Strong Convergence Theorems for Countable Families of Uniformly Quasi-𝜙-Asymptotically Nonexpansive Mappings and a System of Generalized Mixed Equilibrium Problems

Abstract

The purpose of this paper is to present a new hybrid block iterative scheme by the generalized 𝑓- projection method for finding a common element of the fixed point set for a countable family of uniformly quasi-𝜙-asymptotically nonexpansive mappings and the set of solutions of the system of generalized mixed equilibrium problems in a strictly convex and uniformly smooth Banach space with the Kadec-Klee property. Furthermore, we prove that our new iterative scheme converges strongly to a common element of the aforementioned sets. The results presented in this paper improve and extend important recent results in the literature.

1. Introduction

Let 𝐸 be a Banach space with it’s dual space 𝐸^∗, and let 𝐶 be a nonempty closed convex subset of 𝐸. It is well known that the metric projection operator 𝑃_𝐶∶𝐸→𝐶 plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming problems, game theory, variational inequality, complementarity problems, and so forth (see, e.g., [1, 2] and the references therein). In 1994, Alber [3] introduced and studied the generalized projections Π_𝐸∶𝐸→𝐶 and 𝜋_𝐶∶𝐸^∗→𝐶 from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections for approximately solving the variational inequalities and von Neumann’s intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solving the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized 𝑓-projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by Abler [3] and proved some properties of the generalized 𝑓-projection operator. In 2009, Fan et al. [5] presented some basic results for the generalized 𝑓-projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces.

Block iterative method is a method which is often used by many authors to solve the convex feasibility problem (CFP) (see, [6, 7], etc.). In 2008, Plubtieng and Ungchittrakool [8] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Later, Saewan and Kumam [9, 10] introduced a new modified block hybrid projection algorithm for finding a common element of the set of solutions of the generalized equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property.

On the other hand, let {𝜃_𝑖}_𝑖∈Γ∶𝐶×𝐶→ℝ be a bifunction, let {𝜑_𝑖}_𝑖∈Γ∶𝐶→ℝ be a real-valued function, and let {𝐴_𝑖}_𝑖∈Γ∶𝐶→𝐸^∗ be a monotone mapping, where Γ is an arbitrary index set. The system of generalized mixed equilibrium problems is to find 𝑥∈𝐶 such that 𝜃_𝑖𝑥,𝑦+𝐴_𝑖𝑥,𝑦−𝑥+𝜑_𝑖𝑦−𝜑_𝑖(𝑥)≥0,𝑖∈Γ,∀𝑦∈𝐶. (1.1) If Γ is a singleton, then problem (1.1) reduces into the generalized mixed equilibrium problem, which is to find 𝑥∈𝐶 such that𝜃𝑥,𝑦+𝐴𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.2) The set of solutions (1.2) is denoted by GMEP(𝜃,𝐴,𝜑), that is,GMEP𝜃,𝐴,𝜑=𝑥∈𝐶∶𝜃𝑥,𝑦+𝐴𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.3) If 𝐴≡0, the problem (1.2) reduces into the mixed equilibrium problem for 𝜃, denoted by MEP(𝜃,𝜑), which is to find 𝑥∈𝐶 such that𝜃𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.4) If 𝜃≡0, the problem (1.2) reduces into the mixed variational inequality of Browder type, denoted by VI(𝐶,𝐴,𝜑), which is to find 𝑥∈𝐶 such that𝐴𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.5) If 𝐴≡0 and 𝜑≡0, the problem (1.2) reduces into the equilibrium problem for 𝜃, denoted by EP(𝜃), which is to find 𝑥∈𝐶 such that𝜃𝑥,𝑦≥0,∀𝑦∈𝐶. (1.6) If 𝜃≡0, the problem (1.4) reduces into the minimize problem, denoted by argmin(𝜑), which is to find 𝑥∈𝐶 such that𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.7) The above formulation (1.5) was shown in [11] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In other words, the EP(𝜃) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. Some solution methods have been proposed to solve the EP(𝜃); see, for example, [11–24] and references therein.

A point 𝑥∈𝐶 is a fixed point of a mapping 𝑆∶𝐶→𝐶 if 𝑆𝑥=𝑥, by 𝐹(𝑆) denote the set of fixed points of 𝑆; that is, 𝐹(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥}. Recall that 𝑆 is said to be nonexpansive if𝑆𝑥−𝑆𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.8)𝑆 is said to be quasi-nonexpansive if 𝐹(𝑆)≠∅ and𝑥−𝑆𝑦≤𝑥−𝑦,∀𝑥∈𝐹(𝑆),𝑦∈𝐶. (1.9)𝑆 is said to be asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that𝑆^𝑛𝑥−𝑆^𝑛𝑦≤𝑘_𝑛𝑥−𝑦,∀𝑥,𝑦∈𝐶,∀𝑛≥1. (1.10)𝑆 is said to be asymptotically quasi-nonexpansive if 𝐹(𝑆)≠∅ and there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that𝑥−𝑆^𝑛𝑦≤𝑘_𝑛𝑥−𝑦,∀𝑥∈𝐹(𝑆),𝑦∈𝐶,∀𝑛≥1. (1.11)

Recall that a point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑆 [25] if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑆 will be denoted by 𝐹(𝑆).

Let 𝐸 be a real Banach space with norm ‖⋅‖, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝐸^∗ denote the dual of 𝐸. Let ⟨⋅,⋅⟩ denote the duality pairing of 𝐸^∗ and 𝐸. If 𝐸 is a Hilbert space, ⟨⋅,⋅⟩ denotes an inner product on 𝐸. Consider the functional defined by𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,for𝑥,𝑦∈𝐸, (1.12) where 𝐽∶𝐸→2^𝐸∗ is the normalized duality mapping.

A mapping 𝑆 from 𝐶 into itself is said to be relatively nonexpansive [26–28] if 𝐹(𝑆)=𝐹(𝑆)≠∅ and𝜙𝑝,𝑆𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑆). (1.13)𝑆 is said to be relatively asymptotic nonexpansive [29] if 𝐹(𝑆)=𝐹(𝑆)≠∅ and there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that𝜙𝑝,𝑆^𝑛𝑥≤𝑘_𝑛𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑆),𝑛≥1. (1.14) The asymptotic behavior of a relatively nonexpansive mapping was studied in [30–32].

𝑆 is said to be 𝜙-nonexpansive if𝜙𝑆𝑥,𝑆𝑦≤𝜙𝑥,𝑦,∀𝑥,𝑦∈𝐶. (1.15)𝑆 is said to be quasi 𝜙-nonexpansive [17, 33, 34] if 𝐹(𝑆)≠∅ and𝜙𝑝,𝑆𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑆). (1.16)𝑆 is said to be 𝜙-asymptotically nonexpansive if there exists a real sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that𝜙𝑆^𝑛𝑥,𝑆^𝑛𝑦≤𝑘_𝑛𝜙𝑥,𝑦,∀𝑥,𝑦∈𝐶. (1.17)𝑆 is said to be quasi 𝜙-asymptotically nonexpansive [34, 35] if 𝐹(𝑆)≠∅ and there exists a real sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that𝜙𝑝,𝑆^𝑛𝑥≤𝑘_𝑛𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑆),𝑛≥1. (1.18) A mapping 𝑆 is said to be closed if for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑆𝑥_𝑛→𝑦, then 𝑆𝑥=𝑦.

Remark 1.1. It is easy to know that each relatively nonexpansive mapping is closed. The class of quasi-𝜙-asymptotically nonexpansive mappings contains properly the class of quasi-𝜙-nonexpansive mappings as a subclass, and the class of quasi-𝜙-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true (see for more detail [30–32, 36]).

As well known if 𝐶 is a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑃_𝐶∶𝐻→𝐶 is the metric projection of 𝐻 onto 𝐶, then 𝑃_𝐶 is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [1] recently introduced the generalized projection Π_𝐶∶𝐸→𝐶 is a map that assigns to an arbitrary point 𝑥∈𝐸 the minimum point of the functional 𝜙(𝑥,𝑦); that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem𝜙(𝑥,𝑥)=inf_𝑦∈𝐶𝜙𝑦,𝑥. (1.19) The existence and uniqueness of the operator Π_𝐶 follows from the properties of the functional 𝜙(𝑦,𝑥) and the strict monotonicity of the mapping 𝐽 (see, e.g., [1, 37–40]). It is obvious from the definition of function 𝜙 that𝑦−‖𝑥‖²≤𝜙𝑦,𝑥≤𝑦−‖𝑥‖²,∀𝑥,𝑦∈𝐸. (1.20) If 𝐸 is a Hilbert space, then 𝜙(𝑦,𝑥)=‖𝑦−𝑥‖² and Π_𝐶 becomes the metric projection of 𝐸 onto 𝐶.

Next we recall the concept of the generalized 𝑓-projection operator. Let 𝐺∶𝐶×𝐸^∗→ℝ∪{+∞} be a functional defined as follows:𝐺(𝜉,𝜛)=𝜉²−2𝜉,𝜛+‖𝜛‖²+2𝜌𝑓(𝜉), (1.21) where 𝜉∈𝐶,𝜛∈𝐸^∗,𝜌 is positive number, and 𝑓∶𝐶→ℝ∪{+∞} is proper, convex, and lower semicontinuous. From definitions of 𝐺 and 𝑓, it is easy to see the following properties: (1)𝐺(𝜉,𝜛) is convex and continuous with respect to 𝜛 when 𝜉 is fixed; (2)𝐺(𝜉,𝜛) is convex and lower semicontinuous with respect to 𝜉 when 𝜛 is fixed.

Let 𝐸 be a real Banach space with its dual 𝐸^∗. Let 𝐶 be a nonempty closed convex subset of 𝐸. We say that 𝜋^𝑓_𝐶∶𝐸^∗→2^𝐶 is generalized 𝑓-projection operator if𝜋^𝑓_𝐶𝜛=𝑢∈𝐶∶𝐺(𝑢,𝜛)=inf_𝜉∈𝐶𝐺(𝜉,𝜛),∀𝜛∈𝐸^∗. (1.22)

In 2005, Matsushita and Takahashi [36] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping 𝑆 in a Banach space E:𝑥₀∈𝐶,chosenarbitrarily,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑆𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀. (1.23)

They proved that {𝑥_𝑛} converges strongly to Π_𝐹(𝑆)𝑥₀, where Π_𝐹(𝑆) is the generalized projection from 𝐶 onto 𝐹(𝑆).

Motivated by the results of Takahashi and Zembayashi [41], Cholamjiak and Suantai [12] proved the following strong convergence theorem by the hybrid iterative scheme for approximation of common fixed point of countable families of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space: 𝑥₀∈𝐸, 𝑥₁=Π_𝐶1𝑥₀,𝐶₁=𝐶,𝑦_𝑛,𝑖=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑆𝑥_𝑛,𝑢_𝑛,𝑖=𝑇^𝜃𝑚_{𝑟𝑚,𝑛}T^𝜃𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝜃1_𝑟1,𝑛𝑦_𝑛,𝑖,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶sup_𝑖>1𝜙𝑧,𝐽𝑢_𝑛,𝑖≤𝜙𝑧,𝐽𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,𝑛≥1. (1.24) Then, they proved that under certain appropriate conditions imposed on {𝛼_𝑛} and {𝑟_𝑛,𝑖}, the sequence {𝑥_𝑛} converges strongly to Π_𝐶𝑛+1𝑥₀.

In 2010, Li et al. [42] introduced the following hybrid iterative scheme for the approximation of fixed point of relatively nonexpansive mapping using the properties of generalized 𝑓-projection operator in a uniformly smooth real Banach space which is also uniformly convex: 𝑥₀∈𝐶,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑆𝑥_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝐺𝑤,𝐽𝑦_𝑛≤𝐺𝑤,𝐽𝑥_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥0, (1.25) where Π^𝑓_𝐶∶𝐸→2^𝐶 is generalized 𝑓-projection operator. They proved the strong convergence theorem for finding an element in the fixed point set of 𝑆. We remark here that the results of Li et al. [42] extended and improved on the results of Matsushita and Takahashi [36].

Recently, Shehu [43] introduced a new iterative scheme by hybrid methods and proved strong convergence theorem for the approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a solution to a system of generalized mixed equilibrium problems in a uniformly convex and uniformly smooth Banach space by using the properties of the generalized 𝑓-projection operator. Chang et al. [44] used the modified block iterative method to propose an iterative algorithm for solving the convex feasibility problems for an infinite family of quasi-𝜙-asymptotically nonexpansive mappings. Very recently, Kim [45] and Saewan and Kumam [46] considered the shrinking projection methods for asymptotically quasi-𝜙-nonexpansive mappings in a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property.

In this paper, we introduce a new hybrid block iterative scheme of the generalized 𝑓-projection operator for finding a common element of the fixed point set of uniformly quasi-𝜙-asymptotically nonexpansive mappings and the set of solutions of the system of generalized mixed equilibrium problems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Then, we prove that our new iterative scheme converges strongly to a common element of the aforementioned sets. The results presented in this paper improve and extend the results of Shehu [43], Chang et al. [44], Li et al. [42], Takahashi and Zembayashi [41], Cholamjiak and Suantai [12], and many authors.

2. Preliminaries

A Banach space 𝐸 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. Let 𝑈={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. Then a Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0(‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡 exists for each 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit exists uniformly in 𝑥,𝑦∈𝑈. Let 𝐸 be a Banach space. The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(𝑡)=sup{((‖𝑥+𝑦‖+‖𝑥−𝑦‖)/2)−1∶‖𝑥‖=1,‖𝑦‖≤𝑡}. The modulus of convexity of 𝐸 is the function 𝛿_𝐸∶[0,2]→[0,1] defined by 𝛿_𝐸(𝜀)=inf{1−‖(𝑥+𝑦)/2‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀}. The normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is defined by 𝐽(𝑥)={𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²,‖𝑥^∗‖=‖𝑥‖}. If 𝐸 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity mapping.

Remark 2.1. If 𝐸 is a reflexive, strictly convex and smooth Banach space, then for 𝑥,𝑦∈𝐸, 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 (1.12), we have ‖𝑥‖=‖𝑦‖. This implies that ⟨𝑥,𝐽𝑦⟩=‖𝑥‖²=‖𝐽𝑦‖². From the definition of 𝐽, one has 𝐽𝑥=𝐽𝑦. Therefore, we have 𝑥=𝑦; see [38, 40] for more details.

Recall that a Banach space 𝐸 has the Kadec-Klee property [38, 40, 47], if for any sequence {𝑥_𝑛}⊂𝐸 and 𝑥∈𝐸 with 𝑥_𝑛⇀𝑥 and ‖𝑥_𝑛‖→‖𝑥‖, then ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞. It is well known that if 𝐸 is a uniformly convex Banach space, then 𝐸 has the Kadec-Klee property.

Remark 2.2. Let 𝐸 be a Banach space. Then we know that (1)if 𝐸 is an arbitrary Banach space, then 𝐽 is monotone and bounded; (2)if 𝐸 is strictly convex, then 𝐽 is strictly monotone; (3)if 𝐸 is smooth, then 𝐽 is single valued and semicontinuous; (4)if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸; (5)𝐸 is reflexive, smooth, and strictly convex, then the normalized duality mapping 𝐽=𝐽₂ is single valued, one-to-one, and onto; (6)if E is uniformly smooth, then 𝐸 is smooth and reflexive; (7)𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex; see [38] for more details.

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

For solving the equilibrium problem for a bifunction 𝜃∶𝐶×𝐶→ℝ, let us assume that 𝜃 satisfies the following conditions. (A1)𝜃(𝑥,𝑥)=0,forall𝑥∈𝐶. (A2)𝜃 is monotone; that is, 𝜃(𝑥,𝑦)+𝜃(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐶. (A3)for each 𝑥,𝑦,𝑧∈𝐶,lim_𝑡↓0𝜃𝑡𝑧+(1−𝑡)𝑥,𝑦≤𝜃𝑥,𝑦. (2.1)(A4)for each 𝑥∈𝐶, 𝑦↦𝜃(𝑥,𝑦) is convex and lower semicontinuous.

For example, let 𝐴 be a continuous and monotone operator of 𝐶 into 𝐸^∗ and define𝜃𝑥,𝑦=𝐴𝑥,𝑦−𝑥,∀𝑥,𝑦∈𝐶. (2.2)

Then, 𝜃 satisfies (A1)–(A4). The following result is in Blum and Oettli [11].

Motivated by Combettes and Hirstoaga [13] in a Hilbert space and Takahashi and Zembayashi [48] in a Banach space, Zhang [49] obtained the following lemma.

Lemma 2.3 (Liu et al. [50], Zhang [49, Lemma 1.5]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝜃 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, let 𝜑∶𝐶→ℝ be a semicontinuous and convex functional, for 𝑟>0, and let 𝑥∈𝐸. Then, there exists 𝑧∈𝐶 such that 𝑄+1𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶, (2.3) where 𝑄(𝑧,𝑦)=𝜃(𝑧,𝑦)+⟨𝐴𝑧,𝑦−𝑧⟩+𝜑(𝑦)−𝜑(𝑧). Furthermore, define a mapping 𝑇_𝑟∶𝐸→𝐶 as follows: 𝑇_𝑟+1𝑥=𝑧∈𝐶∶𝑄𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (2.4) Then the following holds. (1)𝑇_𝑟 is single-valued. (2)𝑇_𝑟 is firmly nonexpansive; that is, for all 𝑥,𝑦∈𝐸,⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦⟩≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩.(3)F(𝑇_𝑟)=𝐹(𝑇_𝑟)=GMEP(𝜃,𝐴,𝜑). (4)GMEP(𝜃,𝐴,𝜑) is closed and convex. (5)𝜙(𝑝,𝑇_𝑟𝑧)+𝜙(𝑇_𝑟𝑧,𝑧)≤𝜙(𝑝,𝑧),forall𝑝∈𝐹(𝑇_𝑟) and 𝑧∈𝐸.

For the generalized 𝑓-projection operator, Wu and Huang [4] proved the following basic properties.

Lemma 2.4 (Wu and Huang [4]). Let 𝐸 be a reflexive Banach space with its dual 𝐸^∗ and let 𝐶 be a nonempty closed convex subset of 𝐸. The following statements hold. (1)𝜋^𝑓_𝐶𝜛 is nonempty closed convex subset of 𝐶 for all 𝜛∈𝐸^∗.(2)If 𝐸 is smooth, then for all 𝜛∈𝐸^∗, 𝑥∈𝜋^𝑓_𝐶𝜛 if and only if𝑥−𝑦,𝜛−𝐽𝑥+𝜌𝑓𝑦−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐶. (2.5)(3)If 𝐸 is strictly convex and 𝑓∶𝐶→ℝ∪{+∞} is positive homogeneous (i.e., 𝑓(𝑡𝑥)=𝑡𝑓(𝑥) for all 𝑡>0 such that 𝑡𝑥∈𝐶 where 𝑥∈𝐶), then 𝜋^𝑓_𝐶 is single-valued mapping.

Recently, Fan et al. [5] have shown that the condition 𝑓 which is positive homogeneous and appeared in [5, Lemma 2.1(iii)] can be removed.

Lemma 2.5 (Fan et al. [5]). Let 𝐸 be a reflexive Banach space with its dual 𝐸^∗, and let 𝐶 be a nonempty closed convex subset of 𝐸. If 𝐸 is strictly convex, then 𝜋^𝑓_𝐶𝜛 is single valued.

Recall that 𝐽 is single value mapping when 𝐸 is a smooth Banach space. There exists a unique element 𝜛∈𝐸^∗ such that 𝜛=𝐽𝑥 where 𝑥∈𝐸. This substitution for (1.21) gives𝐺(𝜉,𝐽𝑥)=𝜉²−2⟨𝜉,𝐽𝑥⟩+‖𝑥‖²+2𝜌𝑓(𝜉). (2.6)

Now we consider the second generalized 𝑓-projection operator in Banach spaces (see [42]).

Definition 2.6. Let 𝐸 be a real smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. We say that Π^𝑓_𝐶∶𝐸→2^𝐶 is generalized 𝑓-projection operator if Π^𝑓_𝐶𝑥=𝑢∈𝐶∶𝐺(𝑢,𝐽𝑥)=inf_𝜉∈𝐶𝐺(𝜉,𝐽𝑥),∀𝑥∈𝐸. (2.7)

Lemma 2.7 (Deimling [51]). Let 𝐸 be a Banach space, and let 𝑓∶𝐸→ℝ∪{+∞} be a lower semicontinuous convex functional. Then there exist 𝑥^∗∈𝐸^∗ and 𝛼∈ℝ such that 𝑓(𝑥)≥𝑥,𝑥^∗+𝛼,∀𝑥∈𝐸. (2.8)

Lemma 2.8 (Li et al. [42]). Let 𝐸 be a reflexive smooth Banach space, and 𝐶 let be a nonempty closed convex subset of 𝐸. The following statements hold.(1)Π^𝑓_𝐶𝑥 is nonempty closed convex subset of 𝐶 for all 𝑥∈𝐸. (2)For all 𝑥∈𝐸, 𝑥∈Π^𝑓_𝐶𝑥 if and only if 𝑥−𝑦,𝐽𝑥−𝐽𝑥+𝜌𝑓𝑦−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐶. (2.9)(3)If 𝐸 is strictly convex, then Π^𝑓_𝐶 is single-valued mapping.

Lemma 2.9 (Li et al. [42]). Let 𝐸 be a reflexive smooth Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑥∈𝐸, 𝑥∈Π^𝑓_𝐶𝑥. Then 𝜙𝑦,𝑥+𝐺(𝑥,𝐽𝑥)≤𝐺𝑦,𝐽𝑥,∀𝑦∈𝐶. (2.10)

Remark 2.10. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, and let 𝑓(𝑥)=0 for all 𝑥∈𝐸. Then Lemma 2.9 reduces to the property of the generalized projection operator considered by Alber [1].

Lemma 2.11 (Li et al. [42]). Let 𝐸 be a Banach space, and let 𝑓∶𝐸→ℝ∪{+∞} be a proper, convex, and lower semicontinuous mapping with convex domain 𝐷(𝑓). If {𝑥_𝑛} is a sequence in 𝐷(𝑓) such that 𝑥_𝑛⇀𝑥∈𝐷(𝑓) and lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑦)=𝐺(𝑥,𝐽𝑦), then lim_𝑛→∞‖𝑥_𝑛‖=‖𝑥‖.

Lemma 2.12 (Chang et al. [44]). 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 𝑆∶𝐶→𝐶 be a closed and quasi-𝜙-asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1. Then 𝐹(𝑆) is a closed convex subset of 𝐶.

Lemma 2.13 (Chang et al. [44]). Let 𝐸 be a uniformly convex Banach space, let 𝑟>0 be a positive number, and let 𝐵_𝑟(0) be 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 number with ∑^∞_𝑛=1𝜆_𝑛=1, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,2𝑟)→[0,∞) with 𝑔(0)=0 such that, for any positive integer 𝑖,𝑗 with 𝑖<𝑗, ^∞_𝑛=1𝜆_𝑛𝑥_𝑛²≤^∞_𝑛=1𝜆_𝑛𝑥_𝑛²−𝜆_𝑖𝜆_𝑗𝑔𝑥_𝑖−𝑥_𝑗. (2.11)

Definition 2.14. (Chang et al. [44]).(1)Let {𝑆_𝑖}^∞_𝑖=1∶𝐶→𝐶 be a sequence of mappings. {𝑆_𝑖}^∞_𝑖=1 is said to be a family of uniformly quasi-𝜙-asymptotically nonexpansive mappings, if ℱ∶=∩^∞_𝑖=1𝐹(𝑆_𝑖)≠∅, and there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that for each 𝑖≥1𝜙𝑝,𝑆^𝑛_𝑖𝑥≤𝑘_𝑛𝜙𝑝,𝑥,∀𝑝∈ℱ,𝑥∈𝐶,∀𝑛≥1. (2.12)(2)A mapping 𝑆∶𝐶→𝐶 is said to be uniformly 𝐿-Lipschitz continuous, if there exists a constant 𝐿>0 such that𝑆^𝑛𝑥−𝑆^𝑛𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.13)

If 𝑓(𝑥)≥0, it is clearly by the definition of mappings {𝑆_𝑖}^∞_𝑖=1 is a family of uniformly quasi-𝜙-asymptotically nonexpansive is equivalent to if ∩^∞_𝑖=1𝐹(𝑆_𝑖)≠∅ and there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that for each 𝑖≥1,𝐺𝑝,𝐽𝑆^𝑛_𝑖𝑥≤𝑘_𝑛𝐺𝑝,𝐽𝑥,∀𝑝∈ℱ,𝑥∈𝐶,∀𝑛≥1. (2.14)

3. Strong Convergence Theorem

Now we state and prove our main result.

Theorem 3.1. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property. Let {𝑆_𝑖}^∞_𝑖=1∶𝐶→𝐶 be an infinite family of closed uniformly 𝐿_𝑖-Lipschitz continuous and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1, and let 𝑓∶𝐸→ℝ be a convex lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)). For each 𝑗=1,2,…,𝑚, let 𝜃_𝑗 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4), let 𝐴_𝑗∶𝐶→𝐸^∗ be a continuous and monotone mapping, and let 𝜑_𝑗∶𝐶→ℝ be a lower semicontinuous and convex function. Assume that 𝐹∶=(∩^∞_𝑖=1𝐹(𝑆_𝑖))∩(∩^𝑚_𝑗=1GMEP(𝜃_𝑗,𝐴_𝑗,𝜑_𝑗))≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π^𝑓_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑧_𝑛=𝐽⁻¹𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝐺𝑧,𝐽𝑢_𝑛≤𝐺𝑧,𝐽𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.1) where 𝑀_𝑛=sup_𝑞∈𝐹{𝐺(𝑞,𝐽𝑥_𝑛)}, {𝛼_𝑛,𝑖},{𝛽_𝑛} are sequences in [0,1], and ∑^∞_𝑖=0𝛼_𝑛,𝑖=1, for all 𝑛≥0, satisfy the following conditions.(i){𝑟_𝑗,𝑛}⊂[𝑑,∞) for some 𝑑>0.(ii)liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all𝑖≥1, and liminf_𝑛→∞(1−𝛽_𝑛)>0.(iii)𝑓(𝑥)≥0for all𝑥∈𝐶. and 𝑓(0)=0. Then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π^𝑓_𝐹𝑥₀.

Proof. We split the proof into six steps.
Step 1. We first show that 𝐶_𝑛+1 is closed and convex for each 𝑛≥1.
Clearly 𝐶₁=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for each 𝑛∈ℕ. Since for any 𝑧∈𝐶_𝑛, we know that 𝐺(𝑧,𝐽𝑢_𝑛)≤𝐺(𝑧,𝐽𝑥_𝑛)+(𝑘_𝑛−1)𝑀_𝑛 is equivalent to 2𝑧,𝐽𝑥_𝑛−𝐽𝑢_𝑛≤𝑥_𝑛²−𝑢_𝑛²+𝑘_𝑛−1𝑀_𝑛, (3.2) it follow that 𝐶_𝑛+1=𝑧∈𝐶_𝑛∶2𝑧,𝐽𝑥_𝑛−𝐽𝑢_𝑛≤𝑥_𝑛²−𝑢_𝑛²+𝑘_𝑛−1𝑀_𝑛. (3.3) So, 𝐶_𝑛+1 is closed and convex. This implies that Π^𝑓_𝐶𝑛+1𝑥₀ is well defined for all 𝑛≥1.
Step 2. We show that 𝐹⊂𝐶_𝑛 for all 𝑛≥1.
We show by induction that 𝐹⊂𝐶_𝑛 for all 𝑛∈ℕ. It is obvious that 𝐹⊂𝐶₁=𝐶. Suppose that 𝐹⊂𝐶_𝑛 for some 𝑛≥1. Let 𝑞∈𝐹⊂𝐶_𝑛, by the convexity of ‖⋅‖², Lemma 2.13, and the uniformly quasi-𝜙-asymptotically nonexpansive of 𝑆_𝑖, we compute 𝐺𝑞,𝐽𝑧_𝑛=𝐺𝑞,𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛=𝑞²−2𝑞,𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛+𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛²𝑞=𝑞+2𝜌𝑓₂−2𝛼_𝑛,0𝑞,𝐽𝑥_𝑛−2^∞_𝑖=1𝛼_𝑛,𝑖𝑞,𝐽𝑆^𝑛_𝑖𝑥_𝑛+𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛²𝑞≤𝑞+2𝜌𝑓₂−2𝛼_𝑛,0𝑞,𝐽𝑥_𝑛−2^∞_𝑖=1𝛼_𝑛,𝑖𝑞,𝐽𝑆^𝑛_𝑖𝑥_𝑛+𝛼_𝑛,0𝐽𝑥_𝑛²+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_i𝑥_𝑛²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛+2𝜌𝑓𝑞=‖𝑞‖²−2𝛼_𝑛,0𝑞,𝐽𝑥_𝑛+𝛼_𝑛,0𝐽𝑥_𝑛²−2^∞_𝑖=1𝛼_𝑛,𝑖𝑞,𝐽𝑆^𝑛_𝑖𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_i𝑥_𝑛²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛+2𝜌𝑓𝑞=𝛼_𝑛,0𝐺𝑞,𝐽𝑥_𝑛+^∞_𝑖=1𝛼_n,𝑖𝐺𝑞,𝐽𝑆^𝑛_𝑖𝑥_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛≤𝛼_𝑛,0𝑘_𝑛𝐺𝑞,𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝑘_𝑛𝐺𝑞,𝐽𝑥_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛≤𝑘_𝑛𝐺𝑞,𝐽𝑥_𝑛. (3.4) Since 𝑢_𝑛=Ω^𝑚_𝑛𝑦_𝑛, when Ω^𝑗_𝑛=𝑇^𝑄𝑗_{𝑟𝑗,𝑛}𝑇^𝑄𝑗−1_{𝑟𝑗−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛, 𝑗=1,2,3,…,𝑚, Ω⁰_𝑛=𝐼, it follows from (3.4) that 𝐺𝑞,𝐽𝑢_𝑛=𝐺𝑞,𝐽Ω^𝑚_𝑛𝑦_𝑛≤𝐺𝑞,𝐽𝑦_𝑛=𝐺𝑞,𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑧_𝑛=𝑞²−2𝑞,𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑧_𝑛+𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑧_𝑛²𝑞≤𝑞+2𝜌𝑓₂−2𝛽_𝑛𝑞,𝐽𝑥_𝑛−21−𝛽_𝑛𝑞,𝐽𝑧_𝑛+𝛽_𝑛𝑥_𝑛²+1−𝛽_𝑛𝑧_𝑛²+2𝜌𝑓𝑞=𝛽_𝑛𝐺𝑞,𝐽𝑥_𝑛+1−𝛽_𝑛𝐺𝑞,𝐽𝑧_𝑛≤𝛽_𝑛𝐺𝑞,𝐽𝑥_𝑛+1−𝛽_𝑛𝑘_𝑛𝐺𝑞,𝐽𝑥_𝑛≤𝐺𝑞,𝐽𝑥_𝑛+1−𝛽_𝑛𝑘_𝑛−1𝐺𝑞,𝐽𝑥_𝑛=𝐺𝑞,𝐽𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛. (3.5) This shows that 𝑞∈𝐶_𝑛+1 which implies that 𝐹⊂𝐶_𝑛+1, and hence, 𝐹⊂𝐶_𝑛 for all 𝑛≥1. Since 𝐹 is nonempty, 𝐶_𝑛 is a nonempty closed convex subset of 𝐸, and hence, Π^𝑓_𝐶𝑛 exist for all 𝑛≥0. This implies that the sequence {𝑥_𝑛} is well defined.
Step 3. We show that {𝑥_𝑛} is bounded.
Since 𝑓∶𝐸→ℝ is convex and lower semicontinuous mapping, from Lemma 2.7, we have known that there exist 𝑥^∗∈𝐸^∗ and 𝛼∈ℝ such that 𝑓(𝑥)≥𝑥,𝑥^∗+𝛼,∀𝑥∈𝐸. (3.6) Since 𝑥_𝑛∈𝐸, it follows that 𝐺𝑥_𝑛,𝐽𝑥₀=𝑥_𝑛²−2𝑥_𝑛,𝐽𝑥₀+𝑥₀²+2𝜌𝑓𝑥_𝑛≥𝑥_𝑛²−2𝑥_𝑛,𝐽𝑥₀+𝑥₀²+2𝜌𝑥_𝑛,𝑥^∗=𝑥+2𝜌𝛼_𝑛2−2𝑥_𝑛,𝐽𝑥₀−𝜌𝑥^∗+𝑥₀²≥𝑥+2𝜌𝛼_𝑛2−2‖𝑥_𝑛‖‖𝐽𝑥₀−𝜌𝑥^∗‖+𝑥₀²=𝑥+2𝜌𝛼_𝑛−𝐽𝑥₀−𝜌𝑥^∗2+𝑥₀²−𝐽𝑥₀−𝜌𝑥^∗2+2𝜌𝛼. (3.7) For each 𝑞∈𝐹⊂𝐶_𝑛 and by the definition of 𝐶_𝑛 that x_𝑛=Π^𝑓_𝐶𝑛𝑥₀, it follows from (3.7) that 𝐺𝑞,𝐽𝑥₀≥𝐺𝑥_𝑛,𝐽𝑥₀≥𝑥_𝑛−𝐽𝑥₀−𝜌𝑥^∗2+𝑥₀²−𝐽𝑥₀−𝜌𝑥^∗2+2𝜌𝛼. (3.8) This implies that {𝑥_𝑛} is bounded and so are {𝐺(𝑥_𝑛,𝐽𝑥₀)}.
Step 4. We show that lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑥_𝑛‖=0.
By the fact that 𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛 and 𝑥_𝑛=Π^𝑓_𝐶𝑛𝑥₀, followed by Lemma 2.9, we get 0≤𝑥_𝑛+1−𝑥_𝑛²≤𝜙𝑥_𝑛+1,𝑥_𝑛≤𝐺𝑥_𝑛+1,𝐽𝑥₀−𝐺𝑥_𝑛,𝐽𝑥₀. (3.9) This implies that {𝐺(𝑥_𝑛,𝐽𝑥₀)} is nondecreasing. So, we obtain that lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥₀) exist, and taking 𝑛→∞, we obtain that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0. (3.10) Since {𝑥_𝑛} is bounded in 𝐶 and 𝐸 is reflexive, we can assume that 𝑥_𝑛⇀𝑝. From the fact that 𝑥_𝑛=Π^𝑓_𝐶𝑛𝑥₀ when 𝐶_𝑛 is closed and convex for each 𝑛≥1, it is easy to see that 𝑝∈𝐶_𝑛 and we get 𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑝,𝐽𝑥₀,∀𝑛≥1. (3.11) Since 𝑓 is convex and lower semicontinuous, we have liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀=liminf_𝑛→∞𝑥_𝑛²−2𝑥_𝑛,𝐽𝑥₀+𝑥₀²+2𝜌𝑓𝑥_𝑛≥‖𝑝‖²−2𝑝,𝐽𝑥₀+‖𝑥₀‖²+2𝜌𝑓𝑝=𝐺𝑝,𝐽𝑥₀. (3.12) By (3.11) and (3.12), we get 𝐺𝑝,𝐽𝑥₀≤liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤limsup_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑝,𝐽𝑥₀. (3.13) That is lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥₀)=𝐺(𝑝,𝐽𝑥₀), by Lemma 2.11, we have ‖𝑥_𝑛‖→‖𝑝‖, from the Kadec-Klee property of 𝐸, we obtain that lim_𝑛→∞𝑥_𝑛=𝑝, (3.14) and we also have lim_𝑛→∞𝑥_𝑛+1=𝑝. (3.15) Since 𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛 and from the definition of 𝐶_𝑛+1, we have 𝐺𝑥_𝑛+1,𝐽𝑢_𝑛≤𝐺𝑥_𝑛+1,𝐽𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛,∀𝑛∈ℕ, (3.16) is equivalent to 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛,∀𝑛∈ℕ. (3.17) By (3.10) and in view of lim_𝑛→∞(𝑘_𝑛−1)𝑀_𝑛=0, we also have lim_𝑛→∞𝜙𝑥_𝑛+1,𝑢_𝑛=0. (3.18) From (1.20), it follow that 𝑥_𝑛+1−𝑢_𝑛²⟶0. (3.19) Since ‖𝑥_𝑛+1‖→‖𝑝‖, we also have 𝑢_𝑛⟶𝑝as𝑛⟶∞. (3.20) It follows that 𝐽𝑢_𝑛⟶𝐽𝑝as𝑛⟶∞. (3.21) 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 see that 𝐽(𝐸)=𝐸^∗. Hence, there exists 𝑥∈𝐸 such that 𝐽𝑥=𝑥^∗. It follows that 𝜙𝑥_𝑛+1,𝑢_𝑛=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑢_𝑛+𝑢_𝑛²=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑢_𝑛+𝐽𝑢_𝑛². (3.22) Taking liminf_𝑛→∞ on both sides of the equality above and in view of the weak lower semicontinuity of norm ‖⋅‖, it yields that 0≥𝑝²−2𝑝,𝑥^∗+𝑥^∗2=𝑝²−2𝑝,𝐽𝑥+‖𝐽𝑥‖²=𝑝²−2𝑝,𝐽𝑥+‖𝑥‖²=𝜙𝑝,𝑥. (3.23) That is 𝑝=𝑥, which implies that 𝑥^∗=𝐽𝑝. It follows that 𝐽𝑢_𝑛⇀𝐽𝑝∈𝐸^∗. From (3.21) and the Kadec-Klee property of 𝐸^∗; that, is 𝐽𝑢_𝑛→𝐽𝑝 as 𝑛→∞, we known that 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak*-continuous, that is, 𝑢_𝑛⇀𝑝. From (3.20) and the Kadec-Klee property of 𝐸, we have lim_𝑛→∞𝑢_𝑛=𝑝. (3.24) Since ‖𝑥_𝑛−𝑢_𝑛‖≤‖𝑥_𝑛−𝑝‖+‖𝑝−𝑢_𝑛‖, it follows that lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=0. (3.25) From 𝐽 that is uniformly norm-to-norm continuous on bounded subsets of 𝐸, we obtain lim_𝑛→∞𝐽𝑢_𝑛−𝐽𝑥_𝑛=0. (3.26)
Step 5. We will show that 𝑝∈𝐹∶=(∩^∞_𝑖=1𝐹(𝑆_𝑖))∩(∩^𝑚_𝑗=1GMEP(𝜃_𝑗,𝐴_𝑗,𝜑_𝑗)).
(a) We show that 𝑝∈∩^∞_𝑖=1𝐹(𝑆_𝑖).
For 𝑞∈𝐹, we note that 𝜙𝑞,𝑥_𝑛−𝜙𝑞,𝑢_𝑛+𝑘_𝑛−1𝑀_𝑛=𝑥_𝑛²−𝑢_𝑛²−2𝑞,𝐽𝑥_𝑛−𝐽𝑢_𝑛+𝑘_𝑛−1𝑀_𝑛≤𝑥_𝑛−𝑢_𝑛𝑥_𝑛+𝑢_𝑛+2𝑞𝐽𝑥_𝑛−𝐽𝑢_𝑛+𝑘_𝑛−1𝑀_𝑛. (3.27) It follows from ‖𝑥_𝑛−𝑢_𝑛‖→0, ‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖→0 and (𝑘_𝑛−1)𝑀_𝑛→0 as 𝑛→∞ that 𝜙𝑞,𝑥_𝑛−𝜙𝑞,𝑢_𝑛+𝑘_𝑛−1𝑀_𝑛⟶0as𝑛⟶∞. (3.28) For any 𝑖≥1 and any 𝑞∈𝐹, it follows from (3.4) and (3.5) that 𝐺𝑞,𝐽𝑢_𝑛≤𝛽_𝑛𝐺𝑞,𝐽𝑥_𝑛+1−𝛽_𝑛𝐺𝑞,𝐽𝑧_𝑛≤𝛽_𝑛𝐺𝑞,𝐽𝑥_𝑛+1−𝛽_𝑛𝛼_𝑛,0𝑘_𝑛𝐺𝑞,𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝑘_𝑛𝐺𝑞,𝐽𝑥_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛=𝛽_𝑛𝐺𝑞,𝐽𝑥_𝑛+1−𝛽_𝑛𝑘_𝑛𝐺𝑞,𝐽𝑥_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛=𝛽_𝑛𝐺𝑞,𝐽𝑥_𝑛+1−𝛽_𝑛𝑘_𝑛𝐺𝑞,𝐽𝑥_𝑛−1−𝛽_𝑛𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛≤𝐺𝑞,𝐽𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛−1−𝛽_𝑛𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛. (3.29) It follows that 𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛≤11−𝛽_𝑛𝐺𝑞,𝐽𝑥_𝑛−𝐺𝑞,𝐽𝑢_𝑛+𝑘_𝑛−1𝑀_𝑛, (3.30) is in equivalence to 𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛≤11−𝛽_𝑛𝜙𝑞,𝑥_𝑛−𝜙𝑞,𝑢_𝑛+𝑘_𝑛−1𝑀_𝑛. (3.31) From (3.28), liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 and liminf_𝑛→∞(1−𝛽_𝑛)>0, we see that 𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛⟶0,𝑛⟶∞. (3.32) It follows from the property of 𝑔 that lim_𝑛→∞𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛=0,∀𝑗≥1. (3.33) Since 𝑥_𝑛→𝑝 and 𝐽 is uniformly continuous, it yields that 𝐽𝑥_𝑛→𝐽𝑝. Thus from (3.33), we have 𝐽𝑆^𝑛_𝑖𝑥_𝑛⟶𝐽𝑝,∀𝑖≥1. (3.34) Since 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak*-continuous, we also have 𝑆^𝑛_𝑖𝑥_𝑛⇀𝑝,∀𝑖≥1. (3.35) On the other hand, for each 𝑖≥1, we observe that 𝑆^𝑛_𝑖𝑥_𝑛−𝑝=𝐽𝑆^𝑛_𝑖𝑥_𝑛−𝐽𝑝≤𝐽𝑆^𝑛_𝑖𝑥_𝑛−𝐽𝑝. (3.36) In view of (3.34), we obtain ‖𝑆^𝑛_𝑖𝑥_𝑛‖→‖𝑝‖ for each 𝑖≥1. Since 𝐸 has the Kadec-Klee property, we get 𝑆^𝑛_𝑖𝑥_𝑛⟶𝑝foreach𝑖≥1,𝑛∈ℕ. (3.37) By the assumption that for each 𝑖≥1, 𝑆_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous, so we have 𝑆_𝑖^𝑛+1𝑥_𝑛−𝑆^𝑛_𝑖𝑥_𝑛≤𝑆_𝑖^𝑛+1𝑥_𝑛−𝑆_𝑖^𝑛+1𝑥_𝑛+1+𝑆_𝑖^𝑛+1𝑥_𝑛+1−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑆^𝑛_𝑖𝑥_𝑛≤𝐿_𝑖+1𝑥_𝑛+1−𝑥_𝑛+𝑆_𝑖^𝑛+1𝑥_𝑛+1−𝑥_𝑛+1+𝑥_𝑛−𝑆^𝑛_𝑖𝑥_𝑛. (3.38) By (3.14), (3.15), and (3.37), it yields that ‖𝑆_𝑖^𝑛+1𝑥_𝑛−𝑆^𝑛_𝑖𝑥_𝑛‖→0,𝑛→∞,for all𝑖≥1. From 𝑆^𝑛_𝑖𝑥_𝑛→𝑝, we get 𝑆_𝑖^𝑛+1𝑥_𝑛→𝑝, that is, 𝑆_𝑖𝑆^𝑛_𝑖𝑥_𝑛→𝑝. In view of the closeness of 𝑆_𝑖, we have 𝑆_𝑖𝑝=𝑝, for all 𝑖≥1. This imply that 𝑝∈∩^∞_𝑖=1𝐹(𝑆_𝑖).
(b) We show that 𝑝∈∩^𝑚_𝑗=1GMEP(𝜃_𝑗,𝐴_𝑗,𝜑_𝑗).
Since 𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛 and from (3.5), we have 𝐺𝑥_𝑛+1,𝐽𝑦_𝑛≤𝐺𝑥_𝑛+1,𝐽𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛 (3.39) is in equivalence to 𝜙x_𝑛+1,𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛. (3.40) From (3.10) and (𝑘_𝑛−1)𝑀_𝑛→0, as 𝑛→∞, we see that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑦_𝑛=0. (3.41) From (1.20), it follows that 𝑥_𝑛+1−𝑦_𝑛²⟶0. (3.42) Since ‖𝑥_𝑛+1‖→‖𝑝‖, we have 𝑦_𝑛⟶𝑝,as𝑛⟶∞. (3.43) It follow that 𝐽𝑦_𝑛⟶𝐽𝑝as𝑛⟶∞. (3.44) This implies that {‖𝐽𝑦_𝑛‖} is bounded in 𝐸^∗ and 𝐸^∗ is reflexive; we can assume that 𝐽𝑦_𝑛⇀𝑦^∗∈𝐸^∗. In view of 𝐽(𝐸)=𝐸^∗. Hence, there exists 𝑦∈𝐸 such that 𝐽𝑦=𝑦^∗. It follows that 𝜙𝑥_𝑛+1,𝑦_𝑛=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑦_𝑛+𝑦_𝑛²=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑦_𝑛+𝐽𝑦_𝑛². (3.45) Taking liminf_𝑛→∞ on both sides of the equality above and in view of the weak lower semicontinuous of norm ‖⋅‖, it yields that 0≥𝑝²−2𝑝,𝑦^∗+𝑦^∗2=𝑝²−2𝑝,𝐽𝑦+𝐽𝑦²=𝑝²−2𝑝,𝐽𝑦+𝑦²=𝜙𝑝,𝑦. (3.46) That is 𝑝=𝑦, which implies that 𝑦^∗=𝐽𝑝. It follows that 𝐽𝑦_𝑛⇀𝐽𝑝∈𝐸^∗. From (3.44) and the Kadec-Klee property of 𝐸^∗, that is, 𝐽𝑢_𝑛→𝐽𝑝 as 𝑛→∞, note that 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak*-continuous, that is, 𝑦_𝑛⇀𝑝. From (3.43) and the Kadec-Klee property of 𝐸, we have lim_𝑛→∞𝑦_𝑛=𝑝. (3.47) For 𝑞∈𝐹⊂𝐶_𝑛, by nonexpansiveness, we observe that 𝜙𝑞,𝑢_𝑛=𝜙𝑞,Ω^𝑚_𝑛𝑦_𝑛≤𝜙𝑞,Ω_𝑛^𝑚−1𝑦_𝑛≤𝜙𝑞,Ω_𝑛^𝑚−2𝑦_𝑛⋮≤𝜙𝑞,Ω^𝑗_𝑛𝑦_𝑛. (3.48) By Lemma (2.12)(5), we have for 𝑗=1,2,3,…,𝑚𝜙Ω^𝑗_𝑛𝑦_𝑛,𝑦_𝑛≤𝜙𝑞,𝑦_𝑛−𝜙𝑞,Ω^𝑗_𝑛𝑦_𝑛≤𝜙𝑞,𝑥_𝑛−𝜙𝑞,Ω^𝑗_𝑛𝑦_𝑛+𝑘_𝑛−1𝑀_𝑛≤𝜙𝑞,𝑥_𝑛−𝜙𝑞,𝑢_𝑛+𝑘_𝑛−1𝑀_𝑛. (3.49) From (3.28), we get 𝜙(Ω^𝑗_𝑛𝑦_𝑛,𝑦_𝑛)→0𝑎𝑠𝑛→∞, for 𝑗=1,2,3,…,𝑚. From (1.20), it follow that Ω^𝑗_𝑛𝑦_𝑛−𝑦_𝑛²⟶0. (3.50) Since ‖𝑦_𝑛‖→‖𝑝‖, we also have Ω^𝑗_𝑛𝑦_𝑛⟶𝑝as𝑛⟶∞. (3.51) Since {Ω^𝑗_𝑛𝑦_𝑛} is bounded and 𝐸 is reflexive, without loss of generality, we may assume that Ω^𝑗_𝑛𝑦_𝑛⇀ℎ. From the first step, we have known that 𝐶_𝑛 is closed and convex for each 𝑛≥1, it is obvious that ℎ∈𝐶_𝑛. Again since 𝜙Ω^𝑗_𝑛𝑦_𝑛,𝑦_𝑛=Ω^𝑗_𝑛𝑦_𝑛²−2Ω^𝑗_𝑛𝑦_𝑛,𝐽𝑦_𝑛+𝑦_𝑛², (3.52) taking liminf_𝑛→∞ on both sides of the equality above, we have 0≥‖ℎ‖²−2ℎ,𝐽𝑝+𝑝²=𝜙ℎ,𝑝. (3.53) This implies that ℎ=𝑝,for all𝑗=1,2,3,…𝑚, then it follow that Ω^𝑗_𝑛𝑦_𝑛⇀𝑝, (3.54) from (3.51), (3.54) and the Kadec-Klee property, we have lim_𝑛→∞Ω^𝑗_𝑛𝑦_𝑛=𝑝∀𝑗=1,2,3,…,𝑚. (3.55) By using the triangle inequality, we obtain Ω^𝑗_𝑛𝑦_𝑛−Ω_𝑛^𝑗−1𝑦_𝑛≤Ω^𝑗_𝑛𝑦_𝑛−𝑝+𝑝−Ω_𝑛^𝑗−1𝑦_𝑛. (3.56) Hence, we obtain that lim_𝑛→∞Ω^𝑗_𝑛𝑦_𝑛−Ω_𝑛^𝑗−1𝑦_𝑛=0,∀𝑗=1,2,3,…,𝑚. (3.57) Since {𝑟_𝑗,𝑛}⊂[𝑑,∞) and 𝐽 is uniformly norm-to-norm continuous on bounded subsets, so lim_𝑛→∞𝐽Ω^𝑗_𝑛𝑦_𝑛−𝐽Ω_𝑛^𝑗−1𝑦_𝑛𝑟_𝑗,𝑛=0,∀𝑗=1,2,3,…,𝑚. (3.58) From Lemma 2.3, we get for 𝑗=1,2,3,…,𝑚𝑄_𝑗Ω^𝑗_𝑛𝑦_𝑛+1,𝑦𝑟_𝑗,𝑛𝑦−Ω^𝑗_𝑛𝑦_𝑛,𝐽Ω^𝑗_𝑛𝑦_𝑛−𝐽Ω_𝑛^𝑗−1𝑦_𝑛≥0,∀𝑦∈𝐶. (3.59) From (A2), we get 1𝑟_𝑗,𝑛𝑦−Ω^𝑗_𝑛𝑦_𝑛,𝐽Ω^𝑗_𝑛𝑦_𝑛−𝐽Ω_𝑛^𝑗−1𝑦_𝑛≥𝑄_𝑗𝑦,Ω^𝑗_𝑛𝑦_𝑛,∀𝑦∈𝐶,∀𝑗=1,2,3,…,𝑚. (3.60) From (3.55) and (3.58), we have 0≥𝑄_𝑗𝑦,𝑝,∀𝑦∈𝐶,∀𝑗=1,2,3,…,𝑚. (3.61) For 𝑡 with 0<𝑡≤1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝. Then, we get that 𝑦_𝑡∈𝐶. From (3.61), it follows that 𝑄_𝑗𝑦_𝑡,𝑝≤0,∀𝑦∈𝐶,∀𝑗=1,2,3,…,𝑚. (3.62) By the conditions (A1) and (A4), we have for 𝑗=1,2,3,…,𝑚0=𝑄_𝑗𝑦_𝑡,𝑦_𝑡≤𝑡𝑄_𝑗𝑦_𝑡,𝑦+(1−𝑡)𝑄_𝑗𝑦_𝑡,𝑝≤𝑡𝑄_𝑗𝑦_𝑡,𝑦≤𝑄_𝑗𝑦_𝑡,𝑦. (3.63) From (A3), we get 0≤𝑄_𝑗𝑦_𝑡,𝑦=𝑄_𝑗𝑡𝑦+(1−𝑡)𝑝,𝑦, (3.64) and letting 𝑡→0, we have 0=lim_𝑛→∞0≤lim_𝑛→∞𝑄_𝑗𝑡𝑦+(1−𝑡)𝑝,𝑦≤𝑄_𝑗𝑝,𝑦,∀𝑦∈𝐶,∀𝑗=1,2,3,…𝑚. (3.65) This implies that 𝑝∈GMEP(𝜃_𝑗,𝐴_𝑗,𝜑_𝑗),  for all𝑗=1,2,3,…𝑚. Therefore, 𝑝∈∩^𝑚_𝑗=1GMEP(𝜃_𝑗,𝐴_𝑗,𝜑_𝑗). Hence, from (a) and (b), we obtain 𝑝∈𝐹.
Step 6. We show that 𝑝=Π^𝑓_𝐹𝑥₀.
Since 𝐹 is closed and convex set from Lemma 2.8, we have Π^𝑓_𝐹𝑥₀ is single value, denote by 𝑣. From 𝑥_𝑛=Π^𝑓_𝐶𝑛𝑥₀ and 𝑣∈𝐹⊂𝐶_𝑛, we also have 𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑣,𝐽𝑥₀,∀𝑛≥1. (3.66) By definition of 𝐺 and 𝑓, we know that, for each given 𝑥,𝐺(𝜉,𝐽𝑥) is convex and lower semicontinuous with respect to 𝜉. So 𝐺𝑝,𝐽𝑥₀≤liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤limsup_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑣,𝐽𝑥₀. (3.67) From definition of Π^𝑓_𝐹𝑥₀ and 𝑝∈𝐹, we can conclude that 𝑣=𝑝=Π^𝑓_𝐹𝑥₀ and 𝑥_𝑛→𝑝 as 𝑛→∞. This completes the proof.

For a special case that 𝑖=1,2, we can obtain the following results on a pair of quasi-𝜙-asymptotically nonexpansive mappings immediately from Theorem 3.1.

Corollary 3.2. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property, and let 𝑓∶𝐸→ℝ be a convex lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)). Let 𝑆₁,𝑆₂∶𝐶→𝐶 be closed uniformly 𝐿₁,𝐿₂-Lipschitz continuous and quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘¹_𝑛}⊂[1,∞), 𝑘¹_𝑛→1 and {𝑘²_𝑛}⊂[1,∞), 𝑘²_𝑛→1, respectively, and let {𝑘_𝑛}=sup{{𝑘¹_𝑛},{𝑘²_𝑛}}. For each 𝑗=1,2,…,𝑚, let 𝜃_𝑗 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4), let 𝐴_𝑗∶𝐶→𝐸^∗ be a continuous and monotone mapping and let 𝜑_𝑗∶𝐶→ℝ be a lower semicontinuous and convex function. Assume that 𝐹∶=(𝐹(𝑆₁)∩𝐹(𝑆₂))∩(∩^𝑚_𝑗=1GMEP(𝜃_𝑗,𝐴_𝑗,𝜑_𝑗))≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π^𝑓_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑧_𝑛=𝐽⁻¹𝛼_𝑛,0𝐽𝑥_𝑛+𝛼_𝑛,1𝐽𝑆^𝑛₁𝑥_𝑛+𝛼_𝑛,2𝐽𝑆^𝑛₂𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝐺𝑧,𝐽𝑢_𝑛≤𝐺𝑧,𝐽𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.68) where 𝑀_𝑛=sup_𝑞∈𝐹{𝐺(𝑞,𝐽𝑥_𝑛)}, {𝛼_𝑛,𝑖}²_i=0 and {𝛽_𝑛} are sequences in [0,1] such that ∑²_𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0 and satisfy the following conditions. (i){𝑟_𝑗,𝑛}⊂[𝑑,∞) for some 𝑑>0. (ii)liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑖=1,2, and liminf_𝑛→∞(1−𝛽_𝑛)>0.(iii)𝑓(𝑥)≥0 for all 𝑥∈𝐶, and 𝑓(0)=0. Then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π^𝑓_𝐹𝑥₀.

Remark 3.3. Corollary 3.2 improves and extends the result of Shehu [43, Theorem  3.1] following senses: (i)for the mappings, we extend the mappings from two closed weak relatively nonexpansive mappings (or relatively quasi-nonexpansive) mappings to a countable infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings; (ii)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.
If 𝑆_𝑖=𝑆 for each 𝑖∈ℕ, then Theorem 3.1 is reduced to the following corollary.

Corollary 3.4. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property. Let 𝑆∶𝐶→𝐶 be an infinite family of closed uniformly 𝐿-Lipschitz continuous and quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1, and let 𝑓∶𝐸→ℝ be a convex lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)). For each 𝑗=1,2,…,𝑚, let 𝜃_𝑗 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4), let 𝐴_𝑗∶𝐶→𝐸^∗ be a continuous and monotone mapping, and let 𝜑_𝑗∶𝐶→ℝ be a lower semicontinuous and convex function. Assume that 𝐹∶=𝐹(𝑆)∩(∩^𝑚_𝑗=1GMEP(𝜃_𝑗,𝐴_𝑗,𝜑_𝑗))≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π^𝑓_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑆^𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝐺𝑧,𝐽𝑢_𝑛≤𝐺𝑧,𝐽𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.69) where 𝑀_𝑛=sup_𝑞∈𝐹{𝐺(𝑞,𝐽𝑥_𝑛)}, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] and satisfy the following conditions: (i){𝑟_𝑗,𝑛}⊂[𝑑,∞) for some 𝑑>0. (ii)liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0, and liminf_𝑛→∞(1−𝛽_𝑛)>0.(iii)𝑓(𝑥)≥0 for all 𝑥∈𝐶, and 𝑓(0)=0. Then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π^𝑓_𝐹𝑥₀.

If taking 𝜃_𝑗≡0,𝐴_𝑗≡0,𝜑_𝑗≡0,𝑟_𝑗,𝑛=1 and 𝛽_𝑛≡0,forall𝑛∈ℕ in algorithm (3.69) in Corollary 3.4, we obtain the following corollary.

Corollary 3.5. (Li et al. [42]) Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed and convex subset of 𝐸,let 𝑆∶𝐶→𝐶 be a weak relative nonexpansive mapping, and let 𝑓∶𝐸→ℝ be a convex lower semicontinuous mapping with 𝐶⊂int(𝐷(f)). Assume that {𝛼_𝑛}^∞_𝑛=0 is sequence in [0,1) such that limsup_𝑛→∞(𝛼_𝑛)<1. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥_𝑛=𝑥₀∈𝐶,𝐶₀𝑦=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑆𝑥_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝐺𝑤,𝐽𝑦_𝑛≤𝐺𝑤,𝐽𝑥_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥1. (3.70) If 𝐹(𝑆) is nonempty, then {𝑥_𝑛} converges to Π^𝑓_𝐹(𝑆)𝑥₀.

Taking 𝑓(𝑥)=0 for all 𝑥∈𝐸, we have 𝐺(𝜉,𝐽𝑥)=𝜙(𝜉,𝑥) and Π^𝑓_𝐶𝑥=Π_𝐶𝑥. By Theorem 3.1, then we obtain the following Corollaries.

Corollary 3.6. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with the Kadec-Klee property. Let {𝑆_𝑖}^∞_𝑖=1∶𝐶→𝐶 be an infinite family of closed uniformly 𝐿_𝑖-Lipschitz continuous and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1. For each 𝑗=1,2,…,𝑚, let 𝜃_𝑗 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4), let 𝐴_𝑗∶𝐶→𝐸^∗ be a continuous and monotone mapping, and let 𝜑_𝑗∶𝐶→ℝ be a lower semicontinuous and convex function. Assume that 𝐹∶=(∩^∞_𝑖=1𝐹(𝑆_𝑖))∩(∩^𝑚_𝑗=1GMEP(𝜃_𝑗,𝐴_𝑗,𝜑_𝑗))≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑧_𝑛=𝐽⁻¹𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.71) where 𝑀_𝑛=sup_𝑞∈𝐹𝜙(𝑞,𝑥_𝑛), {𝛼_𝑛,𝑖} and {𝛽_𝑛} are sequences in [0,1] such that ∑^∞_𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0. If {𝑥_𝑛} is satisfying the following conditions. (i){𝑟_𝑗,𝑛}⊂[𝑑,∞) for some 𝑑>0.(ii)liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑖≥1 and liminf_𝑛→∞(1−𝛽_𝑛)>0. Then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π_𝐹𝑥₀.

Corollary 3.7. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with Kadec-Klee property. Let 𝑆₁,𝑆₂∶𝐶→𝐶 be closed uniformly 𝐿₁,𝐿₂-Lipschitz continuous and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘¹_𝑛}⊂[1,∞), 𝑘¹_𝑛→1 and {𝑘²_𝑛}⊂[1,∞), 𝑘²_𝑛→1, respectively, such that {𝑘_𝑛}=sup{{𝑘¹_𝑛},{𝑘²_𝑛}}. For each 𝑗=1,2,…,𝑚 let 𝜃_𝑗 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4), let 𝐴_𝑗∶𝐶→𝐸^∗ be a continuous and monotone mapping, and let 𝜑_𝑗∶𝐶→ℝ be a lower semicontinuous and convex function. Assume that 𝐹∶=(𝐹(𝑆₁)∩𝐹(𝑆₂))∩(∩^𝑚_𝑗=1GMEP(𝜃_𝑗,𝐴_𝑗,𝜑_𝑗))≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑧_𝑛=𝐽⁻¹𝛼_𝑛,0𝐽𝑥_𝑛+𝛼_𝑛,1𝐽𝑆^𝑛₁𝑥_𝑛+𝛼_𝑛,2𝐽𝑆^𝑛₂𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝑘_𝑛−1𝑀_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.72) where 𝑀_𝑛=sup_𝑞∈𝐹𝜙(𝑞,𝑥_𝑛), {𝛼_𝑛,𝑖}²_𝑖=0 and {𝛽_𝑛} are sequences in [0,1] such that ∑²_𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0 with the following conditions. (i){𝑟_𝑗,𝑛}⊂[𝑑,∞) for some 𝑑>0.(ii)liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑖=1,2 and liminf_𝑛→∞(1−𝛽_𝑛)>0. Then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π_𝐹𝑥₀.

Remark 3.8. Corollary 3.6 and Corollary 3.7 extend and improve the results of many authors in the literature works.

Acknowledgments

The authors would like to thank the referees for their careful reading and comments leading to the improvements of this paper. This research was supported by a grant from under the program Strategic Scholarships for Frontier Research Network for the Join Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand. Moreover, Mrs. Siwaporn Saewan was supported by King Mongkut's Diamond scholarship for Ph.D. program at King Mongkut's University of Technology Thonburi (KMUTT). Furthermore, the authors also would like to thank the National Research University Project of Thailand's Office of the Higher Education Commission (under the NRU-CSEC project no. 54000267) for their financial support during the preparation of this paper. Poom Kumam was supported by the Commission on Higher Education and the Thailand Research Fund under Grant no. MRG5380044.