A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

Abstract

We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.

1. Introduction

Throughout this paper, we denoted by ℕ and ℝ⁺ the set of all positive integers and all positive real numbers, respectively. We always assume that 𝐻 is a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively, and 𝐶 is a nonempty closed convex subset of 𝐻. Let 𝜑∶𝐶→ℝ be a real-valued function, Θ∶𝐶×𝐶→ℝ be an equilibrium bifunction, and Ψ,Φ∶𝐶→𝐻 be two nonlinear mappings. The generalized mixed equilibrium problem with perturbed mapping is to find 𝑥^∗∈𝐶 such thatΘ𝑥^∗,𝑦+𝜑𝑦−𝜑𝑥^∗+(Ψ+Φ)𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶. (1.1) The set of solutions of the problem (1.1) is denoted by GMEP(Θ,𝜑,Ψ+Φ). As special cases of the problem (1.1), we have the following. (1)If Φ=0, then the problem (1.1) reduces to the generalized mixed equilibrium problem of finding 𝑥^∗∈𝐶 such thatΘ𝑥^∗,𝑦+𝜑𝑦−𝜑𝑥^∗+Ψ𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶, (1.2) which was introduced and studied by Peng and Yao [1]. The set of solutions of the problem (1.2) is denoted by GMEP(Θ,𝜑,Ψ). (2)If Ψ=Φ=0, then the problem (1.1) reduces to the mixed equilibrium problem of finding 𝑥^∗∈𝐶 such thatΘ𝑥^∗,𝑦+𝜑𝑦−𝜑𝑥^∗≥0,∀𝑦∈𝐶, (1.3) which was consider by Ceng and Yao [2]. The set of solutions of the problem (1.3) is denoted by MEP(Θ). (3)If Φ=𝜑=0, then the problem (1.1) reduces to the generalized equilibrium problem of finding 𝑥^∗∈𝐶 such thatΘ𝑥^∗,𝑦+Ψ𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶, (1.4) which was consider by S. Takahashi and W. Takahashi [3]. The set of solutions of the problem (1.4) is denoted by GEP(Θ,Ψ).(4)If Ψ=Φ=𝜑=0, then the problem (1.1) reduces to the equilibrium problem of finding 𝑥^∗∈𝐶 such thatΘ𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (1.5) The set of solutions of the problem (1.5) is denoted by EP(Θ).(5)If Θ=Φ=𝜑=0, then the problem (1.1) reduces to the classical variational inequality problem of finding 𝑥^∗∈𝐶 such thatΨ𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶. (1.6) The set of solutions of the problem (1.6) is denoted by VI(𝐶,Ψ). It is known that 𝑥^∗∈𝐶 is a solution of the problem (1.6) if and only if 𝑥^∗ is a fixed point of the mapping 𝑃_𝐶(𝐼−𝜆Ψ), where 𝜆>0 is a constant and 𝐼 is the identity mapping.

The generalized mixed equilibrium problems with perturbation is very general in the sense that it includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [4, 5]). Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.2). Several methods have been proposed to solve the fixed point problems, variational inequality problems, and equilibrium problems in the literature (see, e.g., [6–34]).

Let 𝐴 be a strongly positive bounded linear operator on 𝐻; that is, there exists a constant 𝛾>0 such that⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (1.7)

Recall that a mapping 𝑓∶𝐶→𝐶 is said to be contractive if there exists a constant 𝛼∈(0,1) such that 𝑓(𝑥)−𝑓𝑦≤𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.8)

A mapping 𝑇∶𝐶→𝐶 is said to be (1)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.9)(2)firmly nonexpansive if 𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑇𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.10)(3)𝑘-strictly pseudocontractive if there exists a constant 𝑘∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (1.11) We denote by 𝐹(𝑇) the set of fixed points of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}.

Recall the following definitions of a nonlinear mapping 𝐵∶𝐶→𝐻; the following is mentioned.

Definition 1.1. The nonlinear mapping 𝐵∶𝐶→𝐻 is said to be (i)monotone if 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶, (1.12)(ii)𝛽-strongly monotone if there exists a constant 𝛽>0 such that 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐶, (1.13)(iii)𝜈-inverse-strongly monotone if there exists a constant 𝜈>0 such that 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥𝜈𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝐶. (1.14)

Let 𝑊∶𝐻→2^𝐻 be a set-valued mapping. The set 𝐷(𝑊) defined by 𝐷(𝑊)={𝑥∈𝐻∶𝑊𝑥≠∅} is said to be the domain of 𝑊. The set 𝑅(𝑊) defined by 𝑅(𝑊)=⋃_𝑥∈𝐻𝑊𝑥 is said to be the range of 𝑊. The set 𝐺(𝑊) defined by 𝐺(𝑊)={(𝑥,𝑦)∈𝐻×𝐻∶𝑥∈𝐷(𝑊),𝑦∈𝑅(𝑊)} is said to be the graph of 𝑊.

Recall that 𝑊 is said to be monotone if 𝑥−𝑦,𝑓−𝑔>0,∀𝑥,𝑓,𝑦,𝑔∈𝐺(𝑊). (1.15)

𝑊 is said to be maximal monotone if it is not properly contained in any other monotone operator. Equivalently, 𝑊 is maximal monotone if 𝑅(𝐼+𝑟𝑊)=𝐻 for all 𝑟>0. For a maximal monotone operator 𝑀 on 𝐻 and 𝑟>0, we may define the single-valued resolvent 𝐽_𝑟=(𝐼+𝑟𝑊)⁻¹∶𝐻→𝐷(𝑊). It is known that 𝐽_𝑟 is firmly nonexpansive 𝑊⁻¹(0)=𝐹(𝐽_𝑟), where 𝐹(𝐽_𝑟) denotes the fixed point set of 𝐽_𝑟.

We discuss the following algorithms for solving the solutions of variational inequality problems and fixed point problems for a nonexpansive mapping (see, e.g., [29, 35–43]).

In 2010, Chantarangsi et al. [44] introduced a new viscosity hybrid steepest descent method for solving the generalized mixed equilibrium problems (1.2), variational inequality problems, and fixed point problems of nonexpansive mappings in a real Hilbert space. More precisely, they proved the following theorem.

Theorem CCK [see [44]]
Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let Θ₁, Θ₂ be two bifunctions satisfying condition (H1)–(H5), let Ψ₁, Ψ₂ be 𝜉-inverse-strongly monotone mapping and 𝛽-inverse-strongly monotone mapping, respectively, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Let 𝐵 be an 𝜔-Lipschitz continuous and relaxed (𝑣,𝜈) cocoercive mapping, 𝑓∶𝐶→𝐶 a contraction mapping with coefficient 𝛼∈(0,1), and 𝐴 a strongly positive linear bounded self-adjoint operator with coefficient 𝛾>0 and 0<𝛾<𝛾/𝛼. Suppose that Ω∶=𝐹(𝑇)∩GMEP(Θ₁,𝜑,Ψ₁)∩GMEP(Θ₂,𝜑,Ψ₂)∩VI(𝐶,𝐵). Let {𝑧_𝑛}, {𝑢_𝑛}, {𝑦_𝑛}, and {𝑥_𝑛} be generated by 𝑢_𝑛=𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂𝑥_𝑛,𝑣_𝑛=𝑉^(Θ1,𝜑1)_𝜇𝑛𝑢_𝑛−𝜇_𝑛Ψ₁𝑢_𝑛,𝑧_𝑛=𝑃_𝐶𝑣_𝑛−𝛼_𝑛𝐵𝑇𝑣_𝑛,𝑦_𝑛=𝜖_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜖_𝑛𝐴𝑧_𝑛,𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,∀𝑛≥1, (1.16) where {𝛾_𝑛}⊂[𝑎,𝑏]⊂[0,2𝜉], {𝑠_𝑛}⊂[𝑐,𝑑]⊂[0,2𝛽], {𝛾_𝑛}⊂[ℎ,𝑗]⊂(0,1), {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, (C3)0<liminf_𝑛→∞𝜇_𝑛≤limsup_𝑛→∞𝜇_𝑛<2𝛽 and lim_𝑛→∞|𝜇_𝑛+1−𝜇_𝑛|=0, (C4)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜉 and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0, (C5){𝛼_𝑛}⊂[𝑒,𝑔]⊂(0,2(𝜈−𝑣𝜔²)/𝜔²), 𝜈>𝑣𝜔² and lim_𝑛→∞|𝛼_𝑛+1−𝛼_𝑛|=0. Then, {𝑥_𝑛} converges strongly to 𝑥^∗=𝑃_Ω(𝛾𝑓+(𝐼−𝐴))(𝑥^∗).

Very recently, Yu and Liang [45] proved the following convergence theorem of finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and inverse strongly monotone mapping in a real Hilbert space.

Theorem YL [see [45]]
Let 𝐻 be a real Hilbert space and 𝐶 a nonempty close and convex subset of 𝐻. Let 𝑊₁∶𝐻→2^𝐻 and 𝑊₂∶𝐻→2^𝐻 be two maximal monotone operators such that 𝐷(𝑊₁)⊂𝐶 and 𝐷(𝑊₂)⊂𝐶, respectively. Let 𝑆∶𝐶→𝐶 be a 𝑘-strict pseudocontraction mapping, 𝐴∶𝐶→𝐻 an 𝛼-inverse-strongly monotone mapping, and 𝐵∶𝐶→𝐻 an 𝛽-inverse-strongly monotone mapping. Assume that Ω∶=𝐹(𝑆)∩(𝐴+𝑊₁)⁻¹(0)∩(𝐵+𝑊₂)⁻¹(0)≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑥₁𝑦∈𝐶,_𝑛=𝐽_𝑡𝑛𝑥_𝑛−𝑡_𝑛𝐵𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝛿_𝑛𝐽_𝑠𝑛𝑦_𝑛−𝑠_𝑛𝐴𝑦_𝑛+1−𝛿_𝑛𝐽_𝑠𝑛𝑦_𝑛−𝑠_𝑛𝐴𝑦_𝑛,∀𝑛≥1, (1.17) where 𝑢∈𝐶 is a fixed element, 𝐽_𝑠𝑛=(𝐼+𝑠_𝑛𝑊₁)⁻¹, 𝐽_𝑡𝑛=(𝐼+𝑡_𝑛𝑊₂)⁻¹,{𝑠_𝑛} is a sequence in (0,2𝛼),{𝑡_𝑛} is a sequence in (0,2𝛽), and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are sequences in [0,1] satisfying the following conditions: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)0<𝑎≤𝑠_𝑛≤𝑏<2𝛼 and lim_𝑛→∞(𝑠_𝑛+1−𝑠_𝑛)=0, (C4)0<𝑐≤𝑡_𝑛≤𝑑<2𝛽 and lim_𝑛→∞(𝑡_𝑛+1−𝑡_𝑛)=0, (C5)0<𝑐≤𝑘≤𝛿_𝑛<𝑒<1 and lim_𝑛→∞(𝛿_𝑛+1−𝛿_𝑛)=0. Then, the sequence {𝑥_𝑛} converges strongly to 𝑥^∗=𝑃_Ω𝑥^∗.

On the other hand, the following optimization problem has been studied extensively by many authors:min_𝑥∈Ω𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥), (1.18) where Ω=⋂^∞_𝑛=1𝐶_𝑛,𝐶₁,𝐶₂,… are infinitely many closed convex subsets of 𝐻 such that ⋂^∞_𝑛=1𝐶_𝑛≠∅, 𝑢∈𝐻, 𝜇≥0 is a real number, 𝐴 is a strongly positive linear bounded operator on 𝐻, and ℎ is a potential function for 𝛾𝑓 (i.e., ℎ′(𝑥)=𝛾𝑓(𝑥) for all 𝑥∈𝐻). This kind of optimization problem has been studied extensively by many authors (see, e.g., [5, 46–52]) for when Ω=⋂^∞_𝑛=1𝐶_𝑛 and ℎ(𝑥)=⟨𝑥,𝑏⟩, where 𝑏 is a given point in 𝐻.

The following questions naturally arise in connection with above the results.

Question 1. Could we weaken the control conditions of Theorems CCK and YL in (C3) and (C4)?

Question 2. Can Theorem YL be extended to finding a common element of the set of solutions of a system generalized mixed equilibrium problems and the set of common fixed points of infinite family of nonexpansive mappings?

The purpose of this paper is to give the affirmative answers to these questions mentioned above. Motivated by the iterative process (1.16) and (1.17), we introduce a new iterative algorithm (3.2) below, for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Then, we prove the strong convergence theorem of these iterative process in a real Hilbert space. The results presented in the paper improve and extend the recent ones announced by many others.

2. Preliminaries

Definition 2.1 (see [53]). Let 𝐶 be a nonempty convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖, 𝑖=1,2,…, be mappings of 𝐶 into itself. For each 𝑗=1,2,…, let 𝛼_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃)∈𝐼×𝐼×𝐼, where 𝐼=[0,1] and 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₃=1. For every 𝑛∈ℕ, we define the mapping 𝑆_𝑛∶𝐶→𝐶 as follows: 𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝛼^𝑛₁𝑇_𝑛𝑈_𝑛,𝑛+1+𝛼^𝑛₂𝑈_𝑛,𝑛+1+𝛼^𝑛₃𝑈𝐼,_{𝑛,𝑛−1}=𝛼₁^𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+𝛼₂^𝑛−1𝑈_𝑛,𝑛+𝛼₃^𝑛−1⋮𝑈𝐼,_𝑛,𝑘+1=𝛼₁^𝑘+1𝑇_𝑘+1𝑈_𝑛,𝑘+2+𝛼₂^𝑘+1𝑈_𝑛,𝑘+2+𝛼₃^𝑘+1𝑈𝐼,_𝑛,𝑘=𝛼^𝑘₁𝑇_𝑘𝑈_𝑛,𝑘+1+𝛼^𝑘₂𝑈_𝑛,𝑘+1+𝛼^𝑘₃⋮𝑈𝐼,_𝑛,2=𝛼²₁𝑇₂𝑈_𝑛,3+𝛼²₂𝑈_𝑛,3+𝛼²₃𝑆𝐼,_𝑛=𝑈_𝑛,1=𝛼¹₁𝑇₁𝑈_𝑛,2+𝛼¹₂𝑈_𝑛,2+𝛼¹₃𝐼. (2.1) Such a mapping 𝑆_𝑛 is nonexpansive from 𝐶 into itself, and it is called 𝑆-mapping generated by 𝑇_𝑛,𝑇_𝑛−1,…,𝑇₁ and 𝛼_𝑛,𝛼_𝑛−1,…,𝛼₁.

Lemma 2.2 (see [53]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑖}^∞_𝑖=1 be nonexpansive mappings of 𝐶 into itself with 𝐹(𝑇_𝑖)≠∅ and let 𝛼_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃)∈𝐼×𝐼×𝐼, where 𝐼=[0,1], 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₃=1, 𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃∈(0,1) for all 𝑗=1,2,…. For all 𝑛∈ℕ, let 𝑆_𝑛 and 𝑆 be 𝑆-mappings generated by 𝑇_𝑛,𝑇_𝑛−1,…,𝑇₁ and 𝛼_𝑛,𝛼_𝑛−1,…,𝛼₁ and 𝑇_𝑛,𝑇_𝑛−1,… and 𝛼_𝑛,𝛼_𝑛−1,…, respectively. Then, (i)𝑆_𝑛 is nonexpansive and 𝐹(𝑆_𝑛)=⋂^𝑛_𝑖=1𝐹(𝑇_𝑖),  for all 𝑛≥1, (ii)for all 𝑥∈𝐶 and for all positive integer 𝑘, the lim_𝑛→∞𝑈_𝑛,𝑘 exists,(iii)the mapping 𝑆∶𝐶→𝐶 defined by 𝑆𝑥∶=lim_𝑛→∞𝑆_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥,∀𝑥∈𝐶 (2.2)is a nonexpansive mapping such that 𝐹(𝑆)=⋂^∞_𝑖=1𝐹(𝑇_𝑖), and it is called the 𝑆-mapping generated by 𝑇_𝑛,𝑇_𝑛−1,… and 𝛼_𝑛,𝛼_𝑛−1,…, (iv) if 𝐾 is any bounded subset of C, then lim_𝑛→∞sup_𝑥∈𝐾𝑆_𝑛𝑥−𝑆𝑥=0. (2.3)

Lemma 2.3 (see [54]). Let {𝑥_𝑛} and {𝑙_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=(1−𝛽_𝑛)𝑙_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑙_𝑛+1−𝑙_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑙_𝑛−𝑥_𝑛‖=0.

Lemma 2.4 (see [55]). Let 𝐻 be a real Hilbert space. Then, the following inequalities hold: (i)‖𝜆𝑥+(1−𝜆)𝑦‖²=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐻and𝜆∈[0,1],(ii)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝐻.

Lemma 2.5 (see [56]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, 𝐴∶𝐶→𝐻 a mapping, and 𝑊∶𝐻→2^𝐻 a maximal monotone mapping. Then, 𝐹𝐽_𝑟(𝐼−𝑟𝐴)=(𝐴+𝑊)⁻¹(0),∀𝑟>0. (2.4)

Lemma 2.6 (see [57]). Let 𝐻 be a real Hilbert space and let 𝑀 be a maximal monotone operator on 𝐻. For 𝑟>0 and 𝑥∈𝐻, define the resolvent 𝐽_𝑟𝑥. Then, the following holds: 𝑠−𝑡𝑠𝐽_𝑠𝑥−𝐽_𝑡𝑥,𝐽_𝑠𝑥−𝑥≥𝐽_𝑠𝑥−𝐽_𝑡𝑥² (2.5) for all 𝑠,𝑡>0 and 𝑥∈𝐻.

For solving the equilibrium problem for bifunction Θ∶𝐶×𝐶→ℝ, let us assume that Θ satisfies the following conditions: (H1)Θ(𝑥,𝑥)=0 for all 𝑥∈𝐶, (H2)Θ is monotone; that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶,(H3) for each 𝑦∈𝐶, 𝑥↦Θ(𝑥,𝑦) is concave and upper semicontinuous, (H4) for each 𝑦∈𝐶, 𝑥↦Θ(𝑥,𝑦) is convex, (H5) for each 𝑦∈𝐶, 𝑥↦Θ(𝑥,𝑦) is lower semicontinuous.

Definition 2.7. A differentiable function 𝐾∶𝐶→ℝ on a convex set 𝐶 is called (i)convex [2] if𝐾𝑦−𝐾(𝑥)≥𝐾^′(𝑥),𝑦−𝑥,∀𝑥,𝑦∈𝐶, (2.6)where 𝐾′(𝑥) is the Fréchet differentiable of 𝐾 at 𝑥; (ii)strongly convex [2] if there exists a constant 𝜎>0 such that𝐾𝑦−𝐾(𝑥)−𝐾^′(𝑥)≥𝜎,𝑥−𝑦2𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (2.7)

It is easy to see that if 𝐾∶𝐶→ℝ is a differentiable strongly convex function with constant 𝜎>0, then 𝐾^′∶𝐶→𝐻 is strongly monotone with constant 𝜎>0.

Let Θ∶𝐶×𝐶→ℝ be an equilibrium bifunction satisfying the conditions (H1)–(H5). Let 𝑟 be any given positive number. For a given point 𝑥∈𝐶, consider the auxiliary mixed equilibrium problem to finding 𝑦∈𝐶 such thatΘ(𝑧)𝑦+1𝑦,𝑧+𝜑−𝜑𝑟𝐾^′𝑦−𝐾^′(𝑥),𝑧−𝑦≥0,∀𝑧∈𝐶, (2.8) where 𝐾^′(𝑥) is the Fréchet differentiable of 𝐾 at 𝑥. Let 𝑉_𝑟^(Θ,𝜑)∶𝐶→𝐶 be the mapping such that for each 𝑥∈𝐶, 𝑉_𝑟^(Θ,𝜑)(𝑥) is the set of solutions of MEP(𝑥,𝑦), that is,𝑉_𝑟^(Θ,𝜑)(𝑥)=(𝑧)𝑦+1𝑦∈𝐶∶Θ𝑦,𝑧+𝜑−𝜑𝑟𝐾^′𝑦−𝐾^′(𝑥),𝑧−𝑦≥0,∀𝑧∈𝐶. (2.9) Then, the following conclusion holds.

Lemma 2.8 (see [58]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝜑∶𝐶→ℝ be a lower semicontinuous and convex functional. Let Θ∶𝐶×𝐶→ℝ be a bifunction satisfying the conditions (H1)–(H5). Assume that (i)𝐾∶𝐶→ℝ is strongly convex with constant 𝜎>0 and the function 𝑥↦⟨𝑦−𝑥,𝐾′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐶;(ii) for each 𝑥∈𝐶, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥 such that for all 𝑦∉𝐷_𝑥, Θ𝑦,𝑧_𝑥+𝜑𝑧_𝑥𝑦+1−𝜑𝑟𝐾^′𝑦−𝐾^′(𝑥),𝑧_𝑥−𝑦<0. (2.10)Then, the following holds: (a)𝑉_𝑟^(Θ,𝜑) is single-valued mapping; (b)𝑉_𝑟^(Θ,𝜑) is nonexpansive if 𝐾^′ is Lipschitz continuous with constant 𝜈>0 and 𝐾^′𝑥₁−𝐾^′𝑥₂,𝑢₁−𝑢₂≥𝐾^′𝑢₁−𝐾^′𝑢₂,𝑢₁−𝑢₂,∀𝑥₁,𝑥₂∈𝐶, (2.11)where 𝑢_𝑖=𝑉_𝑟^(Θ,𝜑)(𝑥_𝑖) for 𝑖=1,2; (c)𝐹(𝑉_𝑟^(Θ,𝜑))=MEP(Θ,𝜑); (d)MEP(Θ,𝜑) is closed and convex.

In particular, whenever Θ∶𝐶×𝐶→ℝ is a bifunction satisfying the conditions (H1)–(H5) and 𝐾(𝑥)=‖𝑥‖²/2, for all 𝑥∈𝐶, then 𝑉_𝑟^(Θ,𝜑) is firmly nonexpansive; that is, for any 𝑥,𝑦∈𝐶, 𝑉_𝑟^(Θ,𝜑)𝑥−𝑉_𝑟^(Θ,𝜑)𝑦²≤𝑉_𝑟^(Θ,𝜑)𝑥−𝑉_𝑟^(Θ,𝜑)𝑦,𝑥−𝑦. (2.12) In this case, 𝑉_𝑟^(Θ,𝜑) is rewritten as 𝑇_𝑟^(Θ,𝜑). If, in addition, 𝜑≡0, then 𝑇_𝑟^(Θ,𝜑) is rewritten as 𝑇^Θ_𝑟 (see [59, Lemma 2.1] for more details).

Remark 2.9. We remark that Lemma 2.8 is not a consequence of [2, Lemma 3.1] because the condition of the sequential continuity from the weak topology to the strong topology for the derivative 𝐾^′ of the function 𝐾∶𝐶→ℝ does not cover the case 𝐾(𝑥)=‖𝑥‖²/2.

Lemma 2.10. Let 𝐶,𝐻,Θ,𝜑, and 𝑉_𝑟^(Θ,𝜑) be as in Lemma 2.8. Then, the following holds: 𝐾^′𝑉_𝑠^(Θ,𝜑)𝑥−𝐾^′𝑉_𝑡^(Θ,𝜑)𝑥,𝑉_𝑠^(Θ,𝜑)𝑥−𝑉_𝑡^(Θ,𝜑)𝑥≤𝑠−𝑡𝑠𝐾^′𝑉_𝑠^(Θ,𝜑)𝑥−𝐾^′(𝑥),𝑉_𝑠^(Θ,𝜑)𝑥−𝑉_𝑡^(Θ,𝜑)𝑥, (2.13) for all 𝑠,𝑡>0 and 𝑥∈𝐶.

Proof. By similar argument as in the proof of Proposition 1 in [58], for all 𝑠,𝑡>0 and 𝑥∈𝐶, let 𝑢=𝑉_𝑠^(Θ,𝜑)𝑥 and 𝑣=𝑉_𝑡^(Θ,𝜑)𝑥; we have Θ𝑦(𝑢)+1𝑢,𝑦+𝜑−𝜑𝑠𝐾^′(𝑢)−𝐾^′(𝑥)Θ𝑦(𝑣)+1,𝑦−𝑢≥0,∀𝑥∈𝐶,𝑣,𝑦+𝜑−𝜑𝑡𝐾^′(𝑣)−𝐾^′(𝑥),𝑦−𝑣≥0,∀𝑥∈𝐶. (2.14)(2.15) Let 𝑦=𝑣 in (2.14) and 𝑦=𝑢 in (2.15); we have Θ()(𝑣)(𝑢)+1𝑢,𝑣+𝜑−𝜑𝑠𝐾^′(𝑢)−𝐾^′(𝑥)Θ()(𝑢)(𝑣)+1,𝑣−𝑢≥0,𝑣,𝑢+𝜑−𝜑𝑡𝐾^′(𝑣)−𝐾^′(𝑥),𝑢−𝑣≥0. (2.16) Adding up the last two inequalities and from the monotonicity of Θ, we obtain that 1𝑠𝐾^′(𝑢)−𝐾^′(𝑥)+1,𝑣−𝑢𝑡𝐾^′(𝑣)−𝐾^′(𝑥),𝑢−𝑣≥0. (2.17) It follows that 𝐾^′(𝑣)−𝐾^′(𝑥)𝑡−𝐾^′(𝑢)−𝐾^′(𝑥)𝑠,𝑢−𝑣≥0. (2.18) We derive from (2.18) that 0≤𝐾^′(𝑣)−𝐾^′(𝑥)−𝑡𝑠𝐾^′(𝑢)−𝐾^′(𝑥)=𝐾,𝑢−𝑣_′(𝑣)−𝐾^′(𝑢)+𝐾^′(𝑢)−𝐾^′(𝑥)−𝑡𝑠𝐾^′(𝑢)−𝐾^′(𝑥)=𝐾,𝑢−𝑣_′(𝑣)−𝐾^′(𝑢)+𝑡1−𝑠𝐾^′(𝑢)−𝐾^′(𝑥),𝑢−𝑣. (2.19) Hence, we obtain that 𝐾^′(𝑢)−𝐾^′(𝑣),𝑢−𝑣≤𝑠−𝑡𝑠𝐾^′(𝑢)−𝐾^′(𝑥),𝑢−𝑣. (2.20)

The following lemma can be found in [60, 61] (see also [62, Lemma 2.2]).

Lemma 2.11. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑔∶𝐶→ℝ∪{+∞} a proper lower semicontinuous differentiable convex function differentiable convex function. If 𝑥^∗ is a solution to the minimization problem 𝑔𝑥^∗=inf_𝑥∈𝐶𝑔(𝑥), (2.21) then 𝑔^′(𝑥),𝑥−𝑥^∗≥0,𝑥∈𝐶. (2.22) In particular, if 𝑥^∗ solves the optimization problem min_𝑥∈Ω𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥), (2.23) then 𝑢+𝛾𝑓−𝐼+𝜇𝐴𝑥^∗,𝑥−𝑥^∗≤0,𝑥∈𝐶, (2.24) where ℎ is a potential function for 𝛾𝑓.

Lemma 2.12 (see [49]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝜎_𝑛𝑎_𝑛+𝛿_𝑛, (2.25) where {𝜎_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that (i)∑^∞_𝑛=0𝜎_𝑛=∞, (ii)limsup_𝑛→∞𝛿_𝑛/𝜎_𝑛≤0 or ∑^∞_𝑛=0|𝛿_𝑛|<∞. Then, lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

Now, we give our main results in this paper.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Let 𝜑₁,𝜑₂∶𝐶→ℝ be two lower semicontinuous and convex functionals, and let Θ₁,Θ₂∶𝐶×𝐶→ℝ be two bifunctions satisfying conditions (H1)–(H5). Let Ψ₁,Ψ₂,Φ₁,Φ₂∶𝐶→𝐻 be 𝛽₁-inverse-strongly monotone mapping, 𝛽₂-inverse-strongly monotone mapping, 𝛾₁-inverse-strongly monotone mapping, and 𝛾₂-inverse-strongly monotone mapping, respectively, and let 𝐵₁,𝐵₂∶𝐶→𝐻 be 𝜉̃₁-inverse-strongly monotone mapping and 𝜉̃₂-inverse-strongly monotone mapping, respectively. Let {𝑇_𝑖}^∞_𝑖=1∶𝐶→𝐶 be an infinite family of nonexpansive mappings, and let 𝛼_𝑖=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃)∈𝐼×𝐼×𝐼, where 𝐼=[0,1], 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₃=1, 𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃∈(0,1) for all 𝑗=1,2,…. For all 𝑛∈ℕ, let 𝑆_𝑛 and 𝑆 be 𝑆-mappings generated by 𝑇_𝑛,𝑇_𝑛−1,…,𝑇₁ and 𝛼_𝑛,𝛼_𝑛−1,…,𝛼₁ and 𝑇_𝑛,𝑇_𝑛−1,… and 𝛼_𝑛,𝛼_𝑛−1,…, respectively. Let 𝑊₁,𝑊₂∶𝐻→2^𝐻 be two maximal monotone operators such that 𝐷(𝑊₁)⊂𝐶 and 𝐷(𝑊₂)⊂𝐶, respectively. Assume that Ω∶=^∞_𝑖=1𝐹𝑇_𝑖∩GMEPΘ₁,𝜑₁,Ψ₁+Φ₁∩GMEPΘ₂,𝜑₂,Ψ₂+Φ₂∩𝐵₁+𝑊₁⁻¹(0)∩𝐵₂+𝑊₂⁻¹(0)≠∅. (3.1) Let 𝑓∶𝐶→𝐻 be a contraction mapping with a coefficient 𝛼∈(0,1), and let 𝐴∶𝐶→𝐻 be a strongly positive linear bounded operator with a coefficient 𝛾∈(0,1). Let 𝜇>0 and 𝛾>0 be two constants such that 0<𝛾<(1+𝜇)𝛾/𝛼. Let {𝑥_𝑛} be a sequence defined by 𝑢,𝑥₁∈𝐻 and 𝑢_𝑛=𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛,𝑣_𝑛=𝑉^(Θ1,𝜑1)_𝜇𝑛𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛,𝑦_𝑛=𝐽_𝑡𝑛𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛,𝑥_𝑛+1=𝜖_𝑛𝑢+𝛾𝑓𝑆_𝑛𝑣_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜖_𝑛𝐼+𝜇𝐴𝑆_𝑛𝐽_𝑠𝑛𝑦_𝑛−𝑠_𝑛𝐵₁𝑦_𝑛,∀𝑛≥1, (3.2) where 𝐽_𝑠𝑛=(𝐼+𝑠_𝑛𝑊₁)⁻¹, 𝐽_𝑡𝑛=(𝐼+𝑡_𝑛𝑊₂)⁻¹, {𝑠_𝑛}⊂(0,2𝜉̃₁), {𝑡_𝑛}⊂(0,2𝜉̃₂), {𝜇_𝑛}⊂(0,min{𝛽₁,𝛾₁}), {𝑟_𝑛}⊂(0,min{𝛽₂,𝛾₂}), {𝜖_𝑛} and {𝛽_𝑛}⊂(0,1). Assume that the following conditions are satisfied: (C1) for all 𝑖=1,2, 𝐾_𝑖∶𝐶→ℝ is strongly convex with constant 𝜎_𝑖>0 and its derivative 𝐾^′_𝑖 is Lipschitz continuous with constant 𝜈_𝑖>0 such that the function 𝑥↦⟨𝑦−𝑥,𝐾^′_𝑖(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐶, (C2)for all 𝑖=1,2 and for each 𝑥∈𝐶, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥∈𝐶 such that, for all 𝑦∉𝐷_𝑥, Θ_𝑖𝑦,𝑧_𝑥+𝜑_𝑖𝑧_𝑥−𝜑_𝑖𝑦+1𝑟_𝑖𝐾^′_𝑖𝑦−𝐾^′_𝑖(𝑥),𝑧_𝑥−𝑦<0, (3.3)(C3)lim_𝑛→∞𝛼¹_𝑛=0, (C4)lim_𝑛→∞𝜖_𝑛=0 and ∑^∞_𝑛=1𝜖_𝑛=∞, (C5)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C6)0<liminf_𝑛→∞𝜇_𝑛≤limsup_𝑛→∞𝜇_𝑛<min{𝛽₁,𝛾₁} and lim_𝑛→∞𝜇_𝑛/𝜇_𝑛+1=1, (C7)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<min{𝛽₂,𝛾₂} and lim_𝑛→∞𝑟_𝑛/𝑟_𝑛+1=1, (C8)0<liminf_𝑛→∞𝑠_𝑛≤limsup_𝑛→∞𝑠_𝑛<2𝜉̃₁ and lim_𝑛→∞𝑠_𝑛/𝑠_𝑛+1=1, (C9)0<liminf_𝑛→∞𝑡_𝑛≤limsup_𝑛→∞𝑡_𝑛<2𝜉̃₂ and lim_𝑛→∞𝑡_𝑛/𝑡_𝑛+1=1. Then, the sequence {𝑥_𝑛} defined by (3.2) converges strongly to 𝑥^∗∈Ω, provided 𝑉^(Θ,𝜑)_𝑟,𝑛 is firmly nonexpansive, where 𝑥^∗ solves the following optimization problem: min_𝑥∈Ω𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥). (3.4)

Proof. By the conditions (C4) and (C5), we may assume, without loss of generality, that 𝜖_𝑛≤(1−𝛽_𝑛)(1+𝜇‖𝐴‖)⁻¹ for all 𝑛∈ℕ. Since 𝐴 is a linear bounded self-adjoint operator on 𝐶, we have ‖𝐴‖=sup{|⟨𝐴𝑢,𝑢⟩|∶𝑢∈𝐶,‖𝑢‖=1}. (3.5) Observe that 1−𝛽_𝑛𝐼−𝜖_𝑛𝐼+𝜇𝐴𝑢,𝑢=1−𝛽_𝑛−𝜖_𝑛−𝜖_𝑛𝜇⟨𝐴𝑢,𝑢⟩≥1−𝛽_𝑛−𝜖_𝑛−𝜖_𝑛𝜇‖𝐴‖≥0. (3.6) This shows that (1−𝛽_𝑛)𝐼−𝜖_𝑛(𝐼+𝜇𝐴) is positive. It follows that 1−𝛽_𝑛𝐼−𝜖_𝑛𝐼+𝜇𝐴=sup1−𝛽_𝑛𝐼−𝜖_𝑛𝐼+𝜇𝐴𝑢,𝑢∶𝑢∈𝐶,‖𝑢‖=1=sup1−𝛽_𝑛−𝜖_𝑛−𝜖_𝑛𝜇⟨𝐴𝑢,𝑢⟩∶𝑢∈𝐶,‖𝑢‖=1≤1−𝛽_𝑛−𝜖_𝑛1+𝜇𝛾<1−𝛽_𝑛−𝜖_𝑛1+𝜇𝛾. (3.7) First, we show that {𝑥_𝑛} is bounded. Take 𝑥∈Ω. Since 0<𝜇_𝑛<min{𝛽₁,𝛾₁} and by Lemma 2.4  (i), we have 𝑣_𝑛−𝑥²=𝑉^(Θ1,𝜑1)_𝜇𝑛𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛−𝑉^(Θ1,𝜑1)_𝜇𝑛𝑥−𝜇_𝑛Ψ₁+Φ₁𝑥²≤𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛−𝑥−𝜇_𝑛Ψ₁+Φ₁𝑥²=𝑢_𝑛−𝑥−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛−Ψ₁+Φ₁𝑥²=12𝑢_𝑛−𝑥−2𝜇_𝑛Ψ₁𝑢_𝑛−Ψ₁𝑥+12𝑢_𝑛−𝑥−2𝜇_𝑛Φ₁𝑢_𝑛−Φ₁𝑥²≤12𝑢_𝑛−𝑥−2𝜇_𝑛Ψ₁𝑢_𝑛−Ψ₁𝑥²+12(𝑢_𝑛−𝑥)−2𝜇_𝑛Φ₁𝑢_𝑛−Φ₁𝑥²=12𝑢_𝑛−𝑥²−4𝜇_𝑛𝑢_𝑛−𝑥,Ψ₁𝑢_𝑛−Ψ₁𝑥+4𝜇²_𝑛Ψ₁𝑢_𝑛−Ψ₁𝑥²+12𝑢_𝑛−𝑥²−4𝜇_𝑛𝑢_𝑛−𝑥,Φ₁𝑢_𝑛−Φ₁𝑥+4𝜇²_𝑛Φ₁𝑢_𝑛−Φ₁𝑥²≤12𝑢_𝑛−𝑥²+4𝜇_𝑛𝜇_𝑛−𝛽₁Ψ₁𝑢_𝑛−Ψ₁𝑥²+12𝑢_𝑛−𝑥²+4𝜇_𝑛𝜇_𝑛−𝛾₁Φ₁𝑢_𝑛−Φ₁𝑥²=𝑢_𝑛−𝑥²+2𝜇_𝑛𝜇_𝑛−𝛽₁Ψ₁𝑢_𝑛−Ψ₁𝑥²+2𝜇_𝑛𝜇_𝑛−𝛾₁Φ₁𝑢_𝑛−Φ₁𝑥²≤𝑢_𝑛−𝑥². (3.8) In a similar way, we can get 𝑢_𝑛−𝑥²=𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛−𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥−𝑟_𝑛Ψ₂+Φ₁𝑥²≤𝑥_𝑛−𝑥²+2𝑟_𝑛𝑟_𝑛−𝛽₂Ψ₂𝑥_𝑛−Ψ₂𝑥²+2𝑟_𝑛𝑟_𝑛−𝛾₂Φ₂𝑥_𝑛−Φ₂𝑥²≤𝑥_𝑛−𝑥². (3.9) It follows from (3.8), and (3.9) that 𝑣_𝑛−𝑥≤𝑢_𝑛−𝑥≤𝑥_𝑛−𝑥. (3.10) Setting 𝑧_𝑛∶=𝐽_𝑠𝑛(𝑦_𝑛−𝑠_𝑛𝐵₁𝑦_𝑛). Since 0<𝑠_𝑛<2𝜉̃₁, we have 𝑧_𝑛−𝑥²=𝐽_𝑠𝑛𝑦_𝑛−𝑠_𝑛𝐵₁𝑦_𝑛−𝐽_𝑠𝑛𝑥−𝑠_𝑛𝐵₁𝑥²≤𝑦_𝑛−𝑠_𝑛𝐵₁𝑦_𝑛−𝑥−𝑠_𝑛𝐵₁𝑥²=𝑦_𝑛−𝑥−𝑠_𝑛𝐵₁𝑦_𝑛−𝐵₁𝑥²=𝑦_𝑛−𝑥²−2𝑠_𝑛𝑦_𝑛−𝑥,𝐵₁𝑦_𝑛−𝐵₁𝑥+𝑠²_𝑛𝐵₁𝑦_𝑛−𝐵₁𝑥²≤𝑦_𝑛−𝑥²+𝑠_𝑛𝑠_𝑛−2𝜉₁𝐵₁𝑦_𝑛−𝐵₁𝑥²≤𝑦_𝑛−𝑥². (3.11) In a similar way, we can get 𝑦_𝑛−𝑥²=𝐽_𝑡𝑛𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛−𝐽_𝑡𝑛𝑥−𝑡_𝑛𝐵₂𝑥²≤𝑣_𝑛−𝑥²+𝑡_𝑛𝑡_𝑛−2𝜉₂𝐵₂𝑣_𝑛−𝐵₂𝑥²≤𝑣_𝑛−𝑥². (3.12) It follows from (3.10), (3.11) and (3.12) that 𝑧_𝑛−𝑥≤𝑦_𝑛−𝑥≤𝑣_𝑛−𝑥≤𝑢_𝑛−𝑥≤𝑥_𝑛−𝑥. (3.13) Since 𝑥_𝑛+1=𝜖_𝑛(𝑢+𝛾𝑓(𝑆_𝑛𝑣_𝑛))+𝛽_𝑛𝑥_𝑛+((1−𝛽_𝑛)𝐼−𝜖_𝑛(𝐼+𝜇𝐴))𝑆_𝑛𝑧_𝑛, it follows from (3.13) that 𝑥_𝑛+1−𝑥=𝜖_𝑛𝑢+𝜖_𝑛𝛾𝑓𝑆_𝑛𝑣_𝑛−𝐼+𝜇𝐴𝑥+𝛽_𝑛𝑥_𝑛−𝑥+1−𝛽_𝑛𝐼−𝜖_𝑛𝐼+𝜇𝐴𝑆_𝑛𝑧_𝑛−𝑥≤𝜖_𝑛‖𝑢‖+𝜖_𝑛𝛾𝑓𝑆_𝑛𝑣_𝑛−𝐼+𝜇𝐴𝑥+𝛽_𝑛𝑥_𝑛−𝑥+1−𝛽_𝑛−𝜖_𝑛1+𝜇𝛾𝑆_𝑛𝑧_𝑛−𝑥≤𝜖_𝑛‖𝑢‖+𝜖_𝑛𝛾𝑓𝑆_𝑛𝑣_𝑛−𝑓(𝑥)+𝜖_𝑛𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥+𝛽_𝑛𝑥_𝑛−𝑥+1−𝛽_𝑛−𝜖_𝑛1+𝜇𝛾𝑧_𝑛−𝑥≤𝜖_𝑛‖𝑢‖+𝜖_𝑛𝛾𝛼𝑆_𝑛𝑣_𝑛−𝑥+𝜖_𝑛𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥+𝛽_𝑛𝑥_𝑛−𝑥+1−𝛽_𝑛−𝜖_𝑛1+𝜇𝛾𝑥_𝑛−𝑥≤1−1+𝜇𝛾−𝛾𝛼𝜖_𝑛𝑥_𝑛−𝑥+𝜖_𝑛𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥+𝜖_𝑛‖𝑢‖=1−1+𝜇𝛾−𝛾𝛼𝜖_𝑛𝑥_𝑛−𝑥+1+𝜇𝛾−𝛾𝛼𝜖_𝑛𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥+‖𝑢‖1+𝜇.𝛾−𝛾𝛼 (3.14) By induction, we have 𝑥_𝑛−𝑥≤max𝑥₁−𝑥,𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥+‖𝑢‖1+𝜇𝛾−𝛾𝛼,∀𝑛≥1. (3.15) Hence, {𝑥_𝑛} is bounded, so are {𝑣_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛}.
Next, we show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖. Since 𝑢_𝑛=𝑉^(Θ2,𝜑2)_𝑟𝑛(𝑥_𝑛−𝑟_𝑛(Ψ₂+Φ₂)𝑥_𝑛) and 𝑢_𝑛+1=𝑉^(Θ2,𝜑2)_𝑟𝑛+1(𝑥_𝑛+1−𝑟_𝑛+1(Ψ₂+Φ₂)𝑥_𝑛+1), we have 𝑢_𝑛+1−𝑢_𝑛=𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛+1−𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+1−𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛≤𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛+1−𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+1−𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛+𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛−𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛≤𝑥_𝑛+1−𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛+𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛−𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+1−Ψ₂+Φ₂𝑥_𝑛+𝑟_𝑛−𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛−𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛=𝑥_𝑛+1−𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+𝑟_𝑛−𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛−𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑟_𝑛+1𝑟1−^𝑛𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛−𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛. (3.16) In a similar way, we can get 𝑣_𝑛+1−𝑣_𝑛=𝑉^(Θ1,𝜑1)_𝜇𝑛+1𝑢_𝑛+1−𝜇_𝑛+1Ψ₁+Φ₁𝑢_𝑛+1−𝑉^(Θ1,𝜑1)_𝜇𝑛𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝜇_𝑛+1𝜇1−^𝑛𝜇_𝑛+1Ψ₁+Φ₁𝑢_𝑛+𝑉^(Θ1,𝜑1)_𝜇𝑛+1𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛−𝑉^(Θ1,𝜑1)_𝜇𝑛𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛. (3.17) Substitution (3.16) into (3.17), we obtain 𝑣_𝑛+1−𝑣_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑟_𝑛+1𝑟1−^𝑛𝑟_𝑛+1Ψ₂+Φ₂𝑥_𝑛+𝜇_𝑛+1𝜇1−^𝑛𝜇_𝑛+1Ψ₁+Φ₁𝑢_𝑛+𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛−𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛+𝑉^(Θ1,𝜑1)_𝜇𝑛+1𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛−𝑉^(Θ1,𝜑1)_𝜇𝑛𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑟1−^𝑛𝑟_𝑛+1+𝜇1−^𝑛𝜇_𝑛+1𝑀₁+𝑉^(Θ2,𝜑2)_𝑟𝑛+1𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛−𝑉^(Θ2,𝜑2)_𝑟𝑛𝑥_𝑛−𝑟_𝑛Ψ₂+Φ₂𝑥_𝑛+𝑉^(Θ1,𝜑1)_𝜇𝑛+1𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛−𝑉^(Θ1,𝜑1)_𝜇𝑛𝑢_𝑛−𝜇_𝑛Ψ₁+Φ₁𝑢_𝑛, (3.18) where 𝑀₁=sup_𝑛≥1{𝜇_𝑛+1‖(Ψ₁+Φ₁)𝑢_𝑛‖,𝑟_𝑛+1‖(Ψ₂+Φ₂)𝑥_𝑛‖}.
On the other hand, notice from Lemma 2.6 that 𝑦_𝑛+1−𝑦_𝑛=𝐽_𝑡𝑛+1𝑣_𝑛+1−𝑡_𝑛+1𝐵₂𝑣_𝑛+1−𝐽_𝑡𝑛𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛≤𝐽_𝑡𝑛+1𝑣_𝑛+1−𝑡_𝑛+1𝐵₂𝑣_𝑛+1−𝐽_𝑡𝑛+1𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛+𝐽_𝑡𝑛+1𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛−𝐽_𝑠𝑛𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛≤𝑣_𝑛+1−𝑡_𝑛+1𝐵₂𝑣_𝑛+1−𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛+𝐽_𝑡𝑛+1𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛−𝐽_𝑠𝑛𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛=𝑣_𝑛+1−𝑡_𝑛+1𝐵₂𝑣_𝑛+1−𝑣_𝑛−𝑡_𝑛+1𝐵₂𝑣_𝑛+𝑡_𝑛−𝑡_𝑛+1𝐵₂𝑣_𝑛+𝐽_𝑡𝑛+1𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛−𝐽_𝑡𝑛𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛≤𝑣_𝑛+1−𝑣_𝑛+𝑡_𝑛+1𝑡1−^𝑛𝑡_𝑛+1𝐵₂𝑣_𝑛+𝑡1−^𝑛𝑡_𝑛+1𝐽_𝑡𝑛+1𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛−𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛≤𝑣_𝑛+1−𝑣_𝑛𝑡+21−^𝑛𝑡_𝑛+1𝑀₂, (3.19) where 𝑀₂>0 is an appropriate constant such that 𝑀₂=sup_𝑛≥1{𝑡_𝑛+1‖𝐵₂𝑣_𝑛‖,‖𝐽_𝑡𝑛+1(𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛)−(𝑣_𝑛−𝑡_𝑛𝐵₂𝑣_𝑛)‖}.
In a similar way, we can get from Lemma 2.6 that 𝑧_𝑛+1−𝑧_𝑛=𝐽_𝑠𝑛+1