A New Iterative Method for Finding Common Solutions of a System of Equilibrium Problems, Fixed-Point Problems, and Variational Inequalities

Abstract

We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize some well-known results in the literature.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and induced norm ‖⋅‖. Let 𝐶 be a nonempty, closed, and convex subset of 𝐻. Let {𝐹_𝑘}_𝑘∈Γ be a countable family of bifunctions from 𝐶×𝐶 to ℝ, where ℝ is the set of real numbers. Combettes and Hirstoaga [1] considered the following system of equilibrium problems: Find𝑥∈𝐶suchthat(∀𝑘∈Γ),(∀𝑦∈𝐶),𝐹_𝑘(𝑥,𝑦)≥0.(1.1)

If Γ is a singleton, problem (1.1) becomes the following equilibrium problem: Finding𝑥∈𝐶suchthat𝐹(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.2)

The solutions set of (1.2) is denoted by EP(𝐹). And clearly the solutions set of problem (1.1) can be written as ⋂_𝑘∈ΓEP(𝐹_𝑘).

Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see for instance, [1–4].

Recall that a mapping 𝑆 of a closed and convex subset 𝐶 into itself is nonexpansive if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖∀𝑥,𝑦∈𝐶.(1.3) We denote fixed-points set of 𝑆 by Fix(𝑆). A mapping 𝑓∶𝐶→𝐶 is called contraction if there exists a constant 𝛼∈(0,1) such that ‖𝑓𝑥−𝑓𝑦‖≤𝛼‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.4) A bounded linear operator 𝐵 on 𝐻 is strongly positive, if there is a constant 𝛾>0 such that ⟨𝐵𝑥,𝑥⟩≥𝛾‖𝑥‖² for all 𝑥∈𝐻.

Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element of the solutions set of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. Peng and Yao [2] introduced a new viscosity approximation scheme based on the extragradient method for finding a common element in the solutions set of the problem (1.1), fixed-points set of an infinite family of nonexpansive mappings and the solutions set of the variational inequality for a monotone and Lipschitz continuous mapping in a Hilbert space and obtained a strong convergence theorem. Colao et al. [3] introduced an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed-points of infinite family of nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem. Saeidi [4] introduced some iterative algorithms for finding a common element of the solutions set of a system of equilibrium problems and of fixed-points set of a finite family and a left amenable semigroup of nonexpansive mappings in a Hilbert space and obtained some strong convergence theorems.

Several algorithms for problem (1.2) have been proposed (see [5–20]). S. Takahashi and W. Takahashi [5] introduced and studied the following iterative scheme by the viscosity approximation method for finding a common element of the solutions set of problem (1.2) and fixed-points set of a nonexpansive mapping in a Hilbert space. Let an arbitrary 𝑥₁∈𝐻 define sequences {𝑥_𝑛} and {𝑢_𝑛} by𝐹(𝑢_𝑛1,𝑦)+𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑆𝑢_𝑛,∀𝑛∈𝑁.(1.5) Shang et al. [6] introduced the following iterative scheme by the viscosity approximation method for finding a common element of the solutions set of problem (1.2) and fixed-points set of a nonexpansive mapping in a Hilbert space. Let an arbitrary 𝑥₁∈𝐻, define sequences {𝑥_𝑛} and {𝑢_𝑛} by𝐹(𝑢_𝑛1,𝑦)+𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝛼_𝑛𝐵)𝑆𝑢_𝑛,∀𝑛∈𝑁.(1.6) They proved that under certain appropriate conditions imposed on {𝛼_𝑛} and {𝛽_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} generated by (1.6) converge strongly to the unique solution of the variational inequality ⟨(𝐵−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈Fix(𝑆)∩EP(𝐹),(1.7) which is the optimality condition for the minimization problemmin_{𝑥∈Fix(𝑆)∩EP(𝐹)}12⟨𝐵𝑥,𝑥⟩−ℎ(𝑥),(1.8) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻). If 𝐶=𝐻, the algorithm (1.6) was also studied by Plubtieng and Punpaeng [7].

Let 𝐴∶𝐶→𝐻 be a monotone mapping. The variational inequality problem is to find a point 𝑥∈𝐶 such that ⟨𝐴𝑥,𝑦−𝑥⟩≥0(1.9) for all 𝑦∈𝐶. The solutions set of the variational inequality problem is denoted by VI(𝐶,𝐴). Qin et al. [8] introduced the following general iterative scheme for finding a common element of the solutions set of problem (1.2), the solutions set of a variational inequality and fixed-points set of a nonexpansive mapping in a Hilbert space. Let an arbitrary 𝑥₁∈𝐻, define sequences {𝑥_𝑛} and {𝑢_𝑛} by𝐹(𝑢_𝑛1,𝑦)+𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝛼_𝑛𝐵)𝑆𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛,∀𝑛∈𝑁.(1.10) They proved that under certain appropriate conditions imposed on {𝛼_𝑛}, {𝑠_𝑛} and {𝛽_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} generated by (1.10) converge strongly to the unique solution of the variational inequality ⟨(𝐵−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈Fix(𝑆)∩VI(𝐶,𝐴)∩EP(𝐹).(1.11)

Qin et al. [9] introduced the following general iterative scheme for finding a common element of the solutions set of problem (1.2) and fixed-points set of a finite family of nonexpansive mappings in a Hilbert space. Let an arbitrary 𝑥₁∈𝐻, define sequences {𝑥_𝑛} and {𝑢_𝑛} by𝐹(𝑢_𝑛1,𝑦)+𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝛾𝑓(𝑊_𝑛𝑥_𝑛)+(1−𝛼_𝑛𝐵)𝑊_𝑛𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛,∀𝑛∈𝑁,(1.12) where 𝑊_𝑛 is the 𝑊-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑁 and 𝜆_𝑛1,𝜆_𝑛2,…,𝜆_𝑛𝑁. They proved that under certain appropriate conditions imposed on {𝛼_𝑛}, {𝑠_𝑛} and {𝛽_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} generated by (1.12) converge strongly to the unique solution of the variational inequality ⟨(𝐵−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈^𝑁_𝑖=1Fix(𝑇_𝑖)∩VI(𝐶,𝐴)∩EP(𝐹).(1.13)

A typical problem is to minimize a quadratic function over the fixed-points set of a nonexpansive mapping 𝑆 on a real Hilbert space 𝐻, that is, min_{𝑥∈Fix(𝑆)}12⟨𝐵𝑥,𝑥⟩−⟨𝑥,𝑏⟩,(1.14) where 𝑏 is a given point in 𝐻. In 2003, Xu [21] proved that the sequence {𝑥_𝑛} defined by the iterative method below, with the initial point 𝑥₀∈𝐻, chosen arbitrarily: 𝑥_𝑛+1=(𝐼−𝛼_𝑛𝐵)𝑆𝑥_𝑛+𝛼_𝑛𝑢,𝑛≥0,(1.15) converges strongly to the unique solution of the minimization problem (1.15) provided the sequence {𝛼_𝑛} satisfies certain conditions. Marino and Xu [22] combine the iterative method (1.15) with the viscosity approximation in [23] and consider the following general iterative method: with the initial point 𝑥₀∈𝐻, chosen arbitrarily: 𝑥_𝑛+1=(1−𝛼_𝑛𝐵)𝑆𝑥_𝑛+𝛼_𝑛𝛾𝑓(𝑥_𝑛),𝑛≥0.(1.16)

They proved that if the sequence {𝛼_𝑛} satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.16) converges strongly to the unique solution of the variational inequality ⟨(𝐵−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,𝑥∈Fix(𝑆)(1.17) which is the optimality condition for the minimization problemmin_{𝑥∈Fix(𝑆)}12⟨𝐵𝑥,𝑥⟩−ℎ(𝑥),(1.18) where ℎ is a potential function for 𝛾𝑓.

Recently, Qin et al. [24] introduced the following general iterative process: with the initial point 𝑥₁∈𝐶, chosen arbitrarily:𝑦_𝑛=𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑊_𝑛𝑥_𝑛)+(𝐼−𝛼_𝑛𝐵)𝑊_𝑛𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛,∀𝑛∈𝑁,(1.19) where 𝑊_𝑛 is the 𝑊-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑁 and 𝜆_𝑛1,𝜆_𝑛2,…,𝜆_𝑛𝑁. They proved that if the sequences of parameters {𝛼_𝑛}, {𝑟_𝑛} and {𝑠_𝑛} satisfies appropriate conditions, then the sequence {𝑥_𝑛}, {𝑦_𝑛} generated by (1.19) converge strongly to a point 𝑥^∗ which is the unique solution of the variational inequality ⟨(𝐵−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈^𝑁_𝑖=1𝐹(𝑇_𝑖)∩VI(𝐶,𝐴).(1.20)

Inspired and motivated by above works, we introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed-points set of a family of infinitely nonexpansive mappings and the solutions set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify, generalize and extend some well-known results in [6–9, 21, 22, 24].

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let 𝐶 be a nonempty, closed, and convex subset of 𝐻. Let symbols → and ⇀ denote strong and weak convergence, respectively. It is well known that‖𝜆𝑥+(1−𝜆)𝑦‖²=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖²(2.1) for all 𝑥,𝑦∈𝐻 and 𝜆∈[0,1].

For any 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶(𝑥), such that ‖𝑥−𝑃_𝐶(𝑥)‖≤‖𝑥−𝑦‖ for all 𝑦∈𝐶. The mapping 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. We know that 𝑃_𝐶 is a nonexpansive mapping from 𝐻 onto 𝐶, 𝑃_𝐶(𝑥)∈𝐶 and ⟨𝑥−𝑃_𝐶(𝑥),𝑃_𝐶(𝑥)−𝑦⟩≥0(2.2) for all 𝑥,𝑦∈𝐻.

It is easy to see that (2.2) is equivalent to ‖𝑥−𝑦‖²≥‖𝑥−𝑃_𝐶(𝑥)‖²+‖𝑦−𝑃_𝐶(𝑥)‖²(2.3) for all 𝑥,𝑦∈𝐻. It is also known that 𝑃_𝐶 has the following firmly nonexpansive property: ⟨𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦⟩≥‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖²(2.4) for all 𝑥,𝑦∈𝐻.

Recall also that a mapping 𝐴 of 𝐶 into 𝐻 is called monotone if ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥0,(2.5) for all 𝑥,𝑦∈𝐶. 𝐴 is said to be 𝜇-cocoercive, if for each 𝑥,𝑦∈𝐶, we have ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝜇‖𝐴𝑥−𝐴𝑦‖²,(2.6) for a constant 𝜇>0. 𝐴 is said to be relaxed (𝑢,𝑣)-cocoercive, if there exist two constants 𝑢,𝑣>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥(−𝑢)‖𝐴𝑥−𝐴𝑦‖²+𝑣‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶.(2.7) Let 𝐴 be a monotone mapping of 𝐶 into 𝐻. In the context of the variational inequality problem the characterization of projection (2.2) implies the following:𝑢∈VI(𝐶,𝐴)⇒𝑢=𝑃_𝐶(𝑢−𝜆𝐴𝑢),𝜆>0,𝑢=𝑃_𝐶(𝑢−𝜆𝐴𝑢)forsome𝜆>0⇒𝑢∈VI(𝐶,𝐴).(2.8)

It is also known that 𝐻 satisfies the Opial's condition [25], that is, for any sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥, the inequalityliminf_𝑛→∞‖𝑥_𝑛−𝑥‖<liminf_𝑛→∞‖𝑥_𝑛−𝑦‖(2.9) holds for every 𝑦∈𝐻 with 𝑥≠𝑦.

A set-valued mapping 𝑇∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻, 𝑓∈𝑇𝑥 and 𝑔∈𝑇𝑦 imply ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone mapping 𝑇∶𝐻→2^𝐻 is maximal if its graph 𝐺(𝑇) of 𝑇 is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑇 is maximal if and only if for (𝑥,𝑓)∈𝐻×𝐻,⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for every (𝑦,𝑔)∈𝐺(𝑇) implies 𝑓∈𝑇𝑥. Let 𝐴 be a monotone and 𝑘-Lipschitz-continuous mapping of 𝐶 into 𝐻 and let 𝑁_𝐶𝑣 be normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,forall𝑢∈𝐶}. Define𝑇𝑣=𝐴𝑣+𝑁_𝐶𝑣if𝑣∈𝐶,∅if𝑣∉𝐶.(2.10) Then 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,𝐴) (see [26]).

For solving the problem (1.1), let us assume that the bifunction 𝐹 satisfies the following condition:(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶;(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for any 𝑥,𝑦∈𝐶;(A3)for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦);(2.11)(A4)for each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex;(A5)for each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is lower semicontinuous.

We recall some lemmas needed later.

Lemma 2.1 (see [1, 10]). Let 𝐶 be a nonempty, closed, and convex subset of 𝐻, and let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A5). For 𝛽>0 and 𝑥∈𝐻, define the mapping 𝑇^𝐹_𝛽∶𝐻→𝐶 as follows: 𝑇^𝐹_𝛽1(𝑥)={𝑧∈𝐶∶𝐹(𝑧,𝑦)+𝛽⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶}(2.12) for all 𝑥∈𝐻. Then, the following statements hold: (1)𝑇^𝐹_𝛽(𝑥)≠∅;(2)𝑇^𝐹_𝛽 is single-valued;(3)𝑇^𝐹_𝛽 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻,‖𝑇^𝐹_𝛽(𝑥)−𝑇^𝐹_𝛽(𝑦)‖²≤⟨𝑇^𝐹_𝛽(𝑥)−𝑇^𝐹_𝛽(𝑦),𝑥−𝑦⟩;(2.13)(4)Fix(𝑇^𝐹_𝛽)=EP(𝐹);(5)EP(𝐹) is closed and convex.

Lemma 2.2 (see [27]). Assume that {𝑠_𝑛} is a sequence of nonnegative real numbers such that 𝑠_𝑛+1≤(1−𝛼_𝑛)𝑠_𝑛+𝛼_𝑛𝛽_𝑛+𝛿_𝑛,𝑛≥1,(2.14) where {𝛼_𝑛}, {𝛽_𝑛} and {𝛿_𝑛} are sequences of numbers which satisfy the conditions:(i){𝛼_𝑛}⊂[0,1],∑^∞_𝑛=1𝛼_𝑛=∞, or equivalently, ∏^∞_𝑖=1(1−𝛼_𝑛)=0;(ii)limsup_𝑛→∞𝛽_𝑛≤0;(iii)𝛿_𝑛≥0(𝑛≥1), ∑^∞_𝑛=1𝛿_𝑛<∞;
Then, lim_𝑛→∞𝑠_𝑛=0.

Lemma 2.3. In a real Hilbert space 𝐻, the following inequality holds: ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩(2.15) for all 𝑥,𝑦∈𝐻.

Lemma 2.4 (see [22]). Assume that 𝐴 is a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Let 𝑆₁,𝑆₂,… be a family of infinitely nonexpansive mappings of 𝐶 into itself and let 𝜉₁,𝜉₂,… be real numbers such that 0≤𝜉_𝑖≤1 for every 𝑖∈𝑁. For any 𝑛∈𝑁, define a mapping 𝑊_𝑛 of 𝐶 into 𝐶 as follows:𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝜉_𝑛𝑆_𝑛𝑈_𝑛,𝑛+1+(1−𝜉_𝑛𝑈)𝐼,_{𝑛,𝑛−1}=𝜉_𝑛−1𝑆_𝑛−1𝑈_𝑛,𝑛+(1−𝜉_𝑛−1⋮𝑈)𝐼,_𝑛,𝑘=𝜉_𝑘𝑆_𝑘𝑈_𝑛,𝑘+1+(1−𝜉_𝑘𝑈)𝐼,_{𝑛,𝑘−1}=𝜉_𝑘−1𝑆_𝑘−1𝑈_𝑛,𝑘+(1−𝜉_𝑘−1⋮𝑈)𝐼,_𝑛,2=𝜉₂𝑆₂𝑈_𝑛,3+(1−𝜉₂𝑊)𝐼,_𝑛=𝑈_𝑛,1=𝜉₁𝑆₁𝑈_𝑛,2+(1−𝜉₁)𝐼.(2.16)

Such a mapping 𝑊_𝑛 is called the 𝑊-mapping generated by 𝑆_𝑛,𝑆_𝑛−1,…,𝑆₁ and 𝜉_𝑛,𝜉_𝑛−1,…,𝜉₁; see [28, 29].

Lemma 2.5 (see [28]). Let 𝐶 be a nonempty, closed, and convex subset of a Banach space 𝐸. Let 𝑆₁,𝑆₂,... be a family of infinitely nonexpansive mappings of 𝐶 into itself such that ⋂^∞_𝑖=1Fix(𝑆_𝑖) is nonempty, and let 𝜉₁,𝜉₂,… be real numbers such that 0<𝜉_𝑖≤𝑑<1 for every 𝑖∈𝑁. For any 𝑛∈𝑁, let 𝑊_𝑛 be the 𝑊-mapping of 𝐶 into itself generated by 𝑆_𝑛,𝑆_𝑛−1,…,𝑆₁ and 𝜉_𝑛,𝜉_𝑛−1,…,𝜉₁. Then 𝑊_𝑛 is asymptotically regular and nonexpansive. Further, if 𝐸 is strict convex, then 𝐹(𝑊_𝑛)=⋂^𝑛_𝑖=1Fix(𝑆_𝑖).

Lemma 2.6 (see [29]). Let 𝐶 be a nonempty, closed, and convex subset of a strictly convex Banach space 𝐸. Let 𝑆₁,𝑆₂,… be a family of infinitely nonexpansive mappings of 𝐶 into itself such that ⋂^∞_𝑖=1Fix(𝑆_𝑖) is nonempty, and let 𝜉₁,𝜉₂,… be real numbers such that 0<𝜉_𝑖≤𝑑<1 for every 𝑖∈𝑁. Then for every 𝑥∈𝐶 and 𝑘∈𝑁, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists.

Remark 2.7. Using Lemma 2.6, one can define mappings 𝑈_∞,𝑘 and 𝑊 of 𝐶 into itself as follows: 𝑈_∞,𝑘𝑥=lim_𝑛→∞𝑈_𝑛,𝑘𝑥,(2.17) and 𝑊𝑥=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥 for every 𝑥∈𝐶. Such a mapping 𝑊 is called the 𝑊-mapping generated by 𝑆₁,𝑆₂,… and 𝜉₁,𝜉₂,…. Since 𝑊_𝑛 is nonexpansive, 𝑊∶𝐶→𝐶 is also nonexpansive. Indeed, observe that for each 𝑥,𝑦∈𝐶‖𝑊𝑥−𝑊𝑦‖=lim_𝑛→∞‖𝑊_𝑛𝑥−𝑊_𝑛𝑦‖≤‖𝑥−𝑦‖.(2.18)

If {𝑥_𝑛} is a bounded sequence in 𝐶, then we have lim_𝑛→∞‖𝑊𝑥−𝑊_𝑛𝑥‖=0.(2.19)

Lemma 2.8 (see [29]). Let 𝐶 be a nonempty, closed and convex subset of a strictly convex Banach space 𝐸. Let 𝑆₁,𝑆₂,… be an infinite family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞_𝑖=1Fix(𝑆_𝑖) is nonempty, and let 𝜉₁,𝜉₂,… be real numbers such that 0<𝜉_𝑖≤𝑑<1 for every 𝑖∈𝑁. Then Fix(𝑊)=⋂^∞_𝑛=1Fix(𝑆_𝑛).

3. Strong Convergence Theorem

In this section, we prove strong convergence theorem which solve the problem of finding a common element of the solutions set of a system of equilibrium problems, fixed-points set of a family of infinitely nonexpansive mappings, and the solutions set of a variational inequality for a relaxed cocoercive mapping in Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty, closed, and convex subset of 𝐻. Let 𝐹₁,𝐹₂,…,𝐹_𝑚 be bifunctions from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A5). Let 𝐴∶𝐶→𝐻 be relaxed (𝑢,𝑣)-cocoercive and 𝜇-Lipschitz continuous and 𝐵 a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let 𝑆₁,𝑆₂,… be a family of infinitely nonexpansive mappings of 𝐶 into itself such that Ω=⋂^∞_𝑖=1Fix(𝑆_𝑖)∩VI(𝐶,𝐴)∩⋂^𝑚_𝑘=1EP(𝐹_𝑘)≠∅ and let 𝜉₁,𝜉₂,… be real numbers such that 0<𝜉_𝑖≤𝛿<1 for every 𝑖∈𝑁, and let 𝑊_𝑛 be the 𝑊-mapping of 𝐶 into itself generated by 𝑆_𝑛,𝑆_𝑛−1,…,𝑆₁ and 𝜉_𝑛,𝜉_𝑛−1,…,𝜉₁. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1) and {𝑥_𝑛}, {𝑢_𝑛}, and {𝑦_𝑛} be sequences generated by 𝑥₁𝑢=𝑥∈𝐻,_𝑛=𝑇^𝐹𝑚_𝛽𝑛𝑇^𝐹𝑚−1_𝛽𝑛⋯𝑇^𝐹2_𝛽𝑛𝑇^𝐹1_𝛽𝑛𝑥_𝑛,𝑦_𝑛=𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑊_𝑛𝑥_𝑛)+(𝐼−𝛼_𝑛𝐵)𝑊_𝑛𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛(3.1) for every 𝑛=1,2,…, where {𝛼_𝑛}, {𝛽_𝑛}, {𝑟_𝑛}, and {𝑠_𝑛} are sequences of numbers which satisfy the conditions:(C1){𝛼_𝑛}⊂[0,1] with lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(C2){𝑟_𝑛}⊂[𝑎,𝑏] and {𝑠_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0≤𝑎≤𝑏≤2(𝜐−𝑢𝜇²)/𝜇², ∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞, and ∑^∞_𝑛=1|𝑠_𝑛+1−𝑠_𝑛|<∞;(C3)liminf_𝑛→∞𝛽_𝑛>0 and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞.
Then, {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} converge strongly to a point 𝑞∈Ω which solves the following variational inequality:⟨𝛾𝑓𝑞−𝐵𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ω.(3.2) Equivalently, one has 𝑞=𝑃_Ω(𝛾𝑓+(𝐼−𝐵))(𝑞).

Proof. Since 𝛼_𝑛→0 from condition (C1), we may assume, with no loss of generality, that 𝛼_𝑛≤‖𝐵‖⁻¹ for all 𝑛. Lemma 2.4 implies ‖𝐼−𝛼_𝑛𝐵‖≤1−𝛼_𝑛𝛾. Next, we will assume that ‖𝐼−𝐵‖≤1−𝛾. Now, we show that the mappings 𝐼−𝑠_𝑛𝐴 and 𝐼−𝑟_𝑛𝐴 are nonexpansive. Indeed, from the relaxed (𝑢,𝑣)-cocoercivity and 𝜇-Lipschitz continuity of 𝐴 and condition (C2), we have ‖(𝐼−𝑠_𝑛𝐴)𝑥−(𝐼−𝑠_𝑛𝐴)𝑦‖²=‖(𝑥−𝑦)−𝑠_𝑛(𝐴𝑥−𝐴𝑦)‖²=‖𝑥−𝑦‖²−2𝑠_𝑛⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩+𝑠²_𝑛‖𝐴𝑥−𝐴𝑦‖²≤‖𝑥−𝑦‖²−2𝑠_𝑛[−𝑢‖𝐴𝑥−𝐴𝑦‖²+𝜐‖𝑥−𝑦‖²]+𝑠²_𝑛‖𝐴𝑥−𝐴𝑦‖²≤‖𝑥−𝑦‖²+2𝑠_𝑛𝜇²𝑢‖𝑥−𝑦‖²−2𝑠_𝑛𝜐‖𝑥−𝑦‖²+𝜇²𝑠²_𝑛‖𝑥−𝑦‖²=(1+2𝑠_𝑛𝜇²𝑢−2𝑠_𝑛𝜐+𝜇²𝑠²_𝑛)‖𝑥−𝑦‖²≤‖𝑥−𝑦‖²,(3.3) which implies the mapping 𝐼−𝑠_𝑛𝐴 is nonexpansive, so is 𝐼−𝑟_𝑛𝐴.
For 𝑘∈{0,1,2,…,𝑚}, and for any positive integer number 𝑛, we define the operator Θ^𝑘_𝛽𝑛∶𝐻→𝐶 as follows:Θ⁰_𝛽𝑛Θ𝑥=𝑥,_𝑘𝛽𝑛𝑥=𝑇^𝐹𝑘_𝛽𝑛𝑇^𝐹𝑘−1_𝛽𝑛⋯𝑇^𝐹2_𝛽𝑛𝑇^𝐹1_𝛽𝑛𝑥,𝑘=1,2,…,𝑚.(3.4)
Next, we show that the sequence {𝑥_𝑛} is bounded. Let 𝑝∈Ω. Then from Lemma 2.1(3), we know that for 𝑘∈{1,2,…,𝑚}, 𝑇^𝐹𝑘_𝛽𝑛 is nonexpansive and 𝑝=𝑇^𝐹𝑘_𝛽𝑛𝑝, and‖𝑢_𝑛−𝑝‖=‖Θ^𝑚_𝛽𝑛𝑥_𝑛−𝑝‖=‖Θ^𝑚_𝛽𝑛𝑥_𝑛−Θ^𝑚_𝛽𝑛𝑝‖≤‖𝑥_𝑛−𝑝‖(3.5) for all 𝑛=1,2,…. By 𝑝=𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑝 and (3.5), we have ‖𝑦_𝑛−𝑝‖=‖𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑝‖≤‖(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−(𝐼−𝑠_𝑛𝐴)𝑝‖≤‖𝑢_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖.(3.6)
Since 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑊_𝑛𝑥_𝑛)+(𝐼−𝛼_𝑛𝐵)𝑊_𝑛𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛 and 𝑝=𝑊_𝑛𝑝, we have‖𝑥_𝑛+1−𝑝‖=‖𝛼_𝑛(𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝)+(𝐼−𝛼_𝑛𝐵)(𝑊_𝑛𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛−𝑝)‖≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖+(1−𝛼_𝑛𝛾)‖𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛−𝑝‖≤𝛼_𝑛𝛾‖𝑓(𝑊_𝑛𝑥_𝑛)−𝑓(𝑝)‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝐵𝑝‖+(1−𝛼_𝑛𝛾)‖𝑦_𝑛−𝑝‖≤[1−𝛼_𝑛(𝛾−𝛼𝛾)]‖𝑥_𝑛−𝑝‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝐵𝑝‖.(3.7) By inductions, we have ‖𝑥_𝑛−𝑝‖≤max{‖𝑥₀−𝑝‖,‖𝛾𝑓(𝑝)−𝐵𝑝‖𝛾−𝛼𝛾},(3.8) which proves that the sequence {𝑥_𝑛} is bounded. It follows from (3.5) and (3.6) that {𝑦_𝑛} and {𝑢_𝑛} are also bounded.
Since Θ^𝑘_𝛽𝑛𝑥_𝑛=𝑇^𝐹𝑘_𝛽𝑛Θ_𝛽^𝑘−1_𝑛𝑥_𝑛 and Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1=𝑇^𝐹𝑘_𝛽𝑛+1Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1 for each 𝑘=1,2,…,𝑚, by Lemma 2.1, we have𝐹_𝑘(Θ^𝑘_𝛽𝑛𝑥_𝑛1,𝑦)+𝛽_𝑛⟨𝑦−Θ^𝑘_𝛽𝑛𝑥_𝑛,Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛𝐹⟩≥0∀𝑦∈𝐶,(3.9)_𝑘(Θ^𝑘_𝛽𝑛+1𝑥_𝑛+11,𝑦)+𝛽_𝑛+1⟨𝑦−Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1,Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1⟩≥0∀𝑦∈𝐶,(3.10)
Setting 𝑦=Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1 in (3.9) and 𝑦=Θ^𝑘_𝛽𝑛𝑥_𝑛 in (3.10), we have𝐹_𝑘(Θ^𝑘_𝛽𝑛𝑥_𝑛,Θ^𝑘_𝛽𝑛+1𝑥_𝑛+11)+𝛽_𝑛⟨Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ^𝑘_𝛽𝑛𝑥_𝑛,Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛𝐹⟩≥0,_𝑘(Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1,Θ^𝑘_𝛽𝑛𝑥_𝑛1)+𝛽_𝑛+1⟨Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1,Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1⟩≥0.(3.11)
Adding the two inequalities and from the monotonicity of 𝐹, we get⟨Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ^𝑘_𝛽𝑛𝑥_𝑛,Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛𝛽_𝑛−Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1𝛽_𝑛+1⟩≥0(3.12)
and hence‖Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ^𝑘_𝛽𝑛𝑥_𝑛‖²≤⟨Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ^𝑘_𝛽𝑛𝑥_𝑛,(Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛𝛽)+(1−^𝑛𝛽_𝑛+1)(Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1)⟩.(3.13)
Without loss of generality, let us assume that there exists a real number 𝑑 such that 𝛽_𝑛>𝑑>0 for all 𝑛=1,2,…. Hence, for each 𝑘=1,2,…,𝑚 we have‖Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ^𝑘_𝛽𝑛𝑥_𝑛‖≤‖Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛1‖+𝛽_𝑛+1|𝛽_𝑛+1−𝛽_𝑛|‖Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1‖≤‖Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛1‖+𝑑|𝛽_𝑛+1−𝛽_𝑛|𝑀₀,(3.14) where 𝑀₀ is an approximate constant such that 𝑀₀≥max{sup_𝑛≥1{‖Θ^𝑘_𝛽𝑛+1𝑥_𝑛+1−Θ_𝛽^𝑘−1_𝑛+1𝑥_𝑛+1‖},𝑘=1,2,…,𝑚}.(3.15)
It follows from (3.14) that‖𝑢_𝑛+1−𝑢_𝑛‖=‖Θ^𝑚_𝛽𝑛+1𝑥_𝑛+1−Θ^𝑚_𝛽𝑛𝑥_𝑛‖≤‖𝑥_𝑛+1−𝑥_𝑛𝑚‖+𝑑|𝛽_𝑛+1−𝛽_𝑛|𝑀₀.(3.16)
Put 𝜌_𝑛=𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛. We have‖𝑦_𝑛−𝑦_𝑛+1‖=‖𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−𝑃_𝐶(𝐼−𝑠_𝑛+1𝐴)𝑢_𝑛+1‖≤‖(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−(𝐼−𝑠_𝑛+1𝐴)𝑢_𝑛+1‖=‖(𝑢_𝑛−𝑠_𝑛𝐴𝑢_𝑛)−(𝑢_𝑛+1−𝑠_𝑛𝐴𝑢_𝑛+1)+(𝑠_𝑛+1−𝑠_𝑛)𝐴𝑢_𝑛+1‖≤‖𝑢_𝑛−𝑢_𝑛+1‖+|𝑠_𝑛+1−𝑠_𝑛|𝑀₁,(3.17) where 𝑀₁ is an approximate constant such that 𝑀₁≥max{sup_𝑛≥1{‖𝐴𝑢_𝑛‖},𝑀₀}.
Substituting (3.16) into (3.17), we have‖𝑦_𝑛−𝑦_𝑛+1‖≤‖𝑥_𝑛+1−𝑥_𝑛𝑚‖+[𝑑|𝛽_𝑛+1−𝛽_𝑛|+|𝑠_𝑛+1−𝑠_𝑛|]𝑀₁.(3.18) It follows from (3.18) that ‖𝜌_𝑛−𝜌_𝑛+1‖=‖𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛−𝑃_𝐶(𝐼−𝑟_𝑛+1𝐴)𝑦_𝑛+1‖≤‖(𝐼−𝑟_𝑛𝐴)𝑦_𝑛−(𝐼−𝑟_𝑛+1𝐴)𝑦_𝑛+1‖=‖(𝑦_𝑛−𝑟_𝑛𝐴𝑦_𝑛)−(𝑦_𝑛+1−𝑟_𝑛𝐴𝑦_𝑛+1)+(𝑟_𝑛+1−𝑟_𝑛)𝐴𝑦_𝑛+1‖≤‖𝑦_𝑛−𝑦_𝑛+1‖+|𝑟_𝑛+1−𝑟_𝑛|𝑀₂≤‖𝑥_𝑛−𝑥_𝑛+1𝑚‖+[𝑑|𝛽_𝑛+1−𝛽_𝑛|+|𝑠_𝑛+1−𝑠_𝑛|+|𝑟_𝑛+1−𝑟_𝑛|]𝑀₂,(3.19) where 𝑀₂ is an approximate constant such that 𝑀₂≥max{𝑀₁,sup_𝑛≥1{‖𝐴𝑦_𝑛+1‖}}.
Observe that𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑊_𝑛𝑥_𝑛)+(𝐼−𝛼_𝑛𝐵)𝑊_𝑛𝜌_𝑛,𝑥_𝑛+2=𝛼_𝑛+1𝛾𝑓(𝑊_𝑛+1𝑥_𝑛+1)+(𝐼−𝛼_𝑛+1𝐵)𝑊_𝑛+1𝜌_𝑛+1,(3.20) we have 𝑥_𝑛+2−𝑥_𝑛+1=𝛼_𝑛+1𝛾[𝑓(𝑊_𝑛+1𝑥_𝑛+1)−𝑓(𝑊_𝑛𝑥_𝑛)]+(𝐼−𝛼_𝑛+1𝐵)(𝑊_𝑛+1𝜌_𝑛+1−𝑊_𝑛𝜌_𝑛)+(𝛼_𝑛+1−𝛼_𝑛)[𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑊_𝑛𝜌_𝑛].(3.21) It follows that ‖𝑥_𝑛+2−𝑥_𝑛+1‖≤𝛼_𝑛+1𝛾𝛼‖𝑊_𝑛+1𝑥_𝑛+1−𝑊_𝑛𝑥_𝑛‖+(1−𝛼_𝑛+1𝛾)‖𝑊_𝑛+1𝜌_𝑛+1−𝑊_𝑛𝜌_𝑛‖+|𝛼_𝑛+1−𝛼_𝑛|‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑊_𝑛𝜌_𝑛‖≤𝛼_𝑛+1𝛾𝛼(‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖)+(1−𝛼_𝑛+1𝛾)(‖𝜌_𝑛+1−𝜌_𝑛‖+‖𝑊_𝑛+1𝜌_𝑛−𝑊_𝑛𝜌_𝑛‖)+|𝛼_𝑛+1−𝛼_𝑛|‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑊_𝑛𝜌_𝑛‖.(3.22)
Next we estimate ‖𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖ and ‖𝑊_𝑛+1𝜌_𝑛−𝑊_𝑛𝜌_𝑛‖. It follows from the definition of 𝑊_𝑛 and nonexpansiveness of 𝑆_𝑖 that‖𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖=‖𝑈_𝑛+1,1𝑥_𝑛−𝑈_𝑛,1𝑥_𝑛‖=‖𝜉₁𝑆₁𝑈_𝑛+1,2𝑥_𝑛+(1−𝜉₁)𝑥_𝑛−{𝜉₁𝑆₁𝑈_𝑛,2𝑥_𝑛+(1−𝜉₁)𝑥_𝑛}‖=𝜉₁‖𝑆₁𝑈_𝑛+1,2𝑥_𝑛−𝑆₁𝑈_𝑛,2𝑥_𝑛‖≤𝜉₁‖𝑈_𝑛+1,2𝑥_𝑛−𝑈_𝑛,2𝑥_𝑛‖=𝜉₁‖𝜉₂𝑆₂𝑈_𝑛+1,3𝑥_𝑛+(1−𝜉₂)𝑥_𝑛−{𝜉₂𝑆₂𝑈_𝑛,3𝑥_𝑛+(1−𝜉₂)𝑥_𝑛}‖=𝜉₁𝜉₂‖𝑆₂𝑈_𝑛+1,3𝑥_𝑛−𝑆₂𝑈_𝑛,3𝑥_𝑛‖≤𝜉₁𝜉₂‖𝑈_𝑛+1,3𝑥_𝑛−𝑈_𝑛,3𝑥_𝑛‖⋮≤Π^𝑛_𝑖=1𝜉_𝑖‖𝑈_{𝑛+1,𝑛+1}𝑥_𝑛−𝑈_𝑛,𝑛+1𝑥_𝑛‖=Π^𝑛_𝑖=1𝜉_𝑖‖𝜉_𝑛+1𝑆_𝑛+1𝑥_𝑛+(1−𝜉_𝑛+1)𝑥_𝑛−𝑥_𝑛‖=Π^𝑛+1_𝑖=1𝜉_𝑖‖𝑆_𝑛+1𝑥_𝑛−𝑥_𝑛‖≤Π^𝑛+1_𝑖=1𝜉_𝑖𝑀₃,(3.23) where 𝑀₃ is an approximate constant such that 𝑀₃≥max{𝑀₂,sup_𝑛≥1{‖𝑆_𝑛+1𝑥_𝑛−𝑥_𝑛‖},sup_𝑛≥1{‖𝑆_𝑛+1𝜌_𝑛−𝜌_𝑛‖}}.(3.24)
Similarly, we have‖𝑊_𝑛+1𝜌_𝑛−𝑊_𝑛𝜌_𝑛‖≤Π^𝑛+1_𝑖=1𝜉_𝑖𝑀₃.(3.25)
Substituting (3.19), (3.23), and (3.25) into (3.22) yields that‖𝑥_𝑛+2−𝑥_𝑛+1‖≤𝛼_𝑛+1𝛾𝛼(‖𝑥_𝑛+1−𝑥_𝑛‖+Π^𝑛+1_𝑖=1𝜉_𝑖𝑀₃)+(1−𝛼_𝑛+1𝛾)(‖𝑥_𝑛+1−𝑥_𝑛𝑚‖+[𝑑|𝛽_𝑛+1−𝛽_𝑛|+|𝑠_𝑛+1−𝑠_𝑛|+|𝑟_𝑛+1−𝑟_𝑛|]𝑀₂+Π^𝑛+1_𝑖=1𝜉_𝑖𝑀₃)+|𝛼_𝑛+1−𝛼_𝑛|‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑊_𝑛𝜌_𝑛‖≤[1−𝛼_𝑛+1(𝛾−𝛾𝛼)]‖𝑥_𝑛+1−𝑥_𝑛‖+𝑀₄(𝑚𝑑|𝛽_𝑛+1−𝛽_𝑛|+|𝑠_𝑛+1−𝑠_𝑛|+|𝑟_𝑛+1−𝑟_𝑛|+|𝛼_𝑛+1−𝛼_𝑛|),+Π^𝑛+1_𝑖=1𝜉_𝑖(3.26) where 𝑀₄ is an approximate constant such that 𝑀₄≥max{𝑀₃,sup_𝑛≥1{‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑊_𝑛𝜌_𝑛‖}}.(3.27) It follows from conditions (C1)–(C3) and Π^𝑛+1_𝑖=1𝜉_𝑖≤𝛿^𝑛+1 and Lemma 2.2 that ‖𝑥_𝑛+1−𝑥_𝑛‖→0.(3.28)
Observe that𝑥_𝑛+1−𝑊_𝑛𝜌_𝑛=𝛼_𝑛(𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑊_𝑛𝜌_𝑛),(3.29) it follows from (C1) that lim_𝑛→∞‖𝑊_𝑛𝜌_𝑛−𝑥_𝑛+1‖=0.(3.30) For 𝑝∈Ω, we have ‖𝑦_𝑛−𝑝‖²=‖𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑝‖²≤‖(𝑢_𝑛−𝑝)−𝑠_𝑛(𝐴𝑢_𝑛−𝐴𝑝)‖²=‖𝑢_𝑛−𝑝‖²−2𝑠_𝑛⟨𝑢_𝑛−𝑝,𝐴𝑢_𝑛−𝐴𝑝⟩+𝑠²_𝑛‖𝐴𝑢_𝑛−𝐴𝑝‖²≤‖𝑥_𝑛−𝑝‖²−2𝑠_𝑛[−𝑢‖𝐴𝑢_𝑛−𝐴𝑝‖²+𝑣‖𝑢_𝑛−𝑝‖²]+𝑠²_𝑛‖𝐴𝑢_𝑛−𝐴𝑝‖²≤‖𝑥_𝑛−𝑝‖²+(2𝑠_𝑛𝑢+𝑠²_𝑛−2𝑠_𝑛𝑣𝜇²)‖𝐴𝑢_𝑛−𝐴𝑝‖².(3.31) Similarly, we have ‖𝜌_𝑛−𝑝‖²≤‖𝑥_𝑛−𝑝‖²+(2𝑟_𝑛𝑢+𝑟²_𝑛−2𝑟_𝑛𝑣𝜇²)‖𝐴𝑦_𝑛−𝐴𝑝‖².(3.32) On the other hand, we have ‖𝑥_𝑛+1−𝑝‖²=‖𝛼_𝑛(𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝)+(𝐼−𝛼_𝑛𝐵)(𝑊_𝑛𝜌_𝑛−𝑝)‖²≤(𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖+(1−𝛼_𝑛𝛾)‖𝜌_𝑛−𝑝‖)²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝜌_𝑛−𝑝‖²+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.33) Substituting (3.32) into (3.33), we have ‖𝑥_𝑛+1−𝑝‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²+(2𝑟_𝑛𝑢+𝑟²_𝑛−2𝑟_𝑛𝑣𝜇²)‖𝐴𝑦_𝑛−𝐴𝑝‖²+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.34) It follows from condition (C2) that (2𝑎𝑣𝜇²−2𝑏𝑢−𝑏²)‖𝐴𝑦_𝑛−𝐴𝑝‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²−‖𝑥_𝑛+1−𝑝‖²+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+(‖𝑥_𝑛−𝑝‖+‖𝑥_𝑛+1−𝑝‖)‖𝑥_𝑛+1−𝑥_𝑛‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.35) As ‖𝑥_𝑛+1−𝑥_𝑛‖→0 and lim_𝑛→∞𝛼_𝑛=0, we have lim_𝑛→∞‖𝐴𝑦_𝑛−𝐴𝑝‖=0(3.36)
It is easy to see that ‖𝜌_𝑛−𝑝‖≤‖𝑦_𝑛−𝑝‖. Using (3.33) again, we have‖𝑥_𝑛+1−𝑝‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑦_𝑛−𝑝‖²+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.37)
Substituting (3.31) into (3.37), we can obtain‖𝑥_𝑛+1−𝑝‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²+(2𝑠_𝑛𝑢+𝑠²_𝑛−2𝑠_𝑛𝑣𝜇²)‖𝐴𝑢_𝑛−𝐴𝑝‖²+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.38) It follows from (C2) that (2𝑎𝑣𝜇²−2𝑏𝑣−𝑏²)‖𝐴𝑢_𝑛−𝐴𝑝‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²−‖𝑥_𝑛+1−𝑝‖²+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+(‖𝑥_𝑛−𝑝‖−‖𝑥_𝑛+1−𝑝‖)‖𝑥_𝑛+1−𝑥_𝑛‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.39) As ‖𝑥_𝑛+1−𝑥_𝑛‖→0 and lim_𝑛→∞𝛼_𝑛=0, we have lim_𝑛→∞‖𝐴𝑢_𝑛−𝐴𝑝‖=0.(3.40) Observe that ‖𝜌_𝑛−𝑝‖²=‖𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛−𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑝‖²≤⟨(𝐼−𝑟_𝑛𝐴)𝑦_𝑛−(𝐼−𝑟_𝑛𝐴)𝑝,𝜌_𝑛=1−𝑝⟩2{‖(𝐼−𝑟_𝑛𝐴)𝑦_𝑛−(𝐼−𝑟_𝑛𝐴)𝑝‖²+‖𝜌_𝑛−𝑝‖²−‖(𝐼−𝑟_𝑛𝐴)𝑦_𝑛−(𝐼−𝑟_𝑛𝐴)𝑝−(𝜌_𝑛−𝑝)‖²}≤12{‖𝑦_𝑛−𝑝‖²+‖𝜌_𝑛−𝑝‖²−‖(𝑦_𝑛−𝜌_𝑛)−𝑟_𝑛(𝐴𝑦_𝑛−𝐴𝑝)‖²}≤12{‖𝑥_𝑛−𝑝‖²+‖𝜌_𝑛−𝑝‖²−‖𝑦_𝑛−𝜌_𝑛‖²−𝑟²_𝑛‖𝐴𝑦_𝑛−𝐴𝑝‖²+2𝑟_𝑛⟨𝑦_𝑛−𝜌_𝑛,𝐴𝑦_𝑛−𝐴𝑝⟩},(3.41) which yields that ‖𝜌_𝑛−𝑝‖²≤‖𝑥_𝑛−𝑝‖²−‖𝑦_𝑛−𝜌_𝑛‖²+2𝑟_𝑛‖𝑦_𝑛−𝜌_𝑛‖‖𝐴𝑦_𝑛−𝐴𝑝‖.(3.42) Substituting (3.42) into (3.33) we have ‖𝑥_𝑛+1−𝑝‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²−‖𝑦_𝑛−𝜌_𝑛‖²+2𝑟_𝑛‖𝑦_𝑛−𝜌_𝑛‖‖𝐴𝑦_𝑛−𝐴𝑝‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖,(3.43) which implies that ‖𝑦_𝑛−𝜌_𝑛‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²−‖𝑥_𝑛+1−𝑝‖²+2𝑟_𝑛‖𝑦_𝑛−𝜌_𝑛‖‖𝐴𝑦_𝑛−𝐴𝑝‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+(‖𝑥_𝑛−𝑝‖+‖𝑥_𝑛+1−𝑝‖)‖𝑥_𝑛+1−𝑥_𝑛‖+2𝑟_𝑛‖𝑦_𝑛−𝜌_𝑛‖‖𝐴𝑦_𝑛−𝐴𝑝‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.44) It follows from (C1), ‖𝑥_𝑛+1−𝑥_𝑛‖→0, and ‖𝐴𝑦_𝑛−𝐴𝑝‖→0 that lim_𝑛→∞‖𝑦_𝑛−𝜌_𝑛‖=0.
For 𝑝∈Ω, we have‖𝑦_𝑛−𝑝‖²=‖𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑝‖²≤⟨𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑝,(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−(𝐼−𝑠_𝑛𝐴)𝑝⟩=⟨𝑦_𝑛−𝑝,(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−(𝐼−𝑠_𝑛=1𝐴)𝑝⟩2(‖𝑦_𝑛−𝑝‖²+‖(𝐼−𝑠_𝑛𝐴)𝑢_𝑛−(𝐼−𝑠_𝑛𝐴)𝑝‖²−‖(𝑦_𝑛−𝑝)−[𝑢_𝑛−𝑝−𝑠_𝑛(𝐴𝑢_𝑛−𝐴𝑝)]‖²)≤12(‖𝑦_𝑛−𝑝‖²+‖𝑢_𝑛−𝑝‖²−‖𝑦_𝑛−𝑝−[𝑢_𝑛−𝑝−𝑠_𝑛(𝐴𝑢_𝑛−𝐴𝑝)]‖²)=12(‖𝑦_𝑛−𝑝‖²+‖𝑢_𝑛−𝑝‖²−‖𝑦_𝑛−𝑢_𝑛‖²+2𝑠_𝑛⟨𝑦_𝑛−𝑢_𝑛,𝐴𝑢_𝑛−𝐴𝑝⟩−𝑠²_𝑛‖𝐴𝑢_𝑛−𝐴𝑝‖²).(3.45) This implies that ‖𝑦_𝑛−𝑝‖²≤‖𝑢_𝑛−𝑝‖²−‖𝑦_𝑛−𝑢_𝑛‖²+2𝑠_𝑛⟨𝑦_𝑛−𝑢_𝑛,𝐴𝑢_𝑛−𝐴𝑝⟩−𝑠²_𝑛‖𝐴𝑢_𝑛−𝐴𝑝‖²≤‖𝑢_𝑛−𝑝‖²−‖𝑦_𝑛−𝑢_𝑛‖²+2𝑠_𝑛‖𝑦_𝑛−𝑢_𝑛‖‖𝐴𝑢_𝑛−𝐴𝑝‖.(3.46)
By (3.46), (3.37), and (3.5), we obtain‖𝑥_𝑛+1−𝑝‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑢_𝑛−𝑝‖²−‖𝑦_𝑛−𝑢_𝑛‖²+2𝑠_𝑛‖𝑦_𝑛−𝑢_𝑛‖‖𝐴𝑢_𝑛−𝐴𝑝‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²−‖𝑦_𝑛−𝑢_𝑛‖²+2𝑠_𝑛‖𝑦_𝑛−𝑢_𝑛‖‖𝐴𝑢_𝑛−𝐴𝑝‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.47) It follows that ‖𝑦_𝑛−𝑢_𝑛‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²−‖𝑥_𝑛+1−𝑝‖²+2𝑠_𝑛‖𝑦_𝑛−𝑢_𝑛‖‖𝐴𝑢_𝑛−𝐴𝑝‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+(‖𝑥_𝑛−𝑝‖+‖𝑥_𝑛+1−𝑝‖)(‖𝑥_𝑛−𝑥_𝑛+1‖)+2𝑠_𝑛‖𝑦_𝑛−𝑢_𝑛‖‖𝐴𝑢_𝑛−𝐴𝑝‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.48)
It follows from (C1), ‖𝐴𝑢_𝑛−𝐴𝑝‖→0, and ‖𝑥_𝑛+1−𝑥_𝑛‖→0 that ‖𝑦_𝑛−𝑢_𝑛‖→0. It follows from ‖𝜌_𝑛−𝑢_𝑛‖≤‖𝜌_𝑛−𝑦_𝑛‖+‖𝑦_𝑛−𝑢_𝑛‖ that lim_𝑛→∞‖𝑢_𝑛−𝜌_𝑛‖=0.
We now show thatlim_𝑛→∞‖Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛‖=0,𝑘=1,2,…,𝑚.(3.49)
Indeed, let 𝑝∈Ω, it follows from the firmly nonexpansiveness of 𝑇^𝐹𝑘_𝛽𝑛, we have for each 𝑘∈{1,2,…,𝑚},‖Θ^𝑘_𝛽𝑛𝑥_𝑛−𝑝‖²=‖𝑇^𝐹𝑘_𝛽𝑛Θ_𝛽^𝑘−1_𝑛𝑥_𝑛−𝑇^𝐹𝑘_𝛽𝑛𝑝‖²≤⟨Θ^𝑘_𝛽𝑛𝑥_𝑛−𝑝,Θ_𝛽^𝑘−1_𝑛𝑥_𝑛=1−𝑝⟩2(‖Θ^𝑘_𝛽𝑛𝑥_𝑛−𝑝‖²+‖Θ_𝛽^𝑘−1_𝑛𝑥_𝑛−𝑝‖²−‖Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛‖²).(3.50) Thus, we get ‖Θ^𝑘_𝛽𝑛𝑥_𝑛−𝑝‖²≤‖Θ_𝛽^𝑘−1_𝑛𝑥_𝑛−𝑝‖²−‖Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛‖²,𝑘=1,2,…,𝑚.(3.51) This implies that for each 𝑘∈{1,2,…,𝑚}, ‖Θ^𝑘_𝛽𝑛𝑥_𝑛−𝑝‖²≤‖Θ⁰_𝛽𝑛𝑥_𝑛−𝑝‖²−‖Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛‖²−‖Θ_𝛽^𝑘−1_𝑛𝑥_𝑛−Θ_𝛽^𝑘−2_𝑛𝑥_𝑛‖²−⋯−‖Θ²_𝛽𝑛𝑥_𝑛−Θ¹_𝛽𝑛𝑥_𝑛‖²−‖Θ¹_𝛽𝑛𝑥_𝑛−Θ⁰_𝛽𝑛𝑥_𝑛‖².(3.52)
It follows from 𝑢_𝑛=Θ^𝑚_𝛽𝑛𝑥_𝑛 that for each 𝑘=1,2,…,𝑚‖𝑢_𝑛−𝑝‖²≤‖𝑥_𝑛−𝑝‖²−‖Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛‖².(3.53) By (3.37), (3.6), and (3.53), we have that for each 𝑘=1,2,…,𝑚‖𝑥_𝑛+1−𝑝‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑢_𝑛−𝑝‖²+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²−‖Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛‖²+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.54) Thus, we have that for each 𝑘=1,2,…,𝑚‖Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛‖²≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+‖𝑥_𝑛−𝑝‖²−‖𝑥_𝑛+1−𝑝‖²+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖≤𝛼_𝑛‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖²+(‖𝑥_𝑛−𝑝‖+‖𝑥_𝑛+1−𝑝‖)‖𝑥_𝑛−𝑥_𝑛+1‖+2𝛼_𝑛‖𝜌_𝑛−𝑝‖‖𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑝‖.(3.55) It follows from (C1) and ‖𝑥_𝑛+1−𝑥_𝑛‖→0 that for each 𝑘=1,2,…,𝑚‖Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛‖→0.(3.56)
Since‖𝑊_𝑛𝜌_𝑛−𝜌_𝑛‖≤‖𝑊_𝑛𝜌_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛−Θ¹_𝛽𝑛𝑥_𝑛‖+‖Θ¹_𝛽𝑛𝑥_𝑛−Θ²_𝛽𝑛𝑥_𝑛‖+⋯+‖Θ_𝛽^𝑚−1_𝑛𝑥_𝑛−Θ^𝑚_𝛽𝑛𝑥_𝑛‖+‖𝑢_𝑛−𝑦_𝑛‖+‖𝑦_𝑛−𝜌_𝑛‖.(3.57) It follows from (3.56) that lim_𝑛→∞‖𝑊_𝑛𝜌_𝑛−𝜌_𝑛‖=0.(3.58) Observe that ‖𝑊𝜌_𝑛−𝜌_𝑛‖≤‖𝑊𝜌_𝑛−𝑊_𝑛𝜌_𝑛‖+‖𝑊_𝑛𝜌_𝑛−𝜌_𝑛‖.(3.59)
It follows from Remark 2.7 thatlim_𝑛→∞‖𝑊𝜌_𝑛−𝜌_𝑛‖=0.(3.60)
We show that 𝑃_Ω(𝛾𝑓+(𝐼−𝐵)) is a contraction. Indeed, for all 𝑥,𝑦∈𝐻, we have‖𝑃_Ω(𝛾𝑓+(𝐼−𝐵))(𝑥)−𝑃_Ω(𝛾𝑓+(𝐼−𝐵))(𝑦)‖≤‖(𝛾𝑓+(𝐼−𝐵))(𝑥)−(𝛾𝑓+(𝐼−𝐵))(𝑦)‖≤𝛾‖𝑓(𝑥)−𝑓(𝑦)‖+‖𝐼−𝐵‖‖𝑥−𝑦‖≤𝛾𝛼‖𝑥−𝑦‖+(1−𝛾)‖𝑥−𝑦‖=(𝛾𝛼+1−𝛾)‖𝑥−𝑦‖.(3.61)
The Banach's Contraction Mapping Principle guarantees that 𝑃_Ω(𝛾𝑓+(𝐼−𝐵)) has a unique fixed point, say 𝑞∈𝐻. That is, 𝑞=𝑃_Ω(𝛾𝑓+(𝐼−𝐵))(𝑞).
Next, we show thatlimsup_𝑛→∞⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛−𝑞⟩≤0.(3.62) To show that, we choose a subsequence {𝑥_𝑛𝑖} of 𝑥_𝑛 such that limsup_𝑛→∞⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛−𝑞⟩=lim_𝑖→∞⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛𝑖−𝑞⟩.(3.63)
As {𝑥_𝑛𝑖} is bounded, we know that there is a subsequence {𝑥_𝑛𝑖𝑗} of {𝑥_𝑛𝑖} which converges weakly to 𝑝. We may assume, without loss of generality, that 𝑥_𝑛𝑖⇀𝑝. From ‖Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛‖→0 for each 𝑘=1,2,…,𝑚, we obtain that Θ^𝑘_𝛽𝑛𝑖𝑥_𝑛𝑖⇀𝑝 for 𝑘=1,2,…,𝑚. From ‖𝑢_𝑛−𝜌_𝑛‖→0, we also obtain that 𝜌_𝑛𝑖⇀𝑝. Since {𝑢_𝑛𝑖}⊂𝐶 and 𝐶 is closed and convex, we obtain 𝑝∈𝐶.
Now we show that 𝑝∈Ω. Indeed, let us first show that 𝑝∈VI(𝐶,𝐴). Put𝑇𝑤₁=𝐴𝑤₁+𝑁_𝐶𝑤₁if𝑤₁∈𝐶,∅if𝑤₁∉𝐶.(3.64) Since 𝐴 is relaxed (𝑢,𝑣)-cocoercive, we have ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥(−𝑢)‖𝐴𝑥−𝐴𝑦‖²+𝑣‖𝑥−𝑦‖²≥(𝑣−𝑢𝜇²)‖𝑥−𝑦‖²≥0,(3.65) which yields that 𝐴 is monotone. Thus 𝑇 is maximal monotone. Let (𝑤₁,𝑤₂)∈𝐺(𝑇). Since 𝑤₂−𝐴𝑤₁∈𝑁_𝐶𝑤₁ and 𝜌_𝑛∈𝐶, we have ⟨𝑤₁−𝜌_𝑛,𝑤₂−𝐴𝑤₁⟩≥0.(3.66) On the other hand, from 𝜌_𝑛=𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛, we have ⟨𝑤₁−𝜌_𝑛,𝜌_𝑛−(𝐼−𝑟_𝑛𝐴)𝑦_𝑛⟩≥0(3.67) and hence ⟨𝑤₁−𝜌_𝑛,𝜌_𝑛−𝑦_𝑛𝑟_𝑛+𝐴𝑦_𝑛⟩≥0.(3.68) It follows that ⟨𝑤₁−𝜌_𝑛𝑖,𝑤₂⟩≥⟨𝑤₁−𝜌_𝑛𝑖,𝐴𝑤₁⟩≥⟨𝑤₁−𝜌_𝑛𝑖,𝐴𝑤₁⟩−⟨𝑤₁−𝜌_𝑛𝑖,𝜌_𝑛𝑖−𝑦_𝑛𝑖𝑟_𝑛𝑖+𝐴𝑦_𝑛𝑖⟩≥⟨𝑤₁−𝜌_𝑛𝑖,𝐴𝑤₁−𝜌_𝑛𝑖−𝑦_𝑛𝑖𝑟_𝑛𝑖−𝐴𝑦_𝑛𝑖⟩=⟨𝑤₁−𝜌_𝑛𝑖,𝐴𝑤₁−𝐴𝜌_𝑛𝑖⟩+⟨𝑤₁−𝜌_𝑛𝑖,𝐴𝜌_𝑛𝑖−𝐴𝑦_𝑛𝑖⟩−⟨𝑤₁−𝜌_𝑛𝑖,𝜌_𝑛𝑖−𝑦_𝑛𝑖𝑟_𝑛𝑖⟩≥⟨𝑤₁−𝜌_𝑛𝑖,𝐴𝜌_𝑛𝑖−𝐴𝑦_𝑛𝑖⟩−⟨𝑤₁−𝜌_𝑛𝑖,𝜌_𝑛𝑖−𝑦_𝑛𝑖𝑟_𝑛𝑖⟩,(3.69) which implies that ⟨𝑤₁−𝑝,𝑤₂⟩≥0. We have 𝑝∈𝑇⁻¹0 and hence 𝑝∈VI(𝐶,𝐴).
We next show that 𝑝∈⋂^𝑚_𝑘=1EP(𝐹_𝑘). Indeed, by Lemma 2.1, we have that for each 𝑘=1,2,…,𝑚,𝐹_𝑘(Θ^𝑘_𝛽𝑛𝑥_𝑛1,𝑦)+𝛽_𝑛⟨𝑦−Θ^𝑘_𝛽𝑛𝑥_𝑛,Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛⟩≥0,∀𝑦∈𝐶(3.70).
It follows from (A2) that1𝛽_𝑛⟨𝑦−Θ^𝑘_𝛽𝑛𝑥_𝑛,Θ^𝑘_𝛽𝑛𝑥_𝑛−Θ_𝛽^𝑘−1_𝑛𝑥_𝑛⟩≥𝐹_𝑘(𝑦,Θ^𝑘_𝛽𝑛𝑥_𝑛),∀𝑦∈𝐶.(3.71)
Hence,⟨𝑦−Θ^𝑘_𝛽𝑛𝑖𝑥_𝑛𝑖,Θ^𝑘_𝛽𝑛𝑖𝑥_𝑛𝑖−Θ_𝛽^𝑘−1_𝑛𝑖𝑥_𝑛𝑖𝛽_𝑛𝑖⟩≥𝐹_𝑘(𝑦,Θ^𝑘_𝛽𝑛𝑖𝑥_𝑛𝑖),∀𝑦∈𝐶.(3.72)
It follows from (A4), (A5), (Θ^𝑘_𝛽𝑛𝑖𝑥_𝑛𝑖−Θ_𝛽^𝑘−1_𝑛𝑖𝑥_𝑛𝑖)/𝛽_𝑛𝑖→0, and Θ^𝑘_𝛽𝑛𝑖𝑥_𝑛𝑖⇀𝑝 that for each 𝑘=1,2,…,𝑚,𝐹_𝑘(𝑦,𝑝)≤0,∀𝑦∈𝐶.(3.73)
For 𝑡 with 0<𝑡≤1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝. Since 𝑦∈𝐶 and 𝑝∈𝐶, we obtain 𝑦_𝑡∈𝐶 and hence 𝐹_𝑘(𝑦_𝑡,𝑝)≤0. So by (A4), we have0=𝐹_𝑘(𝑦_𝑡,𝑦_𝑡)≤𝑡𝐹_𝑘(𝑦_𝑡,𝑦)+(1−𝑡)𝐹_𝑘(𝑦_𝑡,𝑝)≤𝑡𝐹_𝑘(𝑦_𝑡,𝑦).(3.74)
Dividing by 𝑡, we get that for each 𝑘=1,2,…,𝑚,𝐹_𝑘(𝑦_𝑡,𝑦)≥0.(3.75) Letting 𝑡→0, it follows from (A3) that for each 𝑘=1,2,…,𝑚, 𝐹_𝑘(𝑝,𝑦)≥0(3.76) for all 𝑦∈𝐶 and hence 𝑝∈EP(𝐹_𝑘) for 𝑘=1,2,…,𝑚. That is, 𝑝∈⋂^𝑚_𝑘=1EP(𝐹_𝑘).
We now show that 𝑝∈Fix(𝑊). Assume that 𝑝∉Fix(𝑊). Since 𝜌_𝑛𝑖⇀𝑝 and 𝑝≠𝑊𝑝, from (3.60) and the Opial condition we haveliminf_𝑖→∞‖𝜌_𝑛𝑖−𝑝‖<liminf_𝑖→∞‖𝜌_𝑛𝑖−𝑊𝑝‖≤liminf_𝑖→∞{‖𝜌_𝑛𝑖−𝑊𝜌_𝑛𝑖‖+‖𝑊𝜌_𝑛𝑖−𝑊𝑝‖}≤liminf_𝑖→∞‖𝜌_𝑛𝑖−𝑝‖,(3.77) which is a contradiction. So, we get 𝑝∈Fix(𝑊)=⋂^∞_𝑖=1Fix(𝑆_𝑖). This implies that 𝑝∈Ω.
Since 𝑞=𝑃_Ω(𝛾𝑓+(𝐼−𝐵))(𝑞), we havelimsup_𝑛→∞⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛−𝑞⟩=lim_𝑖→∞⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛𝑖−𝑞⟩=⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑝−𝑞⟩≤0.(3.78) That is, (3.62) holds. Next, we consider ‖𝑥_𝑛+1−𝑞‖²=‖𝛼_𝑛(𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑞)+(𝐼−𝛼_𝑛𝐵)(𝑊_𝑛𝜌_𝑛−𝑞)‖²≤(1−𝛼_𝑛𝛾)²‖𝑊_𝑛𝜌_𝑛−𝑞‖²+2𝛼_𝑛⟨𝛾𝑓(𝑊_𝑛𝑥_𝑛)−𝐵𝑞,𝑥_𝑛+1−𝑞⟩≤(1−𝛼_𝑛𝛾)²‖𝑥_𝑛−𝑞‖²+2𝛼_𝑛𝛾⟨𝑓(𝑊_𝑛𝑥_𝑛)−𝑓(𝑞),𝑥_𝑛+1−𝑞⟩+2𝛼_𝑛⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛+1−𝑞⟩≤(1−𝛼_𝑛𝛾)²‖𝑥_𝑛−𝑞‖²+𝛼_𝑛𝛾𝛼(‖𝑥_𝑛−𝑞‖²+‖𝑥_𝑛+1−𝑞‖²)+2𝛼_𝑛⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛+1−𝑞⟩(3.79) So, we can obtain ‖𝑥_𝑛+1−𝑞‖²≤(1−𝛼_𝑛𝛾)²+𝛼_𝑛𝛾𝛼1−𝛼_𝑛𝛾𝛼‖𝑥_𝑛−𝑞‖²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛼⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛+1=−𝑞⟩1−2𝛼_𝑛𝛾+𝛼_𝑛𝛾𝛼1−𝛼_𝑛𝛾𝛼‖𝑥_𝑛−𝑞‖²+𝛼²_𝑛𝛾²1−𝛼_𝑛𝛾𝛼‖𝑥_𝑛−𝑞‖²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛼⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛+1−𝑞⟩≤[1−2𝛼_𝑛(𝛾−𝛼𝛾)1−𝛼_𝑛𝛾𝛼]‖𝑥_𝑛−𝑞‖²+2𝛼_𝑛(𝛾−𝛼𝛾)1−𝛼_𝑛[1𝛾𝛼𝛾−𝛼𝛾⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛+1𝛼−𝑞⟩+^𝑛𝛾²2(𝛾−𝛼𝛾)𝑀],(3.80) where 𝑀 is an approximate constant such that 𝑀≥sup_𝑛≥1{‖𝑥_𝑛−𝑞‖²}}.
Put 𝑙_𝑛=2𝛼_𝑛(𝛾−𝛼𝛾)/(1−𝛼_𝑛𝛾𝛼) and 𝑡_𝑛=(1/(𝛾−𝛼𝛾))⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛+1−𝑞⟩+(𝛼_𝑛𝛾²/2(𝛾−𝛼𝛾))𝑀. That is,‖𝑥_𝑛+1−𝑞‖²≤(1−𝑙_𝑛)‖𝑥_𝑛−𝑞‖+𝑙_𝑛𝑡_𝑛.(3.81) From condition (C1) and Lemma 2.2, we concluded that 𝑥_𝑛→𝑞∈Ω. It is easy to see that 𝑢_𝑛→𝑞 and 𝑦_𝑛→𝑞. This completes the proof.

Corollary 3.2. Let 𝐶 be a nonempty, closed and convex subset of 𝐻. Let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ satisfies conditions (A1)–(A5). Let 𝐴∶𝐶→𝐻 be relaxed (𝑢,𝑣)-cocoercive and 𝜇-Lipschitz continuous and 𝐵 a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let 𝑆₁,𝑆₂,… be a family of infinitely nonexpansive mappings of 𝐶 into itself such that Γ=⋂^∞_𝑖=1Fix(𝑆_𝑖)∩VI(𝐶,𝐴)∩EP(𝐹)≠∅, let 𝜉₁,𝜉₂,… be real numbers such that 0<𝜉_𝑖≤𝛿<1 for every 𝑖∈𝑁 and 𝑊_𝑛 be the 𝑊-mapping of 𝐶 into itself generated by 𝑆_𝑛,𝑆_𝑛−1,…,𝑆₁ and 𝜉_𝑛,𝜉_𝑛−1,…,𝜉₁. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1) and {𝑥_𝑛}, {𝑢_𝑛} and {𝑦_𝑛} be sequences generated by 𝑥₁=𝑥∈𝐻,𝐹(𝑢_𝑛1,𝑦)+𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦⟩≥0,∀𝑦∈𝐶,_𝑛=𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑊_𝑛𝑥_𝑛)+(𝐼−𝛼_𝑛𝐵)𝑊_𝑛𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛(3.82) for every 𝑛=1,2,…, where {𝛼_𝑛}, {𝛽_𝑛}, {𝑟_𝑛} and {𝑠_𝑛} are sequences of numbers satisfying the conditions:(C1){𝛼_𝑛}⊂[0,1] with lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(C2){𝑟_𝑛}⊂[𝑎,𝑏] and {𝑠_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0≤𝑎≤𝑏≤2(𝜐−𝑢𝜇²)/𝜇², ∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞, and ∑^∞_𝑛=1|𝑠_𝑛+1−𝑠_𝑛|<∞;(C3)liminf_𝑛→∞𝛽_𝑛>0 and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞.
Then, {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑞∈Γ, which solves the following variational inequality:⟨𝛾𝑓𝑞−𝐵𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Γ.(3.83)

Proof. Let 𝑚=1, by Theorem 3.1, we obtain the desired result.

Corollary 3.3. Let 𝐶 be a nonempty, closed, and convex subset of 𝐻. Let 𝐴∶𝐶→𝐻 be relaxed (𝑢,𝑣)-cocoercive and 𝜇-Lipschitz continuous and let 𝐵 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let 𝑆₁,𝑆₂,… be a family of infinitely nonexpansive mappings of 𝐶 into itself such that Δ=⋂^∞_𝑖=1Fix(𝑆_𝑖)∩VI(𝐶,𝐴)≠∅, let 𝜉₁,𝜉₂,… be real numbers such that 0<𝜉_𝑖≤𝛿<1 for every 𝑖∈𝑁, and let 𝑊_𝑛 be the 𝑊-mapping of 𝐶 into itself generated by 𝑆_𝑛,𝑆_𝑛−1,…,𝑆₁ and 𝜉_𝑛,𝜉_𝑛−1,…,𝜉₁. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1) and {𝑥_𝑛}, {𝑢_𝑛}, and {𝑦_𝑛} be sequences generated by 𝑥₁𝑦=𝑥∈𝐶,_𝑛=𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑊_𝑛𝑥_𝑛)+(𝐼−𝛼_𝑛𝐵)𝑊_𝑛𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛(3.84) for every 𝑛=1,2,…, where {𝛼_𝑛}, {𝛽_𝑛}, {𝑟_𝑛}, and {𝑠_𝑛} are sequences of numbers satisfying the conditions:(C1){𝛼_𝑛}⊂[0,1] with lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(C2){𝑟_𝑛}⊂[𝑎,𝑏] and {𝑠_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0≤𝑎≤𝑏≤2(𝜐−𝑢𝜇²)/𝜇², ∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞ and ∑^∞_𝑛=1|𝑠_𝑛+1−𝑠_𝑛|<∞.
Then, {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑞∈Δ, which solves the following variational inequality:⟨𝛾𝑓𝑞−𝐵𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Δ.(3.85)

Proof. Let 𝐹(𝑥,𝑦)=0 for 𝑥,𝑦∈𝐶, by Corollary 3.2 we obtain the desired result.

Corollary 3.4. Let 𝐶 be a nonempty, closed and convex subset of 𝐻. Let 𝐹₁,𝐹₂,…,𝐹_𝑚 be bifunctions from 𝐶×𝐶 to ℝ satisfies conditions (A1)–(A5). Let 𝐴∶𝐶→𝐻 be relaxed (𝑢,𝑣)-cocoercive and 𝜇-Lipschitz continuous and 𝐵 a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 such that Ξ=⋂^𝑚_𝑘=1EP(𝐹_𝑘)∩VI(𝐶,𝐴)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1) and {𝑥_𝑛}, {𝑢_𝑛} and {𝑦_𝑛} be sequences generated by 𝑥₁𝑢=𝑥∈𝐻,_𝑛=𝑇^𝐹𝑚_𝛽𝑛𝑇^𝐹𝑚−1_𝛽𝑛⋯𝑇^𝐹2_𝛽𝑛𝑇^𝐹1_𝛽𝑛𝑥_𝑛,𝑦_𝑛=𝑃_𝐶(𝐼−𝑠_𝑛𝐴)𝑢_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝛼_𝑛𝐵)𝑃_𝐶(𝐼−𝑟_𝑛𝐴)𝑦_𝑛(3.86) for every 𝑛=1,2,…, where {𝛼_𝑛}, {𝛽_𝑛}, {𝑟_𝑛} and {𝑠_𝑛} are sequences of numbers satisfying the conditions:(C1){𝛼_𝑛}⊂[0,1] with lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(C2){𝑟_𝑛}⊂[𝑎,𝑏] and {𝑠_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0≤𝑎≤𝑏≤2(𝜐−𝑢𝜇²)/𝜇², ∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞, and ∑^∞_𝑛=1|𝑠_𝑛+1−𝑠_𝑛|<∞;(C3)liminf_𝑛→∞𝛽_𝑛>0 and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞.
Then, {𝑥_𝑛}, {𝑦_𝑛} and {𝑢_𝑛} converge strongly to 𝑞∈Ω, which solves the following variational inequality:⟨𝛾𝑓𝑞−𝐵𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ξ.(3.87)

Remark 3.5. (i) If 𝑠_𝑛=0 for all 𝑛≥0, by Corollary 3.2, we get Theorem 2.1 in [9]. If 𝑠_𝑛=0 and 𝑆_𝑖=𝐼 for all 𝑛≥0, by Corollary 3.2, we get Theorem 2.1 in [8] with 𝑆=𝐼. If 𝑠_𝑛=0, 𝑟_𝑛=0 and 𝑆_𝑖=𝐼 for all 𝑛≥0, by Corollary 3.2, we get Theorem 3.1 in [6] with 𝑆=𝐼 and Theorem 3.3 in [7] with 𝑆=𝐼 and 𝐶=𝐻.
(ii) Corollary 3.3 extends, generalizes and improves the main results in [21, 22, 24].
(iii) It is easy to see that Theorem 3.1 is different from the main results in [1–4].

Acknowledgments

The authors are grateful to the referee and Proffessor S. Reich for the detailed comments and helpful suggestions which improved the original manuscript greatly. This research was supported by the National Center of Theoretical Sciences (South) of Taiwan, the National Natural Science Foundation of China (Grants 10771228 and 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), the Research Project of Chongqing Normal University (Grant 08XLZ05), and the project of the Grant NSC 98-2115-M-110-001.