑛(𝑀−1)}=≤𝜙𝜙𝐾𝑢,𝑢^{Γ𝑀𝑟𝑛}𝑢^{𝑛(𝑀−1)}_{𝑛(𝑀−1)},𝑢^{𝑛(𝑀−1)}≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝐾^{Γ𝑀𝑟𝑛}𝑢^{𝑛(𝑀−1)}=𝜙𝑢,𝑥_𝑛+𝜂_𝑛−𝜙𝑢,Γ^𝑀𝑟_𝑛𝑢^{𝑛(𝑀−1)}=𝜙𝑢,𝑥_𝑛+𝜂_𝑛−𝜙𝑢,𝑢^𝑛(𝑀)+𝜂_𝑛−𝜙𝑢,𝑢_𝑛⟶0(as𝑛⟶∞).(3.42) From (1.10) it yields (‖𝑢^𝑛(𝑀)‖−‖𝑢^{𝑛(𝑀−1)}‖)²→0. Since ‖𝑢^𝑛(𝑀)‖=‖𝑢_𝑛‖→‖𝑝‖, we have ‖‖𝑢^{𝑛(𝑀−1)}‖‖⟶‖𝑝‖(as𝑛⟶∞).(3.43)
Hence we have ‖‖𝐽𝑢^{𝑛(𝑀−1)}‖‖→‖𝐽𝑝‖(as𝑛⟶∞).(3.44)
This implies that {𝐽𝑢^{𝑛(𝑀−1)}} is bounded in 𝐸^∗. Since 𝐸 is reflexive, and so 𝐸^∗ is reflexive, there exists a subsequence {𝐽𝑢^{𝑛(𝑀−1)}_𝑖}⊂{𝐽𝑢^{𝑛(𝑀−1)}} such that 𝐽𝑢^{𝑛(𝑀−1)}_𝑖⇀𝑝₀∈𝐸^∗. In view of the reflexive of 𝐸, we see that 𝐽(𝐸)=𝐸^∗. Hence there exists 𝑥∈𝐸 such that 𝐽𝑥=𝑝₀. Since 𝜙𝑢^𝑛(𝑀)_𝑖,𝑢^{𝑛(𝑁−1)}_𝑖=‖‖𝑢^{𝑛(𝑀)𝑖}‖‖²−2𝑢^𝑛(𝑀)_𝑖,𝐽𝑢^{𝑛(𝑀−1)}_𝑖+‖‖𝑢^{𝑛(𝑀−1)𝑖}‖‖²=‖‖𝑢^𝑛(𝑀)_𝑖‖‖²−2𝑢^𝑛(𝑀)_𝑖,𝐽𝑢^{𝑛(𝑀−1)}_𝑖+‖‖𝐽𝑢^{𝑛(𝑀−1)𝑖}‖‖²(3.45) taking liminf_{𝑛𝑖→∞} on the both sides of above equality and in view of the weak lower semicontinuity of norm ‖⋅‖, it yields that 0≥‖𝑝‖²−2⟨𝑝,𝑝₀⟩+‖‖𝑝₀‖‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝐽𝑥‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝑥‖²=𝜙(𝑝,𝑥).(3.46) This is, 𝑝=𝑥. This implies that 𝑝₀=𝐽𝑝, and so 𝐽𝑢^{𝑛(𝑀−1)}_𝑖⇀𝐽𝑝. It follows from (3.44) and the Kadec-Klee property of 𝐸^∗ that 𝐽𝑢^{𝑛(𝑀−1)}_𝑖→𝐽𝑝 (as 𝑛_𝑖→∞). Note that 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous it yields that 𝑢^{𝑛(𝑀−1)}_𝑖⇀𝑝. It follows from (3.43) and the Kadec-Klee property of 𝐸 that lim_{𝑛𝑖→∞}𝑢^{𝑛(𝑀−1)}_𝑖=𝑝.
By the similar way as given in the proof of (3.20), we can also prove that lim_𝑛→∞𝑢^{𝑛(𝑀−1)}=𝑝.(3.47) From (3.27) and (3.47) we have that ‖‖𝑢^𝑛(𝑀)−𝑢^{𝑛(𝑀−1)}‖‖⟶0(as𝑛⟶∞).(3.48) Since 𝐽 is uniformly continuous on any bounded subset of 𝐸, we have ‖‖𝐽𝑢^𝑛(𝑀)−𝐽𝑢^{𝑛(𝑀−1)}‖‖⟶0(as𝑛⟶∞).(3.49)
Since 𝑢^𝑛(𝑖)=𝐾^Γ𝑖_𝑟𝑛𝑢^{𝑛(𝑖−1)},𝑖=2,3,…,𝑀,𝑢^𝑛(0)=𝑦_𝑛,𝑢^𝑛(𝑀)=𝑢_𝑛.(3.50)
By the similar way as above, we can also prove that 𝑢^𝑛(𝑖)→𝑝,‖‖𝑦‖‖𝑢^𝑛(𝑖)−𝑢^{𝑛(𝑖−1)}‖‖⟶0,‖‖𝐽𝑢^𝑛(𝑖)−𝐽𝑢^{𝑛(𝑖−1)}‖‖⟶0,𝑖=2,3,…,𝑀_𝑛−𝑢^𝑛(1)‖‖⟶0,‖‖𝐽𝑦_𝑛−𝐽𝑢^𝑛(1)‖‖⟶0(as𝑛⟶∞).(3.51) From (3.51) and the assumption that 𝑟_𝑛≥𝑑,∀𝑛≥0, we have lim_𝑛→∞‖‖𝐽𝑢^𝑛(𝑖)−𝐽𝑢^{𝑛(𝑖−1)}‖‖𝑟_𝑖,𝑛=0,𝑖=2,3,…,𝑀;lim_𝑛→∞‖‖𝐽𝑦_𝑛−𝐽𝑢^𝑛(1)‖‖𝑟_1,𝑛=0.(3.52) In the proof of Lemma 2.6 we have proved that the function Γ_𝑖,𝑖=1,2,…,𝑀 defined by (3.6) satisfies the condition (A1)–(A4) and Γ_𝑖𝑢^𝑛(𝑖),𝑦+1𝑟_𝑖,𝑛𝑦−𝑢^𝑛(𝑖),𝐽𝑢^𝑛(𝑖)−𝐽𝑢^{𝑛(𝑖−1)}≥0,∀𝑦∈𝐶.(3.53) Therefore for any 𝑦∈𝐶 we have 1𝑟_𝑖,𝑛𝑦−𝑢^𝑛(𝑖),𝐽𝑢^𝑛(𝑖)−𝐽𝑢^{𝑛(𝑖−1)}≥−Γ^𝑖𝑢^𝑛(𝑖),𝑦≥Γ_𝑖𝑦,𝑢^𝑛(𝑖).(3.54) This implies that Γ_𝑖𝑦,𝑢^𝑛(𝑖)≤1𝑟_𝑖,𝑛𝑦−𝑢^𝑛(𝑖)≤𝑀₁,𝐽𝑢^𝑛(𝑖)−𝐽𝑢^{𝑛(𝑖−1)}+‖𝑦‖‖‖𝐽𝑢^𝑖𝑛−𝐽𝑢^{𝑛(𝑖−1)}‖‖𝑟_𝑖,𝑛(3.55) for some constant 𝑀₁>0. Since the function 𝑦↦Γ_𝑖(𝑥,𝑦) is convex and lower semi-continuous, letting 𝑛→∞ in (3.55), from (3.52) and (3.55), for each 𝑖, we have Γ_𝑖(𝑦,𝑝)≤0,forall𝑦∈𝐶.
For 𝑡∈(0,1] and 𝑦∈𝐶, letting 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝, there are 𝑦_𝑡∈𝐶 and Γ_𝑖(𝑦_𝑡,𝑝)≤0. By condition (A1) and (A4), we have 0=Γ_𝑖𝑦_𝑡,𝑦_𝑡≤𝑡Γ_𝑖𝑦_𝑡,𝑦+(1−𝑡)Γ_𝑖𝑦_𝑡,𝑝≤𝑡Γ_𝑖𝑦_𝑡,𝑦.(3.56) Dividing both sides of the above equation by 𝑡, we have Γ_𝑖(𝑦_𝑡,𝑦)≥0,forall𝑦∈𝐶. Letting 𝑡↓0, from condition (A3), we have Γ_𝑖(𝑝,𝑦)≥0,forall𝑦∈𝐶,forall𝑖=1,2,…,𝑀, that is, for each 𝑖=1,2,…,𝑀, we have 𝐹_𝑖(𝑝,𝑦)+⟨𝐴_𝑖𝑝,𝑦−𝑝⟩+𝜓_𝑖(𝑦)−𝜓_𝑖(𝑝)≥0,∀𝑦∈𝐶.(3.57) This implies that 𝑝∈⋂^𝑀𝑗=1GMEP(𝐹_𝑗,𝐴_𝑗,𝜓_𝑗). Therefore, we have that 𝑝∈ℱ.(3.58)
(V) Now, we prove 𝑥_𝑛→Π_ℱ𝑥₀.
Let 𝑤=Π_ℱ𝑥₀. From 𝑤∈ℱ⊂𝐶_𝑛+1 and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, we have 𝜙(𝑥_𝑛+1,𝑥₀)≤𝜙(𝑤,𝑥₀),forall𝑛≥0. This implies that 𝜙𝑝,𝑥₀=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤𝜙𝑤,𝑥₀.(3.59)
By the definition of Π_ℱ𝑥₀ and (3.59), we have 𝑝=𝑤. Therefore, 𝑥_𝑛→Π_ℱ𝑥₀. This completes the proof of Theorem 3.1.

Theorem 3.2. Let 𝐸,𝐶,{𝐾^Γ𝑖_{𝑟𝑖,𝑛}}^𝑀𝑖=1,{𝐴_𝑖}^𝑀𝑖=1,{𝐹_𝑖}^𝑀𝑖=1,{𝜓_𝑖}^𝑀𝑖=1,{GMEP(𝐹_𝑗,𝐴_𝑗,𝜓_𝑗)}^𝑀𝑗=1 be the same as above. Let {𝑆_𝑖}^∞𝑖=1∶𝐶→𝐶 be an infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that for each 𝑖≥1,𝑆_𝑖 is uniformly L_𝑖-Lipschitz continuous and that 𝒢∶=^∞_𝑖=1𝐹𝑆_𝑖^𝑀_𝑗=1GMEP𝐹_𝑗,𝐴_𝑗,𝜓_𝑗(3.60) is a nonempty and bounded subset of 𝐶. For any given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑧𝑥₀∈𝐶₀=𝐶,_𝑛=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛𝑦_𝑛,=𝐽⁻