Strong Convergence Theorem for Solving Generalized Mixed Equilibrium Problems and Fixed Point Problems for Total Quasi-𝜙-Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

We introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points for countable families of total quasi-𝜙-asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results.

1. Introduction

Let 𝐸 be a real Banach space with the dual 𝐸^∗ and let 𝐶 be a nonempty closed convex subset of 𝐸. We denote by 𝑅⁺ and 𝑅 the set of all nonnegative real numbers and the set of all 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 norm-to-weak* continuous, and that if 𝐸 is uniformly smooth then 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸. We will denote by 𝐽 the single-value duality mapping.

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 limitlim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(1.2)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 of Banach space 𝐸 can be founded in [1].(i)If 𝐸 is an uniformly smooth Banach space, then 𝐽 is uniformly continuous on each bounded subset of 𝐸.(ii)If 𝐸 is a reflexive and strictly convex Banach space, then 𝐽⁻¹ is norm-weak*-continuous.(iii)If 𝐸 is a smooth, reflexive and strictly convex Banach space, then the normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is single-valued, one-to-one, and surjective.(iv)A Banach space 𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex.(v)Each uniformly convex Banach space 𝐸 has the Kadec-Klee property, that is, for any sequence {𝑥_𝑛}⊂𝐸, if 𝑥_𝑛⇀𝑥∈𝐸 and ‖𝑥_𝑛‖→‖𝑥‖, then 𝑥_𝑛→𝑥. (See [1, 2]) for more details.

Next, we assume that 𝐸 is a smooth, reflexive, and strictly convex Banach space. Consider the functional defined as in [3, 4] by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸.(1.3) It is clear that in a Hilbert space 𝐻, (1.3) reduces to 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖², for all 𝑥,𝑦∈𝐻.

It is obvious from the definition of 𝜙 that(‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)²,∀𝑥,𝑦∈𝐸,(1.4) and 𝜙𝑥,𝐽⁻¹(𝜆𝐽𝑦+(1−𝜆)𝐽𝑧)≤𝜆𝜙(𝑥,𝑦)+(1−𝜆)𝜙(𝑥,𝑧),∀𝑥,𝑦∈𝐸.(1.5)

Following Alber [3], the generalized projection Π_𝐶∶𝐸→𝐶 is defined byΠ_C(𝑥)=arginf_𝑦∈𝐶𝜙(𝑦,𝑥),∀𝑥∈𝐸.(1.6) That is, Π_C𝑥=𝑥, where 𝑥 is the unique solution to the minimization problem 𝜙(𝑥,𝑥)=inf_𝑦∈𝐶𝜙(𝑦,𝑥).

The existence and uniqueness of the operator Π_𝐶 follows from the properties of the functional 𝜙(𝑥,𝑦) and strict monotonicity of the mapping 𝐽 (see, e.g., [1–5]). In Hilbert space 𝐻, Π_𝐶=𝑃_𝐶.

Let 𝐶 be a nonempty closed convex subset of 𝐸, let 𝑇 be a mapping from 𝐶 into itself, and let 𝐹(𝑇) be the set of fixed points of 𝑇. A point 𝑝∈𝐶 is called an asymptotically fixed point of 𝑇 [6] if there exists a sequence {𝑥_𝑛}⊂𝐶 such that 𝑥_𝑛⇀𝑝 and ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0. The set of asymptotical fixed points of 𝑇 will be denoted 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. The set of strong asymptotical fixed points of 𝑇 will be denoted by 𝐹(𝑇).

A mapping 𝑇∶𝐶→𝐶 is said to be relatively nonexpansive [7–9], if 𝐹(𝑇)≠∅, 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥), for all 𝑥∈𝐶, 𝑝∈𝐹(𝑇).

A mapping 𝑇∶𝐶→𝐶 is said to be quasi-𝜙-nonexpansive, if 𝐹(𝑇)≠∅ and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥), for all 𝑥∈𝐶, 𝑝∈𝐹(𝑇).

A mapping 𝑇∶𝐶→𝐶 is said to be quasi-𝜙-asymptotically nonexpansive, if 𝐹(𝑇)≠∅ and there exists a real sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that𝜙(𝑝,𝑇^𝑛𝑥)≤𝑘_𝑛𝜙(𝑝,𝑥),∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹(𝑇).(1.7)

A mapping 𝑇∶𝐶→𝐶 is said to be total quasi-𝜙-asymptotically nonexpansive, if 𝐹(𝑇)≠∅ and there exists nonnegative real sequences {𝜈_𝑛}, {𝜇_𝑛} with 𝜈_𝑛→0, 𝜇_𝑛→0(as𝑛→∞) and a strictly increasing continuous function 𝜉∶𝑅⁺→𝑅⁺ with 𝜉(0)=0 such that𝜙(𝑝,𝑇^𝑛𝑥)≤𝜙(𝑝,𝑥)+𝜈_𝑛𝜉(𝜙(𝑝,𝑥))+𝜇_𝑛,∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹(𝑇).(1.8)

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 exists nonnegative real sequences {𝜈_𝑛}, {𝜇_𝑛} with 𝜈_𝑛→0, 𝜇_𝑛→0(as𝑛→∞) and a strictly increasing continuous function 𝜉∶𝑅⁺→𝑅⁺ with 𝜉(0)=0 such that for each 𝑖≥1 and each 𝑥∈𝐶, 𝑝∈⋂^∞𝑖=1𝐹(𝑇_𝑖)𝜙𝑝,𝑇^𝑛𝑖𝑥≤𝜙(𝑝,𝑥)+𝜈_𝑛𝜉(𝜙(𝑝,𝑥))+𝜇_𝑛,∀𝑛≥1.(1.9)

Remark 1.2. From the definition, it is easy to know that:(i)each relatively nonexpansive mapping is closed;(ii)taking 𝜉(𝑡)=𝑡, 𝑡≥0, 𝜈_𝑛=(𝑘_𝑛−1) and 𝜇_𝑛=0 then 𝜈_𝑛→0  (as𝑛→∞) and (1.7) can be rewritten as 𝜙(𝑝,𝑇^𝑛𝑥)≤𝜙(𝑝,𝑥)+𝜈_𝑛𝜉(𝜙(𝑝,𝑥))+𝜇_𝑛,∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹(𝑇),(1.10) this implies that each quasi-𝜙-asymptotically nonexpansive mapping must be a total quasi-𝜙-asymptotically nonexpansive mapping, but the converse is not true;(iii)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;(iv)the class of quasi-𝜙-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true. (See more details [10–14]).

Let 𝑓∶𝐶×𝐶→𝑅 be a bifunction, where 𝑅 is the set of real numbers. The equilibrium problem (For short, EP) is to find 𝑥^∗∈𝐶 such that𝑓𝑥^∗,𝑦≥0,∀𝑦∈𝐶.(1.11) The set of solutions of EP (1.11) is denoted by EP(𝑓).

Let 𝐵∶𝐶→𝐻 be a nonlinear mapping. The generalized equilibrium problem (for short, GEP) is to find 𝑥^∗∈𝐶 such that 𝑓𝑥^∗,𝑦+⟨𝐵𝑥^∗,𝑦−𝑥^∗⟩≥0,∀𝑦∈𝐶.(1.12) The set of solutions of GEP (1.12) is denoted by GEP(𝑓,𝐵), that is, GEP(𝑓,𝐵)=𝑥^∗∈𝐶∶𝑓𝑥^∗,𝑦+⟨𝐵𝑥^∗,𝑦−𝑥^∗⟩≥0,∀𝑦∈𝐶.(1.13)

Let 𝜑∶𝐶→𝑅∪{+∞} be a function. The mixed equilibrium problem (for short, MEP) is to find 𝑥^∗∈𝐶 such that𝑓𝑥^∗,𝑦+𝜑(𝑦)−𝜑𝑥^∗≥0,∀𝑦∈𝐶.(1.14) The set of solutions of MEP (1.14) is denoted by MEP(𝑓).

The concept generalized mixed equilibrium problem (for short, GMEP) was introduced by Peng and Yao [15] in 2008. GMEP is to find 𝑥^∗∈𝐶 such that𝑓𝑥^∗,𝑦+𝜑(𝑦)−𝜑𝑥^∗+⟨𝐵𝑥^∗,𝑦−𝑥^∗⟩≥0,∀𝑦∈𝐶.(1.15) The set of solutions of GMEP (1.15) is denoted by GMEP(𝑓,𝐵,𝜑), that is, GMEP(𝑓,𝐵,𝜑)=𝑥^∗∈𝐶∶𝑓𝑥^∗,𝑦+𝜑(𝑦)−𝜑𝑥^∗+⟨𝐵𝑥^∗,𝑦−𝑥^∗⟩≥0,∀𝑦∈𝐶.(1.16)

The equilibrium problem is an unifying model for several problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems (e. g., [16, 17]). Many authors have proposed some useful methods to solve the EP, GEP, MEP, GMEP; see, for instance, [15–23] and the references therein.

In 2005, Matsushita and Takahashi [13] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping 𝑇 in a Banach space 𝐸: 𝑦_𝑛𝑥₀∈𝐶chosenarbitrary,𝐶_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛,𝑄=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛,=𝑧𝑥∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥0.(1.17) They prove that {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₀, where Π_𝐹(𝑇) is the generalized projection from 𝐶 onto 𝐹(𝑇).

Recently, Qin et al. [24] proposed a shrinking projection method to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of quasi-𝜙-nonexpansive mappings in the framework of Banach spaces:𝑥₀=𝑥c𝐶h₁osenarbitrary,𝑥=𝐶,₁𝑦_𝑛=Π_𝐶1𝑥₀,=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^𝑁_𝑖=1𝛼_𝑛,𝑖𝐽𝑇_𝑖𝑥_𝑛𝑢_𝑛,∈𝐶suchthat𝑓𝑢_𝑛,𝑦+1𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝑥𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛+1,=Π_𝐶𝑛+1𝑥₀,𝑛≥0,(1.18) where Π_𝐶𝑛+1 is the generalized projection from 𝐸 onto 𝐶_𝑛+1. They prove that the sequence {𝑥_𝑛} converges strongly to Π_∩^𝑁𝑖=1_{𝐹(𝑇𝑖)∩EP(𝑓)}𝑥₀.

In [25], Saewan and Kumam introduced a modified new hybrid projection method to find a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings in an uniformly smooth and strictly convex Banach spaces 𝐸 with Kadec-Klee property: 𝑥₀∈𝐶𝑥₁chosenarbitrary,𝐶=₁Π_𝐶1𝑥₀,𝑦=𝐶,_𝑛𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,=𝐽⁻¹𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛𝑢_𝑛,𝐶_𝑛+1∈𝐶suchthat𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛,=𝑧∈𝑥𝐶_𝑛𝑛+1∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝜉_𝑛,=Π_𝐶𝑛+1𝑥₀,𝑛≥0,(1.19) where 𝜉_𝑛=sup_𝑝∈𝐹(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛), Π_𝐶𝑛+1 is the generalized projection of 𝐸 onto 𝐶_𝑛+1. They prove that the sequence {𝑥_𝑛} converges strongly to Π_∩^∞𝑖=1_{𝐹(𝑇𝑖)∩GMEP(𝑓)}𝑥₀.

Very recently, Chang et al. [26] proposed the following iterative algorithm for solving fixed point problems for total quasi-𝜙-asymptotically nonexpansive mappings: 𝑦𝑥₀∈𝐶chosenarbitrary,𝐶₀=𝐶,_𝑛𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,=𝐽⁻¹𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛𝐶_𝑛+1,=𝑧𝑥∈𝐶_𝑛𝑛+1∶𝜙𝜈,𝑦_𝑛≤𝜙𝜈,𝑥_𝑛+𝜉_𝑛,=Π_𝐶𝑛+1𝑥₀,𝑛≥0,(1.20) where 𝜉_𝑛=𝜈_𝑛sup_𝑝∈𝐹𝜉(𝜙(𝑝,𝑥_𝑛))+𝜇_𝑛, Π_𝐶𝑛+1 is the generalized projection of 𝐸 onto 𝐶_𝑛+1. They prove that the sequence {𝑥_𝑛} converges strongly to Π_∩^∞𝑖=1_{𝐹(𝑇𝑖)}𝑥₀.

Inspired and motivated by the recent work of Matsushita and Takahashi [13], Qin et al. [24], Saewan and Kumam [25], Chang et al. [26], and so forth, we introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points of a countable families of total quasi-𝜙-asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results in [13, 24–29].

2. Preliminaries

Throughout this paper, let 𝐸 be a real Banach space with the dual 𝐸^∗ and let 𝐶 be a nonempty closed convex subset of 𝐸. We denote the strong convergence, weak convergence of a sequence {𝑥_𝑛} to a point 𝑥∈𝐸 by 𝑥_𝑛→𝑥, 𝑥_𝑛⇀𝑥, respectively, and 𝐹(𝑇) is the fixed point set of a mapping 𝑇.

In this paper, for solving generalized mixed equilibrium problems, we assume that bifunction 𝑓∶𝐶×𝐶→𝑅 satisfies the following conditions:(A1)𝑓(𝑥,𝑥)=0, for all 𝑥∈𝐶;(A2)𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝐶;(A3) for all 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦);(A4) for each 𝑥∈𝐶, the function 𝑦↦𝑓(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.1 (see [16]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4), and let 𝑟>0 and 𝑥∈𝐸, then there exists 𝑧∈𝐶 such that 𝑓(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.1)

Lemma 2.2 (see [30]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping, let 𝜑∶𝐶→𝑅 be convex and lower semicontinuous and let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4). For 𝑟>0 and 𝑥∈𝐸, then there exists 𝑢∈𝐶 such that 𝑓(𝑢,𝑦)+𝜑(𝑦)−𝜑(𝑢)+⟨𝐵𝑢,𝑦−𝑢⟩+1𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.2) Define a mapping 𝐾_𝑟∶𝐶→𝐶 as follows: 𝐾_𝑟(𝑥)=𝑢∈𝐶∶𝑓(𝑢,𝑦)+𝜑(𝑦)−𝜑(𝑢)+⟨B𝑢,𝑦−𝑢⟩+1𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶(2.3) for all 𝑥∈𝐶. Then, the following hold:(1)𝐾_𝑟 is single-valued;(2)𝐾_𝑟 is firmly nonexpansive, that is, forall𝑥,𝑦∈𝐸, ⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝐽𝐾_𝑟𝑥−𝐽𝐾_𝑟𝑦⟩≤⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩;(2.4)(3)𝐹(𝐾_𝑟)=GMEP(𝑓,𝐵,𝜑);(4)GMEP(𝑓,𝐵,𝜑) is closed and convex;(5)𝜙(𝑝,𝐾_𝑟𝑧)+𝜙(𝐾_𝑟𝑧,𝑧)≤𝜙(𝑝,𝑧), for all 𝑝∈𝐹(𝐾_𝑟) and 𝑧∈𝐸.

Lemma 2.3 (see [28]). 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 {𝑥_𝑛} and {𝑦_𝑛} be two sequences in 𝐶 such that 𝑥_𝑛→𝑝 and 𝜙(𝑥_𝑛,𝑦_𝑛)→0, where 𝜙 is the function defined by (1.3), then 𝑦_𝑛→𝑝.

Lemma 2.4 (see [3]). Let 𝐸 be a smooth, strictly convex and reflexive Banach space and let 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,𝑦∈𝐸;(b)if 𝑥∈𝐸  and 𝑧∈𝐶, then 𝑧=Π_𝐶𝑥 if and only if ⟨𝑧−𝑦,𝐽𝑥−𝐽𝑧⟩≥0, for all 𝑦∈𝐶;(c)for 𝑥,𝑦∈𝐸, 𝜙(𝑥,𝑦)=0  if and only if 𝑥=𝑦.

Lemma 2.5 (see [28]). 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 𝜉∶𝑅⁺→𝑅⁺ such that 𝜈_𝑛→0, 𝜇_𝑛→0 (as 𝑛→∞) and 𝜉(0)=0. If 𝜇₁=0, then the fixed point set 𝐹(𝑇) of 𝑇 is a closed and convex subset of 𝐶.

Lemma 2.6 (see [31]). Let 𝐸 be an uniformly convex Banach space, let 𝑟 be a positive number, and let 𝐵_𝑟(0) be a closed ball of 𝐸. Then, for any sequence {𝑥_𝑖}^∞𝑖=1⊂𝐵_𝑟(0) and for any sequence {𝜆_𝑖}^∞𝑖=1 of positive numbers with ∑^∞𝑛=1𝜆_𝑛=1, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,2𝑟]→[0,∞), 𝑔(0)=0 such that, for any positive integer 𝑖≠1, the following holds: ‖‖‖‖^∞_𝑛=1𝜆_𝑛𝑥_𝑛‖‖‖‖²≤^∞_𝑛=1𝜆_𝑛‖‖𝑥_𝑛‖‖²−𝜆₁𝜆_𝑖𝑔‖‖𝑥₁−𝑥_𝑖‖‖.(2.5)

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty, closed, and convex subset of an uniformly smooth and strictly convex Banach Banach space 𝐸 with Kadec-Klee property. Let 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping and let 𝜑∶𝐶→𝑅 be a lower semicontinuous and convex function. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4). Let {𝑇_𝑖}^∞𝑖=1∶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 function 𝜁∶𝑅⁺→𝑅⁺ such that 𝜇₁=0, 𝜈_𝑛→0, 𝜇_𝑛→0 (as 𝑛→∞), and 𝜁(0)=0, and for each 𝑖≥1, 𝑇_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous. {𝑥_𝑛} is defined by 𝑦𝑥₀∈𝐶chosenarbitrary,𝐶₀=𝐶,_𝑛𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,=𝐽⁻¹𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛𝑢_𝑛,𝐶_𝑛+1∈𝐶suchthat𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛,=𝜈𝑥∈𝐶_𝑛𝑛+1∶𝜙𝜈,𝑢_𝑛≤𝜙𝜈,𝑥_𝑛+𝜉_𝑛,=Π_𝐶𝑛+1𝑥₀,𝑛≥0,(3.1) where 𝜉_𝑛=𝜈_𝑛sup_𝑞∈Θ𝜁(𝜙(𝑞,𝑥_𝑛))+𝜇_𝑛, Π_𝐶𝑛+1is the generalized projection of 𝐸  onto 𝐶_𝑛+1, {𝑟_𝑛}⊂[𝑎,∞)  for some 𝑎>0, {𝛽_𝑛,0,𝛽_𝑛,𝑖}  and {𝛼_𝑛}  are sequences in [0,1] satisfying the following conditions: (1)for each 𝑛≥0,𝛽_𝑛,0+∑^∞𝑖=1𝛽_𝑛,𝑖=1;(2)liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛,𝑖>0 for any 𝑖≥1;(3)0≤𝛼_𝑛≤𝛼<1 for some 𝛼∈(0,1). If Θ∶=⋂^∞𝑖=1𝐹(𝑇_𝑖)∩GMEP(𝑓,𝐵,𝜑) is a nonempty and bounded subset in 𝐶, then the sequence {𝑥_𝑛}  converges strongly to 𝑝∈𝐹, where 𝑝=Π_Θ𝑥₀.

Proof. We will divide the proof into seven steps.Step 1. We first show that Θ and 𝐶_𝑛 are closed and convex for each 𝑛≥0.
It follows from Lemma 2.5 that 𝐹(𝑇_𝑖) is closed and convex subset of 𝐶 for each 𝑖≥1. Therefore, Θ is closed and convex in 𝐶.
Again by the assumption, 𝐶₀=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for some 𝑛≥1. Since for any 𝑧∈𝐶_𝑛, we know that 𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝜉_𝑛⟺2⟨𝑧,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖‖𝑥_𝑛‖‖²−‖‖𝑢_𝑛‖‖²+𝜉_𝑛.(3.2) Hence, the set 𝐶_𝑛+1={𝑧∈𝐶_𝑛∶2⟨𝑧,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖𝑥_𝑛‖²−‖𝑢_𝑛‖²+𝜉_𝑛} is closed and convex. Therefore, Π_𝐶𝑛𝑥₀ and Π_Θ𝑥₀ are well defined.Step 2. We show that Θ⊂𝐶_𝑛 for all 𝑛≥0.
It is obvious that Θ⊂𝐶₀=𝐶. Suppose that Θ⊂𝐶_𝑛 for some 𝑛≥1. Since 𝐸 is uniformly smooth, 𝐸^∗ is uniformly convex. By the convexity of ‖⋅‖², property of 𝜙, for any given 𝑞∈Θ⊂𝐶_𝑛, we observe that 𝜙𝑞,𝑢_𝑛≤=𝜙𝜙𝑞𝑞,,𝐾𝑦_𝑟𝑛𝑛𝑦_𝑛=𝜙𝑞,𝐽₋₁=‖𝑞‖₂𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛−2𝑞,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛+‖‖𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛‖‖²≤‖𝑞‖²−2𝛼_𝑛⟨𝑞,𝐽𝑥_𝑛⟩−21−𝛼_𝑛⟨𝑞,𝐽𝑧_𝑛⟩+𝛼_𝑛‖‖𝑥_𝑛‖‖²+1−𝛼_𝑛‖‖𝑧_𝑛‖‖²=𝛼_𝑛𝜙𝑞,𝑥_𝑛+1−𝛼_𝑛𝜙𝑞,𝑧_𝑛.(3.3) Furthermore, it follows from Lemma 2.6 that, for any positive integers 𝑙>1 and for any 𝑞∈Θ, we have 𝜙𝑞,𝑧_𝑛=𝜙𝑞,𝐽⁻¹𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛=‖𝑞‖₂−2𝑞,𝛽_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛+‖‖‖‖𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖‖‖²≤‖𝑞‖²−2𝛽_𝑛,0⟨𝑞,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛽_𝑛,𝑖𝑞,𝐽𝑇^𝑛𝑖𝑥_𝑛+𝛽_𝑛,0‖‖𝑥_𝑛‖‖²+^∞_𝑖=1=𝛽_𝑛,0𝛽_𝑛,𝑖‖‖𝑇^𝑛𝑖𝑥_𝑛‖‖²−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖𝜙𝑞,𝑥_𝑛+^∞_𝑖=1≤𝛽_𝑛,0𝛽_𝑛,𝑖𝜙𝑞,𝑇^𝑛𝑖𝑥_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖𝜙𝑞,𝑥_𝑛+^∞_𝑖=1−𝛽_𝑛,0𝛽_𝑛,𝑖𝜙𝑞,𝑥_𝑛+𝜈_𝑛𝜁𝜙𝑞,𝑥_𝑛+𝜇_𝑛≤𝜙𝑞,𝛽𝑥_{𝑛,𝑙𝑛}𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖=𝜙𝑞,𝑥_𝑛+𝜈_𝑛sup_𝑝∈Θ𝜙𝑝,𝑥_𝑛+𝜇_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖+𝜉_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖.(3.4) Substituting (3.4) into (3.3), we get 𝜙𝑞,𝑢_𝑛≤≤𝛼𝛼_𝑛𝑛𝜙𝑞,𝑥_𝑛+1−𝛼_𝑛𝜙𝑞,𝑧_𝑛≤𝜙𝜙𝑞,𝑥𝑞,_𝑛𝑥_𝑛+1−𝛼_𝑛𝜙𝑞,𝑥_𝑛+𝜉_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖+1−𝛼_𝑛𝜉_𝑛.(3.5) This shows that 𝑞∈𝐶_𝑛+1. Further, this implies that Θ⊂𝐶_𝑛+1 and hence Θ⊂𝐶_𝑛 for all 𝑛≥0. Since Θ is nonempty, 𝐶_𝑛 is a nonempty closed convex subset of 𝐸, and hence Π_𝐶𝑛 exists for all 𝑛≥0. This implies that the sequence {𝑥_𝑛} is well defined.
Moreover, by the assumption of {𝜈_𝑛}, {𝜇_𝑛}, and Θ, from (1.4), we have 𝜉_𝑛=𝜈_𝑛sup_𝑝∈Θ𝜁𝜙𝑝,𝑥_𝑛+𝜇_𝑛⟶0,𝑛⟶∞.(3.6)Step 3. {𝑥_𝑛} is bounded and {𝜙(𝑥_𝑛,𝑥₀)} is a convergent sequence.
It follows from (3.1) and Lemma 2.4 that 𝜙𝑥_𝑛,𝑥₀=≤𝜙𝜙Π𝑝,𝑥_𝐶𝑛0𝑥₀,𝑥₀≤𝜙𝑝,𝑥₀−𝜙𝑝,𝑥_𝑛,∀𝑝∈𝐶_𝑛+1,∀𝑛≥0.(3.7) From definition of 𝐶_𝑛+1 that 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, we have 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀,∀𝑛≥0.(3.8) Therefore, {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing and bounded. So, {𝜙(𝑥_𝑛,𝑥₀)} is a convergent sequence, without loss of generality, we can assume that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀)=𝑑≥0. In particular, by (1.4), the sequence {(‖𝑥_𝑛‖−‖𝑥₀‖)²} is bounded. This implies {𝑥_𝑛} is also bounded.Step 4. We prove that {𝑥_𝑛} converges strongly to some point 𝑝∈𝐶.
Since {𝑥_𝑛} is bounded and 𝐸 is reflexive, there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑝 (some point in 𝐶). Since 𝐶_𝑛 is closed and convex and 𝐶_𝑛+1⊂𝐶_𝑛, this implies that 𝐶_𝑛 is weakly closed and 𝑝∈𝐶_𝑛 for each 𝑛≥0. From 𝑥_𝑛𝑖=Π_𝐶𝑛𝑖𝑥₀, we have 𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝,𝑥₀,∀𝑛_𝑖≥0.(3.9) Since the norm ‖⋅‖ is weakly lower semicontinuous, we have 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,(3.10) and so 𝜙𝑝,𝑥₀≤liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤limsup_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝,𝑥₀.(3.11) This implies that lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛𝑖,𝑥₀)→𝜙(𝑝,𝑥₀), and so ‖𝑥_𝑛‖→‖𝑝‖. Since 𝑥_𝑛𝑖⇀𝑝, by virtue of the Kadec-Klee property of 𝐸, we obtain that lim_{𝑛𝑖→∞}𝑥_𝑛𝑖=𝑝.(3.12) Since {𝜙(𝑥_𝑛,𝑥₀)} is convergent, this together with lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛𝑖,𝑥₀)=𝜙(𝑝,𝑥₀), we have lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀)=𝜙(𝑝,𝑥₀). If there exists some subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑗→𝑞, then from Lemma 2.4, we have that 𝜙(𝑝,𝑞)==lliimm_{𝑛𝑖,𝑛𝑗→∞𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥_𝑛𝑗≤lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,Π_𝐶𝑛𝑗𝑥₀=lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥^𝑛_𝑖,𝑥₀−𝜙Π_𝐶𝑛𝑗𝑥₀,𝑥₀=𝜙𝑝,𝑥₀𝜙𝑥^𝑛_𝑖,𝑥₀−𝜙𝑥_𝑛𝑗,𝑥₀=0.−𝜙𝑝,𝑥₀(3.13) This implies that 𝑝=𝑞 and lim_𝑛→∞𝑥_𝑛=𝑝.(3.14)Step 5. We prove that lim_𝑛→∞‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖=0.
By definition of Π_𝐶𝑛𝑥₀, 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.(3.15) Since lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists, we have lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0.(3.16) Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛 and the definition of 𝐶_𝑛+1, we get 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝜉_𝑛.(3.17) It follows from (3.6) and (3.16) that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑢_𝑛=0.(3.18) From (1.4), we have lim_𝑛→∞‖‖𝑢_𝑛‖‖=𝑝.(3.19) So, lim_𝑛→∞‖‖𝐽𝑢_𝑛‖‖=𝐽𝑝.(3.20) This implies that {𝐽𝑢_𝑛} is bounded in 𝐸^∗. Note that 𝐸 is reflexive and 𝐸^∗ is also reflexive, we can assume that 𝐽𝑢_𝑛⇀𝑥^∗∈𝐸^∗. In view of the reflexive of 𝐸, we know that 𝐽(𝐸)=𝐸^∗. Hence, there exist 𝑥∈𝐶 such that 𝐽𝑥=𝑥^∗. It follows that 𝜙𝑥_𝑛+1,𝑢_𝑛=‖‖𝑥_𝑛+1‖‖²−2𝑥_𝑛+1,𝐽𝑢_𝑛+‖‖𝑢_𝑛‖‖²=‖‖𝑥_𝑛+1‖‖²−2𝑥_𝑛+1,𝐽𝑢_𝑛+‖‖𝐽𝑢_𝑛‖‖².(3.21) Taking liminf_𝑛→∞ on the both sides of equality above and by the weak lower semicontinuity of norm ‖⋅‖, we have 0≥‖𝑝‖²−2⟨𝑝,𝑥^∗⟩+‖𝑥^∗‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝐽𝑥‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝑥‖²=𝜙(𝑝,𝑥).(3.22) That is, 𝑝=𝑥, which implies that 𝑥^∗=𝐽𝑝. It follows that 𝐽𝑢_𝑛⇀𝐽𝑝∈𝐸^∗. From (1.4) and the Kadec-Klee property of 𝐸, we have lim_𝑛→∞𝑢_𝑛=𝑝.(3.23) Since ‖𝑥_𝑛−𝑢_𝑛‖≤‖𝑥_𝑛−𝑝‖+‖𝑝−𝑢_𝑛‖, so, liminf_𝑛→∞‖‖𝑥_𝑛−𝑢_𝑛‖‖=0.(3.24) Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, we obtain liminf_𝑛→∞‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖‖=0,(3.25)Step 6. We show that 𝑝∈Θ∶=⋂^∞𝑖=1𝐹(𝑇_𝑖)⋂GMEP(𝑓,𝐵,𝜑).
First, we show that 𝑝∈⋂^∞𝑖=1𝐹(𝑇_𝑖).
Since 𝑥_𝑛+1∈𝐶_𝑛+1, it follows from (3.1) and (3.14) that 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝜉_𝑛⟶0(as𝑛⟶∞).(3.26) Since 𝑥_𝑛→𝑝, by Lemma 2.3, lim_𝑛→∞𝑢_𝑛=𝑝.(3.27) By (3.3) and (3.4), for any 𝑞∈Θ, we have 𝜙𝑞,𝑢_𝑛≤𝜙𝑞,𝑥_𝑛+𝜉_𝑛−1−𝛼_𝑛𝛽_𝑛,0𝛽_𝑛,𝑙𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖.(3.28) So, 1−𝛼_𝑛𝛽_𝑛,0𝛽_𝑛,𝑙𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖≤𝜙𝑞,𝑥_𝑛+𝜉_𝑛−𝜙𝑞,𝑢_𝑛⟶0(as𝑛⟶∞).(3.29) Therefore, lim_𝑛→∞1−𝛼_𝑛𝛽_𝑛,0𝛽_𝑛,𝑙𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖=0.(3.30) In view of the property of 𝑔, we have ‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖⟶0(as𝑛⟶∞).(3.31) Since 𝐽𝑥_𝑛→𝐽𝑝, this implies that lim_𝑛→∞𝐽𝑇^𝑛𝑙𝑥_𝑛=𝐽𝑝. From Remark 1.1(ii), it yields 𝑇^𝑛𝑙𝑥_𝑛⇀𝑝(as𝑛⟶∞).(3.32) Again since ‖‖𝑇^𝑛𝑙𝑥_𝑛‖‖−‖𝑝‖=‖‖𝐽𝑇^𝑛𝑙𝑥_𝑛‖‖−‖𝐽𝑝‖≤‖‖𝐽𝑇^𝑛𝑙𝑥_𝑛−𝐽𝑝‖‖⟶0(as𝑛⟶∞),(3.33) this together with (3.32) and the Kadec-Klee-property of 𝐸 shows that lim_𝑛→∞𝑇^𝑛𝑙𝑥_𝑛=𝑝.(3.34) By the assumption that 𝑇_𝑙 is uniformly 𝐿_𝑙-Lipschitz continuous, we have ‖‖𝑇^𝑙𝑛+1𝑥_𝑛−𝑇^𝑛𝑙𝑥_𝑛‖‖≤+‖‖‖‖𝑇𝑥^𝑙𝑛+1_𝑛+1𝑥_𝑛−𝑇^𝑙𝑛+1𝑥_𝑛+1‖‖+‖‖𝑇^𝑙𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖≤𝐿_𝑙−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑇^𝑛𝑙𝑥_𝑛‖‖+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑇^𝑙𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛−𝑇^𝑛𝑙𝑥_𝑛‖‖.(3.35) This together with (3.34) and 𝑥_𝑛→𝑝 shows that lim_𝑛→∞‖𝑇^𝑙𝑛+1𝑥_𝑛−𝑇^𝑛𝑙𝑥_𝑛‖=0 and lim_𝑛→∞𝑇^𝑙𝑛+1𝑥_𝑛=𝑝, that is, lim_𝑛→∞𝑇_𝑙𝑇^𝑛𝑙𝑥_𝑛=𝑝. In view of the closeness of 𝑇_𝑙, it follows that 𝑇_𝑙𝑝=𝑝, that is, 𝑝∈𝐹(𝑇_𝑙). By the arbitrariness of 𝑙≥1, we have 𝑝∈∩^∞𝑖=1𝐹(𝑇_𝑖).
Now, we show that 𝑝∈GMEP(𝑓,𝐵,𝜑).
It follows from (3.2), (3.3), (3.6), Lemma 2.4, and 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛 that 𝜙𝑢_𝑛,𝑦_𝑛≤=𝜙𝜙𝐾𝑝,𝑦_𝑟𝑛𝑦_𝑛𝑛,𝑦_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝐾_𝑟𝑛𝑦_𝑛−𝜙𝑝,𝐾_𝑟𝑛𝑦_𝑛+𝜉_𝑛=𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢_𝑛+𝜉_𝑛⟶0,(as𝑛⟶∞).(3.36) By (1.4), we have ‖‖𝑢_𝑛‖‖⟶‖‖𝑦_𝑛‖‖,(as𝑛⟶∞).(3.37) Since 𝑢_𝑛→𝑝 as 𝑛→∞, so ‖‖𝑦_𝑛‖‖⟶‖𝑝‖,(as𝑛⟶∞).(3.38) Therefore, ‖‖𝐽𝑢_𝑛‖‖⟶‖𝐽𝑝‖,(as𝑛⟶∞).(3.39) Since 𝐸^∗ is reflexive, we may assume that 𝐽𝑦_𝑛⇀𝑧^∗∈𝐸^∗. In view of the reflexive of 𝐸, we have 𝐽(𝐸)=𝐸^∗. Hence, there exist 𝑧∈𝐸 such that 𝐽𝑧=𝑧^∗. It follows that 𝜙𝑢_𝑛,𝑦_𝑛=‖‖𝑢_𝑛‖‖²−2⟨𝑢_𝑛,𝐽𝑦_𝑛⟩+‖‖𝑦_𝑛‖‖²=‖‖𝑢_𝑛‖‖²−2⟨𝑢_𝑛,𝐽𝑦_𝑛⟩+‖‖𝐽𝑦_𝑛‖‖².(3.40) Taking liminf_𝑛→∞ on the both sides of equality above yields that 0≥‖𝑝‖²−2⟨𝑝,𝑧^∗⟩+‖𝑧^∗‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑧⟩+‖𝐽𝑧‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑧⟩+‖𝑧‖²=𝜙(𝑝,𝑥).(3.41) That is, 𝑝=𝑧, which implies that 𝑧^∗=𝐽𝑝. It follows that 𝐽𝑦_𝑛⇀𝐽𝑝∈𝐸^∗. Since 𝐽⁻¹ is norm-weak*-continuous, it follows that 𝑦_𝑛⇀𝑝. From (3.38) and 𝐸 with the Kadec-Klee property, we obtain 𝑦_𝑛⟶𝑝(as𝑛⟶∞).(3.42) It follows from (3.23) and (3.42) that lim_𝑛→∞‖‖𝑢_𝑛−𝑦_𝑛‖‖=0.(3.43) Since 𝐽 is uniformly norm-to-norm continuous, we have lim_𝑛→∞‖‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖‖=0.(3.44) By Lemma 2.2, we have 𝑓𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶.(3.45) From (A2), we have 𝜑(𝑦)−𝜑𝑢_𝑛+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥−𝑓𝑢_𝑛,𝑦≥𝑓𝑦,𝑢_𝑛,∀𝑦∈𝐶.(3.46)
Put 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝑝 for all 𝑡∈(0,1] and 𝑦∈𝐶. Consequently, we get 𝑧_𝑡∈𝐶. It follows from (3.46) that ⟨𝐵𝑧_𝑡,𝑧_𝑡−𝑢_𝑛⟩≥−⟨𝐵⟨𝐵𝑧_𝑡𝑦,_𝑛𝑧_𝑡−𝑢_𝑛⟩−𝜑𝑧_𝑡+𝜑𝑢_𝑛−𝑧_𝑡,𝑧_𝑡−𝑢_𝑛⟩+𝑓𝑧_𝑡,𝑢_𝑛−𝑢_𝑛,𝐽𝑢^𝑛−𝐽𝑦_𝑛=⟨𝐵𝑧_𝑡𝑟_𝑛+⟨𝐵𝑢−_𝑛𝐵𝑢_𝑛,𝑧_𝑡−𝑢_𝑛⟩−𝜑𝑧_𝑡+𝜑𝑢_𝑛−𝑧_𝑡−𝐵𝑦_𝑛,𝑧_𝑡−𝑢_𝑛⟩+𝑓𝑧_𝑡,𝑢_𝑛−𝑢_𝑛,𝐽𝑢^𝑛−𝐽𝑦_𝑛𝑟_𝑛.(3.47) Since 𝐵 is continuous, and from (3.43), and 𝑢_𝑛→𝑝, 𝑦_𝑛→𝑝, as 𝑛→∞, therefore ‖𝐵𝑢_𝑛−𝐵𝑦_𝑛‖→0. Since 𝐵 is monotone, we know that ⟨𝐵𝑧_𝑡−𝐵𝑢_𝑛,𝑧_𝑡−𝑢_𝑛⟩≥0. Further, lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖/𝑟_𝑛=0. So, it follows from (A4), and the weak lower semicontinuity of 𝜑 and (3.43) that 𝑓𝑧_𝑡,𝑝−𝜑𝑧_𝑡+𝜑(𝑝)≤=⟨l𝐵im𝑧_{𝑛→∞𝑡}⟨𝐵𝑧_𝑡,𝑧_𝑡−𝑢_𝑛⟩,𝑧_𝑡−𝑝⟩.(3.48) From (A1) and (3.48), we have 0=≤𝑓𝑡𝑓𝑧𝑧_𝑡,_𝑡𝑧_𝑡−𝜑𝑧_𝑡+𝜑𝑧_𝑡=𝑡𝑓𝑧,_𝑡𝑦+(1−𝑡)𝑓𝑧_𝑡,𝑝+𝑡𝜑(𝑦)+(1−𝑡)𝜑(𝑝)−𝜑𝑧_𝑡≤𝑡𝑓𝑧_𝑡,𝑦+𝜑(𝑦)−𝜑𝑧_𝑡+(1−𝑡)𝑓𝑧_𝑡,𝑝+𝜑(𝑝)−𝜑𝑧_𝑡≤𝑡𝑓𝑧_𝑡,𝑦+𝜑(𝑦)−𝜑𝑧_𝑡+(1−𝑡)⟨𝐵𝑧_𝑡,𝑧_𝑡−𝑝⟩,𝑦+𝜑(𝑦)−𝜑𝑧_𝑡+(1−𝑡)𝑡⟨𝐵𝑧_𝑡,𝑦−𝑝⟩,(3.49) and hence 𝑓𝑧_𝑡,𝑦+𝜑(𝑦)−𝜑𝑧_𝑡+(1−𝑡)⟨𝐵𝑧_𝑡,𝑦−𝑝⟩≥0.(3.50) Letting 𝑡→0, we have 𝑓(𝑝,𝑦)+𝜑(𝑦)−𝜑(𝑝)+⟨𝐵𝑝,𝑦−𝑝⟩≥0.(3.51) This implies that 𝑝∈GMEP(𝑓,𝐵,𝜑). Hence, 𝑝∈∩^∞𝑖=1𝐹(𝑇_𝑖)∩GMEP(𝑓,𝐵,𝜑).Step 7. We prove that 𝑥_𝑛→𝑝=Π_Θ𝑥₀.
Let 𝑞=Π_Θ𝑥₀. From 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑞∈Θ⊂𝐶_𝑛, we have 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑞,𝑥₀,∀𝑛≥0.(3.52) This implies that 𝜙𝑝,𝑥₀=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤𝜙𝑞,𝑥₀.(3.53) By definition of 𝑝=Π_Θ𝑥₀, we have 𝑝=𝑞. Therefore, 𝑥_𝑛→𝑝=Π_Θ𝑥₀. This completes the proof.

Taking 𝜑=0, 𝑇_𝑖=𝑇 for each 𝑖∈𝑁 in Theorem 3.1, we have the following result.

Corollary 3.2. Let 𝐶 be a nonempty, closed, and convex subset of an uniformly smooth and strictly convex Banach Banach space 𝐸 with Kadec-Klee property. Let 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4). Let 𝑇∶𝐶→𝐶 be a closed uniformly 𝐿-Lipschitz continuous and uniformly total quasi-𝜙-asymptotically nonexpansive mappings with nonnegative real sequences {𝜈_𝑛}, {𝜇_𝑛} and a strictly increasing continuous function 𝜁∶𝑅⁺→𝑅⁺ such that 𝜇₁=0, 𝜈_𝑛→0, 𝜇_𝑛→0 (as 𝑛→∞), and 𝜁(0)=0. Let {𝑥_𝑛} be the sequence generated by 𝑦𝑥₀∈𝐶chosenarbitrary,𝐶₀=𝐶,_𝑛𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇^𝑛𝑥_𝑛,∈𝐶suchthat𝑓𝑢_𝑛,𝑦+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_n𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝜈𝑥∈𝐶_𝑛𝑛+1∶𝜙𝜈,𝑢_𝑛≤𝜙𝜈,𝑥_𝑛+𝜉_𝑛,=Π_𝐶𝑛+1𝑥₀,𝑛≥0,(3.54) where 𝜉_𝑛=𝜈_𝑛sup_𝑞∈Θ𝜁(𝜙(𝑞,𝑥_𝑛))+𝜇_𝑛, Π_𝐶𝑛+1is the generalized projection of 𝐸  onto 𝐶_𝑛+1, {𝛽_𝑛}  and {𝛼_𝑛}are sequences in [0,1], liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0, {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. If Θ∶=∩^∞𝑖=1𝐹(𝑇_𝑖)∩GEP(𝑓,𝐵)  is a nonempty and bounded subset in 𝐶, then the sequence {𝑥_𝑛} converges strongly to 𝑝∈Θ, where 𝑝=Π_Θ𝑥₀.

In Theorem 3.1, as 𝜑=0, 𝐵=0, 𝑇_𝑖=𝑇 for each 𝑖∈𝑁, we can obtain the following corollary.

Corollary 3.3. Let 𝐶 be a nonempty, closed and convex subset of an uniformly smooth and strictly convex Banach Banach space 𝐸 with Kadec-Klee property. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4), and 𝑇∶𝐶→𝐶 be a closed uniformly 𝐿-Lipschitz continuous and uniformly total quasi-𝜙-asymptotically nonexpansive mappings with nonnegative real sequences {𝜈_𝑛}, {𝜇_𝑛} and a strictly increasing continuous function 𝜁∶𝑅⁺→𝑅⁺ such that 𝜇₁=0, 𝜈_𝑛→0, 𝜇_𝑛→0 (as 𝑛→∞) and 𝜁(0)=0. Let {𝑥_𝑛} be the sequence generated by 𝑦𝑥₀∈𝐶chosenarbitrary,𝐶₀=𝐶,_𝑛𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_n+1−𝛼_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇^𝑛𝑥_𝑛,∈𝐶suchthat𝑓𝑢_𝑛,𝑦+1𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝜈𝑥∈𝐶_𝑛𝑛+1∶𝜙𝜈,𝑢_𝑛≤𝜙𝜈,𝑥_𝑛+𝜉_𝑛,=Π_𝐶𝑛+1𝑥₀,𝑛≥0,(3.55) where 𝜉_𝑛=𝜈_𝑛sup_𝑞∈Θ𝜉(𝜙(𝑞,𝑥_𝑛))+𝜇_𝑛, Π_𝐶𝑛+1is the generalized projection of 𝐸  onto 𝐶_𝑛+1, {𝛽_𝑛} and {𝛼_𝑛} are sequences in [0,1], liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0, {𝑟_𝑛}⊂[𝑎,∞)  for some 𝑎>0. If Θ∶=∩^∞𝑖=1𝐹(𝑇_𝑖)∩EP(𝑓) is a nonempty and bounded subset in 𝐶, then the sequence {𝑥_𝑛} converges strongly to 𝑝∈Θ, where 𝑝=Π_Θ𝑥₀.

Definition 3.4. A countable family of mapping {𝑇_𝑛}∶𝐶→𝐶 is said to be uniformly quasi-𝜙-asymptotically nonexpansive, if ∩^∞𝑖=1𝐹(𝑇_𝑖)≠∅ and there exist real sequences {𝑘_𝑛}⊂[1,∞),𝑘_𝑛→1 such that for each i≥1, 𝜙𝑝,𝑇^𝑛𝑖𝑥≤𝑘_𝑛𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈^∞_𝑖=1𝐹𝑇_𝑖.(3.56) The following Corollary can be directly obtained from Theorem 3.1.

Corollary 3.5. Let 𝐶 be a nonempty, closed and convex subset of an uniformly smooth and strictly convex Banach Banach space 𝐸 with Kadec-Klee property. Let 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping and let 𝜑∶𝐶→𝑅 be a lower semicontinuous and convex function. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4). Let {𝑇_𝑖}^∞𝑖=1∶𝐶→𝐶 be an infinite family of closed and uniformly 𝐿_𝑖-Lipschitz continuous and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1 such that Θ∶=∩^∞𝑖=1𝐹(𝑇_𝑖)∩GMEP(𝑓,𝐵,𝜑) is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑦𝑥₀∈𝐶chosenarbitrary,𝐶₀=𝐶,_𝑛𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,=𝐽⁻¹𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛𝑢_𝑛,𝐶_𝑛+1∈𝐶suchthat𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛,=𝑧∈𝑥𝐶_𝑛𝑛+1∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝜉_𝑛,=Π_𝐶𝑛+1𝑥₀,𝑛≥0,(3.57) where 𝜉_𝑛=sup_𝑝∈𝐹(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛), Π_𝐶𝑛+1is the generalized projection of 𝐸  onto 𝐶_𝑛+1, {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0, {𝛽_𝑛,0,𝛽_𝑛,𝑖} and {𝛼_𝑛} are sequences in [0,1]. If ∑^∞𝑖=1𝛽_𝑛,𝑖=1 for all 𝑛≥0, and liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛,𝑖>0 for all 𝑖≥1, then the sequence {𝑥_𝑛} converges strongly to 𝑝∈Θ, where 𝑝=Π_Θ𝑥₀.

Remark 3.6. Theorem 3.1 improves and extend the corresponding results in [13, 24–29] in the following aspects:(1)for the mappings, extend the mappings from relatively nonexpansive mappings, quasi-𝜙-nonexpansive mappings, and quasi-𝜙-asymptotically nonexpansive mappings to a countable family of total quasi-𝜙-asymptotically nonexpansive mappings;(2)for the framework of spaces, extend the space from an uniformly smooth and uniformly convex Banach space to an uniformly smooth and strictly convex Banach space with the Kadec-Klee property.

Acknowledgment

This work was supported by the Natural Science Foundation of Yunnan Province (no. 2011FB074).