A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces

Abstract

We present an iterative method for fixed point problems, generalized mixed equilibrium problems, and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some well-known strong convergence theorems in the literature.

1. Introduction

Throughout this paper, unless otherwise specified, we consider 𝐻 to be a real Hilbert space with inner product ⟨⋅,⋅⟩ and its induced norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝑃_𝐶 be the metric projection of 𝐻 onto the closed convex subset 𝐶. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping, that is, ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. The fixed point set of 𝑆 is defined by 𝐹(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥}.(1.1) If 𝐶⊂𝐻 is nonempty, bounded, closed, and convex and 𝑆 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑆) is nonempty; see, for example, [1, 2]. A mapping 𝑓∶𝐶→𝐶 is a contraction on 𝐶 if there exists a constant 𝜂∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜂‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. In addition, let Ψ∶𝐶→𝐻 be a nonlinear mapping. Let 𝜑∶𝐶→ℝ∪{+∞} be a real-valued function and let Θ∶𝐶×𝐶→ℝ be a bifunction such that 𝐶∩dom𝜑≠∅, where ℝ is the set of real numbers and dom𝜑={𝑥∈𝐶∶𝜑(𝑥)<+∞}.

The generalized mixed equilibrium problem for finding 𝑥∈𝐶Θ(𝑥,𝑦)+⟨Ψ𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶.(1.2) The set of solutions of (1.2) is denoted by GMEP(Θ,𝜑,Ψ), that is, GMEP(Θ,𝜑,Ψ)={𝑥∈𝐶∶Θ(𝑥,𝑦)+⟨Ψ𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶}.(1.3) We see that if 𝑥 is a solution of a problem (1.2), then 𝑥∈dom𝜑.

Special Examples
(1)If Ψ=0, then the problem (1.2) is reduced into the mixed equilibrium problem for finding 𝑥∈𝐶 such that Θ(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶.(1.4) The set of solutions of (1.4) is denoted by MEP(Θ,𝜑).(2)If 𝜑=0, then the problem (1.2) is reduced into the generalized equilibrium problem for finding 𝑥∈𝐶 such that Θ(𝑥,𝑦)+⟨Ψ𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶.(1.5) The set of solutions of (1.5) is denoted by GEP(Θ,Ψ).(3)If Ψ=0 and 𝜑=0, then the problem (1.2) is reduced into the equilibrium problem for finding 𝑥∈𝐶 such that Θ(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.6) The set of solutions of (1.6) is denoted by EP(Θ).(4)If Θ=0,𝜑=0, and Ψ=𝐵, then the problem (1.2) is reduced into the variational inequality problem for finding 𝑥∈𝐶 such that ⟨𝐵𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶.(1.7) The set of solutions of (1.7) is denoted by VI(𝐶,𝐵).

The generalized mixed equilibrium problem is very general in the sense that it includes, as special cases, fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems in noncooperative games, the equilibrium problem, and Numerous problems in physics, economics, and others. Some methods have been proposed to solve problem (1.2); see, for instance, [3, 4] and the references therein.

Let 𝐵∶𝐶→𝐻 be a nonlinear mapping. Now, we recall the following definitions.(d1)𝐵 is said to be monotone if for each 𝑥,𝑦∈𝐶⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥0.(1.8)(d2)𝐵 is said to be 𝜌-strongly monotone if there exists a positive real number 𝜌 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝜌‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶.(1.9)(d3)𝐵 is said to be 𝜔-Lipschitz continuous if there exists a positive real number 𝜔 such that ‖𝐵𝑥−𝐵𝑦‖≤𝜔‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.10)(d4)𝐵 is said to be 𝜉-inverse-strongly monotone if there exists a constant 𝜉>0 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝜉‖𝐵𝑥−𝐵𝑦‖²,∀𝑥,𝑦∈𝐶.(1.11)(d5)𝐵 is said to be relaxed (𝑢,𝑣)-cocoercive if there exist positive real numbers 𝑢,𝑣 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥(−𝑢)‖𝐵𝑥−𝐵𝑦‖²+𝑣‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶.(1.12)(d6) A set-valued mapping 𝑄∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻, 𝑓∈𝑄𝑥 and 𝑔∈𝑄𝑦 imply ⟨𝑥−𝑦,𝑓−𝑔⟩≥0.(d7) A monotone mapping 𝑄∶𝐻→2^𝐻 is called maximal if the graph 𝐺(𝑄) of 𝑄 is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping 𝑄 is maximal if and only if for (𝑥,𝑓)∈𝐻×𝐻, ⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for every (𝑦,𝑔)∈𝐺(𝑄) implies 𝑓∈𝑄𝑥.

For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequalities for a 𝜉-inverse-strongly monotone mapping, Takahashi and Toyoda [5] introduced the following iterative scheme: 𝑥𝑥₀∈𝐶chosenarbitrary,_𝑛+1=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝛼_𝑛𝐵𝑥_𝑛,∀𝑛≥0,(1.13) where 𝐵 is a 𝜉-inverse-strongly monotone mapping, {𝛾_𝑛} is a sequence in (0,1), and {𝛼_𝑛} is a sequence in (0,2𝜉). They showed that if 𝐹(𝑆)∩VI(𝐶,𝐵) is nonempty, then the sequence {𝑥_𝑛} generated by (1.13) converges weakly to some 𝑧∈𝐹(𝑆)∩VI(𝐶,𝐵).

For finding an element of VI(𝐶,𝐵), Iiduka et al. [6] introduced the following iterative scheme:𝑥𝑥₀∈𝐶chosenarbitrary,_𝑛+1=𝑃_𝐶𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝛼_𝑛𝐵𝑥_𝑛,∀𝑛≥0,(1.14) where 𝐵 is a 𝜉-inverse-strongly monotone mapping, {𝛾_𝑛} is a sequence in (−1,1), and {𝛼_𝑛} is a sequence in (0,2𝜉). They showed that if VI(𝐶,𝐵) is nonempty, then the sequence {𝑥_𝑛} generated by (1.14) converges weakly to some 𝑧∈VI(𝐶,𝐵).

For finding a common element of 𝐹(𝑆)∩VI(𝐶,𝐵), let 𝑆∶𝐻→𝐻 be a nonexpansive mapping. Yamada [7] introduced the following iterative scheme called the hybrid steepest descent method: 𝑥_𝑛+1=𝑆𝑥_𝑛−𝛼_𝑛𝜇𝐵𝑆𝑥_𝑛,∀𝑛≥1,(1.15) where 𝑥₁=𝑥∈𝐻,{𝛼_𝑛}⊂(0,1),𝐵∶𝐻→𝐻 is a strongly monotone and Lipschitz continuous mapping, and 𝜇 is a positive real number. He proved that the sequence {𝑥_𝑛} generated by (1.15) converges strongly to the unique solution of the 𝐹(𝑆)∩VI(𝐶,𝐵).

The hybrid steepest descent method is constructed by blending important ideas in the steepest descent method and in the fixed point theory. The remarkable applicability of this method to the convexly constrained generalized pseudoinverse problem as well as to the convex feasibility problem is demonstrated by constructing nonexpansive mappings whose fixed point sets are the feasible sets of the problems.

On the other hand, Shang et al. [8] introduced a new iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequalities for relaxed (𝑢,𝑣)-cocoercive mappings in a real Hilbert space by using viscosity approximation method. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping and let 𝑓∶𝐶→𝐶 be a contraction mapping. Starting with arbitrary initial 𝑥₁∈𝐶 and define sequences {𝑥_𝑛} recursively by 𝑥_𝑛+1=𝜖_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝛼_𝑛𝐵𝑥_𝑛,∀𝑛≥1.(1.16) They proved that under certain appropriate conditions imposed on {𝜖_𝑛},{𝛽_𝑛},{𝛾_𝑛}, and {𝛼_𝑛}, the sequence {𝑥_𝑛} converges strongly to 𝑧∈𝐹(𝑆)∩VI(𝐶,𝐵), where 𝑧=𝑃_{𝐹(𝑆)∩VI(𝐶,𝐵)}𝑓(𝑧).

For finding a common element of 𝐹(𝑆)∩GEF(Θ,Ψ), let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Ψ be a 𝜉-inverse-strongly monotone mapping of 𝐶 into 𝐻 and let 𝑆 be a nonexpansive mapping of 𝐶 into itself. S. Takahashi and W. Takahashi [9] introduced the following iterative scheme: Θ𝑢_𝑛,𝑦+⟨Ψ𝑥_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑢_𝑛,=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑆𝑦_𝑛,(1.17) where {𝛼_𝑛}⊂[0,1],{𝛾_𝑛}⊂[0,1], and {𝑟_𝑛}⊂[0,2𝜉] satisfy some parameters controlling conditions. They proved that the sequence {𝑥_𝑛} defined by (1.17) converges strongly to a common element of 𝐹(𝑆)∩GEF(Θ,Ψ).

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [7, 10–12] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences.

A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping defined on a real Hilbert space 𝐻: min_𝑥∈𝐹12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩,(1.18) where 𝐹 is the fixed point set of a nonexpansive mapping 𝑆 defined on 𝐻 and 𝑏 is a given point in 𝐻.

A linear bounded operator 𝐴 is strongly positive if there exists a constant 𝛾>0 with the property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻.(1.19)

Recently, Marino and Xu [13] introduced a new iterative scheme by the viscosity approximation method: 𝑥_𝑛+1=𝜖_𝑛𝛾𝑓𝑥_𝑛+1−𝜖_𝑛𝐴𝑆𝑥_𝑛.(1.20) They proved that the sequence {𝑥_𝑛} generated by (1.20) converges strongly to the unique solution of the variational inequality: ⟨𝛾𝑓𝑧−𝐴𝑧,𝑥−𝑧⟩≤0,∀𝑥∈𝐹(𝑆),(1.21) which is the optimality condition for the minimization problem: min_{𝑥∈𝐹(𝑆)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥),(1.22) where ℎ is a potential function for 𝛾𝑓.

In 2008, Qin et al. [14] proposed the following iterative algorithm: Θ𝑢_𝑛,𝑦+1𝑟_𝑛𝑥⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐻,_𝑛+1=𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑆𝑃_𝐶𝑢_𝑛−𝛼_𝑛𝐵𝑢_𝑛,(1.23) where 𝐴 is a strongly positive linear bounded operator and 𝐵 is a relaxed cocoercive mapping of 𝐶 into 𝐻. They proved that if the sequences {𝜖_𝑛},{𝛼_𝑛}, and {𝑟_𝑛} of parameters satisfy appropriate condition, then the sequence {𝑥_𝑛} defined by (1.23) converges strongly to the unique solution 𝑧 of the variational inequality: ⟨𝛾𝑓𝑧−𝐴𝑧,𝑥−𝑧⟩≤0,∀𝑥∈𝐹(𝑆)∩VI(𝐶,𝐵)∩EP(Θ),(1.24) which is the optimality condition for the minimization problem: min_{𝑥∈𝐹(𝑆)∩VI(𝐶,𝐵)∩EP(Θ)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥),(1.25) where ℎ is a potential function for 𝛾𝑓.

In this paper, we introduce an iterative scheme by using a viscosity hybrid steepest descent method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the set of solutions of variational inequality problem for a relaxed cocoercive mapping in a real Hilbert space. The results shown in this paper improve and extend the recent ones announced by many others.

2. Preliminaries

Throughout this paper, we always assume that 𝐻 is a real Hilbert space and 𝐶 is a nonempty closed convex subset of 𝐻. For a sequence {𝑥_𝑛}, the notation of 𝑥_𝑛⇀𝑥 and 𝑥_𝑛→𝑥 means that the sequence {𝑥_𝑛} converges weakly and strongly to 𝑥, respectively.

The following lemmata give some characterizations and useful properties of the metric projection 𝑃_𝐶 in a real Hilbert space. The metric (or nearest point) projection from 𝐻 onto 𝐶 is the mapping 𝑃_𝐶∶𝐻→𝐶 which assigns to each point 𝑥∈𝐻 the unique point 𝑃_𝐶𝑥∈𝐶 satisfying the following property: ‖‖𝑥−𝑃_𝐶𝑥‖‖=inf_𝑦∈𝐶‖𝑥−𝑦‖.(2.1)

Lemma 2.1. It is well known that the metric projection 𝑃_𝐶 has the following properties: (m1) for each 𝑥∈𝐻 and 𝑧∈𝐶, 𝑧=𝑃_𝐶𝑥⟺⟨𝑥−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐶;(2.2)(m2)𝑃_𝐶∶𝐻→𝐶 is nonexpansive, that is, ‖‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐻;(2.3)(m3)𝑃_𝐶 is firmly nonexpansive, that is, ‖‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖²≤⟨𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦⟩∀𝑥,𝑦∈𝐻.(2.4)

In order to prove our main results, we also need the following lemmata.

Lemma 2.2 (see [2]). Let 𝐻 be a Hilbert space, let 𝐶 be a nonempty closed convex subset of 𝐻, and let 𝐵 be a mapping of 𝐶 into 𝐻. Let 𝑥^∗∈𝐶. Then, for 𝜆>0, 𝑥^∗∈VI(𝐶,𝐵)⟺𝑥^∗=𝑃_𝐶𝑥^∗−𝜆𝐵𝑥^∗,(2.5) that is, 𝑥^∗∈VI(𝐶,𝐵)⟺𝑥^∗∈𝐹𝑃_𝐶(𝐼−𝜆𝐵),(2.6) where 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶.

Lemma 2.3 (see [15]). Let 𝐵 be a monotone mapping of 𝐶 into 𝐻 and let 𝑁_𝐶𝑤₁ be the normal cone to 𝐶 at 𝑤₁∈𝐶, that is, 𝑁_𝐶𝑤₁=𝑤∈𝐻∶⟨𝑤₁−𝑤₂,𝑤⟩≥0,∀𝑤₂∈𝐶,(2.7) and define a mapping 𝑄 on 𝐶 by 𝑄𝑤₁=𝐵𝑤¹∅,+𝑁_𝐶𝑤₁,𝑤𝑤₁₁∈𝐶,∉𝐶.(2.8) Then 𝑄 is maximal monotone and 0∈𝑄𝑤₁ if and only if ⟨𝐵𝑤₁,𝑤₂−𝑤₁⟩≥0 for all 𝑤₂∈𝐶.

Lemma 2.4 (see [16]). Each Hilbert space 𝐻 satisfies Opials condition; that is, for any sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞‖‖𝑥_𝑛−𝑥‖‖<liminf_𝑛→∞‖‖𝑥_𝑛−𝑦‖‖(2.9) holds for each 𝑦∈𝐻 with 𝑦≠𝑥.

Lemma 2.5 (see [13]). Let 𝐶 be a nonempty closed convex subset of 𝐻, let 𝑓 be a contraction of 𝐻 into itself with coefficient 𝜂∈(0,1), and let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0. Then, for 0<𝛾<𝛾/𝜂, ⟨𝑥−𝑦,(𝐴−𝛾𝑓)𝑥−(𝐴−𝛾𝑓)𝑦⟩≥𝛾−𝜂𝛾‖𝑥−𝑦‖²,𝑥,𝑦∈𝐻.(2.10) That is, 𝐴−𝛾𝑓 is strongly monotone with coefficient 𝛾−𝜂𝛾.

Lemma 2.6 (see [13]). Assume that 𝐴 is a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 2.7 (see [17]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛾_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1.(2.11) Suppose lim𝑥sup_{𝑛+1𝑛→∞}=1−𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝑥_𝑛,∀𝑛≥0,‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(2.12) Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.8 (see [18]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝜚_𝑛𝑎_𝑛+𝜎_𝑛,𝑛≥0,(2.13) where {𝜚_𝑛} is a sequence in (0,1) and {𝜎_𝑛} is a sequence in ℝ such that (1)∑^∞𝑛=1𝜚_𝑛=∞, (2)limsup_𝑛→∞(𝜎_𝑛/𝜚_𝑛)≤0 or ∑^∞𝑛=1|𝜎_𝑛|<∞.Then lim_𝑛→∞𝑎_𝑛=0.

For solving the generalized mixed equilibrium problem and the mixed equilibrium problem, let us give the following assumptions for the bifunction Θ, the function 𝜑, and the set 𝐶: (H1)Θ(𝑥,𝑥)=0,∀𝑥∈𝐶;(H2)Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0,∀𝑥,𝑦∈𝐶;(H3) for each 𝑦∈𝐶,𝑥↦Θ(𝑥,𝑦) is weakly upper semicontinuous; (H4) for each 𝑥∈𝐶,𝑦↦Θ(𝑥,𝑦) is convex; (H5) for each 𝑥∈𝐶,𝑦↦Θ(𝑥,𝑦) is lower semicontinuous; (B1) for each 𝑥∈𝐻 and 𝑟>0, there exist abounded subsets 𝐷_𝑥⊆𝐶 and 𝑦_𝑥∈𝐶 such that for any 𝑧∈𝐶⧵𝐷_𝑥, Θ𝑧,𝑦_𝑥+𝜑𝑦_𝑥−𝜑(𝑧)+1𝑟⟨𝑦^𝑥−𝑧,𝑧−𝑥⟩<0;(2.14)(B2)𝐶 is a bounded set.

Lemma 2.9 (see [19]). Let 𝐶 be a nonempty closed convex subset of 𝐻. Let Θ∶𝐶×𝐶→ℝ be a bifunction that satisfies (H1)–(H5) and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇^{𝑟(Θ,𝜑)}∶𝐻→𝐶 as follows: 𝑇^{𝑟(Θ,𝜑)}(𝑥)=𝑧∈𝐶∶Θ(𝑧,𝑦)+𝜑(𝑦)−𝜑(𝑧)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶(2.15) for all 𝑧∈𝐻. Then, the following properties hold: (i)for each 𝑥∈𝐻,𝑇^{𝑟(Θ,𝜑)}(𝑥)≠∅; (ii)𝑇^{𝑟(Θ,𝜑)} is single-valued; (iii)𝑇^{𝑟(Θ,𝜑)} is firmly nonexpansive; that is, for any 𝑥,𝑦∈𝐻,‖‖𝑇^{𝑟(Θ,𝜑)}𝑥−𝑇^{𝑟(Θ,𝜑)}𝑦‖‖²≤𝑇^{𝑟(Θ,𝜑)}𝑥−𝑇^{𝑟(Θ,𝜑)}𝑦,𝑥−𝑦;(2.16)(iv)𝐹(𝑇^{𝑟(Θ,𝜑)})=MEP(Θ,𝜑);(v)MEP(Θ,𝜑) is closed and convex.

Remark 2.10. If 𝜑=0, then 𝑇^{𝑟(Θ,𝜑)} is rewritten as 𝑇^Θ𝑟.

Lemma 2.11 (see [9]). Let 𝐶,𝐻,Θ, and 𝑇^Θ𝑟 be as in Remark 2.10. Then the following holds: ‖‖𝑇^Θ𝑠𝑥−𝑇^Θ𝑡𝑥‖‖²≤𝑠−𝑡𝑠𝑇^Θ𝑠𝑥−𝑇^Θ𝑡𝑥,𝑇^Θ𝑠𝑥−𝑥(2.17) for all 𝑠,𝑡>0 and 𝑥∈𝐻.

The following lemma is an immediate consequence of an inner product.

Lemma 2.12. Let 𝐻 be a real Hilbert space, let 𝑥 and 𝑦 be elements in 𝐻, and let 𝜆∈[0,1]. Then (1)‖𝜆𝑥+(1−𝜆)𝑦‖²=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖², (2)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩.

3. Main Results

In this section, we will introduce an iterative scheme by using a viscosity hybrid steepest descent method for finding a common element of the set of fixed points for nonexpansive mappings, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of variational inequality problem for a relaxed cocoercive mapping in a real Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets.

In order to prove our main results, we first prove the following lemmata.

Lemma 3.1. Let 𝐶,𝐻,Θ,𝜑, and 𝑇^{𝑟(Θ,𝜑)} be as in Lemma 2.9. Then the following holds: ‖‖𝑇^{𝑠(Θ,𝜑)}𝑥−T^{𝑡(Θ,𝜑)}𝑥‖‖²≤𝑠−𝑡𝑠𝑇^{𝑠(Θ,𝜑)}𝑥−𝑇^{𝑡(Θ,𝜑)}𝑥,𝑇^{𝑠(Θ,𝜑)}𝑥−𝑥(3.1) for all 𝑠,𝑡>0 and 𝑥∈𝐻.

Proof. By similar argument as in the proof of Lemma  2.11 in [9], for 𝑠,𝑡>0 and 𝑥∈𝐻. Observing that ℑ=𝑇^{𝑠(Θ,𝜑)}𝑥 and 𝜗=𝑇^{𝑡(Θ,𝜑)}𝑥, we have Θ(ℑ,𝑦)+𝜑(𝑦)−𝜑(ℑ)+1Θ(𝜗,𝑦)+𝜑(𝑦)−𝜑(𝜗)+1𝑠⟨𝑦−ℑ,ℑ−𝑥⟩≥0,∀𝑦∈𝐶,(3.2)𝑡⟨𝑦−𝜗,𝜗−𝑥⟩≥0,∀𝑦∈𝐶.(3.3) Putting 𝑦=𝜗 in (3.2) and 𝑦=ℑ in (3.3), we obtain Θ(ℑ,𝜗)+𝜑(𝜗)−𝜑(ℑ)+1Θ(𝜗,ℑ)+𝜑(ℑ)−𝜑(𝜗)+𝑠1⟨𝜗−ℑ,ℑ−𝑥⟩≥0,𝑡⟨ℑ−𝜗,𝜗−𝑥⟩≥0.(3.4) So, summing up these two equalities and using the monotonicity of Θ (H2), we get 1𝑠⟨𝜗−ℑ,ℑ−𝑥⟩+1𝑡⟨ℑ−𝜗,𝜗−𝑥⟩≥0,(3.5) and hence ℑ−𝜗,𝜗−𝑡𝑥−ℑ−𝑠𝑥≥0.(3.6) We derive from (3.6) that ℑ−𝜗,𝜗−𝑥−𝑠𝑡(ℑ−𝑥)≥0,(3.7) and so −‖ℑ−𝜗‖²+ℑ−𝜗,1−𝑡𝑠(ℑ−𝑥)≥0.(3.8) This indicates that 1−𝑠𝑡⟨ℑ−𝜗,ℑ−𝑥⟩≥‖ℑ−𝜗‖².(3.9) In other words, ‖ℑ−𝜗‖²≤𝑠−𝑡𝑠⟨ℑ−𝜗,ℑ−𝑥⟩,(3.10) and thus the claim holds.

Lemma 3.2. Let 𝐻 be a real Hilbert space, let 𝐶 be a nonempty closed convex subset of 𝐻, let 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and let 𝐵∶𝐶→𝐻 be an 𝜔-Lipschitz continuous and relaxed (𝑢,𝑣)-cocoercive mappings with 𝑣>𝑢𝜔². If 0≤𝛼_𝑛≤2(𝑣−𝑢𝜔²)/𝜔², then 𝑆−𝛼_𝑛𝐵𝑆 is a nonexpansive mapping in 𝐻.

Proof. Let 𝛼_𝑛≤2(𝑣−𝑢𝜔²)/𝜔²,𝑣>𝑢𝜔². Then, for every 𝑥,𝑦∈𝐶, we have ‖‖𝑆−𝛼_𝑛𝐵𝑆𝑥−𝑆−𝛼_𝑛𝐵𝑆𝑦‖‖²=‖‖(𝑆𝑥−𝑆𝑦)−𝛼_𝑛(𝐵𝑆𝑥−𝐵𝑆𝑦)‖‖²=‖𝑆𝑥−𝑆𝑦‖²−2𝛼_𝑛⟨𝑆𝑥−𝑆𝑦,𝐵𝑆𝑥−𝐵𝑆𝑦⟩+𝛼^2𝑛‖𝐵𝑆𝑥−𝐵𝑆𝑦‖²≤‖𝑆𝑥−𝑆𝑦‖²−2𝛼_𝑛−𝑢‖𝐵𝑆𝑥−𝐵𝑆𝑦‖²+𝑣‖𝑆𝑥−𝑆𝑦‖²+𝛼^2𝑛‖𝐵𝑆𝑥−𝐵𝑆𝑦‖²≤‖𝑆𝑥−𝑆𝑦‖²+2𝛼_𝑛𝑢𝜔²‖𝑆𝑥−𝑆𝑦‖²−2𝛼_𝑛𝑣‖𝑆𝑥−𝑆𝑦‖²+𝛼^2𝑛𝜔²‖𝑆𝑥−𝑆𝑦‖²=1+2𝛼_𝑛𝑢𝜔²−2𝛼_𝑛𝑣+𝛼^2𝑛𝜔²‖𝑆𝑥−𝑆𝑦‖²≤1−𝛼_𝑛𝜔²2𝑣−𝑢𝜔²𝜔²−𝛼_𝑛‖𝑥−𝑦‖².(3.11) Now, since (1−𝛼_𝑛𝜔²[2(𝑣−𝑢𝜔²)/𝜔²−𝛼_𝑛])<1, thus ‖(𝑆−𝛼_𝑛𝐵𝑆)𝑥−(𝑆−𝛼_𝑛𝐵𝑆)𝑦‖≤‖𝑥−𝑦‖.
Thus, 𝑆−𝛼_𝑛𝐵𝑆 is a nonexpansive mapping of 𝐶 into 𝐻.

Now we can prove that a strong convergence theorem is a real Hilbert space.

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ₁ and Θ₂ be two bifunctions from 𝐶×𝐶 to ℝ satisfying (H1)–(H5) and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let (i)Ψ₁∶𝐶→𝐻 be a 𝜉-inverse-strongly monotone mapping, (ii)Ψ₂∶𝐶→𝐻 be a 𝛽-inverse-strongly monotone mapping, (iii)𝐵∶𝐶→𝐻 be an 𝜔-Lipschitz continuous and relaxed (𝑢,𝑣)-cocoercive mappings, (iv)𝑓∶𝐶→𝐶 be a contraction mapping with coefficient 𝜂∈(0,1) and let 𝐴 be a strongly positive linear bounded self-adjoint operator with the coefficient 𝛾>0 and 0<𝛾<𝛾/𝜂. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping with 𝐹(𝑆)≠∅.
Assume that ℱ∶=𝐹(𝑆)∩GMEPΘ₁,𝜑,Ψ₁∩GMEPΘ₂,𝜑,Ψ₂∩VI(𝐶,𝐵)≠∅.(3.12) Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, {𝑣_𝑛}, and {𝑢_𝑛} be the sequences generated by 𝑢𝑣_𝑛𝑛=𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛,𝑧_𝑛=𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛,𝑦_𝑛=𝑃_𝐶𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛,=𝜖𝑥_𝑛𝛾𝑓_𝑛+1𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛,=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,∀𝑛≥1,(3.13) where {𝑟_𝑛}⊂[𝑎,𝑏]⊂[0,2𝜉],{𝑠_𝑛}⊂[𝑐,𝑑]⊂[0,2𝛽],{𝛾_𝑛}⊂[ℎ,𝑗]⊂(0,1), and {𝛾_𝑛},{𝜖_𝑛}, and {𝛽_𝑛} are three sequences in (0,1) satisfying the following conditions: (C1)lim_𝑛→∞𝜖_𝑛=0 and ∑^∞𝑛=1𝜖_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, and lim_𝑛→∞𝛽_𝑛=0, (C3)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜉, and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0, (C4)0<liminf_𝑛→∞𝑠_𝑛≤limsup_𝑛→∞𝑠_𝑛<2𝛽, and lim_𝑛→∞|𝑠_𝑛+1−𝑠_𝑛|=0, (C5){𝛼_𝑛}⊂[𝑒,𝑔]⊂(0,2(𝑣−𝑢𝜔²)/𝜔²),𝑣>𝑢𝜔², and lim_𝑛→∞|𝛼_𝑛+1−𝛼_𝑛|=0. Then, {𝑥_𝑛} converges strongly to 𝑧=𝑃_ℱ(𝛾𝑓+(𝐼−𝐴))(𝑧), which is the unique solution of the variational inequality: ⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑥−𝑧⟩≤0,∀𝑥∈ℱ.(3.14)

Proof. From the restrictions on control sequences, we may assume, without loss of generality, that 𝜖_𝑛≤(1−𝛽_𝑛)‖𝐴‖⁻¹ for all 𝑛≥1. Since 𝐴 is a strongly positive linear bounded self-adjoint operator on 𝐻, we have ‖𝐴‖=sup||⟨𝐴𝑥,𝑥⟩||∶𝑥∈𝐻,‖𝑥‖=1.(3.15) Observe that 1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑥,𝑥=1−𝛽_𝑛−𝜖_𝑛⟨𝐴𝑥,𝑥⟩≥1−𝛽_𝑛−𝜖_𝑛‖𝐴‖≥0.(3.16) That is, (1−𝛽_𝑛)𝐼−𝜖_𝑛𝐴 is positive. It follows that ‖‖1−𝛽_𝑛𝐼−𝜖_𝑛𝐴‖‖==ssuupp||1−𝛽1_𝑛−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑥,𝑥||∶𝑥∈𝐻,‖𝑥‖=1≤1−𝛽_𝑛−𝜖_𝑛⟨𝐴𝑥,𝑥⟩∶𝑥∈𝐻,‖𝑥‖=1−𝜖_𝑛𝛾.(3.17) We will split the proof of Theorem 3.3 into six steps.Step 1. We claim that the sequence {𝑥_𝑛} is bounded.
Indeed, let 𝑥^∗∈ℱ, by Lemmas 2.2 and 2.9, we obtain 𝑥^∗=𝑆𝑥^∗=𝑃_𝐶𝑥^∗−𝛼_𝑛𝐵𝑥^∗=𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥^∗−𝑟_𝑛Ψ₁𝑥^∗=𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑥^∗−𝑠_𝑛Ψ₂𝑥^∗.(3.18) Since 𝑢_𝑛=𝑇^{(Θ1𝑟,𝜑)}_𝑛(𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛)∈dom𝜑,Ψ₁ is 𝜉-inverse-strongly monotone, and 0≤𝑟_𝑛≤2𝜉, we know that, for any 𝑛∈ℕ, we have ‖‖𝑢_𝑛−𝑥^∗‖‖²=‖‖𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥^∗−𝑟_𝑛Ψ₁𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑥^∗−𝑟_𝑛Ψ₁𝑥^∗‖‖²=‖‖𝑥_𝑛−𝑥^∗−𝑟_𝑛Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖²=‖‖𝑥_𝑛−𝑥^∗‖‖²−2𝑟_𝑛⟨𝑥_𝑛−𝑥^∗,Ψ₁𝑥_𝑛−Ψ₁𝑥^∗⟩+𝑟^2𝑛‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−2𝑟_𝑛𝜉‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖²+𝑟^2𝑛‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖²=‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑟_𝑛𝑟_𝑛−2𝜉‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖².(3.19) Similarly, from (3.19), 𝑣_𝑛=𝑇^{(Θ2𝑠,𝜑)}_𝑛(𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛)∈dom𝜑, and 0≤𝑠_𝑛≤2𝛽, we can prove that ‖‖𝑣_𝑛−𝑥^∗‖‖²=‖‖𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑥^∗−𝑠_𝑛Ψ₂𝑥^∗‖‖²≤‖‖𝑢_𝑛−𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²,(3.20) and hence ‖‖𝑣_𝑛−𝑥^∗‖‖≤‖‖𝑢_n−𝑥^∗‖‖≤‖‖𝑥_𝑛−𝑥^∗‖‖.(3.21) Let 𝑥^∗∈ℱ, and form Lemma 3.2  𝑆−𝛼_𝑛𝐵𝑆 is a nonexpansive mapping and from Lemma  2.2 𝑥^∗=𝑃_𝐶(𝑥^∗−𝛼_𝑛𝐵𝑥^∗), we have ‖‖𝑧_𝑛−𝑥^∗‖‖≤=‖‖‖‖𝑃_𝐶𝑆𝑣𝑆_𝑛𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝑃_𝐶𝑥^∗−𝛼_𝑛𝐵𝑥^∗‖‖=‖‖𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝑥^∗−𝛼_𝑛𝐵𝑥^∗‖‖=‖‖𝑆−−𝛼_𝑛𝛼_𝑛𝐵𝑆𝑣_𝑛−𝑆𝑥^∗−𝛼_𝑛𝐵𝑆𝑥^∗‖‖≤‖‖𝑣_𝑛𝐵𝑆𝑣_𝑛−𝑆−𝛼_𝑛𝐵𝑆𝑥^∗‖‖‖‖𝑦_𝑛−𝑥^∗‖‖≤‖‖𝑥_𝑛−𝑥^∗‖‖,−𝑥^∗‖‖=≤‖‖𝜖1_𝑛−𝛾𝛽𝑓_𝑛𝑥_𝑛−𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗‖‖≤1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑧_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝜖_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖≤1−𝜖_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝜖_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖≤1−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖+𝜖_𝑛𝛾‖‖𝑓𝑥_𝑛−𝑓𝑥^∗‖‖+𝜖_𝑛‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖=1−𝛾‖‖𝑥_𝑛−𝑥^∗‖‖+𝜖_𝑛𝛾𝜂‖‖𝑥_𝑛−𝑥^∗‖‖+𝜖_𝑛‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖𝛾−𝜂𝛾𝜖_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝜖_𝑛‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖,(3.22) which yields that ‖‖≤𝑥𝛾_𝑛+1𝑛−𝑥^∗‖‖≤𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛾_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖=𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛾_𝑛1−𝛾−𝜂𝛾𝜖_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝜖_𝑛‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖=1‖‖−𝑥_𝑛1−−𝑥𝛾^∗‖‖_𝑛+1−𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖−1−𝛾_𝑛𝛾−𝜂𝛾𝜖_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛾_𝑛𝜖_𝑛‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖𝛾−𝜂𝛾𝜖_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛾_𝑛𝜖_𝑛‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖.(3.23) By mathematical induction, putting 𝐷=max{‖𝑥₁−𝑥^∗‖,‖𝛾𝑓(𝑥^∗)−𝐴𝑥^∗‖/𝛾−𝜂𝛾}, we have that ‖𝑥_𝑛−𝑥^∗‖≤𝐷 for all 𝑛≥1. Indeed, we can easily see that ‖𝑥₁−𝑥^∗‖≤𝐷. Suppose that ‖𝑥_𝑘−𝑥^∗‖≤𝐷 for some positive integral 𝑘. Then we have that ‖‖𝑥_𝑘+1−𝑥^∗‖‖≤≤11−−11−−𝛾𝛾_𝑘𝑘𝛾−𝜂𝛾𝜖_𝑘‖‖𝑥_𝑘−𝑥^∗‖‖+1−𝛾_𝑘𝜖_𝑘‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖=1−1−𝛾_𝑘𝛾−𝜂𝛾𝜖_𝑘𝐷+1−𝛾_𝑘𝜖_𝑘‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖𝛾−𝜂𝛾𝜖_𝑘𝐷+1−𝛾_𝑘𝛾−𝜂𝛾𝜖_𝑘‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖≤1−1−𝛾_𝑘𝛾−𝜂𝛾=𝐷.𝛾−𝜂𝛾𝜖_𝑘𝐷+1−𝛾_𝑘𝛾−𝜂𝛾𝜖_𝑘𝐷(3.24) This shows that {𝑥_𝑛} is bounded in 𝐻. From (3.21), we know that {𝑢_𝑛} and {𝑣_𝑛} are bounded in 𝐶 and so {𝑦_𝑛}, {Ψ₁𝑢_𝑛}, {Ψ₂𝑥_𝑛}, {𝑆𝑣_𝑛}, {𝐵𝑆𝑣_𝑛}, and {𝑓(𝑥_𝑛)} are bounded sequence in 𝐻.Step 2. We claim that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Since 𝑆−𝛼_𝑛𝐵𝑆 is nonexpansive, we have ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖=≤‖‖‖‖𝑃_𝐶𝑆𝑣𝑆𝑣_𝑛+1𝑛+1−𝛼_𝑛+1𝐵𝑆𝑣_𝑛+1−𝑃_𝐶𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛‖‖=‖‖𝑆𝑣_𝑛+1−𝛼_𝑛+1𝐵𝑆𝑣_𝑛+1−𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛‖‖≤‖‖𝑆𝑣_𝑛+1−𝛼_𝑛+1𝐵𝑆𝑣_𝑛+1−𝑆𝑣_𝑛−𝛼_𝑛+1𝐵𝑆𝑣_𝑛+𝛼_𝑛−𝛼_𝑛+1𝐵𝑆𝑣_𝑛‖‖≤‖‖𝑣_𝑛+1−𝛼_𝑛+1𝐵𝑆𝑣_𝑛+1−𝑆𝑣_𝑛−𝛼_𝑛+1𝐵𝑆𝑣_𝑛‖‖+||𝛼_𝑛−𝛼_𝑛+1||‖‖𝐵𝑆𝑣_𝑛‖‖−𝑣_𝑛‖‖+||𝛼_𝑛−𝛼_𝑛+1||‖‖𝐵𝑆𝑣_𝑛‖‖.(3.25) Next, we estimate ‖𝑢_𝑛+1−𝑢_𝑛‖. Observing that 𝑢_𝑛=𝑇^{(Θ1𝑟,𝜑)}_𝑛(𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛) and 𝑢_𝑛+1=𝑇^{(Θ1𝑟,𝜑)}_𝑛+1(𝑥_𝑛+1−𝑟_𝑛+1Ψ₁𝑥_𝑛+1), we have ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖==‖‖‖‖𝑇𝑇^{(Θ1𝑟,𝜑)}_{𝑛+1(Θ1𝑟,𝜑)𝑛+1}𝑥_𝑛+1−𝑟_𝑛+1Ψ₁𝑥_𝑛+1−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖+𝑇_{(Θ1𝑟,𝜑)𝑛+1}𝑥_𝑛+1−𝑟_𝑛+1Ψ₁𝑥_𝑛+1−𝑇^{(Θ1𝑟,𝜑)}_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛≤‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖+‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}𝑥_𝑛+1−𝑟_𝑛+1Ψ₁𝑥_𝑛+1−𝑇^{(Θ1𝑟,𝜑)}_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖≤‖‖𝑥_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖+‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝑟_𝑛+1Ψ₁𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖=‖‖𝑥_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖+‖‖𝑇−_{(Θ1𝑟,𝜑)𝑛+1}𝑥_𝑛−𝑟_𝑛+1Ψ₁𝑥_𝑛+1−Ψ₁𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛Ψ₁𝑥_𝑛‖‖≤‖‖𝑥_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖+‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝑥_𝑛−𝑟_𝑛+1Ψ₁𝑥_𝑛+1−Ψ₁𝑥_𝑛‖‖+||𝑟_𝑛+1−𝑟_𝑛||‖‖Ψ₁𝑥_𝑛‖‖≤‖‖𝑥_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖+‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝑥_𝑛‖‖+||𝑟_𝑛+1−𝑟_𝑛||‖‖Ψ₁𝑥_𝑛‖‖𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖.(3.26) Similarly, we can prove that ‖‖𝑣_𝑛+1−𝑣_𝑛‖‖≤+‖‖‖‖𝑢𝑇_𝑛+1−_{(Θ2𝑠,𝜑)𝑛+1}𝑢_𝑛‖‖+||𝑠_𝑛+1−𝑠_𝑛||‖‖Ψ₂𝑢_𝑛‖‖𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛‖‖.(3.27) Substitution (3.26) into (3.27), we derive ‖‖𝑣_𝑛+1−𝑣_𝑛‖‖≤+‖‖‖‖𝑥𝑇_{𝑛+1(Θ1𝑟,𝜑)𝑛+1}−𝑥_𝑛‖‖+||𝑟_𝑛+1−𝑟_𝑛||‖‖Ψ₁𝑥_𝑛‖‖+||𝑠_𝑛+1−𝑠_𝑛||‖‖Ψ₂𝑢_𝑛‖‖+‖‖𝑇_{(Θ2𝑠,𝜑)𝑛+1}𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛‖‖.(3.28) Since {Ψ₁𝑥_𝑛},{Ψ₂𝑢_𝑛}, and {𝐵𝑆𝑣_𝑛} are bounded, 𝐾 is an appropriate constant such that 𝐾≥maxsup_𝑛≥1‖‖Ψ₁𝑥_𝑛‖‖,sup_𝑛≥1‖‖Ψ₂𝑢_𝑛‖‖,sup_𝑛≥1‖‖𝐵𝑆𝑣_𝑛‖‖.(3.29) Substitution (3.28) into (3.25), we obtain ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤≤‖‖‖‖𝑣𝑥_𝑛+1𝑛+1−𝑣_𝑛‖‖+||𝛼_𝑛−𝛼_𝑛+1||‖‖𝐵𝑆𝑣_𝑛‖‖+‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝑥_𝑛‖‖+||𝑟_𝑛+1−𝑟_𝑛||‖‖Ψ₁𝑥_𝑛‖‖+||𝑠_𝑛+1−𝑠_𝑛||‖‖Ψ₂𝑢_𝑛‖‖+||𝛼_𝑛−𝛼_𝑛+1||‖‖𝐵𝑆𝑣_𝑛‖‖+‖‖𝑇_{(Θ2𝑠,𝜑)𝑛+1}𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖≤‖‖𝑥_𝑛+1𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛‖‖+‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝑥_𝑛‖‖+𝐾||𝑟_𝑛+1−𝑟_𝑛||+||𝑠_𝑛+1−𝑠_𝑛||+||𝛼_𝑛−𝛼_𝑛+1||+‖‖𝑇_{(Θ2𝑠,𝜑)𝑛+1}𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛‖‖.(3.30) From (3.13), we have 𝑦_𝑛+1𝑦_𝑛=𝜖_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛,=𝜖_𝑛+1𝛾𝑓𝑥_𝑛+1+𝛽_𝑛+1𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝜖_𝑛+1𝐴𝑧_𝑛+1.(3.31) Simple calculations show that 𝑦_𝑛+1−𝑦_𝑛=+𝜖𝛽_𝑛+1𝛾_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝜖_𝑛+1−𝜖_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛−𝛽_𝑛+1−𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛+1𝐼−𝜖_𝑛+1𝐴𝑧_𝑛+1−𝑧_𝑛−𝛽_𝑛𝑧_𝑛−𝜖_𝑛+1−𝜖_𝑛𝐴𝑧_𝑛=+𝜖𝛽_𝑛+1𝛾_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝜖_𝑛+1−𝜖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛+1−𝑥𝛽_𝑛+1𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑧_𝑛𝐼−𝜖_𝑛+1𝐴𝑧_𝑛+1−𝑧_𝑛,(3.32) which yields that ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤+𝜖𝛽_𝑛+1𝛾_𝑛+1‖‖𝑓𝑥_𝑛+1−𝑓𝑥_𝑛‖‖+||𝜖_𝑛+1−𝜖_𝑛||‖‖𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛‖‖≤𝜖_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝛽_𝑛+1−𝛽_𝑛||‖‖𝑥_𝑛−𝑧_𝑛‖‖+1−𝛽_𝑛+1−𝜖_𝑛+1𝛾‖‖𝑧_𝑛+1−𝑧_𝑛‖‖+𝛽𝛾_𝑛+1𝜂‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝜖_𝑛+1−𝜖_𝑛||‖‖𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝛽_𝑛+1−𝛽_𝑛||‖‖𝑥_𝑛−𝑧_𝑛‖‖+1−𝛽_𝑛+1−𝜖_𝑛+1𝛾‖‖𝑧_𝑛+1−𝑧_𝑛‖‖.(3.33) Substitution (3.30) into (3.33) yields that ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤+𝜖𝛽_𝑛+1𝛾_𝑛+1𝜂‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝜖_𝑛+1−𝜖_𝑛||‖‖𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛‖‖×‖‖𝑥‖‖𝑥_𝑛+1𝑛+1−𝑥_𝑛‖‖+||𝛽_𝑛+1−𝛽_𝑛||‖‖𝑥_𝑛−𝑧_𝑛‖‖+1−𝛽_𝑛+1−𝜖_𝑛+1𝛾+‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝑥_𝑛‖‖+𝐾||𝑟_𝑛+1−𝑟_𝑛||+||𝑠_𝑛+1−𝑠_𝑛||+||𝛼_𝑛−𝛼_𝑛+1||+‖‖𝑇_{(Θ2𝑠,𝜑)𝑛+1}𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖=1−𝜖_𝑛+1𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛‖‖+||𝛽_𝑛+1𝛾−𝛾𝜂‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝜖_𝑛+1−𝜖_𝑛||‖‖𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛‖‖×||𝑟_𝑛+1−𝛽_𝑛||‖‖𝑥_𝑛−𝑧_𝑛‖‖+1−𝛽_𝑛+1−𝜖_𝑛+1𝛾𝐾×‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝑟_𝑛||+||𝑠_𝑛+1−𝑠_𝑛||+||𝛼_𝑛−𝛼_𝑛+1||+1−𝛽_𝑛+1−𝜖_𝑛+1𝛾+1−𝛽_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖≤1−𝜖_𝑛+1−𝜖_𝑛+1𝛾‖‖𝑇^{(Θ2𝑠,𝜑)}_𝑛+1𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛‖‖+𝑀||𝜖_𝑛+1𝛾−𝛾𝜂‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝜖_𝑛||+||𝛽_𝑛+1−𝛽_𝑛||+||𝑟_𝑛+1−𝑟_𝑛||+||𝑠_𝑛+1−𝑠_𝑛||+||𝛼_𝑛−𝛼_𝑛+1||+‖‖𝑇_{(Θ2𝑠,𝜑)𝑛+1}𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛‖‖,(3.34) where 𝑀 is an appropriate constant such that 𝑀≥maxsup_𝑛≥1‖‖𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛‖‖,sup_𝑛≥1‖‖𝑥_𝑛−𝑧_𝑛‖‖,𝐾.(3.35) Since 𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑦_𝑛 and ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤+𝑀‖‖𝑇||𝜖_{𝑛+1(Θ1𝑟,𝜑)𝑛+1}−𝜖_𝑛||+||𝛽_𝑛+1−𝛽_𝑛||+||𝑟_𝑛+1−𝑟_𝑛||+||𝑠_𝑛+1−𝑠_𝑛||+||𝛼_𝑛−𝛼_𝑛+1||+‖‖𝑇_{(Θ2𝑠,𝜑)𝑛+1}𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛‖‖,(3.36) next, we estimate ‖‖𝑇^{(Θ1𝑟,𝜑)}_𝑛+1𝑥_n−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖.(3.37) Note that liminf_𝑛→∞𝑟_𝑛>0; there exists a constant 𝑟̃>0 such that 𝑟_𝑛≥𝑟̃>0 for all 𝑛≥1. From Lemma 3.1, we get ‖‖𝑇^{(Θ1𝑟,𝜑)}_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖²≤𝑟^𝑛+1−𝑟_𝑛𝑟_𝑛+1𝑇^{(Θ1𝑟,𝜑)𝑛+1}𝑥_𝑛𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛,=𝑟_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑟_𝑛𝑟_𝑛+1𝑇^{(Θ1𝑟,𝜑)𝑛+1}𝑥_𝑛𝑇_{(Θ1𝑟,𝜑)𝑛+1}−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛,≤||𝑟_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑟_𝑛||𝑟_𝑛+1‖‖𝑇_{(Θ1𝑟,𝜑)𝑛+1}‖‖𝑇^{(Θ1𝑟,𝜑)}_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖.(3.38) It follows that ‖‖𝑇≤^{(Θ1𝑟,𝜑)}_𝑛+1||𝑟𝑥_𝑛𝑛+1−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖−𝑟_𝑛||𝑟̃‖‖𝑇^{(Θ1𝑟,𝜑)𝑛+1}𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖.(3.39) Since lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0, we obtain lim_𝑛→∞‖‖𝑇^{(Θ1𝑟,𝜑)}_𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛‖‖=0.(3.40) Similarly, we can prove that lim_𝑛→∞‖‖𝑇^{(Θ2𝑠,𝜑)}_𝑛+1𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛‖‖=0.(3.41) Consequently, from (3.40), (3.41), and conditions in Theorem 3.3, we obtain lim_𝑛→∞‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.42) It follows from Lemma 2.7 that lim_𝑛→∞‖‖𝑦_𝑛−𝑥_𝑛‖‖=0.(3.43) In view of (3.13), we see that ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=1−𝛾_𝑛‖‖𝑥_𝑛−𝑦_𝑛‖‖,∀𝑛≥1,(3.44) which, combining with (3.43) and 0<ℎ≤𝛾_𝑛≤𝑗<1, yields that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=0.(3.45)Step 3. We claim that lim_𝑛→∞‖𝑆𝑧_𝑛−𝑧_𝑛‖=0.
Observing that 𝑦_𝑛=𝜖_𝑛𝛾𝑓(𝑥_𝑛)+𝛽_𝑛𝑥_𝑛+((1−𝛽_𝑛)𝐼−𝜖_𝑛𝐴)𝑧_𝑛, we have ‖‖𝑦_𝑛−𝑧_𝑛‖‖≤𝜖_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑧_𝑛‖‖,(3.46) which, combining with the conditions (C1) and (C2), gives lim_𝑛→∞‖‖𝑦_𝑛−𝑧_𝑛‖‖=0.(3.47) From (3.43) and (3.47), we have lim_𝑛→∞‖‖𝑥_𝑛−𝑧_𝑛‖‖=0.(3.48) For any 𝑥^∗∈ℱ, we see that ‖‖𝑧_𝑛−𝑥^∗‖‖²=‖‖𝑃_𝐶𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝑃_𝐶𝑥^∗−𝛼_𝑛𝐵𝑥^∗‖‖²≤‖‖𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝑥^∗−𝛼_𝑛𝐵𝑥^∗‖‖²=‖‖𝑆𝑣_𝑛−𝑥^∗−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²=‖‖𝑆𝑣_𝑛−𝑥^∗‖‖²−2𝛼_𝑛⟨𝑆𝑣_𝑛−𝑥^∗,𝐵𝑆𝑣_𝑛−𝐵𝑥^∗⟩+𝛼^2𝑛‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²≤‖‖𝑣_𝑛−𝑥^∗‖‖²−2𝛼_𝑛⟨𝑆𝑣_𝑛−𝑥^∗,𝐵𝑆𝑣_𝑛−𝐵𝑥^∗⟩+𝛼^2𝑛‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−2𝛼_𝑛−𝑢‖‖𝐵𝑆𝑣^𝑛−𝐵𝑥^∗‖‖²+𝑣‖‖𝑆𝑣_𝑛−𝑥^∗‖‖²+𝛼^2𝑛‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑢‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²−2𝛼_𝑛𝑣‖‖𝑆𝑣_𝑛−𝑥^∗‖‖²+𝛼^2𝑛‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑢‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²−2𝛼^𝑛𝑣𝜔²‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²+𝛼^2𝑛‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²=‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑢+𝛼^2𝑛−2𝛼^𝑛𝑣𝜔²‖‖𝐵𝑆𝑣^𝑛−𝐵𝑥^∗‖‖²,(3.49)‖‖𝑦_𝑛−𝑥^∗‖‖²=‖‖1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+𝜖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²=‖‖1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗‖‖²++2𝜖^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝛽_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝜖_𝑛𝛾‖‖𝑧_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²=+12𝛽−_𝑛𝛽𝜖_𝑛𝑛⟨𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗⟩+2𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝜖_2𝑛−𝜖_𝑛𝛾²‖‖𝑧_𝑛−𝑥^∗‖‖²+𝛽^2𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+21−𝛽_𝑛−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑧_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+2𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝛽_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝜖_2𝑛−𝜖_𝑛𝛾²‖‖𝑧_𝑛−𝑥^∗‖‖²+𝛽^2𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑧^𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗=1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾²−21−𝜖_𝑛𝛾𝛽_𝑛+𝛽^2𝑛‖‖𝑧^𝑛−𝑥^∗‖‖²+𝛽^2𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾𝛽_𝑛−𝛽^2𝑛‖‖𝑧_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²=+21𝛽−_𝑛𝜖𝜖_𝑛𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+2𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾²−1−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑧^𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²++2𝜖^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗=1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑧_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²++2𝜖^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑢+𝛼^2𝑛−2𝛼^𝑛𝑣𝜔²‖‖𝐵𝑆𝑣^𝑛+1−𝜖_𝑛−𝐵𝑥^∗‖‖+2𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗=1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝛼_𝑛𝑢+𝛼^2𝑛−2𝛼^𝑛𝑣𝜔²‖‖𝐵𝑆𝑣^𝑛−𝐵𝑥^∗‖‖+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²++22𝜖𝛽_𝑛𝑛𝜖_𝑛⟨𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗⟩=1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝛼_𝑛𝑢+𝛼^2𝑛−2𝛼^𝑛𝑣𝜔²‖‖𝐵𝑆𝑣^𝑛+𝜖_2𝑛−𝐵𝑥^∗‖‖+2𝜀‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤‖‖𝑥_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝑥^∗‖‖²+1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾2𝛼_𝑛𝑢+𝛼^2𝑛−2𝛼^𝑛𝑣𝜔²‖‖𝐵𝑆𝑣^𝑛+𝜀_2𝑛−𝐵𝑥^∗‖‖+2𝜀‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.50) Furthermore, from (3.13) and Lemma 2.12(1), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛−𝑥^∗‖‖²≤𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛾_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²≤×𝛾_𝑛‖‖𝑥‖‖_𝑛𝑥−_𝑛𝑥^∗‖‖²+1−𝛾_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝛼_𝑛𝑢+𝛼^2𝑛−2𝛼^𝑛𝑣𝜔²‖‖𝐵𝑆𝑣^𝑛+𝜖_2𝑛−𝐵𝑥^∗‖‖+2𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤‖‖𝑥_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝑥^∗‖‖²+1−𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝛼_𝑛𝑢+𝛼^2𝑛−2𝛼^𝑛𝑣𝜔²‖‖𝐵𝑆𝑣^𝑛+1−𝛾_𝑛−𝐵𝑥^∗‖‖+21−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+21−𝛾_𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.51) It follows that (1−𝑗)1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝑒𝑣𝜔²≤1−𝛾_𝑛−2𝑔𝑢−𝑔²‖‖𝐵𝑆𝑣^𝑛−𝐵𝑥^∗‖‖1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝛼^𝑛𝑣𝜔²−2𝛼_𝑛𝑢−𝛼^2𝑛‖‖𝐵𝑆𝑣^𝑛≤‖‖𝑥_𝑛−𝐵𝑥^∗‖‖−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²+1−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²++2211−−𝛾𝛾_𝑛𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤‖‖𝑥_𝑛𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+1−−𝑥𝛾_𝑛+1𝑛‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖+21−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+21−𝛾_𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.52) From the condition (C1) and (3.45), we arrive at lim_𝑛→∞‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖=0.(3.53) On the other hand, we have ‖‖𝑧_𝑛−𝑥^∗‖‖²=‖‖𝑃_𝐶𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝑃_𝐶𝑥^∗−𝛼_𝑛𝐵𝑥^∗‖‖²≤=1𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝑥^∗−𝛼_𝑛𝐵𝑥^∗,𝑧_𝑛−𝑥^∗2‖‖𝑆𝑣^𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝑥^∗−𝛼_𝑛𝐵𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑥^∗‖‖²≤1−‖‖𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝑥^∗−𝛼_𝑛𝐵𝑥^∗−𝑧_𝑛−𝑥^∗‖‖²2‖‖𝑣^𝑛≤1−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑥^∗‖‖²−‖‖𝑆𝑣_𝑛−𝑧_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²2‖‖𝑥^𝑛−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑥^∗‖‖²−‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖²−𝛼^2𝑛‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖²+2𝛼_𝑛⟨𝑆𝑣_𝑛−𝑧_𝑛,𝐵𝑆𝑣_𝑛−𝐵𝑥^∗⟩,(3.54) which yields that ‖‖𝑧_𝑛−𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖²+2𝛼_𝑛‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖.(3.55) Substituting (3.55) into (3.50), we have ‖‖𝑦_𝑛−𝑥^∗‖‖²≤1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑧_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²++2𝜖^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+1−𝜖𝛾_𝑛1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖²+2𝛼_𝑛‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖+2𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗=1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²−1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖²++𝜖1_2𝑛−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝛼_𝑛‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖‖‖𝐵𝑆𝑥_𝑛−𝐵𝑥^∗‖‖+2𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤‖‖𝑥_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝑥^∗‖‖²−1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖²++𝜖1_2𝑛−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝛼_𝑛‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖+2𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.56) Using (3.51) and (3.56), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛾_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²≤×𝛾_𝑛‖‖𝑥‖‖_𝑛𝑥−_𝑛𝑥^∗‖‖²+1−𝛾_𝑛−𝑥^∗‖‖²−1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖²++𝜖1_2𝑛−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝛼_𝑛‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖+2𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗=‖‖𝑥_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑆𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝑥^∗‖‖²−1−𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖²++211−−𝛾_𝑛𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝛼_𝑛‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖+21−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+21−𝛾_𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑆𝑣_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.57) It follows that 1−𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²++211−−𝛾_𝑛𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝛼_𝑛‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖+21−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+21−𝛾_𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤‖‖𝑥_𝑛𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑆𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+21−−𝑥𝛾_𝑛+1𝑛‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖+1−𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝛼_𝑛‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖‖‖𝐵𝑆𝑣_𝑛−𝐵𝑥^∗‖‖+21−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+21−𝛾_𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑆𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.58) From condition (C1), (3.45), and (3.53), we obtain lim_𝑛→∞‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖=0.(3.59) Consequently, from (3.50) we derive that ‖‖𝑦_𝑛−𝑥^∗‖‖²≤1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑧_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²++2𝜖^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑣_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²++2𝜖^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗=1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑥^∗−𝑠_𝑛Ψ₂𝑥^∗‖‖²++21𝜖_𝑛−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛−𝑥^∗−𝑠_𝑛Ψ₂𝑥^∗‖‖²++21𝜖_𝑛−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+1−𝜖𝛾_𝑛1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑢_𝑛−𝑥^∗‖‖²+𝑠_𝑛𝑠_𝑛−2𝛽‖‖Ψ₂𝑢_𝑛−Ψ₂𝑥^∗‖‖²+2𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+1−𝜖𝛾_𝑛1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑠_𝑛𝑠_𝑛−2𝛽‖‖Ψ₂𝑢_𝑛−Ψ₂𝑥^∗‖‖²+2𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤‖‖𝑥_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝑥^∗‖‖²+1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝑠_𝑛𝑠_𝑛−2𝛽‖‖Ψ₂𝑢_𝑛−Ψ₂𝑥^∗‖‖²++2𝜖^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.60) From (3.51) and (3.60), we obtain ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛾_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²≤×𝛾_𝑛‖‖𝑥‖‖_𝑛𝑥−_𝑛𝑥^∗‖‖²+1−𝛾_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝑠_𝑛𝑠_𝑛−2𝛽‖‖Ψ₂𝑢_𝑛−Ψ₂𝑥^∗‖‖²++𝜖2^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗=‖‖𝑥_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝑥^∗‖‖²+1−𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝑠_𝑛𝑠_𝑛−2𝛽‖‖Ψ₂𝑢_𝑛−Ψ₂𝑥^∗‖‖²++211−−𝛾_𝑛𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+21−𝛾_𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.61) So, we obtain (1−𝑗)1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝑐(2𝛽−𝑑)‖‖Ψ₂𝑢_𝑛−Ψ₂𝑥^∗‖‖²≤1−𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝑠_𝑛2𝛽−𝑠_𝑛‖‖Ψ₂𝑢_𝑛−Ψ₂𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²+1−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²++2211−−𝛾𝛾_𝑛𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤‖‖𝑥_𝑛𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖+1−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²++2211−−𝛾𝛾_𝑛𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.62) Since lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞𝜖_𝑛=0, we obtain lim_𝑛→∞‖‖Ψ₂𝑢_𝑛−Ψ₂𝑥^∗‖‖=0.(3.63) Similarly, we can prove that lim_𝑛→∞‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖=0.(3.64) In addition, from the firmly nonexpansivity of 𝑇^{(Θ1𝑟,𝜑)}_𝑛, we have ‖‖𝑢_𝑛−𝑥^∗‖‖²=‖‖𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥^∗−𝑟_𝑛Ψ₁𝑥^∗‖‖²≤=1𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑥^∗−𝑟_𝑛Ψ₁𝑥^∗,𝑢_𝑛−𝑥^∗2‖‖𝑥^𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑥^∗−𝑟_𝑛Ψ₁𝑥^∗‖‖²+‖‖𝑢_𝑛−𝑥^∗‖‖²≤1−‖‖𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−𝑥^∗−𝑟_𝑛Ψ₁𝑥^∗−𝑢_𝑛−𝑥^∗‖‖²2‖‖𝑥^𝑛=1−𝑥^∗‖‖²+‖‖𝑢_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖²2‖‖𝑥^𝑛−𝑥^∗‖‖²+‖‖𝑢_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²+2𝑟_𝑛⟨𝑥_𝑛−𝑢_𝑛,Ψ₁𝑥_𝑛−Ψ₁𝑥^∗⟩−𝑟^2𝑛‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖².(3.65) So, we obtain ‖‖𝑢_𝑛−𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²+2𝑟_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖.(3.66) Similarly, we can prove that ‖‖𝑣_𝑛−𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑢_𝑛−𝑣_𝑛‖‖²+2𝑠_𝑛‖‖𝑢_𝑛−𝑣_𝑛‖‖‖‖Ψ₂𝑢_𝑛−Ψ₂𝑥^∗‖‖.(3.67) Substituting (3.66) into (3.50), we have ‖‖𝑦_𝑛−𝑥^∗‖‖²≤1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑧_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²++2𝜖^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑢_𝑛−𝑥^∗‖‖²+1−𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²++2𝜖^2𝑛𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+1−𝜖𝛾_𝑛1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²+2𝑟_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖+2𝜖_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗=1−𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²−1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑢_𝑛‖‖²++𝜖1_2𝑛−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝑟_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖+2𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗≤‖‖𝑥_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝑥^∗‖‖²−1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑢_𝑛‖‖²++𝜖1_2𝑛−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝑟_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖+2𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.68) Using (3.51) and (3.68), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛾_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²≤×𝛾_𝑛‖‖𝑥‖‖_𝑛𝑥−_𝑛𝑥^∗‖‖²+1−𝛾_𝑛−𝑥^∗‖‖²−1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑢_𝑛‖‖²++𝜖1_2𝑛−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾2𝑟_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖+2𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+2𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗=‖‖𝑥_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗−𝑥^∗‖‖²−1−𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑢_𝑛‖‖²++211−−𝛾_𝑛𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝑟_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖+21−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+21−𝛾_𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.69) It follows that 1−𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑢_𝑛‖‖²≤+‖‖2𝑥_𝑛1−−𝑥𝛾_𝑛+1𝑛‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖+1−𝛾_𝑛1−𝜖_𝑛𝛾1−𝛽_𝑛−𝜖_𝑛𝛾𝑟_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖‖‖Ψ₁𝑥_𝑛−Ψ₁𝑥^∗‖‖+21−𝛾_𝑛𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥^∗‖‖²+21−𝛾_𝑛𝛽_𝑛𝜖_𝑛𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝜖_𝑛1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑥^∗,𝛾𝑓𝑥_𝑛−𝐴𝑥^∗.(3.70) Since lim_𝑛→∞𝜖_𝑛=0,lim_𝑛→∞‖Ψ₁𝑥_n−Ψ₁𝑥^∗‖=0, and lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0, we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑢_𝑛‖‖=0.(3.71) Similarly, we can prove that lim_𝑛→∞‖‖𝑢_𝑛−𝑣_𝑛‖‖=0.(3.72) Furthermore, by the triangular inequality, we also have ‖‖𝑥_𝑛−𝑣_𝑛‖‖≤‖‖𝑥_𝑛−𝑢_𝑛‖‖+‖‖𝑢_𝑛−𝑣_𝑛‖‖.(3.73) Applying (3.71) and (3.72), we have lim_𝑛→∞‖‖𝑥_𝑛−𝑣_𝑛‖‖=0.(3.74) Since ‖‖𝑧_𝑛−𝑣_𝑛‖‖≤‖‖𝑧_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑣_𝑛‖‖,(3.75) so we get lim_𝑛→∞‖‖𝑧_𝑛−𝑣_𝑛‖‖=0.(3.76) Also, observe that ‖‖𝑆𝑧_𝑛−𝑧_𝑛‖‖≤≤‖‖‖‖𝑆𝑧_𝑛𝑧_𝑛−𝑆𝑣_𝑛‖‖+‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖−𝑣_𝑛‖‖+‖‖𝑆𝑣_𝑛−𝑧_𝑛‖‖.(3.77) Consequently, we obtain lim_𝑛→∞‖‖𝑆𝑧_𝑛−𝑧_𝑛‖‖=0.(3.78)Step 4. We prove that the mapping 𝑃_ℱ(𝛾𝑓+(𝐼−𝐴)) has a unique fixed point.
Since 𝑓 is a contraction of 𝐶 into itself with coefficient 𝜂∈(0,1), then, we have ‖‖𝑃_ℱ(𝛾𝑓+(𝐼−𝐴))(𝑥)−𝑃_ℱ(𝛾𝑓+(𝐼−𝐴))(𝑦)‖‖≤≤≤𝛾𝛾‖𝜂‖(‖𝛾𝑓𝑓𝑥(𝑥−+)𝑦(−𝐼‖𝑓−+(𝐴𝑦))1)‖(−𝑥+)‖−𝐼(−𝛾𝑓𝐴+‖(‖𝐼𝑥−−𝐴𝑦)‖)(𝑦)‖=1−𝛾‖𝑥−𝑦‖𝛾−𝜂𝛾‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(3.79) Since 0<1−(𝛾−𝜂𝛾)<1, it follows that 𝑃_ℱ(𝛾𝑓+(𝐼−𝐴)) is a contraction of 𝐶 into itself. Therefore by the Banach Contraction Mapping Principle, it has a unique fixed point, say 𝑧∈𝐶, that is, 𝑧=𝑃_ℱ(𝛾𝑓+(𝐼−𝐴))(𝑧).(3.80)Step 5. We claim that limsup_𝑛→∞⟨𝛾𝑓𝑧−𝐴𝑧,𝑥_𝑛−𝑧⟩≤0, where 𝑧 is the unique solution of the variational inequality ⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑥−𝑧⟩≤0, for all 𝑥∈ℱ.
Since 𝑧=𝑃_ℱ(𝛾𝑓+(𝐼−𝐴))(𝑧) is a unique solution of the variational inequality (3.14), to show this inequality, we choose a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝛾𝑓𝑧−𝐴𝑧,𝑥_𝑛−𝑧⟩=lim_𝑖→∞𝛾𝑓𝑧−𝐴𝑧,𝑥_𝑛𝑖−𝑧.(3.81) Correspondingly, there exists a subsequence {𝑧_𝑛𝑖} of {𝑧_𝑛} such that limsup_𝑛→∞⟨𝛾𝑓𝑧−𝐴𝑧,𝑧_𝑛−𝑧⟩=lim_𝑖→∞𝛾𝑓𝑧−𝐴𝑧,𝑧_𝑛𝑖−𝑧.(3.82) Since {𝑧_𝑛𝑖} is bounded, there exists a subsequence {𝑧_𝑛𝑖𝑗} of {𝑧_𝑛𝑖} which converges weakly to 𝑤. Without loss of generality, we can assume that 𝑧_𝑛𝑖⇀𝑤. From ‖𝑆𝑧_𝑛−𝑧_𝑛‖→0, we obtain 𝑆𝑧_𝑛𝑖⇀𝑤.
Next, we show that 𝑤∈𝐹(𝑆)∩GMEP(Θ₁,𝜑,Ψ₁)∩GMEP(Θ₂,𝜑,Ψ₂)∩VI(𝐶,𝐵).
First, we show that 𝑤∈𝐹(𝑆).
Assume 𝑤∉𝐹(𝑆). Since 𝑧_𝑛𝑖⇀𝑤 and 𝑤≠𝑆𝑤, it follows by Opial's condition (Lemma 2.4) that liminf_𝑖→∞‖‖𝑧_𝑛𝑖−𝑤‖‖<≤lliimmiinnf_𝑖→∞f_𝑖→∞‖‖𝑧_𝑛𝑖−𝑆𝑤‖‖=liminf_𝑖→∞‖‖𝑧_𝑛𝑖−𝑆𝑧_𝑛𝑖‖‖+‖‖𝑆𝑧_𝑛𝑖−𝑆𝑤‖‖≤liminf_𝑖→∞‖‖𝑆𝑧_𝑛𝑖−𝑆𝑤‖‖‖‖𝑧_𝑛𝑖−𝑤‖‖.(3.83) This is a contradiction. Thus, we have 𝑤∈𝐹(𝑆).
Next, we prove that 𝑤∈GMEP(Θ₁,𝜑,Ψ₁).
For any 𝑦∈𝐶, we have 0≤Θ₁𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+⟨Ψ₁𝑥_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛=Θ₁⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩=Θ₁𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+𝑟1_𝑛⟨𝑟_𝑛Ψ₁𝑥_𝑛,𝑦−𝑢_𝑛⟩+𝑟1_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+𝑟1_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛+𝑟_𝑛Ψ₁𝑥_𝑛⟩,(3.84) which yield that Θ₁𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛≥0,∀𝑦∈𝐶.(3.85) It follows from Lemma 2.9 that 𝑢_𝑛=𝑇^{(Θ1𝑟,𝜑)}_𝑛(𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛) for all 𝑛≥1.
Thus, we conclude that 𝑢_𝑛=𝑇^{(Θ1𝑟,𝜑)}_𝑛(𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛) is equivalent to Θ₁𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+⟨Ψ₁𝑥_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶.(3.86) From (H2), we also have 𝜑(𝑦)−𝜑𝑢_𝑛+⟨Ψ₁𝑥_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥−Θ₁𝑢_𝑛,𝑦≥Θ₁𝑦,𝑢_𝑛.(3.87) Replacing 𝑛 by 𝑛_𝑖, we obtain 𝜑(𝑦)−𝜑𝑢_𝑛𝑖+Ψ₁𝑥_𝑛𝑖,𝑦−𝑢_𝑛𝑖+𝑦−𝑢_𝑛𝑖,𝑢^𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖≥Θ¹𝑦,𝑢_𝑛𝑖.(3.88) Let 𝑢_𝑡=𝑡𝑦+(1−𝑡)𝑤 for all 𝑡∈(0,1] and 𝑦∈𝐶. Since 𝑦∈𝐶 and 𝑤∈𝐶, we obtain 𝑢_𝑡∈𝐶. So, from (3.88) we have ⟨𝑢_𝑡−𝑢_𝑛𝑖,Ψ₁𝑢_𝑡⟩≥−𝑢_𝑡𝑢−_𝑡𝑢_𝑛𝑖,Ψ₁𝑢_𝑡−𝜑𝑢_𝑡+𝜑𝑢_𝑛𝑖−Ψ₁𝑥_𝑛𝑖,𝑢_𝑡−𝑢_𝑛𝑖−𝑢_𝑛𝑖,𝑢^𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖+Θ¹𝑢_𝑡≥𝑢_𝑡,𝑢_𝑛𝑖+𝜑−𝑢𝑢_{𝑛𝑖𝑛𝑖},Ψ₁𝑢_𝑡−Ψ₁𝑢_𝑛𝑖+𝑢_𝑡−𝑢_𝑛𝑖,Ψ₁𝑢_𝑛𝑖−Ψ₁𝑥_𝑛𝑖−𝜑𝑢_𝑡−𝑢_𝑡−𝑢_𝑛𝑖,𝑢^𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖+Θ¹𝑢_𝑡,𝑢_𝑛𝑖.(3.89) Since ‖𝑢_𝑛𝑖−𝑥_𝑛𝑖‖→0, we have ‖Ψ₁𝑢_𝑛𝑖−Ψ₁𝑥_𝑛𝑖‖→0. Further, from the inverse strong monotonicity of Ψ₁, we have 𝑢_𝑡−𝑢_𝑛𝑖,Ψ₁𝑢_𝑡−Ψ₁𝑢_𝑛𝑖≥0.(3.90) So, from (H4), (H5), and the weak lower semicontinuity of 𝜑,(𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0, and 𝑢_𝑛𝑖⇀𝑤, we have ⟨𝑢_𝑡−𝑤,Ψ₁𝑢_𝑡⟩≥−𝜑𝑢_𝑡+𝜑(𝑤)+Θ₁𝑢_𝑡,𝑤,as𝑖→∞.(3.91) From (H1), (H4), and (3.91), we also get 0=≤𝑡ΘΘ₁₁𝑢_𝑡,𝑢_𝑡+𝜑𝑢_𝑡−𝜑𝑢_𝑡=𝑡Θ₁𝑢_𝑡,𝑦+(1−𝑡)Θ₁𝑢_𝑡,𝑤+𝑡𝜑(𝑦)+(1−𝑡)𝜑(𝑤)−𝜑𝑢_𝑡≤𝑡Θ₁𝑢_𝑡,𝑦+𝜑(𝑦)−𝜑𝑢_𝑡+(1−𝑡)Θ₁𝑢_𝑡,𝑤+𝜑(𝑤)−𝜑𝑢_𝑡=𝑡Θ₁𝑢_𝑡,𝑦+𝜑(𝑦)−𝜑𝑢_𝑡+(1−𝑡)⟨𝑢_𝑡−𝑤,Ψ₁𝑢_𝑡⟩𝑢_𝑡,𝑦+𝜑(𝑦)−𝜑𝑢_𝑡+(1−𝑡)𝑡⟨𝑦−𝑤,Ψ₁𝑢_𝑡⟩.(3.92) Dividing by 𝑡, we get Θ₁𝑢_𝑡,𝑦+𝜑(𝑦)−𝜑𝑢_𝑡+(1−𝑡)⟨𝑦−𝑤,Ψ₁𝑢_𝑡⟩≥0.(3.93) Letting 𝑡→0 in the above inequality, we arrive that Θ₁(𝑤,𝑦)+𝜑(𝑦)−𝜑(𝑤)+⟨𝑦−𝑤,Ψ₁𝑤⟩≥0.(3.94) Thus, 𝑤∈GMEP(Θ₁,𝜑,Ψ₁). Similarly, we can prove that 𝑤∈GMEP(Θ₂,𝜑,Ψ₂).
Finally, now we prove that 𝑤∈VI(𝐶,𝐵).
We define the maximal monotone operator: 𝑄𝑤₁=𝐵𝑤¹∅,+𝑁_𝐶𝑤₁,𝑤𝑤₁₁∈𝐶,∉𝐶.(3.95) Since 𝐵 is relaxed (𝑢,𝑣)-cocoercive and condition (C5), we have ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥(−𝑢)‖𝐵𝑥−𝐵𝑦‖²+𝑣‖𝑥−𝑦‖²≥𝑣−𝑢𝜔²‖𝑥−𝑦‖²≥0,(3.96) which yields that 𝐵 is monotone. Thus, 𝑄 is maximal monotone. Let (𝑤₁,𝑤₂)∈𝐺(𝑄). Since 𝑤₂−𝐵𝑤₁∈𝑁_𝐶𝑤₁ and 𝑧_𝑛∈𝐶, we have ⟨𝑤₁−𝑧_𝑛,𝑤₂−𝐵𝑤₁⟩≥0.(3.97) On the other hand, from 𝑧_𝑛=𝑃_𝐶(𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛), we have 𝑤₁−𝑧_𝑛,𝑧_𝑛−𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛≥0,(3.98) and hence 𝑤¹−𝑧_𝑛,𝑧^𝑛−𝑆𝑣_𝑛𝛼_𝑛+𝐵𝑆𝑣_𝑛≥0.(3.99) It follows that 𝑤₁−𝑧_𝑛𝑖,𝑤₂≥≥𝑤𝑤₁₁−𝑧_𝑛𝑖,𝐵𝑤₁−𝑧_𝑛𝑖,𝐵𝑤₁−𝑤₁−𝑧_𝑛𝑖,𝑧^𝑛𝑖−𝑆𝑣_𝑛𝑖𝛼_𝑛𝑖+𝐵𝑆𝑣_𝑛𝑖=𝑤₁−𝑧_𝑛𝑖,𝐵𝑤₁−𝐵𝑆𝑣_𝑛𝑖−𝑧^𝑛𝑖−𝑆𝑣_𝑛𝑖=𝑤₁𝛼_𝑛𝑖−𝑤−₁𝑧_𝑛𝑖,𝐵𝑤₁−𝐵𝑧_𝑛𝑖+𝑤₁−𝑧_𝑛𝑖,𝐵𝑧_𝑛𝑖−𝐵𝑆𝑣_𝑛𝑖−𝑧_𝑛𝑖,𝑧^𝑛𝑖−𝑆𝑣_𝑛𝑖≥𝑤₁𝛼_𝑛𝑖−𝑧_𝑛𝑖,𝐵𝑧_𝑛𝑖−𝐵𝑆𝑣_𝑛𝑖−𝑤₁−𝑧_𝑛𝑖,𝑧^𝑛𝑖−𝑆𝑣_𝑛𝑖𝛼_𝑛𝑖,(3.100) which implies that ⟨𝑤₁−𝑤,𝑤₂⟩≥0.(3.101) Since 𝑄 is maximal monotone, we obtain that 𝑤∈𝑄⁻¹0. From Lemma 2.3, we get that 𝑤∈VI(𝐶,𝐵). That is, 𝑤∈ℱ. Since 𝑧=𝑃_ℱ(𝛾𝑓+(𝐼−𝐴))(𝑧), we have limsup_𝑛→∞⟨𝛾𝑓𝑧−𝐴𝑧,𝑥_𝑛−𝑧⟩==lliimm_𝑖→∞sup_𝑛→∞⟨𝛾𝑓𝑧−𝐴𝑧,𝑥_𝑛−𝑧⟩=⟨𝛾𝑓𝑧−𝛾𝑓𝐴𝑧𝑧−,𝑤𝐴−𝑧,𝑧𝑥⟩_𝑛𝑖≤−0𝑧.(3.102) On the other hand, we have ⟨𝛾𝑓𝑧−𝐴𝑧,𝑦_𝑛−𝑧⟩=≤⟨‖𝛾𝛾𝑓𝑓𝑧𝑧−−𝐴𝐴𝑧𝑧,‖𝑦_𝑛‖‖−𝑦_𝑛𝑥_𝑛⟩+⟨𝛾𝑓𝑧−𝐴𝑧,𝑥_𝑛−𝑧⟩−𝑥_𝑛‖‖+⟨𝛾𝑓𝑧−𝐴𝑧,𝑥_𝑛−𝑧⟩.(3.103) From (3.43) and (3.102), we obtain that limsup_𝑛→∞⟨𝛾𝑓𝑧−𝐴𝑧,𝑦_𝑛−𝑧⟩≤0.(3.104)Step 6. Finally, we show that {𝑥_𝑛} converges strongly to 𝑧=𝑃_ℱ(𝛾𝑓+(𝐼−𝐴))(𝑧). Indeed, by (3.13) and using Lemmas 2.6 and 2.12(2), we observe that ‖‖𝑦_𝑛−𝑧‖‖²=‖‖𝜖_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑧‖‖²=‖‖1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑧+𝜖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧‖‖²≤‖‖1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑧‖‖²=+‖‖2𝜖1_𝑛−𝛾𝛽𝑓_𝑛𝑥_𝑛−𝐴𝑧,1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑧+𝜖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑧‖‖²=+‖‖‖‖21𝜖_𝑛−𝛽𝛾𝑓_𝑛𝑥_𝑛−𝐴𝑧,𝜖_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛−𝑧1−𝛽^𝑛𝐼−𝜖_𝑛𝐴1−𝛽_𝑛𝑧^𝑛≤1−𝛽_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑧‖‖‖‖²+2𝜖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧,𝑦_𝑛−𝑧‖‖‖‖1−𝛽^𝑛𝐼−𝜖_𝑛𝐴1−𝛽_𝑛𝑧_𝑛−𝑧‖‖‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²≤+12𝜖−_𝑛𝛾𝛽_𝑛𝑓𝑥_𝑛−𝑓(𝑧),𝑦_𝑛−𝑧+2𝜖_𝑛⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩‖‖‖‖1−𝛽^𝑛𝐼−𝜖_𝑛𝐴1−𝛽_𝑛𝑧_𝑛−𝑧‖‖‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²≤+‖‖2𝜖1_𝑛−𝛾𝜂𝛽‖‖_𝑛𝑥_𝑛−𝑧‖‖‖‖𝑦_𝑛−𝑧‖‖+2𝜖_𝑛⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩𝐼−𝜖_𝑛𝐴‖‖²1−𝛽_𝑛‖‖𝑧_𝑛−𝑧‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²≤+‖‖𝜖_𝑛1𝛾−𝜂𝛽‖‖_𝑛𝑥_𝑛−𝑧‖‖²+‖‖𝑦_𝑛−𝑧‖‖²+2𝜖^𝑛⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩𝐼−𝜖_𝑛𝐴‖‖²1−𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²=+𝜖_𝑛𝛾1𝜂−‖‖𝛽𝑥_𝑛𝑛−𝑧‖‖²+‖‖𝑦_𝑛−𝑧‖‖²+2𝜖^𝑛⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩−𝛾𝜖_𝑛²1−𝛽_𝑛+𝛽_𝑛+𝜖_𝑛𝛾𝜂‖‖𝑥_𝑛−𝑧‖‖²=+𝜖1_𝑛−𝛾𝜂‖‖2𝑦_𝑛−𝑧‖‖²+2𝜖_𝑛⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩𝛾−𝜂𝛾𝜖_𝑛+𝛾²𝜖^2𝑛1−𝛽_𝑛‖‖𝑥^𝑛−𝑧‖‖²+𝜖_𝑛𝛾𝜂‖‖𝑦_𝑛−𝑧‖‖²+2𝜖_𝑛⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩,(3.105) which implies that ‖‖𝑦_𝑛−𝑧‖‖²≤1−2𝛾−𝜂𝛾𝜖_𝑛1−𝜂𝛾𝜖_𝑛‖‖𝑥^𝑛−𝑧‖‖²+𝜖^𝑛1−𝜂𝛾𝜖_𝑛𝛾²𝜖_𝑛1−𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²+2⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩.(3.106) On the other hand, we have ‖‖𝑥_𝑛+1−𝑧‖‖²=‖‖𝛾_𝑛𝑥_𝑛−𝑧+1−𝛾_𝑛𝑦_𝑛−𝑧‖‖²≤𝛾_𝑛‖‖𝑥_𝑛−𝑧‖‖²+1−𝛾_𝑛‖𝑦_𝑛−𝑧‖².(3.107) Substituting (3.106) into (3.107) yields that ‖‖𝑥_𝑛+1−𝑧‖‖²≤𝛾_𝑛‖‖𝑥_𝑛−𝑧‖‖²+1−𝛾_𝑛‖‖𝑦_𝑛−𝑧‖‖²≤×𝛾_𝑛‖‖𝑥_𝑛−1−𝑧‖‖2²+1−𝛾_𝑛𝛾−𝜂𝛾𝜖_𝑛1−𝜂𝛾𝜖_𝑛‖‖𝑥^𝑛−𝑧‖‖²+𝜖^𝑛1−𝜂𝛾𝜖_𝑛𝛾²𝜖_𝑛1−𝛽_𝑛=1−21−𝛾_𝑛‖‖𝑥_𝑛−𝑧‖‖²+2⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩𝛾−𝜂𝛾𝜖_𝑛1−𝜂𝛾𝜖_𝑛‖‖𝑥^𝑛−𝑧‖‖²+𝜖^𝑛1−𝛾_𝑛1−𝜂𝛾𝜖_𝑛𝛾²𝜖_𝑛1−𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²+2⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩.(3.108) Taking 𝜚_𝑛=21−𝛾^𝑛𝛾−𝜂𝛾𝜖_𝑛1−𝜂𝛾𝜖_𝑛𝜎_𝑛,=𝜖^𝑛1−𝛾_𝑛1−𝜂𝛾𝜖_𝑛𝛾²𝜖_𝑛1−𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²+2⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩,(3.109) then, we can rewrite (3.108) as ‖‖𝑥_𝑛+1−𝑧‖‖²≤1−𝜚_𝑛‖‖𝑥_𝑛−𝑧‖‖²+𝜎_𝑛,(3.110) It follows from condition (C1) and (3.104) that lim_𝑛→∞𝜚_𝑛=0,^∞_𝑛=1𝜚_𝑛=∞.(3.111) Since limsup_𝑛→∞⟨𝛾𝑓𝑧−𝐴𝑧,𝑦_𝑛−𝑧⟩≤0.(3.112) and {𝑥_𝑛−𝑧} is bounded, we have limsup_𝑛→∞𝜎_𝑛=limsup_𝑛→∞𝜖_𝑛1−𝛾_𝑛1−𝜂𝛾𝜖_𝑛𝛾²𝜖_𝑛1−𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²+2⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑦_𝑛−𝑧⟩≤0.(3.113) Applying Lemma 2.8 to (3.110), we conclude that {𝑥_𝑛} converges strongly to 𝑧 in norm. This completes the proof.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ₁ and Θ₂ be two bifunctions from 𝐶×𝐶 to ℝ satisfying (H1)–(H5) and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let (i)Ψ₁∶𝐶→𝐻 be a 𝜉-inverse-strongly monotone mapping, (ii)Ψ₂∶𝐶→𝐻 be a 𝛽-inverse-strongly monotone mapping, (iii)𝑓∶𝐶→𝐶 be a contraction mapping with coefficient 𝜂∈(0,1). Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping with 𝐹(𝑆)≠∅.
Assume that ℱ∶=𝐹(𝑆)∩GMEPΘ₁,𝜑,Ψ₁∩GMEPΘ₂,𝜑,Ψ₂≠∅.(3.114) Let {𝑥_𝑛},{𝑦_𝑛},{𝑣_𝑛}, and {𝑢_𝑛} be the sequences generated by 𝑢𝑣_𝑛𝑛=𝑇^{(Θ1𝑟,𝜑)}_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛,𝑦_𝑛=𝑇^{(Θ2𝑠,𝜑)}_𝑛𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛,𝑥_𝑛+1=𝜖_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛿_𝑛𝑆𝑣_𝑛,=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,∀𝑛≥1,(3.115) where {𝑟_𝑛}⊂[𝑎,𝑏]⊂[0,2𝜉],{𝑠_𝑛}⊂[𝑐,𝑑]⊂[0,2𝛽],{𝛾_𝑛}⊂[ℎ,𝑗]⊂(0,1), and {𝛾_𝑛},{𝜖_𝑛},{𝛽_𝑛}, and {𝛿_𝑛} are three sequences in (0,1) satisfying the following conditions: (C1)𝜖_𝑛+𝛽_𝑛+𝛿_𝑛=1, (C2)lim_𝑛→∞𝜖_𝑛=0 and ∑^∞𝑛=1𝜖_𝑛=∞, (C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and lim_𝑛→∞𝛽_𝑛=0, (C4)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜉 and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0, (C5)0<liminf_𝑛→∞𝑠_𝑛≤limsup_𝑛→∞𝑠_𝑛<2𝛽 and lim_𝑛→∞|𝑠_𝑛+1−𝑠_𝑛|=0. Then, {𝑥_𝑛} converges strongly to 𝑧=𝑃_ℱ𝑓(𝑧), which is the unique solution of the variational inequality: ⟨𝑓(𝑧)−𝑧,𝑥−𝑧⟩≤0,∀𝑥∈ℱ.(3.116)

Proof. In Theorem 3.3, put 𝐴=𝐼,𝛾=1, and 𝛿_𝑛=1−𝜖_𝑛−𝛽_𝑛. Let 𝐵=0 in Theorem 3.3; then we have VI(𝐶,𝐵)=𝐶 and 𝑧_𝑛=𝑃_𝐶𝑆𝑣_𝑛−𝛼_𝑛𝐵𝑆𝑣_𝑛=𝑆𝑣_𝑛,(3.117) and we can obtain the desired conclusion from Theorem 3.3 immediately.

Corollary 3.5. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ₁ and Θ₂ be two bifunctions from 𝐶×𝐶 to ℝ satisfying (H1)–(H5) and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let 𝑓∶𝐶→𝐶 be a contraction mapping with coefficient 𝜂∈(0,1) and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping with 𝐹(𝑆)≠∅.
Assume that ℱ∶=𝐹(𝑆)∩MEPΘ₁,𝜑∩MEPΘ₂,𝜑≠∅.(3.118) Let {𝑥_𝑛},{𝑦_𝑛},{𝑣_𝑛}, and {𝑢_𝑛} be the sequences generated by Θ₁𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛Θ⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,₂𝑣_𝑛,𝑦+𝜑(𝑦)𝑦−𝜑𝑣_𝑛+𝑠1_𝑛⟨𝑦−𝑣_𝑛,𝑣_𝑛−𝑢_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑥_𝑛+1=𝜖_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛿_𝑛𝑆𝑣_𝑛,=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,∀𝑛≥1,(3.119) where {𝑟_𝑛}⊂[𝑎,𝑏]⊂(0,∞),{𝑠_𝑛}⊂[𝑐,𝑑]⊂(0,∞),{𝛾_𝑛}⊂[ℎ,𝑗]⊂(0,1), and {𝛾_𝑛},{𝜖_𝑛},{𝛽_𝑛}, and {𝛿_𝑛} are three sequences in (0,1) satisfying the following conditions: (C1)𝜖_𝑛+𝛽_𝑛+𝛿_𝑛=1, (C2)lim_𝑛→∞𝜖_𝑛=0 and ∑^∞𝑛=1𝜖_𝑛=∞, (C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and lim_𝑛→∞𝛽_𝑛=0, (C4)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<∞ and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0, (C5)0<liminf_𝑛→∞𝑠_𝑛≤limsup_𝑛→∞𝑠_𝑛<∞ and lim_𝑛→∞|𝑠_𝑛+1−𝑠_𝑛|=0. Then, {𝑥_𝑛} converges strongly to 𝑧=𝑃_ℱ𝑓(𝑧), which is the unique solution of the variational inequality: ⟨𝑓(𝑧)−𝑧,𝑥−𝑧⟩≤0,∀𝑥∈ℱ.(3.120)

Proof. In Theorem 3.3, put 𝑢_𝑛=𝑇^{(Θ1𝑟,𝜑)}_𝑛(𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛) to be equivalent to Θ₁𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+⟨Ψ₁𝑥_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,(3.121) and put 𝑣_𝑛=𝑇^{(Θ2𝑠,𝜑)}_𝑛(𝑢_𝑛−𝑠_𝑛Ψ₂𝑢_𝑛) to be equivalent to Θ₂𝑣_𝑛,𝑦+𝜑(𝑦)−𝜑𝑣_𝑛+⟨Ψ₂𝑥_𝑛,𝑦−𝑣_𝑛⟩+1𝑠_𝑛⟨𝑦−𝑣_𝑛,𝑣_𝑛−𝑢_𝑛⟩≥0,∀𝑦∈𝐶.(3.122) Now, put Ψ₁=Ψ₂=0. Then, it follows that Θ₁𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛Θ⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,₂𝑣_𝑛,𝑦+𝜑(𝑦)−𝜑𝑣_𝑛+𝑠1_𝑛⟨𝑦−𝑣_𝑛,𝑣_𝑛−𝑢_𝑛⟩≥0,∀𝑦∈𝐶.(3.123) Observe that for all 𝜉>0, we see that ⟨Ψ₁𝑥−Ψ₂𝑦,𝑥−𝑦⟩≥𝜉‖‖Ψ₁𝑥−Ψ₂𝑦‖‖²,∀𝑥,𝑦∈𝐶.(3.124) Thus, let {𝑟_𝑛} be a sequence satisfying the restriction: 𝑎≤𝑟_𝑛≤𝑏, where 𝑎,𝑏∈(0,∞) with 0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<∞. Similarly, we obtain 𝑐≤𝑠_𝑛≤𝑑, where 𝑐,𝑑∈(0,∞) with 0<liminf_𝑛→∞𝑠_𝑛≤limsup_𝑛→∞𝑠_𝑛<∞, and we obtain the desired result by Corollary 3.4.

Acknowledgments

The first author was supported by Kasetsart University Research and Development Institute (KURDI). The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute. The third author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044.