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,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+𝜉_𝑛,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.61) where 𝜉_𝑛=sup_𝑢∈𝒢(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛),𝑟_𝑛∈[𝑑,∞) for some 𝑑>0, and for 𝑖≥0,{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_𝒢𝑥₀.

Proof. Since {𝑆_𝑖}^∞𝑖=1∶𝐶→𝐶 is an infinite family of closed quasi-𝜙-asymptotically nonexpansive mappings, it is an infinite family of closed and uniformly total quasi-𝜙-asymptotically nonexpansive mappings with sequence 𝜁(𝑡)=𝑡,𝑡≥0,𝜈_𝑛=𝑘_𝑛−1,𝜇_𝑛=0. Hence 𝜁_𝑛=𝜈_𝑛sup_𝑢∈𝒢𝜁(𝜙(𝑢,𝑥_𝑛))+𝜇_𝑛=sup_𝑢∈𝒢(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛)→0. Therefore all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 is obtained from Theorem 3.1 immediately.

Remark 3.3. Theorems 3.1 and 3.2 improve and extend the corresponding results in [8, 11, 15, 16, 18–24, 28] and others in the following aspects.(a) For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property (note that each uniformly convex Banach space must have Kadec-Klee property).(b) For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, quasi-𝜙-nonexpansive mapping or quasi-𝜙-asymptotically nonexpansive mappings to a countable family of total quasi-𝜙-asymptotically nonexpansive mappings.(c)We extend a single generalized mixed equilibrium problem to a system of generalized mixed equilibrium problems.

Acknowledgments

This work was supported by Scientific Reserch Fund of Sichuan Provincial Education Department (09zb102, 11zb146) and the Natural Science Foundation of Yibin University and Yunnan University of Finance and Economics.