Modified Hybrid Block Iterative Algorithm for Convex Feasibility Problems and Generalized Equilibrium Problems for Uniformly Quasi-𝜙-Asymptotically Nonexpansive Mappings

Abstract

We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results.

1. Introduction and Preliminaries

The convex feasibility problem (CFP) is the problem of computing points laying in the intersection of a finite family of closed convex subsets 𝐶_𝑗,𝑗=1,2,…,𝑁, of a Banach space 𝐸. This problem appears in various fields of applied mathematics. The theory of optimization [1], Image Reconstruction from projections [2], and Game Theory [3] are some examples. There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [4]. The advantage of a Hilbert space 𝐻 is that the projection 𝑃_𝐶 onto a closed convex subset 𝐶 of 𝐻 is nonexpansive. So projection methods have dominated in the iterative approaches to (CFP) in Hilbert spaces. In 1993, Kitahara and Takahashi [5] deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in a uniformly convex Banach space. It is known that if 𝐶 is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space, then the generalized projection (see, Alber [6] or Kamimura and Takahashi [7]) from 𝐸 onto 𝐶 is relatively nonexpansive, whereas the metric projection from 𝐻 onto 𝐶 is not generally nonexpansive.

We note that the block iterative method is a method which is often used by many authors to solve the convex feasibility problem (CFP) (see, [8, 9], etc.). In 2008, Plubtieng and Ungchittrakool [10] 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.

Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸 with ‖⋅‖ and 𝐸^∗ being the dual space of 𝐸. Let 𝑓 be a bifunction of 𝐶×𝐶 into ℝ and 𝐵∶𝐶→𝐸^∗ a monotone mapping. The generalized equilibrium problem, denoted by GEP, is to find 𝑥∈𝐶 such that𝑓(𝑥,𝑦)+⟨𝐵𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶.(1.1) The set of solutions for the problem (1.1) is denoted by GEP(𝑓,𝐵), that is GEP(𝑓,𝐵)∶={𝑥∈𝐶∶𝑓(𝑥,𝑦)+⟨𝐵𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶}.(1.2) If 𝐵≡0, the problem (1.1) reducing into the equilibrium problem for 𝑓, denoted by EP(𝑓), is to find 𝑥∈𝐶 such that𝑓(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.3) If 𝑓≡0, the problem (1.1) reducing into the classical variational inequality, denoted by VI(𝐵,𝐶), is to find 𝑥^∗∈𝐶 such that⟨𝐵𝑥^∗,𝑦−𝑥^∗⟩≥0,∀𝑦∈𝐶.(1.4) The above formulation (1.3) 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 addition, there are several other problems, for example, the complementarity problem, fixed point problem, and optimization problem, which can also be written in the form of an EP(𝑓). In other words, the EP(𝑓) is a unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP(𝑓); see, for example [11] and references therein. Some solution methods have been proposed to solve the EP(𝑓); see, for example, [12–29] and references therein.

Consider the functional defined by𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸,(1.5) where 𝐽 is the duality mapping from 𝐸 into 𝐸^∗. It is well known that 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. It is obvious from the definition of function 𝜙 that(‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)²,∀𝑥,𝑦∈𝐸.(1.6) If 𝐸 is a Hilbert space, then 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖², for all 𝑥,𝑦∈𝐸. On the other hand, the generalized projection (Alber [6]) 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∈𝐶𝜙(𝑦,𝑥),(1.7) and existence and uniqueness of the operator Π_𝐶 follow from the properties of the functional 𝜙(𝑥,𝑦) and strict monotonicity of the mapping 𝐽 (see, for example, [6, 7, 30–32]).

Remark 1.1. 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 (1.5), 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 [31, 32] for more details.
Let 𝐶 be a closed convex subset of 𝐸; a mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐶. A point 𝑥∈𝐶 is a fixed point of 𝑇 provided 𝑇𝑥=𝑥. Denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. Recall that a point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [33] if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇).
A mapping 𝑇 from 𝐶 into itself is said to be relatively nonexpansive [34–36] if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). The asymptotic behavior of a relatively nonexpansive mapping was studied in [37–39]. 𝑇 is said to be 𝜙-nonexpansive, if 𝜙(𝑇𝑥,𝑇𝑦)≤𝜙(𝑥,𝑦) for 𝑥,𝑦∈𝐶. 𝑇 is said to be relatively quasi-nonexpansive if 𝐹(𝑇)≠∅ and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). 𝑇 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 that 𝜙(𝑝,𝑇^𝑛𝑥)≤𝑘_𝑛𝜙(𝑝,𝑥) for all 𝑛≥1 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). A mapping 𝑇 is said to be closed if for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, 𝑇𝑥=𝑦. 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 may be not true (see more details [37–41]).
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 is attained uniformly for 𝑥,𝑦∈𝑈. Let 𝐸 be a Banach space. The modulus of convexity of 𝐸 is the function 𝛿∶[0,2]→[0,1] defined by 𝛿(𝜀)=inf{1−‖(𝑥+𝑦)/2‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀}. A Banach space 𝐸 is uniformly convex if and only if 𝛿(𝜀)>0 for all 𝜀∈(0,2]. Let 𝑝 be a fixed real number with 𝑝≥2. A Banach space 𝐸 is said to be 𝑝-uniformly convex if there exists a constant 𝑐>0 such that 𝛿(𝜀)≥𝑐𝜀^𝑝 for all 𝜀∈[0,2]; see [42] for more details. Observe that every 𝑝-uniform convex is uniformly convex. One should note that no Banach space is 𝑝-uniform convex for 1<𝑝<2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each 𝑝>1, the generalized duality mapping 𝐽_𝑝∶𝐸→2^𝐸∗ is defined by 𝐽_𝑝(𝑥)={𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖^𝑝,‖𝑥^∗‖=‖𝑥‖^𝑝−1} for all 𝑥∈𝐸. In particular, 𝐽=𝐽₂ is called the normalized duality mapping. If 𝐸 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity mapping. It is also known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.
The following basic properties can be found in Cioranescu [31].(i)If 𝐸 is a 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, strictly convex, and reflexive Banach space, then the normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is single-valued, one-to-one, and onto. (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‖, 𝑥_𝑛→𝑥.

In 2005, Matsushita and Takahashi [40] 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:𝑥₀∈𝐶chosenarbitrarily,𝑦_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑥_𝑛𝐶),_𝑛={𝑧∈𝐶∶𝜙(𝑧,𝑦_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑄)},_𝑛={𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛𝑥⟩≥0},_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀.(1.8) They proved that {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₀, where Π_𝐹(𝑇) is the generalized projection from 𝐶 onto 𝐹(𝑇). In 2007, Plubtieng and Ungchittrakool [43] generalized the processes (1.8) to the new general processes of two relatively nonexpansive mappings in a Banach space. Let 𝐶 be a closed convex subset of a Banach space 𝐸 and 𝑆,𝑇∶𝐶→𝐶 relatively nonexpansive mappings such that 𝐹∶=𝐹(𝑆)∩𝐹(𝑇)≠∅. Define {𝑥_𝑛} in the following way:𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑧_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆𝑥_𝑛𝐻),_𝑛={𝑧∈𝐶∶𝜙(𝑧,𝑦_𝑛)⩽𝜙(𝑧,𝑥_𝑛𝑊)},_𝑛={𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛𝑥⟩⩾0},_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥,𝑛=0,1,2,…,(1.9) where {𝛼_𝑛}, {𝛽_𝑛⁽¹⁾}, {𝛽_𝑛⁽²⁾}, and {𝛽_𝑛⁽³⁾} are sequences in [0,1] with 𝛽_𝑛⁽¹⁾+𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾=1 for all 𝑛∈ℕ∪{0}.

In 2009, Qin et al. [26] introduced a hybrid projection algorithm to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two quasi-𝜙-nonexpansive mappings in the framework of Banach spaces:𝑥₀=𝑥chosenarbitrarily,𝐶₁𝑥=𝐶,₁=Π_𝐶1𝑥₀,𝑦_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑥_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀,(1.10) where Π_𝐶𝑛+1 is the generalized projection from 𝐸 onto 𝐶_𝑛+1. They proved that the sequence {𝑥_𝑛} converges strongly to Π_{𝐹(𝑇)∩𝐹(𝑆)∩EP(𝑓)}𝑥₀. In the same year, Wattanawitoon and Kumam [44] and Petrot et al. [45] using the idea of Takahashi and Zembayashi [46, 47] and Plubtieng and Ungchittrakool [43] extend the notion from relatively nonexpansive mappings or quasi-𝜙-nonexpansive mappings to two relatively quasi-nonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. In 2010, Chang et al. [48] proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mapping; they obtain the strong convergence theorems in a Banach space. Recently, many authors considered the iterative methods for finding a common element of the set of solutions to the problem (1.3) and of the set of fixed points of nonexpansive mappings; see, for instance, [12–27] and the references therein.

Motivated by Chang et al. [48], Qin et al. [26, 49], Wattanawitoon and Kumam [44], Petrot et al. [45], Zegeye [50], and other recent works, in this paper we introduce 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 which is more general than closed quasi-𝜙-nonexpansive mappings in a uniformly smooth and strictly convex Banach space 𝐸 with Kadec-Klee property. The results presented in this paper improve and generalize some well-known results in the literature.

2. Basic Results

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

Lemma 2.1 (Kamimura and Takahashi [7]). Let 𝐸 be a uniformly convex and smooth Banach space and let {𝑥_𝑛} and {𝑦_𝑛} be two sequences of 𝐸. If 𝜙(𝑥_𝑛,𝑦_𝑛)→0 and either {𝑥_𝑛} or {𝑦_𝑛} is bounded, then ‖𝑥_𝑛−𝑦_𝑛‖→0.

Lemma 2.2 (Alber [6]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑥∈𝐸. Then 𝑥₀=Π_𝐶𝑥 if and only if ⟨𝑥₀−𝑦,𝐽𝑥−𝐽𝑥₀⟩≥0,∀𝑦∈𝐶.(2.1)

Lemma 2.3 (Alber [6]). Let 𝐸 be a reflexive, strictly convex and smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑥∈𝐸. Then 𝜙(𝑦,Π_𝐶𝑥)+𝜙(Π_𝐶𝑥,𝑥)≤𝜙(𝑦,𝑥),∀𝑦∈𝐶.(2.2)

For solving the equilibrium problem for a bifunction 𝑓∶𝐶×𝐶→ℝ, let us assume that 𝑓 satisfies the following conditions: (A1)𝑓(𝑥,𝑥)=0for all 𝑥∈𝐶;(A2)𝑓 is monotone, that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶; (A3)for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦);(2.3)(A4)for each 𝑥∈𝐶, 𝑦↦𝑓(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.4 (Blum and Oettli [11]). Let 𝐶 be a 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 1𝑓(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.4)

Lemma 2.5 (Zegeye [50]). Let 𝐶 be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space 𝐸 and let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴1)-(𝐴4) and let 𝐵 be a monotone mapping from 𝐶 into 𝐸^∗. For 𝑟>0 and 𝑥∈𝐸, define a mapping 𝑇_𝑟∶𝐶→𝐶 as follows: 𝑇_𝑟1𝑥={𝑧∈𝐶∶𝑓(𝑧,𝑦)+⟨𝐵𝑥,𝑦−𝑧⟩+𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶}(2.5) for all 𝑥∈𝐶. Then the following hold: (1)𝑇_𝑟 is single-valued; (2)𝑇_𝑟 is a firmly nonexpansive-type mapping, for all 𝑥,𝑦∈𝐸, ⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽T_𝑟𝑥−𝐽𝑇_𝑟𝑦⟩≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩;(2.6)(3)𝐹(𝑇_𝑟)=𝐺𝐸𝑃(𝑓,𝐵);(4)𝐺𝐸𝑃(𝑓,𝐵) is closed and convex.

Lemma 2.6 (Zegeye [50]). (Let 𝐶 be a 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 𝐵 be a monotone mapping from 𝐶 into 𝐸^∗. For 𝑟>0, 𝑥∈𝐸, and 𝑞∈𝐹(𝑇_𝑟), we have that 𝜙(𝑞,𝑇_𝑟𝑥)+𝜙(𝑇_𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥).(2.7)

Lemma 2.7 (Chang et al. [48]). 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 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_𝑛‖²≤^∞_𝑛=1𝜆_𝑛‖𝑥_𝑛‖²−𝜆_𝑖𝜆_𝑗𝑔(‖𝑥_𝑖−𝑥_𝑗‖).(2.8)

Lemma 2.8 (Chang et al. [48]). Let 𝐸 be a real uniformly smooth and strictly convex Banach space, and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a closed and quasi-𝜙-asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1. Then 𝐹(𝑇) is a closed convex subset of 𝐶.

Definition 2.9 (Chang et al. [48]). (1) Let {𝑆_𝑖}^∞_𝑖=1∶𝐶→𝐶 be a sequence of mapping. {𝑆_𝑖}^∞_𝑖=1 is said to be a family of uniformly quasi-𝜙-asymptotically nonexpansive mappings, if ∩^∞_𝑛=1𝐹(𝑆_𝑛)≠∅, and there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that for each 𝑖≥1,𝜙(𝑝,𝑆^𝑛_𝑖𝑥)≤𝑘_𝑛𝜙(𝑝,𝑥),∀𝑝∈⋂^∞_𝑛=1𝐹(𝑆_𝑛),𝑥∈𝐶,∀𝑛≥1.(2.9)
(2) A mapping 𝑆∶𝐶→𝐶 is said to be uniformly 𝐿-Lipschitz continuous if there exists a constant 𝐿>0 such that ‖𝑆^𝑛𝑥−𝑆^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(2.10)

3. Main Results

In this section, we prove the new convergence theorems for finding the set of solutions of a general equilibrium problems and the common fixed point set of a family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space 𝐸 with Kadec-Klee property.

Theorem 3.1. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with Kadec-Klee property. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴1)-(𝐴4). Let 𝐵 be a continuous monotone mapping of 𝐶 into 𝐸^∗. 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 such that 𝐹∶=∩^∞_𝑖=1𝐹(𝑆_𝑖)∩𝐺𝐸𝑃(𝑓,𝐵) is a nonempty and bounded subset in 𝐶. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑦_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑧_𝑛𝑧),_𝑛=𝐽⁻¹(𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛,𝑦)+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛)+𝜃_𝑛𝑥},_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.1) where 𝐽 is the duality mapping on 𝐸,𝜃_𝑛=sup_𝑞∈𝐹(𝑘_𝑛−1)𝜙(𝑞,𝑥_𝑛), {𝛼_𝑛,𝑖},{𝛽_𝑛} are sequences in [0,1] and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. If ∑^∞_𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0 and liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑖≥1, then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π_𝐹𝑥₀.

Proof. We first show that 𝐶_𝑛+1 is closed and convex for each 𝑛≥0. 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 𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛)+𝜃_𝑛⇔2⟨𝑧,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖𝑥_𝑛‖²−‖𝑢_𝑛‖²+𝜃_𝑛.(3.2) So, 𝐶_𝑛+1 is closed and convex. Therefore, Π_𝐹𝑥₀ and Π_𝐶𝑛𝑥₀ are well defined. Next, we show that 𝐹⊂𝐶_𝑛 for all 𝑛≥0. Indeed, put 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛 for all 𝑛≥0. It is clear that 𝐹₁⊂𝐶₁=𝐶. Suppose 𝐹⊂𝐶_𝑛 for 𝑛∈ℕ, by the convexity of ‖⋅‖², property of 𝜙, Lemma 2.7, and uniformly quasi-𝜙-asymptotically nonexpansive of 𝑆_𝑛 for each 𝑞∈𝐹⊂𝐶_𝑛, we observed that 𝜙(𝑞,𝑢_𝑛)=𝜙(𝑞,𝑇_𝑟𝑛𝑦_𝑛)≤𝜙(𝑞,𝑦_𝑛)=𝜙(𝑞,𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑧_𝑛)=‖𝑞‖²−2⟨𝑞,𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑧_𝑛⟩+‖𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑧_𝑛‖²≤‖𝑞‖²−2𝛽_𝑛⟨𝑞,𝐽𝑥_𝑛⟩−2(1−𝛽_𝑛)⟨𝑞,𝐽𝑧_𝑛⟩+𝛽_𝑛‖𝑥_𝑛‖²+(1−𝛽_𝑛)‖𝑧_𝑛‖²=𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝑧_𝑛)(3.3) and  𝜙(𝑞,𝑧_𝑛)=𝜙(𝑞,𝐽⁻¹(𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛))=‖𝑞‖²−2⟨𝑞,𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛⟩+‖𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛‖²≤‖𝑞‖²−2𝛼_𝑛,0⟨𝑞,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖⟨𝑞,𝐽𝑆^𝑛_𝑖𝑥_𝑛⟩+‖𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛‖²≤‖𝑞‖²−2𝛼_𝑛,0⟨𝑞,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖⟨𝑞,𝐽𝑆^𝑛_𝑖𝑥_𝑛⟩+𝛼_𝑛,0‖𝐽𝑥_𝑛‖²+^∞_𝑖=1𝛼_𝑛,𝑖‖𝐽𝑆^𝑛_𝑖𝑥_𝑛‖²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛‖=‖𝑞‖²−2𝛼_𝑛,0⟨𝑞,𝐽𝑥_𝑛⟩+𝛼_𝑛,0‖𝐽𝑥_𝑛‖²−2^∞_𝑖=1𝛼_𝑛,𝑖⟨𝑞,𝐽𝑆^𝑛_𝑖𝑥_𝑛⟩+^∞_𝑖=1𝛼_𝑛,𝑖‖𝐽𝑆^𝑛_𝑖𝑥_𝑛‖²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛‖=𝛼_𝑛,0𝜙(𝑞,𝑥_𝑛)+^∞_𝑖=1𝛼_𝑛,𝑖𝜙(𝑞,𝑆^𝑛_𝑖𝑥_𝑛)−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛‖≤𝛼_𝑛,0𝑘_𝑛𝜙(𝑞,𝑥_𝑛)+^∞_𝑖=1𝛼_𝑛,𝑖𝑘_𝑛𝜙(𝑞,𝑥_𝑛)−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑗𝑥_𝑛‖≤𝑘_𝑛𝜙(𝑞,𝑥_𝑛).(3.4) Substituting (3.4) into (3.3), we get 𝜙(𝑞,𝑢_𝑛)≤𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝑧_𝑛)≤𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝑘_𝑛𝜙(𝑞,𝑥_𝑛)≤𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)[𝜙(𝑞,𝑥_𝑛)+sup_𝑞∈𝐹(𝑘_𝑛−1)𝜙(𝑞,𝑥_𝑛)]≤𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)sup_𝑞∈𝐹(𝑘_𝑛−1)𝜙(𝑞,𝑥_𝑛)≤𝜙(𝑞,𝑥_𝑛)+𝜃_𝑛.(3.5) This show that 𝑞∈𝐶_𝑛+1 implies that 𝐹⊂ 𝐶_𝑛+1 and hence, 𝐹⊂ 𝐶_𝑛 for all 𝑛≥0. Since 𝐹 is nonempty, 𝐶_𝑛 is a nonempty closed convex subset of 𝐸, and hence Π_𝐶𝑛 exist for all 𝑛≥0. This implies that the sequence {𝑥_𝑛} is well defined. 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.(3.6) By Lemma 2.3, we also have 𝜙(𝑥_𝑛,𝑥₀)=𝜙(Π_𝐶𝑛𝑥₀,𝑥₀)≤𝜙(𝑞,𝑥₀)−𝜙(𝑞,𝑥_𝑛)≤𝜙(𝑞,𝑥₀),∀𝑞∈𝐹⊂𝐶_𝑛,∀𝑛≥0.(3.7) From (3.6) and (3.7), then {𝜙(𝑥_𝑛,𝑥₀)} are nondecreasing and bounded. So, we obtain that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists. In particular, by (1.6), the sequence {(‖𝑥_𝑛‖−‖𝑥₀‖)²} is bounded. This implies that {𝑥_𝑛} is also bounded. Denote 𝐾=sup_𝑛≥0{‖𝑥_𝑛‖}<∞.(3.8) Moreover, by the definition of {𝜃_𝑛} and (3.8), it follows that 𝜃_𝑛→0,𝑛→∞.(3.9)
Next, we show that {𝑥_𝑛} is a Cauchy sequence in 𝐶. Since 𝑥_𝑚=Π_𝐶𝑚𝑥₀ ∈𝐶_𝑚⊂𝐶_𝑛, for 𝑚>𝑛, by Lemma 2.3, we have 𝜙(𝑥_𝑚,𝑥_𝑛)=𝜙(𝑥_𝑚,Π_𝐶𝑛𝑥₀)≤𝜙(𝑥_𝑚,𝑥₀)−𝜙(Π_𝐶𝑛𝑥₀,𝑥₀)=𝜙(𝑥_𝑚,𝑥₀)−𝜙(𝑥_𝑛,𝑥₀).(3.10) Since lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists and we take 𝑚,𝑛→∞, then, we get 𝜙(𝑥_𝑚,𝑥_𝑛)→0. From Lemma 2.1, we have lim_𝑛→∞‖𝑥_𝑚−𝑥_𝑛‖=0. Thus {𝑥_𝑛} is a Cauchy sequence and by the completeness of 𝐸, and there exists a point 𝑝∈𝐶 such that 𝑥_𝑛→𝑝 as 𝑛→∞.
Now, we claim that ‖J𝑢_𝑛−𝐽𝑥_𝑛‖→0, as 𝑛→∞. By definition of Π_𝐶𝑛𝑥₀, one has 𝜙(𝑥_𝑛+1,𝑥_𝑛)=𝜙(𝑥_𝑛+1,Π_𝐶𝑛𝑥₀)≤𝜙(𝑥_𝑛+1,𝑥₀)−𝜙(Π_𝐶𝑛𝑥₀,𝑥₀)=𝜙(𝑥_𝑛+1,𝑥₀)−𝜙(𝑥_𝑛,𝑥₀).(3.11) Since lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists, we have lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑥_𝑛)=0.(3.12) By Lemma 2.1, we obtain that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.(3.13) Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, we get lim_𝑛→∞‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖=0.(3.14) From 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛 and the definition of 𝐶_𝑛+1, we have 𝜙(𝑥_𝑛+1,𝑢_𝑛)≤𝜙(𝑥_𝑛+1,𝑥_𝑛)+𝜃_𝑛.(3.15) By (3.9) and (3.12), we also have lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑢_𝑛)=0.(3.16) Applying Lemma 2.1, we obtain lim_𝑛→∞‖𝑥_𝑛+1−𝑢_𝑛‖=0.(3.17) Since ‖𝑢_𝑛−𝑥_𝑛‖≤‖𝑢_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛‖, we get lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0.(3.18) Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, we have lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑥_𝑛‖=0.(3.19)
Next, we will show that 𝑥_𝑛→𝑝∈𝐹∶=GEP(𝑓,𝐵)∩(∩^∞_𝑖=1𝐹(𝑆_𝑖)).
(i) First, we show that 𝑥_𝑛→𝑝∈GEP(𝑓,𝐵). It follows from (3.3) and (3.4) we observe that 𝜙(𝑝,𝑦_𝑛)≤𝜙(𝑝,𝑥_𝑛)+𝜃_𝑛. By Lemma 2.6 and 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛, we obtain 𝜙(𝑢_𝑛,𝑦_𝑛)=𝜙(𝑇_𝑟𝑛𝑦_𝑛,𝑦_𝑛)≤𝜙(𝑝,𝑦_𝑛)−𝜙(𝑝,𝑇_𝑟𝑛𝑦_𝑛)≤𝜙(𝑝,𝑥_𝑛)−𝜙(𝑝,𝑇_𝑟𝑛𝑦_𝑛)+𝜃_𝑛=𝜙(𝑝,𝑥_𝑛)−𝜙(𝑝,𝑢_𝑛)+𝜃_𝑛=‖𝑝‖²−2⟨𝑝,𝐽𝑥_𝑛⟩+‖𝑥_𝑛‖²−(‖𝑝‖²−2⟨𝑝,𝐽𝑢_𝑛⟩+‖𝑢_𝑛‖²)+𝜃_𝑛=‖𝑥_𝑛‖²−‖𝑢_𝑛‖²−2⟨𝑝,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩+𝜃_𝑛≤‖𝑥_𝑛−𝑢_𝑛‖(‖𝑥_𝑛‖+‖𝑢_𝑛‖)+2‖𝑝‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖+𝜃_𝑛.(3.20) By Lemma 2.1, (3.9), (3.18), and (3.19), we have lim_𝑛→∞‖𝑢_𝑛−𝑦_𝑛‖=0.(3.21) Again since 𝐽 is uniformly norm-to-norm continuous, we also have lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖=0.(3.22) From (A2), we note that ⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥−𝑓(𝑢_𝑛,𝑦)≥𝑓(𝑦,𝑢_𝑛),∀𝑦∈𝐶,(3.23) and hence ⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛⟩+⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝑟_𝑛⟩≥𝑓(𝑦,𝑢_𝑛),∀𝑦∈𝐶.(3.24) For 𝑡 with 0<𝑡<1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝. Then 𝑦_𝑡∈𝐶 and hence 0≥−⟨𝐵𝑦_𝑛,𝑦_𝑡−𝑢_𝑛⟩−⟨𝑦_𝑡−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝑟_𝑛⟩+𝑓(𝑦_t,𝑢_𝑛),∀𝑦∈𝐶.(3.25) It follows that ⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩≥⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩−⟨𝐵𝑦_𝑛,𝑦_𝑡−𝑢_𝑛⟩−⟨𝑦_𝑡−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝑟_𝑛⟩+𝑓(𝑦_𝑡,𝑢_𝑛),∀𝑦_𝑡∈𝐶=⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩−⟨𝐵𝑢_𝑛,𝑦_𝑡−𝑢_𝑛⟩+⟨𝐵𝑢_𝑛,𝑦_𝑡−𝑢_𝑛⟩−⟨𝐵𝑦_𝑛,𝑦_𝑡−𝑢_𝑛⟩−⟨𝑦_𝑡−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝑟_𝑛⟩+𝑓(𝑦_𝑡,𝑢_𝑛),∀𝑦_𝑡∈𝐶=⟨𝐵𝑦_𝑡−𝐵𝑢_𝑛,𝑦_𝑡−𝑢_𝑛⟩+⟨𝐵𝑢_𝑛−𝐵𝑦_𝑛,𝑦_𝑡−𝑢_𝑛⟩−⟨𝑦_𝑡−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝑟_𝑛⟩+𝑓(𝑦_𝑡,𝑢_𝑛),∀𝑦_𝑡∈𝐶.(3.26) Since 𝑥_𝑛→𝑝 as 𝑛→∞ from (3.18) and (3.21), we can get 𝑢_𝑛→𝑝 and 𝑦_𝑛→𝑝 as 𝑛→∞. Furthermore, it follows from the continuity of 𝐵 that 𝐵𝑢_𝑛−𝐵𝑦_𝑛→0 as 𝑛→∞. From 𝑟_𝑛>0 and (3.22), we have ‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖/𝑟_𝑛→0 as 𝑛→∞. Since 𝐵 is monotone, we know that ⟨𝐵𝑦_𝑡−𝐵𝑢_𝑛,𝑦_𝑡−𝑢_𝑛⟩≥0. Thus, it follows from (A4) that 𝑓(𝑦_𝑡,𝑝)≤liminf_𝑛→∞𝑓(𝑦_𝑡,𝑢_𝑛)≤lim_𝑛→∞⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩=⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑝⟩.(3.27) From the conditions (A1) and (A4), we have 0=𝑓(𝑦_𝑡,𝑦_𝑡)≤𝑡𝑓(𝑦_𝑡,𝑦)+(1−𝑡)𝑓(𝑦_𝑡,𝑝)≤𝑡𝑓(𝑦_𝑡,𝑦)+(1−𝑡)⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑝⟩≤𝑡𝑓(𝑦_𝑡,𝑦)+(1−𝑡)𝑡⟨𝐵𝑦_𝑡,𝑦−𝑝⟩,(3.28) and hence 0≤𝑓(𝑦_𝑡,𝑦)+(1−𝑡)⟨𝐵𝑦_𝑡,𝑦−𝑝⟩.(3.29) Letting 𝑡→0, we get 0≤𝑓(𝑝,𝑦)+⟨𝐵𝑝,𝑦−𝑝⟩,∀𝑦∈𝐶.(3.30) This implies that 𝑝∈GEP(𝑓,𝐵).
(ii) We show that 𝑥_𝑛→𝑝∈∩^∞_𝑖=1𝐹(𝑆_𝑖). From definition of 𝐶_𝑛+1 and since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1, we have 𝜙(𝑥_𝑛+1,𝑧_𝑛)≤𝜙(𝑥_𝑛+1,𝑥_𝑛)+𝜃_𝑛.(3.31) Form Lemma 2.1 and (3.9), we obtain that lim_𝑛→∞‖𝑥_𝑛+1−𝑧_𝑛‖=0.(3.32) Since 𝐽 is uniformly norm-to-norm continuous, we also have lim_𝑛→∞‖𝐽𝑥_𝑛+1−𝐽𝑧_𝑛‖=0.(3.33) From (3.1), we compute ‖𝐽𝑥_𝑛+1−𝐽𝑧_𝑛‖=‖𝐽𝑥_𝑛+1−(𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛)‖=‖𝛼_𝑛,0𝐽𝑥_𝑛+1−𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛‖=‖𝛼_𝑛,0(𝐽𝑥_𝑛+1−𝐽𝑥_𝑛)+^∞_𝑖=1𝛼_𝑛,𝑖(𝐽𝑥_𝑛+1−𝐽𝑆^𝑛_i𝑥_𝑛)‖=‖^∞_𝑖=1𝛼_𝑛,𝑖(𝐽𝑥_𝑛+1−𝐽𝑆^𝑛_𝑖𝑥_𝑛)−𝛼_𝑛,0(𝐽𝑥_𝑛−𝐽𝑥_𝑛+1≥)‖_∞𝑖=1𝛼_𝑛,𝑖‖𝐽𝑥_𝑛+1−𝐽𝑆^𝑛_𝑖𝑥_𝑛‖−𝛼_𝑛,0‖𝐽𝑥_𝑛−𝐽𝑥_𝑛+1‖,(3.34) and hence ‖𝐽𝑥_𝑛+1−𝐽𝑆^𝑛_𝑖𝑥_𝑛1‖≤∑^∞_𝑖=1𝛼_𝑛,𝑖(‖𝐽𝑥_𝑛+1−𝐽𝑧_𝑛‖+𝛼_𝑛,0‖𝐽𝑥_𝑛−𝐽𝑥_𝑛+1‖).(3.35) From (3.14), (3.33), and liminf_𝑛→∞∑^∞_𝑖=1𝛼_𝑛,𝑖>0, we get lim_𝑛→∞‖𝐽𝑥_𝑛+1−𝐽𝑆^𝑛_𝑖𝑥_𝑛‖=0.(3.36) Since 𝐽⁻¹ is uniformly norm-to-norm continuous on bounded sets, we obtain that lim_𝑛→∞‖𝑥_𝑛+1−𝑆^𝑛_𝑖𝑥_𝑛‖=0.(3.37) By the triangle inequality, ‖𝑥_𝑛−𝑆^𝑛_𝑖𝑥_𝑛‖=‖𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆^𝑛_𝑖𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑆^𝑛_𝑖𝑥_𝑛‖.(3.38) Hence from (3.13) and (3.37), we have lim_𝑛→∞‖𝑥_𝑛−𝑆^𝑛_𝑖𝑥_𝑛‖=0.(3.39) Since 𝐽 is uniformly continuous on any bounded subset of 𝐸, we obtain lim_𝑛→∞‖𝐽𝑥_𝑛−𝐽𝑆^𝑛_𝑖𝑥_𝑛‖=0,∀𝑖≥1.(3.40) Since 𝑥_𝑛→𝑝 and 𝐽 is uniformly continuous, it yields 𝐽𝑥_𝑛→𝐽𝑝. Thus from (3.40), we get 𝐽𝑆^𝑛_𝑖𝑥_𝑛→𝐽𝑝,∀𝑖≥1.(3.41) Since 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak^∗-continuous, we have 𝑆^𝑛_𝑖𝑥_𝑛⇀𝑝,∀𝑖≥1.(3.42) On the other hand, for each 𝑖≥1, we observe that |‖𝑆^𝑛_𝑖𝑥_𝑛‖−‖𝑝‖|=|‖𝐽(𝑆^𝑛_𝑖𝑥_𝑛)‖−‖𝐽𝑝‖|≤‖𝐽(𝑆^𝑛_𝑖𝑥_𝑛)−𝐽𝑝‖.(3.43) In view of (3.41), we obtain ‖𝑆^𝑛_𝑖𝑥_𝑛‖→‖𝑝‖ for each 𝑖≥1. Since 𝐸 has the Kadec-Klee property, we get 𝑆^𝑛_𝑖𝑥_𝑛→𝑝,foreach𝑖≥1.(3.44) By the assumption that for each 𝑖≥1, 𝑆_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous, 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+1)‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑆_𝑖^𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖+‖𝑥_𝑛−𝑆^𝑛_𝑖𝑥_𝑛‖.(3.45) From (3.13) and (3.39), it yields that ‖𝑆_𝑖^𝑛+1𝑥_𝑛−𝑆^𝑛_𝑖𝑥_𝑛‖→0. 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 closeness of 𝑆_𝑖, we have 𝑆_𝑖𝑝=𝑝, for all 𝑖≥1. This implies that 𝑝∈∩^∞_𝑖=1𝐹(𝑆_𝑖).
Finally, we show that 𝑥_𝑛→𝑝=Π_𝐹𝑥₀. Let 𝑢=Π_𝐹𝑥₀. From 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑢∈𝐹⊂𝐶_𝑛, we have𝜙(𝑥_𝑛,𝑥₀)≤𝜙(𝑢,𝑥₀),∀𝑛≥0.(3.46) This implies that 𝜙(𝑝,𝑥₀)=lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀)≤𝜙(𝑢,𝑥₀).(3.47) By definition of 𝑝=Π_𝐹𝑥₀, we have 𝑝=𝑢. Therefore, 𝑥_𝑛→𝑝=Π_𝐹𝑥₀. This completes the proof.

If 𝑆_𝑖=𝑆 for each 𝑖∈ℕ, then Theorem 3.1 is reduced to the following corollary.

Corollary 3.2. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with Kadec-Klee property. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴1)-(𝐴4). Let 𝐵 be a continuous monotone mapping of 𝐶 into 𝐸^∗. Let 𝑆∶𝐶→𝐶 be a closed uniformly 𝐿-Lipschitz continuous and quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1, such that 𝐹∶=𝐹(𝑆)∩𝐺𝐸𝑃(𝑓,𝐵) is a nonempty and bounded subset in 𝐶. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑦_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑧_𝑛𝑧),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆^𝑛𝑥_𝑛𝑢),_𝑛∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑓(𝑢_𝑛,𝑦)+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛)+𝜃_𝑛𝑥},_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.48) where 𝐽 is the duality mapping on 𝐸, 𝜃_𝑛=sup_𝑞∈𝐹(𝑘_𝑛−1)𝜙(𝑞,𝑥_𝑛), {𝛼_𝑛},{𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. If liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0, then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π_𝐹𝑥₀.

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 3.3. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with Kadec-Klee property. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴1)-(𝐴4). Let 𝐵 be a continuous monotone mapping of 𝐶 into 𝐸^∗. Let 𝑆,𝑇∶𝐶→𝐶 be two closed quasi-𝜙-asymptotically nonexpansive mappings and uniformly 𝐿_𝑆,𝐿_𝑇-Lipschitz continuous, respectively with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1 such that 𝐹∶=𝐹(𝑆)∩𝐹(𝑇)∩𝐺𝐸𝑃(𝑓,𝐵) is a nonempty and bounded subset in 𝐶. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑦_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑧_𝑛𝑧),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑆^𝑛𝑥_𝑛+𝛾_𝑛𝐽𝑇^𝑛𝑥_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛,𝑦)+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛)+𝜃_𝑛𝑥},_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.49) where 𝐽 is the duality mapping on 𝐸, 𝜃_𝑛=sup_𝑞∈𝐹(𝑘_𝑛−1)𝜙(𝑞,𝑥_𝑛), {𝛼_𝑛,𝑖},{𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. If 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1 for all 𝑛≥0 and liminf_𝑛→∞𝛼_𝑛𝛽_𝑛>0 and liminf_𝑛→∞𝛼_𝑛𝛾_𝑛>0, then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π_𝐹𝑥₀.

Remark 3.4. Corollary 3.3 improves and extends [44, Theorem 3.1] in the following senses: (i)for the mappings, we extend the mappings from two closed relatively quasi-nonexpansive mappings to an infinite family of closed and uniformly quasi-𝜙-asymptotically mappings, (ii)from a solution of the classical equilibrium problem to the generalized equilibrium problem, (iii)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.

Corollary 3.5. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with Kadec-Klee property. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴1)-(𝐴4). Let 𝐵 be a continuous monotone mapping of 𝐶 into 𝐸^∗. Let {𝑆_𝑖}^∞_𝑖=1∶𝐶→𝐶 be an infinite family of closed quasi-𝜙-nonexpansive mappings such that 𝐹∶=∩^∞_𝑖=1𝐹(𝑆_𝑖)∩𝐺𝐸𝑃(𝑓,𝐵)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑦_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑧_𝑛𝑧),_𝑛=𝐽⁻¹(𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆_𝑖𝑥_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛,𝑦)+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥,∀𝑦∈𝐶,_𝑛+1={𝑧∈C_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.50) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛,𝑖},{𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. If ∑^∞_𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0 and liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑖≥1, then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π_𝐹𝑥₀.

Proof. Since {𝑆_𝑖}^∞_𝑖=1∶𝐶→𝐶 is an infinite family of closed quasi-𝜙-nonexpansive mappings, it is an infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with sequence 𝑘_𝑛=1. Hence the conditions appearing in Theorem 3.1𝐹 is a bounded subset in 𝐶 and for each 𝑖≥1,𝑆_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous are of no use here. By virtue of the closeness of mapping 𝑆_𝑖 for each 𝑖≥1, it yields that 𝑝∈𝐹(𝑆_𝑖) for each 𝑖≥1, that is, 𝑝∈∩^∞_𝑖=1𝐹(𝑆_𝑖). Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Corollary 3.5 is obtained from Theorem 3.1 immediately.

Corollary 3.6. Let 𝐶 be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space 𝐸 with Kadec-Klee property. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴1)-(𝐴4). Let 𝐵 be a continuous monotone mapping of 𝐶 into 𝐸^∗. 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,∞), 𝑘_𝑛→1 and uniformly 𝐿_𝑖-Lipschitz continuous such that 𝐹∶=∩^∞_𝑖=1𝐹(𝑆_𝑖)∩𝐺𝐸𝑃(𝑓,𝐵) is a nonempty and bounded subset in 𝐶. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑦_𝑛=𝐽⁻¹(𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛_𝑖𝑥_𝑛𝑢),_𝑛∈𝐶suchthat𝑓(𝑢_𝑛,𝑦)+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨y−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛)+𝜃_𝑛𝑥},_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.51) where 𝐽 is the duality mapping on 𝐸, 𝜃_𝑛=sup_𝑞∈𝐹(𝑘_𝑛−1)𝜙(𝑞,𝑥_𝑛), {𝛼_𝑛,𝑖} is a sequence in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. If ∑^∞_𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0 and liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑖≥1, then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π_𝐹𝑥₀.

Proof. Setting 𝛽_𝑛≡0 in Theorem 3.1, then, we get that 𝑦_𝑛=𝑧_𝑛. Thus, the method of proof of Theorem 3.1 we obtain Corollary 3.6 immediately.

Remark 3.7. Theorem 3.1, Corollary 3.3, and Corollary 3.5, improve and extend the corresponding results in Qin et al. [49] and Zegeye [50] in the following senses: (i)from a solution of the classical equilibrium problem to the generalized equilibrium problem with an infinite family of quasi-𝜙-asymptotically mappings, (ii)for the mappings, we extend the mappings from nonexpansive mappings, relatively quasi-nonexpansive mappings or quasi-𝜙-nonexpansive mappings and a finite family of closed relatively quasi-nonexpansive mappings to an infinite family of quasi-𝜙-asymptotically nonexpansive mappings, (iii)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.

4. Applications

If 𝐸=𝐻, a Hilbert space, then 𝐻 is uniformly smooth and strictly convex Banach space 𝐸 with Kadec-Klee property and closed relatively quasi-nonexpansive mappings reducing to closed quasi-nonexpansive mappings. Moreover, 𝐽=𝐼, identity operator on 𝐻 and Π_𝐶=𝑃_𝐶, projection mapping from 𝐻 into 𝐶. Thus, the following corollaries hold.

Theorem 4.1. Let 𝐶 be a nonempty closed and convex subset of a Hilbert space 𝐻. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴1)-(𝐴4). Let 𝐵 be a continuous monotone mapping of 𝐶 into 𝐻. 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,∞), 𝑘_𝑛→1 and uniformly 𝐿_𝑖-Lipschitz continuous such that 𝐹∶=∩^∞_𝑖=1𝐹(𝑆_𝑖)∩𝐺𝐸𝑃(𝑓,𝐵) is a nonempty and bounded subset in 𝐶. For an initial point 𝑥₀∈𝐻 with 𝑥₁=𝑃_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑧_𝑛,𝑧_𝑛=𝛼_𝑛,0𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝑆^𝑛_𝑖𝑥_𝑛,𝑢_𝑛∈𝐶suchthat𝑓(𝑢_𝑛,y)+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶‖𝑧−𝑢_𝑛‖≤𝛼_𝑛‖𝑧−𝑥_𝑛‖+(1−𝛼_𝑛)‖𝑧−𝑧_𝑛‖≤‖𝑧−𝑥_𝑛‖+𝜃_𝑛𝑥},_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,∀𝑛≥0,(4.1) where 𝜃_𝑛=sup_𝑞∈𝐹(𝑘_𝑛−1)‖𝑞−𝑥_𝑛‖, {𝛼_𝑛,𝑖},{𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. If ∑^∞_𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0 and liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑖≥1, then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π_𝐹𝑥₀.

Corollary 4.2. Let 𝐶 be a nonempty closed and convex subset of a Hilbert space 𝐻. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (𝐴1)-(𝐴4). Let 𝐵 be a continuous monotone mapping of 𝐶 into 𝐻. Let {𝑆_𝑖}^∞_𝑖=1∶𝐶→𝐶 be an infinite family of closed and quasi-𝜙-nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1 and uniformly 𝐿_𝑖-Lipschitz continuous such that 𝐹∶=∩^∞_𝑖=1𝐹(𝑆_𝑖)∩𝐺𝐸𝑃(𝑓,𝐵) is a nonempty and bounded subset in 𝐶. For an initial point 𝑥₀∈𝐻 with 𝑥₁=𝑃_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑧_𝑛,𝑧_𝑛=𝛼_𝑛,0𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝑆^𝑛_𝑖𝑥_𝑛,𝑢_𝑛∈𝐶suchthat𝑓(𝑢_𝑛,𝑦)+⟨𝐵𝑦_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶‖𝑧−𝑢_𝑛‖≤𝛼_𝑛‖𝑧−𝑥_𝑛‖+(1−𝛼_𝑛)‖𝑧−𝑧_𝑛‖≤‖𝑧−𝑥_𝑛𝑥‖},_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,∀𝑛≥0,(4.2) where {𝛼_𝑛,𝑖},{𝛽_𝑛} are sequences in [0,1] and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. If ∑^∞_𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0 and liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0 for all 𝑖≥1, then {𝑥_𝑛} converges strongly to 𝑝∈𝐹, where 𝑝=Π_𝐹𝑥₀.

Remark 4.3. Theorem 4.1 improves and extends the Corollary  3.7 in Zegeye [50] in the aspect for the mappings; we extend the mappings from a finite family of closed relatively quasi-nonexpansive mappings to more general a infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings.

Acknowledgments

The authors would like to thank the referees for their careful readings and valuable suggestions to improve the writing of this paper. The first author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Commission on Higher Education and the second author is supported by the Thailand Research Fund and the Commission on Higher Education under the project no. MRG5380044.