The Modified Block Iterative Algorithms for Asymptotically Relatively Nonexpansive Mappings and the System of Generalized Mixed Equilibrium Problems

Abstract

The propose of this paper is to present a modified block iterative algorithm for finding a common element between the set of solutions of the fixed points of two countable families of asymptotically relatively nonexpansive mappings and the set of solution of the system of generalized mixed equilibrium problems in a uniformly smooth and uniformly convex Banach space. Our results extend many known recent results in the literature.

1. Introduction

The equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, networks, elasticity, and optimization, and it has been extended and generalized in many directions.

In the theory of equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. This theory combines theoretical and algorithmic advances with novel domain of applications. Analysis of these problems requires a blend of techniques from convex analysis, functional analysis, and numerical analysis.

Let 𝐸 be a Banach space with norm ‖⋅‖, 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝐸^∗ denote the dual of 𝐸. Let 𝑓_𝑖∶𝐶×𝐶→ℝ be a bifunction, 𝜓_𝑖∶𝐶→ℝ be a real-valued function, where ℝ is denoted by the set of real numbers, and 𝐴_𝑖∶𝐶→𝐸^∗ be a nonlinear mapping. The goal of the system of generalized mixed equilibrium problem is to find 𝑢∈𝐶 such that 𝑓𝑓₁(𝑢,𝑦)+⟨𝐴₁𝑢,𝑦−𝑢⟩+𝜓₁(𝑦)−𝜓₁(𝑢)≥0,∀𝑦∈𝐶,₂𝑓(𝑢,𝑦)+⟨𝐴₂𝑢,𝑦−𝑢⟩+𝜓⋮₂(𝑦)−𝜓₂(𝑢)≥0,∀𝑦∈𝐶,_𝑁(𝑢,𝑦)+⟨𝐴_𝑁𝑢,𝑦−𝑢⟩+𝜓_𝑁(𝑦)−𝜓_𝑁(𝑢)≥0,∀𝑦∈𝐶.(1.1) If 𝑓_𝑖=𝑓, 𝐴_𝑖=𝐴, and 𝜓_𝑖=𝜓, the problem (1.1) is reduced to the generalized mixed equilibrium problem, denoted by GEMP(𝑓,𝐴,𝜓), to find 𝑢∈𝐶 such that 𝑓(𝑢,𝑦)+⟨𝐴𝑢,𝑦−𝑢⟩+𝜓(𝑦)−𝜓(𝑢)≥0,∀𝑦∈𝐶.(1.2) The set of solutions to (1.2) is denoted by Ω, that is, Ω={𝑥∈𝐶∶𝑓(𝑢,𝑦)+⟨𝐴𝑢,𝑦−𝑢⟩+𝜑(𝑦)−𝜑(𝑢)≥0,∀𝑦∈𝐶}.(1.3) If 𝐴=0, the problem (1.2) is reduced to the mixed equilibrium problem for 𝑓, denoted by MEP(𝑓,𝜓), to find 𝑢∈𝐶 such that 𝑓(𝑢,𝑦)+𝜓(𝑦)−𝜓(𝑢)≥0,∀𝑦∈𝐶.(1.4) If 𝑓≡0, the problem (1.2) is reduced to the mixed variational inequality of Browder type, denoted by VI(𝐶,𝐴,𝜓), is to find 𝑢∈𝐶 such that ⟨𝐴𝑢,𝑦−𝑢⟩+𝜓(𝑦)−𝜓(𝑢)≥0,∀𝑦∈𝐶.(1.5) If 𝐴=0 and 𝜓=0, the problem (1.2) is reduced to the equilibrium problem for 𝑓, denoted by EP(𝑓), to find 𝑢∈𝐶 such that 𝑓(𝑢,𝑦)≥0,∀𝑦∈𝐶.(1.6)

The above formulation (1.6) was shown in [1] to cover monotone inclusion problems, saddle-point problems, variational inequality problems, minimization 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, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions to EP(𝑓); see, for example [1–4] and references therein. Some solution methods have been proposed to solve the EP(𝑓); see, for example, [2, 4–15] and references therein. In 2005, Combettes and Hirstoaga [5] introduced an iterative scheme of finding the best approximation to the initial data when EP(𝑓) is nonempty, and they also proved a strong convergence theorem.

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 the Banach space 𝐸 is said to be smooth, provided lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(1.7) exists for each 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦∈𝐸. The modulus of convexity of 𝐸 is the function 𝛿∶[0,2]→[0,1] defined by 𝛿(𝜀)=inf1−‖‖‖𝑥+𝑦2‖‖‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀.(1.8) A Banach space 𝐸 is uniformly convex, if and only if 𝛿(𝜀)>0 for all 𝜀∈(0,2].

Let 𝐸 be a Banach space, 𝐶 be a closed convex subset of 𝐸, a mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖(1.9) for all 𝑥,𝑦∈𝐶. We denote by 𝐹(𝑇) the set of fixed points of 𝑇. If 𝐶 is a bounded closed convex set and 𝑇 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑇) is nonempty (see [16]). A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [17] 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 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 asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 from 𝐶 into itself is said to be relatively nonexpansive [18–20] 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 [21, 22]. 𝑇 is said to be 𝜙-nonexpansive, if 𝜙(𝑇𝑥,𝑇𝑦)≤𝜙(𝑥,𝑦) for 𝑥,𝑦∈𝐶. 𝑇 is said to be quas-ϕ-nonexpansive if 𝐹(𝑇)≠∅ and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). A mapping 𝑇 is said to be asymptotically relatively 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), forall𝑛≥1,𝑥∈𝐶, and 𝑝∈𝐹(𝑇). {𝑇_𝑛}^∞𝑛=0 is said to be a countable family of weak relatively nonexpansive mappings [23] if the following conditions are satisfied: (i)𝐹({𝑇_𝑛}^∞𝑛=0)≠∅; (ii)𝜙(𝑢,𝑇_𝑛𝑥)≤𝜙(𝑢,𝑥),forall𝑢∈𝐹(𝑇_𝑛),𝑥∈𝐶,𝑛≥0; (iii)𝐹({𝑇_𝑛}^∞𝑛=0)=∩^∞𝑛=0𝐹(𝑇_𝑛). A mapping 𝑇∶𝐶→𝐶 is said to be uniformly L-Lipschitz continuous, if there exists a constant 𝐿>0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶,∀𝑛≥1.(1.10) A mapping 𝑇∶𝐶→𝐶 is said to be closed if for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, then 𝑇𝑥=𝑦. Let {𝑇_𝑖}^∞𝑖=1∶𝐶→𝐶 be a sequence of mappings. {𝑇_𝑖}^∞𝑖=1 is said to be a countable family of uniformly asymptotically relatively nonexpansive mappings, if ∩^∞𝑛=1𝐹(𝑇_𝑛)≠∅, and there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that for each 𝑖>1𝜙𝑝,𝑇^𝑛𝑖𝑥≤𝑘_𝑛𝜙(𝑝,𝑥),∀𝑝∈^∞_𝑛=1𝐹𝑇_𝑛,𝑥∈𝐶,∀𝑛≥1.(1.11)

In 2009, Petrot et al. [24], introduced a hybrid projection method for approximating a common element of the set of solutions of fixed points of hemirelatively nonexpansive (or quasi-𝜙-nonexpansive) mappings in a uniformly convex and uniformly smooth Banach space:𝑦𝑥₀∈𝐶,𝐶₀=𝐶,_𝑛𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇_𝑛𝑧_𝑛,𝐶_𝑛+1=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_𝑛𝑥_𝑛,=𝑣∈𝑥𝐶_𝑛∶𝜙𝑣,𝑦_𝑛≤𝜙𝑣,𝑥_𝑛_𝑛+1,=Π_𝐶𝑛+1𝑥₀.(1.12) They proved that the sequence {𝑥_𝑛} converges strongly to 𝑝∈𝐹(𝑇), where 𝑝∈Π_𝐹(𝑇)𝑥 and Π_𝐶 is the generalized projection from 𝐸 onto 𝐹(𝑇). Kumam and Wattanawitoon [25], introduced a hybrid iterative scheme for finding a common element of the set of common fixed points of two quasi-𝜙-nonexpansive mappings and the set of solutions of an equilibrium problem in Banach spaces, by the following manner: 𝑦𝑥₀∈𝐶,𝐶₀=𝐶_𝑛𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑆𝑧_𝑛,𝑢_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇𝑥_𝑛,∈𝐶suchthat𝑓𝑢_𝑛,𝑦+1𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝑧∈𝑥𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛+1,=Π_𝐶𝑛+1𝑥₀.(1.13) They proved that the sequence {𝑥_𝑛} converges strongly to 𝑝∈𝐹(𝑇)∩𝐹(𝑆)∩EP(𝑓), where 𝑝∈Π_{𝐹(𝑇)∩𝐹(𝑆)∩EP(𝑓)}𝑥 under the assumptions (C1) limsup_𝑛→∞𝛼_𝑛<1, (C2) lim_𝑛→∞𝛽_𝑛<1, and (C3) liminf_𝑛→∞(1−𝛼_𝑛)𝛽_𝑛(1−𝛽_𝑛)>0.

Recently, Chang et al. [26], introduced the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi-𝜙-asymptotically nonexpansive mappings,𝑦𝑥₀∈𝐶chosenarbitrary,𝐶₀=𝐶,_𝑛=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛𝐶_𝑛+1,=𝑣𝑥∈𝐶_𝑛𝑛+1∶𝜙𝑣,𝑦_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(1.14)where 𝜉_𝑛=sup_𝑢∈𝐹(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛). Then, they proved that under appropriate control conditions the sequence {𝑥_𝑛} converges strongly to Π_∩^∞𝑛=1_{𝐹(𝑆𝑖)}𝑥₀.

Very recently, Tan and Chang [27], introduced a new hybrid iterative scheme for finding a common element between set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi-𝜙-asymptotically nonexpansive mappings (which is more general than quasi-𝜙-nonexpansive mappings), and null spaces of finite family of 𝛾-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space.

In this paper, motivated and inspired by Petrot et al. [24], Kumam and Wattanawitoon [25], Chang et al. [26], and Tan and Chang [27], we introduce the new hybrid block algorithm for two countable families of closed and uniformly Lipschitz continuous and uniformly asymptotically relatively nonexpansive mappings in a Banach space. Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐶, 𝐶₀=𝐶 and 𝑦_𝑛=𝐽⁻¹𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝑧_𝑛𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛,=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝑢_𝑛(𝑖)𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑦_𝑛,𝐶=𝐾_{𝑓𝑖,𝑟𝑖}𝐾_{𝑓𝑖−1,𝑟𝑖−1}⋯𝐾_𝑓1,𝑟1𝑧_𝑛,𝑖=1,2,…,𝑁,_𝑛+1=𝑧∈𝐶_𝑛∶max_{𝑖=1,2,…,𝑁}𝜙𝑧,𝑢^𝑛(𝑖)≤𝜙𝑧,𝑥^𝑛+𝜃_𝑛,𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+𝜉_𝑛𝑥_𝑛+1,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0.(1.15) Under appropriate conditions, we will prove that the sequence {𝑥_𝑛} generated by algorithms (1.15) converges strongly to the point Π_(∩^𝑁𝑖=1_{Ω𝑖)∩(∩}^∞𝑖=1_{𝐹(𝑇𝑖))∩(∩}^∞𝑖=1_{𝐹(𝑆𝑖))}𝑥₀. Our results extend many known recent results in the literature.

2. Preliminaries

Let 𝐸 be a real Banach space with norm ‖⋅‖, and let 𝐽 be the normalized duality mapping from 𝐸 into 2^𝐸∗ given by 𝐽𝑥=𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖‖𝑥^∗‖,‖𝑥‖=‖𝑥^∗‖(2.1) for all 𝑥∈𝐸, where 𝐸^∗ denotes the dual space of 𝐸 and ⟨⋅,⋅⟩ the generalized duality pairing between 𝐸 and 𝐸^∗. It is also known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.

We know the following (see [28, 29]):(i)if 𝐸 is smooth, then 𝐽 is single valued;(ii)if 𝐸 is strictly convex, then 𝐽 is one-to-one and ⟨𝑥−𝑦,𝑥^∗−𝑦^∗⟩>0 holds for all (𝑥,𝑥^∗),(𝑦,𝑦^∗)∈𝐽 with 𝑥≠𝑦;(iii)if 𝐸 is reflexive, then 𝐽 is surjective;(iv)if 𝐸 is uniformly convex, then it is reflexive;(v)if 𝐸 is a reflexive and strictly convex, then 𝐽⁻¹ is norm-weak-continuous;(vi)𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex;(vii)if 𝐸^∗ is uniformly convex, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸;(viii)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.

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Throughout this paper, we denote by 𝜙 the function defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,for𝑥,𝑦∈𝐸.(2.2) Following Alber [30], the generalized projection Π_𝐶∶𝐸→𝐶 is a map that assigns to an arbitrary point 𝑥∈𝐸 the minimum point of the function 𝜙(𝑥,𝑦), that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem 𝜙𝑥,𝑥=inf_𝑦∈𝐶𝜙(𝑦,𝑥).(2.3) 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. It is obvious from the definition of function 𝜙 that (see [30]) (‖𝑦‖−‖𝑥‖)²≤𝜙(𝑦,𝑥)≤(‖𝑦‖+‖𝑥‖)²,∀𝑥,𝑦∈𝐸.(2.4) If 𝐸 is a Hilbert space, then 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖².

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 (2.4), 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 [28, 29] for more details.

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

Lemma 2.1 (see Kamimura and Takahashi [31]). Let 𝐸 be a uniformly convex and smooth real Banach space, and let {𝑥_𝑛},{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 (see Alber [30]). 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.(2.5)

Lemma 2.3 (see Alber [30]). 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.(2.6)

Lemma 2.4 (see Chang et al. [26]). 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 integers 𝑖,𝑗 with 𝑖<𝑗, ‖‖‖‖^∞_𝑛=1𝜆_𝑛𝑥_𝑛‖‖‖‖²≤^∞_𝑛=1𝜆_𝑛‖‖𝑥_𝑛‖‖²−𝜆_𝑖𝜆_𝑗𝑔‖‖𝑥_𝑖−𝑥_𝑗‖‖.(2.7)

Lemma 2.5 (see Chang et al. [26]). Let 𝐸 be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a closed and asymptotically relatively nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞),𝑘_𝑛→1. Then 𝐹(𝑇) is closed and convex subset of 𝐶.

For solving the generalized mixed equilibrium problem (or a system of generalized mixed equilibrium problem), let us assume that the bifunction 𝑓∶𝐶×𝐶→ℝ and 𝜓∶𝐶→ℝ is convex and lower semicontinuous satisfies the following conditions: (A1)𝑓(𝑥,𝑥)=0 for 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, limsup_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦);(2.8)(A4) for each 𝑥∈𝐶, 𝑦↦𝑓(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.6 (see Chang et al. [26]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜓∶𝐶→ℝ is convex and lower semicontinuous and 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)+⟨𝐴𝑢,𝑦−𝑢⟩+𝜓(𝑦)−𝜓(𝑢)+1𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.9) Define a mapping 𝐾_𝑓,𝑟∶𝐶→𝐶 as follows: 𝐾_𝑓,𝑟(𝑥)=𝑢∈𝐶∶𝑓(𝑢,𝑦)+⟨𝐴𝑢,𝑦−𝑢⟩+𝜓(𝑦)−𝜓(𝑢)+1𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶(2.10) for all 𝑥∈𝐸. Then, the following hold: (i)𝐾_𝑓,𝑟 is singlevalued; (ii)𝐾_𝑓,𝑟 is firmly nonexpansive, that is, for all 𝑥,𝑦∈𝐸, ⟨𝐾_𝑓,𝑟𝑥−𝐾_𝑓,𝑟𝑦,𝐽𝐾_𝑓,𝑟𝑥−𝐽𝐾_𝑓,𝑟𝑦⟩≤⟨𝐾_𝑓,𝑟𝑥−𝐾_𝑓,𝑟𝑦,𝐽𝑥−𝐽𝑦⟩;(iii)𝐹(𝐾_𝑓,𝑟)=𝐹(𝐾_𝑓,𝑟); (iv)𝑢∈𝐶 is a solution of variational equation (2.9) if and only if 𝑢∈𝐶 is a fixed point of 𝐾_𝑓,𝑟; (v)𝐹(𝐾_𝑓,𝑟)=Ω; (vi)Ω is closed and convex; (vii)𝜙(𝑝,𝐾_𝑓,𝑟𝑧)+𝜙(𝐾_𝑓,𝑟𝑧,𝑧)≤𝜙(𝑝,𝑧), forall𝑝∈𝐹(𝐾_𝑓,𝑟), 𝑧∈𝐸.

3. Main Results

Theorem 3.1. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝐴_𝑖∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜓_𝑖∶𝐶→ℝ be a lower semi-continuous and convex function, 𝑓_𝑖 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), 𝐾_{𝑓𝑖,𝑟𝑖} is the mapping defined by (2.10) where 𝑟_𝑖≥𝑟>0, and let {𝑇_𝑖}^∞𝑖=1, {𝑆_𝑖}^∞𝑖=1 be countable families of closed and uniformly 𝐿_𝑖, 𝜇_𝑖-Lipschitz continuous and asymptotically relatively nonexpansive mapping with sequence {𝑘_𝑛},{𝜁_𝑛}⊂[1,∞);  𝑘_𝑛→1,𝜁_𝑛→1 such that ℱ∶=(∩^𝑁𝑖=1Ω_𝑖)∩(∩^∞𝑖=1𝐹(𝑇_𝑖))∩(∩^∞𝑖=1𝐹(𝑆_𝑖))≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶 and 𝐶₀=𝐶, 𝑦_𝑛=𝐽⁻¹𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝑧_𝑛𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛,=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝑢_𝑛(𝑖)𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑦_𝑛,𝐶=𝐾_{𝑓𝑖,𝑟𝑖}𝐾_{𝑓𝑖−1,𝑟𝑖−1}⋯𝐾_𝑓1,𝑟1𝑧_𝑛,𝑖=1,2,…,𝑁,_𝑛+1=𝑧∈𝐶_𝑛∶max_{𝑖=1,2,…,𝑁}𝜙𝑧,𝑢^𝑛(𝑖)≤𝜙𝑧,𝑥^𝑛+𝜃_𝑛,𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+𝜉_𝑛𝑥_𝑛+1,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.1) where 𝜉_𝑛=sup_𝑝∈ℱ(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛), 𝜃_𝑛=𝛿_𝑛+𝜉_𝑛𝜁_𝑛, and 𝛿_𝑛=sup_𝑝∈ℱ(𝜁_𝑛−1)𝜙(𝑝,𝑥_𝑛). The coefficient sequences {𝛼_𝑛,𝑖} and {𝛽_𝑛,𝑖}⊂[0,1] satisfy the following: (i)∑^∞𝑖=0𝛼_𝑛,𝑖=1; (ii)∑^∞𝑖=0𝛽_𝑛,𝑖=1; (iii)liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0, forall𝑖≥1; (iv)liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛,𝑖>0, forall𝑖≥1, Ω_𝑖,𝑖=1,2,…,𝑁 is the set of solutions to the following generalized mixed equilibrium problem: 𝑓_𝑖(𝑧,𝑦)+⟨𝐴_𝑖𝑧,𝑦−𝑧⟩+𝜓_𝑖(𝑦)−𝜓_𝑖(𝑧)≥0,∀𝑦∈𝐶,𝑖=1,2,…,𝑁.(3.2) Then the sequence {𝑥_𝑛} converges strongly to Π_ℱ𝑥₀.

Proof. We first show that 𝐶_𝑛, forall𝑛≥0 is closed and convex. Clearly 𝐶₀=𝐶 is closed and convex. Suppose that 𝐶_𝑘 is closed and convex for some 𝑘>1. For each 𝑧∈𝐶_𝑘, we see that 𝜙(𝑧,𝑢^𝑘(𝑖))≤𝜙(𝑧,𝑥_𝑘) is equivalent to 2⟨𝑧,𝑥_𝑘⟩−𝑧,𝑢^𝑘(𝑖)≤‖‖𝑥^𝑘‖‖²−‖‖𝑢^𝑘(𝑖)‖‖².(3.3) By the set of 𝐶_𝑘+1, we have 𝐶_𝑛+1=𝑧∈𝐶_𝑛∶max_{𝑖=1,2,…,𝑁}𝜙𝑧,𝑢^𝑛(𝑖)≤𝜙𝑧,𝑥^𝑛+𝜃_𝑛=_𝑁_𝑖=1𝑧∈𝐶∶𝜙𝑧,𝑢^𝑛(𝑖)≤𝜙𝑧,𝑥^𝑛+𝜃_𝑛.(3.4) Hence, 𝐶_𝑛+1 is also closed and convex.
By taking Θ^𝑗𝑛=𝐾_{𝑟𝑗,𝑓𝑖}𝐾_{𝑟𝑗−1,𝑓𝑗−1}⋯𝐾_𝑟1,𝑓1 for any 𝑗∈{1,2,…,𝑖} and Θ^0𝑛=𝐼 for all 𝑛≥1. We note that 𝑢^𝑛(𝑖)=Θ^𝑖𝑛𝑧_𝑛.
Next, we show that ℱ⊂𝐶_𝑛,forall𝑛≥1. For 𝑛≥1, we have ℱ⊂𝐶=𝐶₁. For any given 𝑝∈ℱ∶=(∩^𝑁𝑖=1Ω_𝑖)∩(∩^∞𝑖=1𝐹(𝑇_𝑖))∩(∩^∞𝑖=1𝐹(𝑆_𝑖)). By (3.1) and Lemma 2.4, we have 𝜙𝑝,𝑦_𝑛=𝜙𝑝,𝐽⁻¹^∞_𝑖=0𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛=‖𝑝‖₂−^∞_𝑖=0𝛽_𝑛,𝑖2𝑝,𝐽𝑇^𝑛𝑖𝑥_𝑛+‖‖‖‖^∞_𝑖=0𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖‖‖²≤‖𝑝‖²−^∞_𝑖=0𝛽_𝑛,𝑖2𝑝,𝐽𝑇^𝑛𝑖𝑥_𝑛+^∞_𝑖=0=‖𝑝‖₂𝛽_𝑛,𝑖‖‖𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖²−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔‖‖𝐽𝑇^𝑛0𝑥_𝑛−𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖−^∞_𝑖=0𝛽_𝑛,𝑖2𝑝,𝐽𝑇^𝑛𝑖𝑥_𝑛+^∞_𝑖=0=_∞𝛽_𝑛,𝑖‖‖𝑇^𝑛𝑖𝑥_𝑛‖‖²−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖_𝑖=0≤𝑘_𝑛𝛽_𝑛,𝑖𝜙𝑝,𝑇^𝑛𝑖𝑥_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖≤𝜙𝜙𝑝,𝑥𝑝,_𝑛𝑥_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖≤𝜙𝑝,𝑥_𝑛+sup_𝑝∈𝐹𝑘_𝑛−1𝜙𝑝,𝑥_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖≤𝜙𝑝,𝑥_𝑛+𝜉_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖+𝜉_𝑛,(3.5) where 𝜉_𝑛=sup_𝑝∈ℱ(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛).
By (3.1) and (3.5), we note that 𝜙𝑝,𝑢^𝑛(𝑖)=≤𝜙𝜙𝑝𝑝,,𝑧Θ^𝑖𝑛_𝑛𝑧_𝑛≤𝜙𝑝,𝐽₋₁𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝐽𝑆^𝑛𝑖𝑦_𝑛=‖𝑝‖₂−2𝑝,𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝐽𝑆^𝑛𝑖𝑦_𝑛+‖‖‖‖𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖‖‖²≤‖𝑝‖²−2𝛼_𝑛,0⟨𝑝,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖𝑝,𝐽𝑆^𝑛𝑖𝑦_𝑛+𝛼_𝑛,0‖‖𝑥_𝑛‖‖²+^∞_𝑖=1‖‖𝑆^𝑛𝑖𝑦_𝑛‖‖²≤−𝛼𝛼_𝑛,0𝑛,0𝛼_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖𝜙𝑝,𝑥_𝑛+^∞_𝑖=1≤𝛼_𝑛,0𝛼_𝑛,𝑖𝜙𝑝,𝑆^𝑛𝑖𝑦_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖𝜙𝑝,𝑥_𝑛+𝜁_𝑛∞_𝑖=1≤𝛼_𝑛,0𝛼_𝑛,𝑖𝜙𝑝,𝑦_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖𝜙𝑝,𝑥_𝑛+𝜁_𝑛∞_𝑖=1≤𝛼_𝑛,0𝛼_𝑛,𝑖𝜙𝑝,𝑥_𝑛+𝜉_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖𝜙𝑝,𝑥_𝑛+𝜁_𝑛∞_𝑖=1𝛼_𝑛,𝑖𝜙𝑝,𝑥_𝑛+𝜉_𝑛𝜁_𝑛∞_𝑖=1≤𝜁_𝑛𝛼_𝑛,𝑖−𝛼_𝑛,0𝛼_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖𝜙𝑝,𝑥_𝑛+𝜉_𝑛𝜁_𝑛∞_𝑖=1≤𝜙𝑝,𝑥_𝑛𝛼_𝑛,𝑖−𝛼_𝑛,0𝛼_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖≤𝜙𝑝,𝑥_𝑛+sup_𝑝∈𝐹𝜁_𝑛−1𝜙𝑝,𝑥_𝑛+𝜉_𝑛𝜁^𝑛∞_𝑖=1𝛼_𝑛,𝑖−𝛼_𝑛,0𝛼_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖≤𝜙𝑝,𝑥_𝑛+𝛿_𝑛+𝜉_𝑛𝜁_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖+𝜃_𝑛,(3.6) where 𝛿_𝑛=sup_𝑝∈ℱ(𝜁_𝑛−1)𝜙(𝑝,𝑥_𝑛), 𝜃_𝑛=𝛿_𝑛+𝜉_𝑛𝜁_𝑛. By assumptions on {𝑘_𝑛} and {𝜁_𝑛}, we have 𝜉_𝑛=≤ssuup_𝑝∈ℱp_𝑝∈ℱ𝑘_𝑛−1𝜙𝑝,𝑥_𝑛𝛿𝑘_𝑛−1(‖𝑝‖+𝑀)²⟶0as𝑛⟶∞,(3.7)_𝑛=≤ssuupp_{𝑝∈ℱ𝑝∈ℱ}𝜁_𝑛−1𝜙𝑝,𝑥_𝑛𝜁_𝑛−1(‖𝑝‖+𝑀)²⟶0as𝑛⟶∞,(3.8) where 𝑀=sup_𝑛≥0‖𝑥_𝑛‖.
So, we have 𝑝∈𝐶_𝑛+1. This implies that ℱ∈𝐶_𝑛,forall𝑛≥0 and also {𝑥_𝑛} is well defined.
From Lemma 2.2 and 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we have ⟨⟨𝑥𝑥_𝑛𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0,∀𝑧∈𝐶_𝑛,−𝑝,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0,∀𝑝∈𝐶_𝑛.(3.9) From Lemma 2.3, one has 𝜙𝑥_𝑛,𝑥₀=𝜙Π_𝐶𝑛𝑥₀,𝑥₀≤𝜙𝑝,𝑥₀−𝜙𝑝,𝑥_𝑛≤𝜙𝑝,𝑥₀(3.10) for all 𝑝∈ℱ⊂𝐶_𝑛 and 𝑛≥1. Then, the sequence {𝜙(𝑥_𝑛,𝑥₀)} is also bounded. Thus {𝑥_𝑛} is bounded. Since 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀,∀𝑛∈ℕ.(3.11) Therefore, {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing. Hence, the limit of {𝜙(𝑥_𝑛,𝑥₀)} exists. By the construction of 𝐶_𝑛, one has that 𝐶_𝑚⊂𝐶_𝑛 and 𝑥_𝑚=Π_𝐶𝑚𝑥₀∈𝐶_𝑛 for any positive integer 𝑚≥𝑛. It follows that 𝜙𝑥_𝑚,𝑥_𝑛=≤𝜙𝜙𝑥𝑥_𝑚𝑚,Π_𝐶𝑛𝑥₀=𝜙𝑥_𝑚,𝑥₀−𝜙Π_𝐶𝑛𝑥₀,𝑥₀,𝑥₀−𝜙𝑥_𝑛,𝑥₀.(3.12) Letting 𝑚,𝑛→0 in (3.12), we get 𝜙(𝑥_𝑚,𝑥_𝑛)→0. It follows from Lemma 2.1, that ‖𝑥_𝑚−𝑥_𝑛‖→0 as 𝑚,𝑛→∞. That is, {𝑥_𝑛} is a Cauchy sequence.
Since {𝑥_𝑛} is bounded and 𝐸 is reflexive, there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑢. Since 𝐶_𝑛 is closed and convex and 𝐶_𝑛+1⊂𝐶_𝑛, this implies that 𝐶_𝑛 is weakly closed and 𝑢∈𝐶_𝑛 for each 𝑛≥0. since 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we have 𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑢,𝑥₀,∀𝑛_𝑖≥0.(3.13) Since liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀=≤‖li𝑢m‖i₂nf_{𝑛𝑖→∞}‖‖𝑥^𝑛_𝑖‖‖²−2𝑥_𝑛𝑖,𝐽𝑥₀+‖‖𝑥₀‖‖²−2⟨𝑢,𝐽𝑥₀⟩+‖‖𝑥₀‖‖²=𝜙𝑢,𝑥₀.(3.14) We have 𝜙𝑢,𝑥₀≤liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤limsup_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑢,𝑥₀.(3.15) This implies that lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛𝑖,𝑥₀)=𝜙(𝑢,𝑥₀). That is, ‖𝑥_𝑛𝑖‖→‖𝑢‖. Since 𝑥_𝑛𝑖⇀𝑢, by the Kadec-klee property of 𝐸, we obtain that lim_𝑛→∞𝑥_𝑛𝑖=𝑢.(3.16) If there exists some subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑗→𝑞, then we have 𝜙(𝑢,𝑞)==lliimm_{𝑛𝑖→∞,𝑛𝑗→∞𝑛𝑖→∞,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥_𝑛𝑗≤lim_{𝑛𝑖→∞,𝑛𝑗→∞}𝜙𝑥^𝑛_𝑖,𝑥₀−𝜙Π_𝐶𝑛𝑗𝑥₀,𝑥₀𝜙𝑥^𝑛_𝑖,𝑥₀−𝜙𝑥_𝑛𝑗𝑥₀,𝑥₀=0.(3.17) Therefore, we have 𝑢=𝑞. This implies that lim_𝑛→∞𝑥_𝑛=𝑢.(3.18) Since 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(3.19) for all 𝑛∈ℕ, we also have lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0.(3.20) Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1 and by the definition of 𝐶_𝑛+1, for 𝑖=1,2,…,𝑁, we have 𝜙𝑥_𝑛+1,𝑢^𝑖𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝜃_𝑛.(3.21) Noticing that lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑥_𝑛)=0, we obtain lim_𝑛→∞𝜙𝑥_𝑛+1,𝑢^𝑖𝑛=0,for𝑖=1,2,…,𝑁.(3.22) It then yields that lim_𝑛→∞(‖𝑥_𝑛+1‖−‖𝑢^𝑖𝑛‖)=0,forall𝑖=1,2,…,𝑁. Since lim_𝑛→∞‖𝑥_𝑛+1‖=‖𝑢‖, we have lim_𝑛→∞‖‖𝑢^𝑖𝑛‖‖=‖𝑢‖,∀𝑖=1,2,…,𝑁.(3.23) Hence, lim_𝑛→∞‖‖𝐽𝑢^𝑖𝑛‖‖=‖𝐽𝑢‖,∀𝑖=1,2,…,𝑁.(3.24) From Lemma 2.1 and (3.22), we have lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛+1−𝑢^𝑖𝑛‖‖=0,∀𝑖=1,2,…,𝑁.(3.25) By the triangle inequality, we get lim_𝑛→∞‖‖𝑥_𝑛−𝑢^𝑖𝑛‖‖=0,∀𝑖=1,2,…,𝑁.(3.26) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim_𝑛→∞‖‖𝐽𝑥_𝑛−𝐽𝑢^𝑖𝑛‖‖=lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑢^𝑖𝑛‖‖=0,∀𝑖=1,2,…,𝑁.(3.27)
Now, we prove that 𝑢∈(∩^∞𝑖=1𝐹(𝑇_𝑖))∩(∩^∞𝑖=1𝐹(𝑆_𝑖)). From the construction of 𝐶_𝑛, we obtain that 𝜙𝑥_𝑛+1,𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝜉_𝑛.(3.28) From (3.7) and (3.20), we have lim_𝑛→∞𝜙𝑥_𝑛+1,𝑦_𝑛=0.(3.29) By Lemma 2.1, we also have lim_𝑛→∞‖‖𝑥_𝑛+1−𝑦_𝑛‖‖=0.(3.30) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖=0.(3.31) From (2.4) and (3.29), we have (‖𝑥_𝑛+1‖−‖𝑦_𝑛‖)²→0. Since ‖𝑥_𝑛+1‖→‖𝑢‖, it yields that ‖‖𝑦_𝑛‖‖⟶‖𝑢‖as𝑛⟶∞.(3.32) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, it follows that ‖‖J𝑦_𝑛‖‖⟶‖𝐽𝑢‖as𝑛⟶∞.(3.33) This implies that {𝐽𝑦_𝑛} is bounded in 𝐸^∗. Since 𝐸 is reflexive, there exists a subsequence {𝐽𝑦_𝑛𝑖}⊂{𝐽𝑦_𝑛} such that 𝐽𝑦_𝑛𝑖⇀𝑟∈𝐸^∗. Since 𝐸 is reflexive, we see that 𝐽(𝐸)=𝐸^∗. Hence, there exists 𝑥∈𝐸 such that 𝐽𝑥=𝑟. We note that 𝜙𝑥_𝑛𝑖+1,𝑦_𝑛𝑖=‖‖𝑥_𝑛𝑖+1‖‖²−2𝑥_𝑛𝑖+1,𝐽𝑦_𝑛𝑖+‖‖𝑦_𝑛𝑖‖‖²=‖‖𝑥_𝑛𝑖+1‖‖²−2𝑥_𝑛𝑖+1,𝐽𝑦_𝑛𝑖+‖‖𝐽𝑦_𝑛𝑖‖‖².(3.34) Taking the limit interior of both side and in view of weak lower semicontinuity of norm ‖⋅‖, we have 0≥‖𝑢‖²−2⟨𝑢,𝑟⟩+‖𝑟‖²=‖𝑢‖²−2⟨𝑢,𝐽𝑥⟩+‖𝐽𝑥‖²=‖𝑢‖²−2⟨𝑢,𝐽𝑥⟩+‖𝑥‖²=𝜙(𝑢,𝑥),(3.35) that is, 𝑢=𝑥. This implies that 𝑟=𝐽𝑢 and so 𝐽𝑦_𝑛⇀𝐽𝑝. It follows from lim_𝑛→∞‖𝐽𝑦_𝑛‖=‖𝐽𝑢‖, as 𝑛→∞ and the Kadec-Klee property of 𝐸^∗ that 𝐽𝑦_𝑛𝑖→𝐽𝑢 as 𝑛→∞. Note that 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous, it yields that 𝑦_𝑛𝑖⇀𝑢. It follows from lim_𝑛→∞‖𝑢_𝑛‖=‖𝑢‖, as 𝑛→∞ and the Kadec-Klee property of 𝐸 that lim_{𝑛𝑖→∞}𝑦_𝑛𝑖=𝑢.
By similar, we can prove that lim_𝑛→∞𝑦_𝑛=𝑢.(3.36) By (3.20) and (3.30), we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑦_𝑛‖‖=0.(3.37) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim_𝑛→∞‖‖𝐽𝑥_𝑛−𝐽𝑦_𝑛‖‖=0.(3.38) So, from (3.27) and (3.31), by the triangle inequality, we get lim_𝑛→∞‖‖𝐽𝑦_𝑛−𝐽𝑢^𝑖𝑛‖‖=0,for𝑖=1,2,…,𝑁.(3.39) Since 𝐽⁻¹ is uniformly norm-to-norm continuous on bounded sets, we note that lim_𝑛→∞‖‖𝑦_𝑛−𝑢^𝑖𝑛‖‖=0,for𝑖=1,2,…,𝑁.(3.40) Since 𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑦_𝑛=≤‖‖‖‖𝑥𝑥_𝑛𝑛‖‖²−‖‖𝑦_𝑛‖‖²−2⟨𝑝,𝐽𝑥_𝑛−𝐽𝑦_𝑛⟩≤‖‖𝑥_𝑛‖‖²−‖‖𝑦_𝑛‖‖²+2‖𝑝‖‖‖𝐽𝑥_𝑛−𝐽𝑦_𝑛‖‖−𝑦_𝑛‖‖‖‖𝑥_𝑛‖‖+‖‖𝑦_𝑛‖‖+2‖𝑝‖‖‖𝐽𝑥_𝑛−𝐽𝑦_𝑛‖‖.(3.41) From (3.37) and (3.38), we obtain 𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑦_𝑛⟶0,𝑛⟶∞.(3.42) On the other hand, we observe that, for 𝑖=1,2,…,𝑁. 𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢^𝑖𝑛=≤‖‖‖‖𝑥𝑥_𝑛𝑛‖‖²−‖‖𝑢^𝑖𝑛‖‖²−2𝑝,𝐽𝑥_𝑛−𝐽𝑢^𝑖𝑛≤‖‖𝑥_𝑛‖‖²−‖‖𝑢^𝑖𝑛‖‖²+2‖𝑝‖‖‖𝐽𝑥_𝑛−𝐽𝑢^𝑖𝑛‖‖−𝑢^𝑖𝑛‖‖‖‖𝑥_𝑛‖‖+‖‖𝑢^𝑖𝑛‖‖+2‖𝑝‖‖‖𝐽𝑥_𝑛−𝐽𝑢^𝑖𝑛‖‖.(3.43) From (3.22) and (3.27), we have 𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢^𝑖𝑛⟶0,𝑛⟶∞,∀𝑖=1,2,…,𝑁.(3.44) For any 𝑝∈∩^𝑁𝑖=1Ω_𝑖∩(∩^∞𝑖=1F(𝑇_𝑖))∩(∩^∞𝑖=1𝐹(𝑆_𝑖)), it follows from (3.5) that 𝛽_𝑛,0𝛽_𝑛,𝑖𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖≤𝜙𝑝,𝑥_𝑛+𝜉_𝑛−𝜙𝑝,𝑦_𝑛.(3.45)From condition, liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛,𝑖>0, property of 𝑔, (3.7), and (3.42), we have that ‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑖𝑥_𝑛‖‖⟶0,𝑛⟶∞,∀𝑖=1,2,…,𝑁.(3.46) Since 𝑥_𝑛→𝑢 and 𝐽 is uniformly norm-to-norm continuous. It yields 𝐽𝑥_𝑛→𝐽𝑝. Hence from (3.46), we have ‖‖𝑥_𝑛−𝑇^𝑛𝑖𝑥_𝑛‖‖⟶0,𝑛⟶∞,∀𝑖=1,2,…,𝑁.(3.47) Since 𝑥_𝑛→𝑢, this implies that lim_𝑛→∞𝐽𝑇^𝑛𝑖𝑥_𝑛→𝐽𝑢 as 𝑛→∞. Since 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous, it follows that 𝑇^𝑛𝑖𝑥_𝑛⇀𝑢,foreach𝑖≥1.(3.48) On the other hand, for each 𝑖≥1, we have ‖‖𝑇^𝑛𝑖𝑥_𝑛‖‖−‖𝑢‖=≤||‖‖‖‖𝑇𝑇^𝑛𝑖𝑥_𝑛‖‖−‖𝑢‖_𝑛𝑖||𝑥_𝑛−𝑢‖‖⟶0,𝑛⟶∞.(3.49) from this, together with (3.48) and the Kadec-Klee property of 𝐸, we obtain 𝑇^𝑛𝑖𝑥_𝑛⟶𝑢,foreach𝑖≥1.(3.50) On the other hand, by the assumption that 𝑇_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous, we have ‖‖𝑇^𝑖𝑛+1𝑥_𝑛−𝑇^𝑛𝑖𝑥_𝑛‖‖≤+‖‖‖‖𝑇𝑥^𝑖𝑛+1_𝑛+1𝑥_𝑛−𝑇^𝑖𝑛+1𝑥_𝑛+1‖‖+‖‖𝑇^𝑖𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖≤𝐿_𝑖−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑇^𝑛𝑖𝑥_𝑛‖‖+‖‖𝑥_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑇^𝑖𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖−𝑇^𝑛𝑖𝑥_𝑛‖‖.(3.51) By (3.18) and (3.50), we obtain lim_𝑛→∞‖‖𝑇^𝑖𝑛+1𝑥_𝑛−𝑇^𝑛𝑖𝑥_𝑛‖‖=0,∀𝑖≥1,(3.52) and lim_𝑛→∞𝑇^𝑖𝑛+1𝑥_𝑛=𝑢, that is, 𝑇_𝑖𝑇^𝑛𝑥_𝑛→𝑢, forall𝑖≥1. By the closeness of 𝑇_𝑖, we have 𝑇_𝑖𝑢=u, forall𝑖≥1. This implies that 𝑢∈∩^∞𝑖=1𝐹(𝑇_𝑖).
By the similar way, we can prove that for each 𝑖≥1‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑦_𝑛‖‖⟶0,𝑛⟶∞.(3.53) Since 𝑥_𝑛→𝑢 and 𝐽 is uniformly norm-to-norm continuous. it yields 𝐽𝑥_𝑛→𝐽𝑝. Hence from (3.53), we have ‖‖𝑥_𝑛−𝑆^𝑛𝑖𝑦_𝑛‖‖⟶0,𝑛⟶∞.(3.54) Since 𝑥_𝑛→𝑢, this implies that lim_𝑛→∞𝐽𝑆^𝑛𝑖𝑦_𝑛→𝐽𝑢 as 𝑛→∞. Since 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous, it follows that 𝑆^𝑛𝑖𝑦_𝑛⇀𝑢,foreach𝑖≥1.(3.55) On the other hand, for each 𝑖≥1, we have ‖‖𝑆^𝑛𝑖𝑦_𝑛‖‖−‖𝑢‖=≤||‖‖‖‖𝑆𝑆^𝑛𝑖𝑦_𝑛‖‖−‖𝑢‖_𝑛𝑖||𝑦_𝑛−𝑢‖‖⟶0,𝑛⟶∞.(3.56) From this, together with (3.54) and the Kadec-Klee property of 𝐸, we obtain 𝑆^𝑛𝑖𝑦_𝑛⟶𝑢,foreach𝑖≥1.(3.57) On the other hand, by the assumption that 𝑆_𝑖 is uniformly 𝜇_𝑖-Lipschitz continuous, we have ‖‖𝑆^𝑖𝑛+1𝑦_𝑛−𝑆^𝑛𝑖𝑦_𝑛‖‖≤+‖‖‖‖𝑆𝑦^𝑖𝑛+1_𝑛+1𝑦_𝑛−𝑆^𝑖𝑛+1𝑦_𝑛+1‖‖+‖‖𝑆^𝑖𝑛+1𝑦_𝑛+1−𝑦_𝑛+1‖‖≤𝜇_𝑖−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑆^𝑛𝑖𝑦_𝑛‖‖+‖‖𝑦+_𝑛1‖‖𝑦_𝑛+1−𝑦_𝑛‖‖+‖‖𝑆^𝑖𝑛+1𝑦_𝑛+1−𝑦_𝑛+1‖‖−𝑆^𝑛𝑖𝑦_𝑛‖‖.(3.58) By (3.36) and (3.57), we obtain lim_𝑛→∞‖‖𝑆^𝑖𝑛+1𝑦_𝑛−𝑆^𝑛𝑖𝑦_𝑛‖‖=0(3.59) and lim_𝑛→∞𝑆^𝑖𝑛+1𝑦_𝑛=𝑢, that is, 𝑆_𝑖𝑇^𝑛𝑦_𝑛→𝑢. By the closeness of 𝑆_𝑖, we have 𝑆_𝑖𝑢=𝑢, forall𝑖≥1. This implies that 𝑢∈∩^∞𝑖=1𝐹(𝑆_𝑖). Hence 𝑢∈(∩^∞𝑖=1𝐹(𝑇_𝑖))∩(∩^∞𝑖=1𝐹(𝑆_𝑖)).
Next, we prove that 𝑢∈∩^𝑁𝑖=1Ω_𝑖. For any 𝑝∈ℱ, for each 𝑖=1,2,…,𝑁, we have 𝜙𝑢^𝑖𝑛,𝑧_𝑛≤=𝜙𝜙Θ𝑝,^𝑖𝑛𝑧𝑧_𝑛𝑛,𝑧_𝑛=𝜙𝑝,𝑧_𝑛−𝜙𝑝,Θ^𝑖𝑛𝑧_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢^𝑖𝑛+𝜃_𝑛−𝜙𝑝,𝑢^𝑖𝑛→0,as𝑛→∞.(3.60) It then yields that lim_𝑛→∞(‖𝑢^𝑖𝑛‖−‖𝑧_𝑛‖)=0. Since lim_𝑛→∞‖𝑢^𝑖𝑛‖=‖𝑢‖, forall𝑖≥1, we have lim_𝑛→∞‖‖𝑧_𝑛‖‖=‖𝑢‖.(3.61) Hence, lim_𝑛→∞‖‖𝐽𝑧_𝑛‖‖=‖𝐽𝑢‖.(3.62) This together with lim_𝑛→∞‖𝑢^𝑖𝑛‖=‖𝑢‖ show that for each 𝑖=1,2,…,𝑁, lim_𝑛→∞‖‖𝑢^𝑖𝑛−𝑢^𝑛𝑖−1‖‖=lim_𝑛→∞‖‖𝐽𝑢^𝑖𝑛−𝐽𝑢^𝑛𝑖−1‖‖=0,(3.63) where 𝑢^0𝑛=𝑧_𝑛. On the other hand, we have 𝑢^𝑖𝑛=𝐾_{𝑓𝑖,𝑟𝑖}𝑢^𝑛𝑖−1,foreach𝑖=2,3,…,𝑁,(3.64) and 𝑢^𝑖𝑛 is a solution of the following variational equation 𝑓_𝑖𝑢^𝑖𝑛,𝑦+𝐴_𝑖𝑢^𝑖𝑛,𝑦−𝑢^𝑖𝑛+𝜓_𝑖(𝑦)−𝜓_𝑖𝑢^𝑖𝑛+1𝑟_𝑖𝑦−𝑢^𝑖𝑛,𝐽𝑢^𝑖𝑛−𝐽𝑢^𝑛𝑖−1≥0,∀𝑦∈𝐶.(3.65) By condition (A2), we note that 𝐴_𝑖𝑢≥^𝑖𝑛,𝑦−−𝑓_𝑖𝑢^𝑖𝑛+𝜓_𝑖(𝑦)−𝜓_𝑖𝑢^𝑖𝑛+𝑟1_𝑖𝑦−𝑢^𝑖𝑛,𝐽𝑢^𝑖𝑛−𝐽𝑢^𝑛𝑖−1𝑢^𝑖𝑛,𝑦≥𝑓_𝑖𝑦,𝑢^𝑖𝑛,∀𝑦∈𝐶.(3.66) By (A4), (3.63), and 𝑢^𝑖𝑛→𝑢 for each 𝑖=2,3,…,𝑁, we have ⟨𝐴_𝑖𝑢,𝑦−𝑢⟩+𝜓_𝑖(𝑦)−𝜓_𝑖(𝑢)≥𝑓_𝑖(𝑦,𝑢),∀𝑦∈𝐶.(3.67) For 0<𝑡<1 and 𝑦∈𝐶, define 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑢. Noticing that 𝑦,𝑢∈𝐶, we obtain 𝑦_𝑡∈𝐶, which yields that ⟨𝐴_𝑖𝑢,𝑦_𝑡−𝑢⟩+𝜓_𝑖𝑦_𝑡−𝜓_𝑖(𝑢)≥𝑓_𝑖𝑦_𝑡,𝑢.(3.68) In view of the convexity of 𝜙 it yields 𝑡⟨𝐴_𝑖𝑢,𝑦−𝑢⟩+𝑡𝜓_𝑖(𝑦)−𝜓_𝑖(𝑢)≥𝑓_𝑖𝑦_𝑡,𝑢.(3.69) It follows from (A1) and (A4) that 0=≤𝑡𝑓𝑓_𝑖𝑖𝑦_𝑡,𝑦_𝑡≤𝑡𝑓_𝑖𝑦_𝑡,𝑦+(1−𝑡)𝑓_𝑖𝑦_𝑡,𝑢𝑦_𝑡,𝑦+(1−𝑡)𝑡⟨𝐴_𝑖𝑢,𝑦−𝑢⟩+𝜓_𝑖(𝑦)−𝜓_𝑖(𝑢).(3.70) Let 𝑡→0, from (A3), we obtain the following: 𝑓_i(𝑢,𝑦)+⟨𝐴_𝑖𝑢,𝑦−𝑢⟩+𝜓_𝑖(𝑦)−𝜓_𝑖(𝑢)≥0,∀𝑦∈𝐶,𝑖=1,2,…,𝑁.(3.71) This implies that 𝑢 is a solution of the system of generalized mixed equilibrium problem (3.2), that is, 𝑢∈∩^𝑁𝑖=1Ω_𝑖. Hence, 𝑢∈ℱ∶=(∩^𝑁𝑖=1Ω_𝑖)∩(∩^∞𝑖=1𝐹(𝑇_𝑖))∩(∩^∞𝑖=1𝐹(𝑆_𝑖)).
Finally, we show that 𝑥_𝑛→𝑢=Π_𝐹𝑥₀. Indeed from 𝑤∈𝐹⊂𝐶_𝑛 and 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we have the following: 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑤,𝑥₀,∀𝑛≥0.(3.72) This implies that 𝜙𝑢,𝑥₀=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤𝜙𝑤,𝑥₀.(3.73) From the definition of Π_𝐹𝑥₀ and (3.73), we see that 𝑢=𝑤. This completes the proof.

Since every asymptotically relatively nonexpansive mappings is quasi-𝜙-nonexpansive mappings, hence we obtain the following corollary.

Corollary 3.2. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝐴_𝑖∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜓_𝑖∶𝐶→ℝ be a lower semi-continuous and convex function, 𝑓_𝑖 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), 𝐾_{𝑓𝑖,𝑟𝑖} is the mapping defined by (2.10) where 𝑟_𝑖≥𝑟>0, and let {𝑇_𝑖}^∞𝑖=1, {𝑆_𝑖}^∞𝑖=1 be countable families of closed and quasi-𝜙-nonexpansive mapping such that ℱ∶=(∩^𝑁𝑖=1Ω_𝑖)∩(∩^∞𝑖=1𝐹(𝑇_𝑖))∩(∩^∞𝑖=1𝐹(𝑆_𝑖))≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶 and 𝐶₀=𝐶, such that 𝑦_𝑛=𝐽⁻¹𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝑧_𝑛𝛽_𝑛,𝑖𝐽𝑇_𝑖𝑥_𝑛,=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝑢_𝑛(𝑖)𝛼_𝑛,𝑖𝐽𝑆_𝑖𝑦_𝑛,𝐶=𝐾_{𝑓𝑖,𝑟𝑖}𝐾_{𝑓𝑖−1,𝑟𝑖−1}⋯𝐾_𝑓1,𝑟1𝑧_𝑛,𝑖=1,2,…,𝑁,_𝑛+1=𝑧∈𝐶_𝑛∶max𝑥_{𝑖=1,2,…,𝑁}𝜙_𝑛+1𝑧,𝑢^𝑛(𝑖)≤𝜙𝑧,𝑥^𝑛,𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.74) where Π_𝐶 is the generalized projection from 𝐸 onto 𝐶, 𝐽 is the duality mapping on 𝐸. The coefficient sequences {𝛼_𝑛,𝑖} and {𝛽_𝑛,𝑖}⊂[0,1], satisfying: (i)∑^∞𝑖=0𝛼_𝑛,i=1; (ii)∑^∞𝑖=0𝛽_𝑛,𝑖=1; (iii)liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0, forall𝑖≥1; (iv)liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛,𝑖>0, forall𝑖≥1. Ω_𝑖,𝑖=1,2,…,𝑁 is the set of solutions to the following generalized mixed equilibrium problem: 𝑓_𝑖(𝑧,𝑦)+⟨𝐴_𝑖𝑧,𝑦−𝑧⟩+𝜓_𝑖(𝑦)−𝜓_𝑖(𝑧)≥0,∀𝑦∈𝐶,𝑖=1,2,…,𝑁.(3.75) Then the sequence {𝑥_𝑛} converges strongly to Π_ℱ𝑥₀.

If 𝐴_𝑖=𝐴,𝜓_𝑖=𝜓, and 𝑓_𝑖=𝑓 for all 𝑖≥1 in Theorem 3.1, we obtain the following corollary.

Corollary 3.3. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜓∶𝐶→ℝ be a lower semicontinuous and convex function, 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), 𝐾_𝑓,𝑟 be the mapping define by (2.10) where 𝑟>0, and let {𝑇_𝑖}^∞𝑖=1, {𝑆_𝑖}^∞𝑖=1 be countable families of closed and uniformly 𝐿_𝑖, 𝜇_𝑖-Lipschitz continuous, and asymptotically relatively nonexpansive mappings with sequence {𝑘_𝑛},{𝜁_𝑛}⊂[1,∞);  𝑘_𝑛→1,𝜁_𝑛→1 such that ℱ∶=Ω∩(∩^∞𝑖=1𝐹(𝑇_𝑖))∩(∩^∞𝑖=1𝐹(𝑆_𝑖))≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶 and 𝐶₀=𝐶, such that 𝑦_𝑛=𝐽⁻¹𝛽^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝑧_𝑛𝛽_𝑛,𝑖𝐽𝑇^𝑛𝑖𝑥_𝑛,=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝑢_𝑛𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑦_𝑛,𝐶_𝑛+1=𝐾_𝑓,𝑟𝑧_𝑛,=𝑧∈𝐶_𝑛∶max_{𝑖=1,2,…,𝑁}𝜙𝑧,𝑢^𝑛(𝑖)≤𝜙𝑧,𝑥^𝑛+𝜃_𝑛,𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+𝜉_𝑛𝑥_𝑛+1,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.76) where 𝜉_𝑛=sup_𝑝∈ℱ(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛), 𝜃_𝑛=𝛿_𝑛+𝜉_𝑛𝜁_𝑛, and 𝛿_𝑛=sup_𝑝∈ℱ(𝜁_𝑛−1)𝜙(𝑝,𝑥_𝑛). The coefficient sequences {𝛼_𝑛,𝑖} and {𝛽_𝑛,𝑖}⊂[0,1], satisfying: (i)∑^∞𝑖=0𝛼_𝑛,𝑖=1; (ii)∑^∞𝑖=0𝛽_𝑛,𝑖=1; (iii)liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑖>0, forall𝑖≥1; (iv)liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛,𝑖>0, forall𝑖≥1. Then the sequence {𝑥_𝑛} converges strongly to Π_ℱ𝑥₀.

If 𝑖=1 in Theorem 3.1, then we obtain the following corollary.

Corollary 3.4. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜓∶𝐶→ℝ be a lower semicontinuous and convex function, 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), 𝐾_𝑓,𝑟 is the mapping define by (2.10) where 𝑟>0, and let 𝑇,𝑆 are two closed and uniformly 𝐿, 𝜇-Lipschitz continuous and asymptotically relatively nonexpansive mappings with sequence {𝑘_𝑛},{𝜁_𝑛}⊂[1,∞);  𝑘_𝑛→1,𝜁_𝑛→1 such that ℱ∶=Ω∩𝐹(𝑇)∩𝐹(𝑆)≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶 and 𝐶₀=𝐶, we have𝑧𝑦_𝑛𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇^𝑛𝑥_𝑛,=𝐽⁻¹𝛼_𝑛𝑢𝐽_𝑛𝑥_𝑛+1−𝛼_𝑛𝐽𝑆^𝑛𝑦_𝑛,𝐶_𝑛+1=𝐾_𝑓,𝑟𝑧_𝑛,=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢𝑥_𝑛_𝑛+1≤𝜙𝑧,𝑥_𝑛+𝜃_𝑛,𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+𝜉_𝑛,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.77) where 𝜉_𝑛=sup_𝑝∈ℱ(𝑘_𝑛−1)𝜙(𝑝,𝑥_𝑛), 𝜃_𝑛=𝛿_𝑛+𝜉_𝑛𝜁_𝑛, and 𝛿_𝑛=sup_𝑝∈ℱ(𝜁_𝑛−1)𝜙(𝑝,𝑥_𝑛). The coefficient sequences {𝛼_𝑛} and {𝛽_𝑛}⊂[0,1], satisfying (D1)liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0; (D2)liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0. Then the sequence {𝑥_𝑛} converges strongly to Π_ℱ𝑥₀.

Remark 3.5. Theorem 3.1 and Corollary 3.3 improve and extend the corresponding results of Petrot et al. [24], Kumam and Wattanawitoon [25], and Chang et al. [26] in the following senses: (i)for the mappings, we extend the mappings from nonexpansive mappings, hemirelatively nonexpansive mappings to two infinite family of closed asymptotically relatively nonexpansive mappings; (ii)from a solution of the classical equilibrium problem to a system of generalized mixed equilibrium problems and the generalized mixed equilibrium problem with an infinite family of closed relatively nonexpansive mappings.

Remark 3.6. Corollary 3.4 improves and extends the corresponding results of Theorem 3.1 in Kumam and Wattanawitoon [25] and Corollary 3.3 in Saewan et al. [11] in the following senses:(i)the mapping in [11] and [25] (ii)the conditions (D1) and (D2) of the parameters {𝛼_𝑛} and {𝛽_𝑛} are weaker and not complicated than the conditions (C1)–(C3) in [[25], Theorem  3.1] and [[11], Theorem  3.1] which are easy to compute.

Acknowledgments

This first author was supported by The Hands-On Research and Development, Rajamangala University of Technology Lanna (Grant no. UR2L003). Moreover, this work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC Grant no. 55000613).