A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

Abstract

In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and induced norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝐹∶𝐶×𝐶→𝑅 be a bifunction. We consider the following equilibrium problem (EP) which is to find 𝑥^∗∈𝐶 such that EP:𝐹𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (1) The set of solutions of EP is denoted by EP(𝐹).

Let {𝐹_𝑖,𝑖=1,2,…,𝑁} be a finite family of bifunctions from 𝐶×𝐶 into 𝑅, where 𝑅 is the set of real numbers. The system of equilibrium problems for {𝐹₁,𝐹₂,…,𝐹_𝑁} is to find a common element 𝑥^∗∈𝐶 such that 𝐹₁𝑥^∗𝐹,𝑦≥0,∀𝑦∈𝐶,₂𝑥^∗⋮𝐹,𝑦≥0,∀𝑦∈𝐶,_𝑁𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (2) We denote the set of solutions of (2) by ∩^𝑁_𝑘=1SEP(𝐹_𝑘), where SEP(𝐹_𝑘) is the set of solutions to the equilibrium problems, that is,𝐹_𝑘𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (3)

If 𝑁=1, then the problem (2) is reduced to the equilibrium problems.

If 𝑁=1 and 𝐹(𝑥^∗,𝑦)=⟨𝑇𝑥^∗,𝑦−𝑥^∗⟩, then the problem (2) is reduced to the variational inequality problems of finding 𝑥^∗∈𝐶 such that 𝑇𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶. (4) The set of solutions of (4) is denoted by VI(𝐶,𝑇).

The 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, and numerous problems in physics, economics, and others. Some methods have been proposed to solve VI(𝐶,𝑇), EP(𝐹), and SEP(𝐹_𝑘); see, for example, [1–29] and references therein. Formulations (2) extend this formulism to such problems, covering in particular various forms of feasibility problems [30, 31].

Definition 1. One-parameter family mapping Γ={𝑇(𝑡)∶𝑡∈𝑅⁺} from 𝐶 into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions:(i)𝑇(0)𝑥=𝑥 for all 𝑥∈𝐶,(ii)𝑇(𝑠+𝑡)=𝑇(𝑠)𝑇(𝑡) for all 𝑠,𝑡∈𝑅⁺,(iii)for each 𝑥∈𝐶, the mapping 𝑇(𝑡)𝑥 is continuous,(iv)‖𝑇(𝑡)𝑥−𝑇(𝑡)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 and 𝑡∈𝑅⁺.

Remark 2. We denote by 𝐹(Γ) the set of all common fixed points of Γ, that is, 𝐹(Γ)∶=∩_𝑡∈𝑅⁺𝐹(𝑇(𝑡))={𝑥∈𝐶∶𝑇(𝑡)𝑥=𝑥}.

Let 𝐵∶𝐶→𝐻 be a nonlinear mapping. Now, we recall the following definitions.(1)𝐵 is said to be monotone if 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶. (5)(2)𝐵∶𝐶→𝐶 is called 𝜔-Lipschitzian if there exists a positive constant 𝜔 such that 𝐵𝑥−𝐵𝑦≤𝜔𝑥−𝑦,∀𝑥,𝑦∈𝐶. (6)(3)𝐵 is said to be 𝜂-strongly monotone if there exists a positive constant 𝜂 such that 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥𝜂𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (7)(4)𝐵 is said to be nonexpansive if 𝐵𝑥−𝐵𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (8) And 𝐹(𝐵) denotes the set of fixed points of the mapping 𝐵, that is, 𝐹(𝐵)={𝑥∈𝐶∶𝐵𝑥=𝑥}.(5)𝐵∶𝐶→𝐶 is said to be 𝑘-strictly pseudocontractive mapping if there exists a constant 0≤𝑘<1 such that 𝐵𝑥−𝐵𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝐵)𝑥−(𝐼−𝐵)𝑦²,∀𝑥,𝑦∈𝐶. (9)(6)𝐵 is said to be 𝛼-inverse-strongly monotone if there exists a constant 𝛼>0 such that 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥𝛼𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝐶. (10)(7)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−𝐵𝑦²+𝑣𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (11)(8)A linear bounded operator 𝐵 is strong positive if there exists a constant 𝛾>0 with the property ⟨𝐵𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐶. (12)(9)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.(10)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.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space 𝐻: 𝜃(𝑥)=12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩,∀𝑥∈𝐹(𝑆), (13) where 𝐴 is a linear bounded operator, 𝐹(𝑆) is the fixed point set of a nonexpansive mapping 𝑆, and 𝑏 is a given point in 𝐻 [16].

For finding a common element of the set of fixed points of nonexpansive mappings and the set of the variational inequalities, in 2006, Marino and Xu [16] introduced the general iterative method and proved that for a given 𝑥₀∈𝐻, the sequence {𝑥_𝑛} generated by the algorithm 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐵𝑇𝑥_𝑛, (14) where 𝑇 is a self-nonexpansive mapping on 𝐻, 𝑓 is an 𝛼-contraction of 𝐻 into itself (i.e., ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐻 and 𝛼∈(0,1)), {𝛼_𝑛}⊂(0,1) satisfies certain conditions, and 𝐵 is strongly positive bounded linear operator on 𝐻 and converges strongly to fixed point 𝑥^∗ of 𝑇 which is the unique solution to the following variational inequality: 𝛾𝑓−𝐵𝑥^∗,𝑥^∗−𝑥≤0,∀𝑥∈𝐹(𝑇), (15) which is the optimality condition for the minimization problem min_{𝑥∈𝐹(𝑆)∩EP}12⟨𝐵𝑥,𝑥⟩−ℎ(𝑥), (16) where ℎ is a potential function for 𝑟𝑓(i.e., ℎ^′(𝑥)=𝑟𝑓(𝑥) for 𝑧∈𝐻).

Takahashi and Toyoda [32] introduced the following iterative scheme: 𝑥₀𝑥∈𝐶,_𝑛+1=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛SP_𝐶𝑥_𝑛−𝛼_𝑛𝐵𝑥_𝑛, (17) 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)≠∅, then the sequence {𝑥_𝑛} generated by (17) converges weakly to some 𝑧∈𝐹(𝑆)∩VI(𝐶,𝐵).

Yamada [33] introduced the following iterative scheme called the hybrid steepest descent method: 𝑥_𝑛+1=𝑆𝑥_𝑛+𝛼_𝑛𝜇𝐵𝑆𝑥_𝑛,𝑛∈𝑁, (18) where 𝑥₁=𝑥∈𝐻, {𝛼_𝑛}⊂(0,1), and let 𝐵∶𝐻→𝐻 be a strongly monotone and Lipschitz continuous mapping and 𝜇 is a positive real number. He proved that the sequence {𝑥_𝑛} generated by (18) converges strongly to the unique solution of 𝐹(𝑆)∩VI(𝐶,𝐵).

Let 𝐶 be a nonempty closed convex subset of 𝐻. Given 𝑟>0 the operators 𝐽^𝐹_𝑟∶𝐻→𝐶 defined by 𝐽^𝐹_𝑟+1𝑥=𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶 (19) are called the resolvent of 𝐹 (see [19]). It is shown in [19] that under suitable hypotheses on 𝐹 (to be stated precisely in Section 2), 𝐽^𝐹_𝑟∶𝐻→𝐶 is single-valued and firmly nonexpansive and satisfied 𝐹(𝐽^𝐹_𝑟)=EP(𝐹),∀𝑟>0.

For finding a common element of EP(𝐹)∩𝐹(𝑆), S. Takahashi and W. Takahashi [23] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let 𝑆∶𝐶→𝐻 be a nonexpansive mapping. Starting with arbitrary initial point 𝑥₁∈𝐻, define sequences {𝑥_𝑛} and {𝑢_𝑛} recursively by 𝐹𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑢_𝑛,∀𝑛∈𝑁. (20) They proved that under certain appropriate conditions imposed on {𝛼_𝑛} and {𝑟_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈𝐹(𝑆)∩EP(𝐹), where 𝑧=𝑃_{𝐹(𝑆)∩EP(𝐹)}𝑓(𝑧).

In 2012, Chamnarnpan and Kumam [34] introduced the following explicit viscosity scheme with respect to 𝑊-mappings for an infinite family of nonexpansive mappings 𝑥_𝑛+1=𝜀_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑊_𝑛𝐽^𝐹_𝑟𝑥_𝑛. (21) They prove that sequence {𝑥_𝑛} and 𝐽^𝐹_𝑟𝑛 converge strongly to 𝑧∈(∩^∞_𝑛=1𝐹(𝑇_𝑛))∩EP(𝐹), where 𝑧 is an equilibrium point for 𝐹 and is the unique solution of the variational inequality 𝑟𝑓−𝐴𝑧,𝑥−𝑧≤0,∀𝑥∈^∞_𝑛=1𝐹𝑇_𝑛∩EP(𝐹). (22)

In 2012, Kangtunyakarn [35] modify the set of variational inequality to construct a new iterative scheme for finding a common element of the set of fixed point problems of infinite family of 𝑘_𝑖 pseudocontractive mappings and the set of equilibrium problem and two sets of variational inequality problems. Let 𝐹∶=^∞_𝑖=1𝐹𝑇_𝑖∩^𝑀_𝑘=1SEP𝐹_𝑘∩VI(𝐶,𝐴)∩VI(𝐶,𝐵). (23) Starting with arbitrary initial point 𝑥₁∈𝐶, define sequences {𝑥_𝑛} and {𝑢_𝑛} recursively by 𝐹𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑆_𝑛𝑃_𝐶𝐼−𝛾(𝑎𝐴+(1−𝑎)𝐵)𝑢_𝑛,∀𝑛∈𝑁, (24) where {𝑆_𝑛∶𝐶→𝐶} is the sequence defined by (37), 𝐴,𝐵 is 𝛼 and 𝛽-inverse-strongly monotone mapping, respectively, 𝑎∈(0,1), 0<𝑟<min{2𝛼,2𝛽} and {𝑟_𝑛}⊂[𝑎,𝑏]⊂(0,min{2𝛼,2𝛽}). Under certain appropriate conditions they proved that the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈𝐹, where 𝑧=𝑃_𝐹𝑢.

Let 𝐴_𝑖∶𝐶→𝐻 be a mapping, for 𝑖=1,2,…,𝑁. By modification of (4), for 𝛿_𝑖∈(0,1), we have VI𝐶,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖=𝑥^∗∈𝐶∶𝑦−𝑥^∗,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑥^∗≥0,∀𝑦∈𝐶,^𝑁_𝑖=1𝛿_𝑖=1. (25)

In this paper, motivated by the above results, we extend the nonexpansive mapping of hybrid steepest descent method (18) to nonexpansive semigroups and introduce a new iterative scheme for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of solutions of variational inequality problems for relaxed cocoercive mapping in a real Hilbert space by the hybrid steepest descent method. 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 𝐶 is a nonempty closed convex subset of a Hilbert space 𝐻. We write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. 𝑥_𝑛→𝑥 implies that {𝑥_𝑛} converges strongly to 𝑥. We denote by 𝑁 and 𝑅 the sets of positive integers and real numbers, respectively. For any 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶. (26) Such a 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is known that 𝑃_𝐶 is nonexpansive. Furthermore, for 𝑥∈𝐻 and 𝑢∈𝐶, 𝑢=𝑃_𝐶𝑥⟺𝑥−𝑢,𝑢−𝑦≥0,∀𝑦∈𝐶. (27) It is widely known that 𝐻 satisfies Opial’s condition [8], that is, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, the inequality lim_𝑛→∞inf𝑥_𝑛−𝑥<lim_𝑛→∞inf𝑥_𝑛−𝑦 (28) holds for every 𝑦∈𝐻 with 𝑦≠𝑥.

In order to solve the equilibrium problem for a bifunction 𝐹∶𝐶×𝐶→𝑅, we assume that 𝐹 satisfies the following conditions:(A1)𝐹(𝑥,𝑥)=0,∀𝑥∈𝐶,(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0,∀𝑥,𝑦∈𝐶,(A3)lim_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦),∀𝑥,𝑦,𝑧∈𝐶,(A4) For each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

Let us recall the following lemmas which will be useful for our paper.

Lemma 3 (see [19]). Let 𝐹 be a bifunction from 𝐶×𝐶 into 𝑅 satisfying (A1), (A2), (A3), and (A4). Then, for any 𝑟>0 and 𝑥∈𝐻, there exists 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (29) Furthermore, if 𝐽^𝐹_𝑟𝑥={𝑧∈𝐶∶𝐹(𝑧,𝑦)+(1/𝑟)(𝑦−𝑧,𝑧−𝑥)≥0,∀𝑦∈𝐶}, then the following hold:(1) 𝐽^𝐹_𝑟 is single-valued,(2) 𝐽^𝐹_𝑟 is firmly nonexpansive, that is, 𝐽^𝐹_𝑟𝑥−𝐽^𝐹_𝑟𝑦²≤𝐽^𝐹_𝑟𝑥−𝐽^𝐹_𝑟𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐻, (30)(3) 𝐹(𝐽^𝐹_𝑟)=EP(𝐹),(4) EP(𝐹) is closed and convex.

Lemma 4 (see [12]). Let 𝐶 be a nonempty bounded closed and convex subset of a real Hilbert space 𝐻. Let Γ={𝑇(𝑠)∶𝑠∈𝑅⁺} from 𝐶 be a nonexpansive semigroup on C, then for all ℎ>0, lim_𝑡→∞sup_𝑥∈𝐶1𝑡^𝑡₀𝑇(𝑠)(ℎ)1𝑥𝑑𝑠−𝑇𝑡^𝑡₀𝑇(𝑠)𝑥𝑑𝑠=0. (31)

Lemma 5 (see [13]). Let 𝐶 be a nonempty bounded closed and convex subset of a real Hilbert space 𝐻, let {𝑥_𝑛} be a sequence, and let Γ={𝑇(𝑠)∶𝑠∈𝑅⁺} from 𝐶 be a nonexpansive semigroup on C, if the following conditions are satisfied:(i)𝑥_𝑛⇀𝑧,(ii)limsup_𝑠→∞limsup_𝑛→∞‖𝑇(𝑠)𝑥_𝑛−𝑥_𝑛‖=0,
then, 𝑧∈𝐹(Γ).

Lemma 6 (see [36]). In a Hilbert space 𝐻, there holds the inequality 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈𝐻. (32)

Lemma 7 (see [16]). Assume 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and 0≤𝜌≥‖𝐴‖⁻¹, then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 8 (see [37]). Let 𝐵 be a monotone mapping of 𝐶 into 𝐻 and let 𝑁_𝐶𝜔₁ be the normal cone to 𝐶 at 𝜔₁∈𝐶, that is, 𝑁_𝐶𝜔₁=𝜔∈𝐻∶𝜔₁−𝜔₂,𝜔≥0,∀𝜔₂∈𝐶, (33) and define a mapping 𝑄 on 𝐶 by 𝑄𝜔₁=𝐵𝜔₁+𝑁_𝐶𝜔₁,𝜔₁∈𝐶,∅,𝜔₁∉𝐶. (34) Then 𝑄 is maximal monotone and 0∈𝑄𝜔₁ if and only if, ⟨𝐵𝜔₁,𝜔₁−𝜔₂⟩≥0 for all 𝜔₂∈𝐶.

Lemma 9 (see [27]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝐸 and {𝛾_𝑛} be a sequence in [0,1] satisfying the following condition: 0<lim_𝑛→∞inf𝛾_𝑛≤lim_𝑛→∞sup𝛾_𝑛<1. (35) Suppose that 𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑦_𝑛,𝑛≥0 and lim_𝑛→∞sup(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 10 (see [28]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝑏_𝑛𝑎_𝑛+𝑐_𝑛,𝑛≥0, (36) where {𝑏_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in 𝑅, such that(i)∑^∞_𝑖=1𝑏_𝑖=∞,(ii)lim_𝑛→∞sup(𝑐_𝑛/𝑏_𝑛)≤0 or ∑^∞_𝑖=1|𝑐_𝑛|<∞.
Then, lim_𝑛→∞𝑎_𝑛=0.

Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Let {𝑇_𝑖}^∞_𝑖=1 be mapping of 𝐶 into self. For all 𝑗=1,2,…, let 𝜌_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂)∈𝐼×𝐼×𝐼 where 𝐼=[0,1] and 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₂=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^𝑛₃⋮𝑈𝐼,_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. (37) This mapping is called 𝑆-mapping generated by 𝑇₁,…,𝑇_𝑛 and 𝜌₁,…,𝜌_𝑛.

Lemma 11 (see [38]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Let {𝑇_𝑖}^∞_𝑖=1 be a 𝑘_𝑖-strict pseudocontractive mapping of 𝐶 into self with 𝜅=sup_𝑖𝑘_𝑖 and let 𝜌_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂)∈𝐼×𝐼×𝐼 where 𝐼=[0,1], 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₂=1, 𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂∈(𝜅,1) for all 𝑗=1,2,…. For every 𝑛∈𝑁, let 𝑆_𝑛 and 𝑆-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑛 and 𝜌₁,𝜌₂,…,𝜌_𝑛 and 𝑇₁,𝑇₂,…, and 𝜌₁,𝜌₂,…, respectively. Then, for every 𝑥∈𝐶 and 𝑘∈𝑁, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists.

In view of the previous lemma, we will define the mapping 𝑆∶𝐶→𝐶 as follows: 𝑆𝑥∶=lim_𝑛→∞𝑆_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥,𝑥∈𝐶. (38)

Remark 12 (see [38]). For each 𝑛∈𝑁, 𝑆_𝑛 is nonexpansive and lim_𝑛→∞sup_𝑥∈𝐷‖𝑆_𝑛𝑥−𝑆𝑥‖=0 for every bounded subset 𝐷 of 𝐶.

Lemma 13 (see [38]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Let {𝑇_𝑖}^∞_𝑖=1 be a 𝑘_𝑖-strict pseudocontractive mapping of 𝐶 into self such that ∩^∞_𝑖=1𝐹(𝑇_𝑖)≠∅ with 𝜅=sup_𝑖𝑘_𝑖 and let 𝜌_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂)∈𝐼×𝐼×𝐼 where 𝐼=[0,1], 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₂=1, 𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂∈(𝜅,1) for all 𝑗=1,2,…. For every 𝑛∈𝑁, let 𝑆_𝑛 and 𝑆-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑛 and 𝜌₁,𝜌₂,…,𝜌_𝑛, respectively. Then, 𝐹(𝑆)=∩^∞_𝑖=1𝐹(𝑇_𝑖)≠∅.

3. Main Results

In this section, we will present our main results. To establish our results, we need the following technical lemmas.

Lemma 14. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and let 𝐴_𝑖∶𝐶→𝐻 be 𝜔_𝑖-Lipschitz continuous and relaxed (𝑢_𝑖,𝑣_𝑖)-cocoercive mappings with 𝑣_𝑖−𝑢_𝑖𝜔²_𝑖>0, for 𝑖=1,2,…,𝑁. If ∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖)≠∅, then, for 𝛿_𝑖∈(0,1) and ∑^𝑁_𝑖=1𝛿_𝑖=1, ^𝑁_𝑖=1VI𝐶,𝐴_𝑖=VI𝐶,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖. (39)

Proof. The proof is by induction. This holds for 𝑁=2. In fact, for 𝑎∈(0,1), it is obvious that VI𝐶,𝐴₁∩VI𝐶,𝐴₂⊆VI𝐶,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖. (40) Next, we will show that VI(𝐶,∑^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖)⊆VI(𝐶,𝐴₁)∩VI(𝐶,𝐴₂).
Let 𝑥₀∈VI𝐶,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖,𝑥^∗∈VI𝐶,𝐴₁∩VI𝐶,𝐴₂. (41) It follows that 𝑦−𝑥^∗,𝐴₁𝑥^∗≥0,∀𝑦∈𝐶,𝑦−𝑥^∗,𝐴₂𝑥^∗≥0,∀𝑦∈𝐶. (42)(43) Then, for every 𝑎∈(0,1), one has 𝑦−𝑥^∗,𝑎𝐴₁𝑥^∗≥0,∀𝑦∈𝐶,𝑦−𝑥^∗,(1−𝑎)𝐴₂𝑥^∗≥0,∀𝑦∈𝐶. (44) From 𝑥₀∈VI(𝐶,∑^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖) and (43), one has 𝑥^∗−𝑥₀,𝑎𝐴₁𝑥₀=𝑥^∗−𝑥₀,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑥₀−𝑥^∗−𝑥₀,(1−𝑎)𝐴₂𝑥₀≥(1−𝑎)𝑥₀−𝑥^∗,𝐴₂𝑥₀=(1−𝑎)𝑥₀−𝑥^∗,𝐴₂𝑥₀−𝐴₂𝑥^∗+(1−𝑎)𝑥₀−𝑥^∗,𝐴₂𝑥^∗≥(1−𝑎)𝑣₂−𝑢₂𝜔²₂𝑥₀−𝑥^∗2≥0, (45) which means 𝑥^∗−𝑥₀,𝐴₁𝑥₀≥0. (46) On the other hand, from 𝑥^∗∈VI(𝐶,𝐴₁), we have 𝑥^∗−𝑥₀,𝐴₁𝑥₀=𝑥^∗−𝑥₀,𝐴₁𝑥₀−𝐴₁𝑥^∗+𝑥^∗−𝑥₀,𝐴₁𝑥^∗≤⟨𝑥^∗−𝑥₀,𝐴₁𝑥₀−𝐴₁𝑥^∗⟩≤𝑢₁𝐴₁𝑥₀−𝐴₁𝑥^∗2−𝑣₁𝑥₀−𝑥^∗2≤𝑢₁𝜔²₁𝑥₀−𝑥^∗2−𝑣₁𝑥₀−𝑥^∗2=𝑢₁𝜔²₁−𝑣₁𝑥₀−𝑥^∗2≤0. (47) This together with (46) leads to 𝐴₁𝑥^∗=𝐴₁𝑥₀. (48) Furthermore, for every 𝑦∈𝐶, from (46) and (48), we obtain 𝑦−𝑥₀,𝐴₁𝑥₀=𝑦−𝑥^∗,𝐴₁𝑥₀+⟨𝑥^∗−𝑥₀,𝐴₁𝑥₀⟩≥⟨𝑦−𝑥^∗,𝐴₁𝑥₀⟩=⟨𝑦−𝑥^∗,𝐴₁𝑥^∗⟩≥0, (49) which implies 𝑥₀∈VI𝐶,𝐴₁. (50) It follows from (45) and (42) that 𝑥^∗−𝑥₀,(1−𝑎)𝐴₂𝑥₀≥𝑥₀−𝑥^∗,𝑎𝐴₁𝑥₀=𝑎𝑥₀−𝑥^∗,𝐴₁𝑥₀−𝑥^∗+𝑎𝑥₀−𝑥^∗,𝐴₁𝑥^∗≥𝑎𝑥₀−𝑥^∗,𝐴₁𝑥₀−𝑥^∗≥𝑎𝑣₁−𝑢₁𝜔²₁𝑥₀−𝑥^∗2≥0. (51) It yields that 𝑥^∗−𝑥₀,𝐴₂𝑥₀≥0. (52) From 𝑥^∗∈VI(𝐶,𝐴₂) and (52), one has 0≤𝑥^∗−𝑥₀,𝐴₂𝑥₀=𝑥^∗−𝑥₀,𝐴₂𝑥₀−𝐴₂𝑥^∗+𝑥^∗−𝑥₀,𝐴₂𝑥^∗≤𝑥^∗−𝑥₀,𝐴₂𝑥₀−𝐴₂𝑥^∗≤𝑢₂𝐴₂𝑥₀−𝐴₂𝑥^∗2−𝑣₂𝑥₀−𝑥^∗2≤𝑢₂𝜔²₂𝑥₀−𝑥^∗2−𝑣₁𝑥₀−𝑥^∗2=𝑢₂𝜔²₂−𝑣₂𝑥₀−𝑥^∗2≤0. (53) That is, 𝐴₂𝑥^∗=𝐴₂𝑥₀. (54) Therefore, for every 𝑦∈𝐶, from (52) and (54), we obtain 𝑦−𝑥₀,𝐴₂𝑥₀=𝑦−𝑥^∗,𝐴₂𝑥₀+𝑥^∗−𝑥₀,𝐴₂𝑥₀≥𝑦−𝑥^∗,𝐴₂𝑥₀=𝑦−𝑥^∗,𝐴₂𝑥^∗≥0, (55) which means 𝑥₀∈VI𝐶,𝐴₂. (56) And hence, 𝑥₀∈VI𝐶,𝐴₁∩VI𝐶,𝐴₂. (57) Thus, we have VI𝐶,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖⊆VI𝐶,𝐴₁∩VI𝐶,𝐴₂. (58) Thus, VI𝐶,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖=VI𝐶,𝐴₁∩VI𝐶,𝐴₂. (59)
Assume now that ∩^𝑘_𝑖=1VI(𝐶,𝐴_𝑖)=VI(𝐶,∑^𝑘_𝑖=1𝛿_𝑖𝐴_𝑖) is true for some 𝑘, and we show that it continues to hold for 𝑘+1. For 𝛿_𝑖∈(0,1) and ∑^𝑘+1_𝑖=1𝛿_𝑖=1, we have VI𝐶,^𝑘+1_𝑖=1𝛿_𝑖𝐴_𝑖=VI𝐶,𝛿₁𝐴₁+^𝑘+1_𝑖=2𝛿_𝑖𝐴_𝑖=VI𝐶,𝛿₁𝐴₁+1−𝛿₁^𝑘+1_𝑖=2𝛿_𝑖1−𝛿₁𝐴_𝑖=VI𝐶,𝛿₁𝐴₁∩VI𝐶,1−𝛿₁^𝑘+1_𝑖=2𝛿_𝑖1−𝛿₁𝐴_𝑖=VI𝐶,𝐴₁∩VI𝐶,^𝑘+1_𝑖=2𝛿_𝑖1−𝛿₁𝐴_𝑖=VI𝐶,𝐴₁∩^𝑘+1_𝑖=2VI𝐶,𝐴_𝑖=^𝑘+1_𝑖=1VI𝐶,𝐴_𝑖. (60) By induction, ∩^𝑘_𝑖=1VI(𝐶,𝐴_𝑖)=VI(𝐶,∑^𝑘_𝑖=1𝛿_𝑖𝐴_𝑖) holds for 𝑘=1,2,…,𝑁 and this completes the proof.

Lemma 15. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, let Γ={𝑇(𝑠)∶𝑠∈𝑅⁺} from 𝐶 be a nonexpansive semigroup on 𝐶, and let 𝐴_𝑖∶𝐶→𝐻 be 𝜔_𝑖-Lipschitz continuous and relaxed (𝜇_𝑖,𝑣_𝑖)-cocoercive mappings with 𝑣_𝑖−𝜇_𝑖𝜔²_𝑖>0,  for 𝑖=1,2,…,𝑁. Assume that 𝐷=∑^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖, for 𝛿_𝑖∈(0,1) and ∑^𝑁_𝑖=1𝛿_𝑖=1. If 𝐾_𝑛(𝑥)=(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑥𝑑𝑠, where {𝑆_𝑛∶𝐶→𝐶} is the sequence defined by (37) with 0≤𝛼_𝑛≤(2∑^𝑁_𝑖=1𝛿_𝑖(𝑣_𝑖−𝜇_𝑖𝜔²_𝑖))/(∑^𝑁_𝑖=1𝛿_𝑖𝜔²_𝑖), then 𝐾_𝑛−𝛼_𝑛𝐷𝐾_𝑛 is a nonexpansive mapping in 𝐻. Furthermore, 𝐼−𝛼_𝑛𝐷 is a nonexpansive mapping in 𝐻.

Proof. Since 0≤𝛼_𝑛≤(2∑^𝑁_𝑖=1𝛿_𝑖(𝑣_𝑖−𝜇_𝑖𝜔²_𝑖))/(∑^𝑁_𝑖=1𝛿_𝑖𝜔²_𝑖), for every 𝑥,𝑦∈𝐶, we have 𝐾_𝑛−𝛼_𝑛𝐷𝐾_𝑛𝑥−𝐾_𝑛−𝛼_𝑛𝐷𝐾_𝑛𝑦²=𝐾_𝑛𝑥−𝐾_𝑛𝑦−𝛼_𝑛𝐷𝐾_𝑛𝑥−𝐷𝐾_𝑛𝑦²=𝐾_𝑛𝑥−𝐾_𝑛𝑦²−2𝛼_𝑛𝐾_𝑛𝑥−𝐾_𝑛𝑦,𝐷𝐾_𝑛𝑥−𝐷𝐾_𝑛𝑦+𝛼²_𝑛𝐷𝐾_𝑛𝑥−𝐷𝐾_𝑛𝑦²≤𝐾_𝑛𝑥−𝐾_𝑛𝑦²−2𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝑣_𝑖−𝜇_𝑖𝜔²_𝑖×𝐾_𝑛𝑥−𝐾_𝑛𝑦²+𝛼²_𝑛^𝑁_𝑖=1𝛿_𝑖𝜔²_𝑖𝐾_𝑛𝑥−𝐾_𝑛𝑦²=1−2𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝑣_𝑖−𝜇_𝑖𝜔²_𝑖+𝛼²_𝑛^𝑁_𝑖=1𝛿_𝑖𝜔²_𝑖×𝐾_𝑛𝑥−𝐾_𝑛𝑦²≤𝑥−𝑦². (61) Thus, we obtain that 𝐾_𝑛−𝛼_𝑛𝐷𝐾_𝑛 is a nonexpansive mapping.
Similarly, we can obtain that 𝐼−𝛼_𝑛𝐷 is a nonexpansive mapping in 𝐻 and this completes the proof.

The following main results follow from Lemmas 14 and 15.

Theorem 16. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝐹_𝑘,𝑘∈{1,2,…,𝑀} be a bifunction from 𝐶×𝐶→𝑅 satisfying (A1)–(A4). Let Γ={𝑇(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on 𝐶 and let 𝑡_𝑛 be a positive real divergent sequence. Let {𝑇_𝑖}^∞_𝑖=1 be 𝑘_𝑖-strict pseudo-contractive mappings of 𝐶 into self with 𝜅=sup_𝑖𝑘_𝑖 and let 𝜌_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂)∈𝐼×𝐼×𝐼, where 𝐼=[0,1], 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₂=1,𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂∈(𝜅,1) for all 𝑗=1,2,…. For every 𝑛∈𝑁, let 𝑆_𝑛 and 𝑆-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑛 and 𝜌₁,𝜌₂,…,𝜌_𝑛 with 𝑇_𝑖(𝐹(Γ))⊂𝐹(Γ). Let 𝐴_𝑖∶𝐶→𝐻 be 𝜔_𝑖-Lipschitz continuous and relaxed (𝜇_𝑖,𝑣_𝑖)-cocoercive mappings with 𝑣_𝑖−𝜇_𝑖𝜔²_𝑖>0,  for 𝑖=1,2,…,𝑁, let 𝑓 be a contraction of 𝐻 into itself with 𝜂∈(0,1), and let 𝐴 be is a strongly positive linear bounded self-adjoint operator with the coefficients 𝛾>0 and 0<𝑟<𝛾/𝜂. Assume that Θ∶=𝐹(Γ)∩^∞_𝑖=1𝐹𝑇_𝑖∩^𝑀_𝑘=1SEP𝐹_𝑘∩^𝑁_𝑖=1VI𝐶,𝐴_𝑖. (62) Let {𝑥_𝑛} be a sequence generated by 𝑥₁∈𝐶 and 𝑢_𝑛=𝐽^𝐹𝑀_{𝑟𝑀,𝑛}𝐽^𝐹𝑀−1_{𝑟𝑀−1,𝑛}⋯𝐽^𝐹2_𝑟2,𝑛𝐽^𝐹1_𝑟1,𝑛𝑥_𝑛,𝑧_𝑛=𝑃_𝐶1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛,𝑦𝑑𝑠_𝑛=𝜀_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,∀𝑛∈𝑁, (63) where {𝑆_𝑛∶𝐶→𝐶} is the sequence defined by (37) and 𝛿_𝑖∈(0,1), ∑^𝑁_𝑖=1𝛿_𝑖=1. If {𝜀_𝑛}, {𝛽_𝑛} are two sequences in (0,1) and {𝛾_𝑛}⊂[𝑐₁,𝑐₂]⊂(0,1) and {𝑟_𝑘,𝑛}, for 𝑘∈{1,2,…,𝑀} is a real sequence in (0,∞) satisfy the following conditions:(i)lim_𝑛→∞𝜀_𝑛=0, ∑^∞_𝑖=1𝜀_𝑛=∞, (ii)0<lim_𝑛→∞inf𝛽_𝑛≤lim_𝑛→∞sup𝛽_𝑛<1 and lim_𝑛→∞|𝛽_𝑛+1−𝛽_𝑛|=0,(iii)lim_𝑛→∞inf𝑟_𝑘,𝑛>0 and lim_𝑛→∞|𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛|=0, for 𝑘∈{1,2,…,𝑀},(iv){𝛼_𝑛}⊂[𝑔₁,𝑔₂]⊂(0,(2∑^𝑁_𝑖=1𝛿_𝑖(𝑣_𝑖−𝜇_𝑖𝜔²_𝑖))/(∑^𝑁_𝑖=1𝛿_𝑖𝜔²_𝑖)) and lim_𝑛→∞|𝛼_𝑛+1−𝛼_𝑛|=0,(v)lim_𝑛→∞|𝑡_𝑛−𝑡_𝑛−1|/𝑡_𝑛=0.
Then {𝑥_𝑛} converges strongly to 𝑧∈Θ, where 𝑧 is the unique solution of variational inequality lim_𝑛→∞sup𝑟𝑓−𝐴𝑧,𝑝−𝑧≤0,∀𝑝∈Θ, (64) which is the optimality condition for the minimization problem min_𝑧∈Θ12⟨𝐴𝑧,𝑧⟩−ℎ(𝑧), (65) where ℎ is a potential function for 𝑟𝑓(i.e., ℎ^′(𝑧)=𝑟𝑓(𝑧) for 𝑧∈𝐻).

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. From Lemma 7, we know that if 0≤𝜌≤‖𝐴‖⁻¹, then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾. We will assume that ‖𝐼−𝐴‖≤1−𝛾. Since 𝐴 is a strongly positive linear bounded self-adjoint operator on 𝐻, we have ‖𝐴‖=sup{|⟨𝐴𝑥,𝑥⟩|∶𝑥∈𝐻,‖𝑥‖=1}. (66) Note that 1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑥,𝑥=1−𝛽_𝑛−𝜀_𝑛⟨𝐴𝑥,𝑥⟩≥1−𝛽_𝑛−𝜀_𝑛‖𝐴‖≥0. (67) That is, (1−𝛽_𝑛)𝐼−𝜀_𝑛𝐴 is positive. Furthermore, 1−𝛽_𝑛𝐼−𝜀_𝑛𝐴=sup1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑥,𝑥∶𝑥∈𝐻,‖𝑥‖=1=sup1−𝛽_𝑛−𝜀_𝑛⟨𝐴𝑥,𝑥⟩∶𝑥∈𝐻,‖𝑥‖=1≤1−𝛽_𝑛−𝜀_𝑛𝛾. (68)
Next, We divide the proof of Theorem into five steps.
Step 1. We show that {𝑥_𝑛} is bounded.
Take 𝑝∈Θ. Let ℑ^𝑘_𝑛=𝐽^𝐹𝑘_{𝑟𝑘,𝑛}𝐽^𝐹𝑘−1_{𝑟𝑘−1,𝑛}⋯𝐽^𝐹2_𝑟2,𝑛𝐽^𝐹1_𝑟1,𝑛, for 𝑘∈{1,2,…𝑀} and ℑ⁰_𝑛=𝐼, for any 𝑛∈𝑁. Since 𝐽^𝐹𝑘_{𝑟𝑘,𝑛} is nonexpansive for each 𝑘=1,2,…𝑀 and 𝑢_𝑛=ℑ^𝑀_𝑛𝑥_𝑛, we have 𝑢_𝑛−𝑝=ℑ^𝑀_𝑛𝑥_𝑛−ℑ^𝑀_𝑛𝑝≤𝑥_𝑛−𝑝. (69) From Lemma 15 and (69), one has 𝑧_𝑛=𝑃−𝑝_𝐶1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛=𝑃𝑑𝑠−𝑝_𝐶1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑃_𝐶1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛≤1𝑝𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛≤𝑢𝑝𝑑𝑠_𝑛≤𝑥−𝑝_𝑛−𝑝. (70) It follows that 𝑦_𝑛=𝜀−𝑝_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛=𝜀−𝑝_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑝+𝛾𝑓𝑝−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛−𝜀_𝑛𝛾𝑧_𝑛−𝑝≤𝜀_𝑛𝛾𝜂𝑥_𝑛−𝑝+𝜀_𝑛𝑝+𝛾𝑓−𝐴𝑝1−𝜀_𝑛𝛾𝑧_𝑛≤−𝑝1−𝜀_𝑛𝛾−𝛾𝜂𝑥_𝑛−𝑝+𝜀_𝑛𝛾𝑓𝑝−𝐴𝑝. (71) Furthermore, 𝑥_𝑛+1−𝑝=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛−𝑝≤𝛾_𝑛𝑥_𝑛−𝑝+1−𝛾_𝑛𝑦_𝑛≤−𝑝1−𝜀_𝑛1−𝛾_𝑛𝛾−𝛾𝜂𝑥_𝑛+−𝑝1−𝛾_𝑛𝜀_𝑛𝑝𝑥𝛾𝑓−𝐴𝑝≤max_𝑛,𝑝−𝑝𝛾𝑓−𝐴𝑝.𝛾−𝛾𝜂 (72) By induction, we have 𝑥_𝑛−𝑝≤max𝑥₁,𝑝−𝑝𝛾𝑓−𝐴𝑝𝛾−𝛾𝜂,𝑛≥1. (73) Hence, {𝑥_𝑛} is bounded and we also obtain that {𝑢_𝑛},{𝑧_𝑛},{𝑦_𝑛},{(∑^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖)(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠} and {𝑓(𝑥_𝑛)} are all bounded.
Step 2. We claim that lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0.
From the definition of 𝑧_𝑛 and Lemma 15, for 𝑝∈Θ, we have 𝑧_𝑛+1−𝑧_𝑛=𝑃_𝐶1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑆_𝑛+1𝑢_𝑛+1𝑑𝑠−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑆_𝑛+1𝑢_𝑛+1𝑑𝑠−𝑃_𝐶1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛≤1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑆_𝑛+1𝑢_𝑛+1𝑑𝑠−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑆_𝑛+1𝑢_𝑛+1−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛≤𝑑𝑠𝐼−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑆_𝑛+1𝑢_𝑛+11𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛+𝛼𝑑𝑠_𝑛−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛≤1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑆_𝑛+1𝑢_𝑛+11𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛+𝛼𝑑𝑠_𝑛−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛≤1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑆_𝑛+1𝑢_𝑛+1−𝑆_𝑛𝑢_𝑛+1𝑑𝑠𝑡_𝑛+1−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛+1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1_𝑡𝑛𝑇(𝑠)𝑆_𝑛𝑢_𝑛+𝛼𝑑𝑠_𝑛−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛≤1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝑠)𝑆_𝑛+1𝑢_𝑛+1−𝑆_𝑛𝑢_𝑛+1𝑑𝑠𝑡_𝑛+1−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−𝑆_𝑛𝑝+1𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1_𝑡𝑛𝑇(𝑠)𝑆_𝑛𝑢_𝑛−𝑆_𝑛𝑝+𝛼𝑑𝑠_𝑛−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛≤𝑆𝑑𝑠_𝑛+1𝑢_𝑛+1−𝑆_𝑛𝑢_𝑛+2𝑡_𝑛+1−𝑡_𝑛𝑡_𝑛+1𝑢_𝑛+𝛼−𝑝_𝑛−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛≤𝑢𝑑𝑠_𝑛+1−𝑢_𝑛+𝑆_𝑛+1𝑢_𝑛−𝑆_𝑛𝑢_𝑛+2𝑡_𝑛+1−𝑡_𝑛𝑡_𝑛+1𝑢_𝑛+𝛼−𝑝_𝑛−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠. (74)
First, we will show that if {𝑥_𝑛} is bounded, then lim_𝑛→∞ℑ^𝑘_𝑛𝑥_𝑛−ℑ^𝑘_𝑛+1𝑥_𝑛=0 (75) for 𝑘∈{1,2,…,𝑀}.
From Step 2 of the proof in [4], we have for 𝑘∈{1,2,…,𝑀}lim_𝑛→∞𝐽^𝐹𝑘_{𝑟𝑘,𝑛+1}𝑥_𝑛−𝐽^𝐹𝑘_{𝑟𝑘,𝑛}𝑥_𝑛=0. (76) For 𝑘∈{1,2,…,𝑀}, notice that ℑ^𝑘_𝑛=𝐽^𝐹𝑘_{𝑟𝑘,𝑛}𝐽^𝐹𝑘−1_{𝑟𝑘−1,𝑛}⋯𝐽^𝐹2_𝑟2,𝑛𝐽^𝐹1_𝑟1,𝑛=𝐽^𝐹𝑘_{𝑟𝑘,𝑛}ℑ_𝑛^𝑘−1. (77) It follows that ℑ^𝑘_𝑛𝑥_𝑛−ℑ^𝑘_𝑛+1𝑥_𝑛=𝐽^𝐹𝑘_{𝑟𝑘,𝑛}ℑ_𝑛^𝑘−1𝑥_𝑛−𝐽^𝐹𝑘_{𝑟𝑘,𝑛+1}ℑ^𝑘−1_𝑛+1𝑥_𝑛≤𝐽^𝐹𝑘_{𝑟𝑘,𝑛}ℑ_𝑛^𝑘−1𝑥_𝑛−𝐽^𝐹𝑘_{𝑟𝑘,𝑛+1}ℑ_𝑛^𝑘−1𝑥_𝑛+𝐽^𝐹𝑘_{𝑟𝑘,𝑛+1}ℑ_𝑛^𝑘−1𝑥_𝑛−𝐽^𝐹𝑘_{𝑟𝑘,𝑛+1}ℑ^𝑘−1_𝑛+1𝑥_𝑛≤𝐽^𝐹𝑘_{𝑟𝑘,𝑛}ℑ_𝑛^𝑘−1𝑥_𝑛−𝐽^𝐹𝑘_{𝑟𝑘,𝑛+1}ℑ_𝑛^𝑘−1𝑥_𝑛+𝐽^𝐹𝑘−1_{𝑟𝑘−1,𝑛}ℑ_𝑛^𝑘−2𝑥_𝑛−𝐽^𝐹𝑘−1_{𝑟𝑘−1,𝑛+1}ℑ_𝑛^𝑘−2𝑥_𝑛+ℑ_𝑛^𝑘−2𝑥_𝑛−ℑ^𝑘−2_𝑛+1𝑥_𝑛≤𝐽^𝐹𝑘_{𝑟𝑘,𝑛}ℑ_𝑛^𝑘−1𝑥_𝑛−𝐽^𝐹𝑘_{𝑟𝑘,𝑛+1}ℑ_𝑛^𝑘−1𝑥_𝑛+𝐽^𝐹𝑘−1_{𝑟𝑘−1,𝑛}ℑ_𝑛^𝑘−2𝑥_𝑛−𝐽^𝐹𝑘−1_{𝑟𝑘−1,𝑛+1}ℑ_𝑛^𝑘−2𝑥_𝑛+⋯+𝐽^𝐹2_𝑟2,𝑛ℑ¹_𝑛𝑥_𝑛−𝐽^𝐹2_𝑟2,𝑛+1ℑ¹_𝑛𝑥_𝑛+𝐽^𝐹1_𝑟1,𝑛𝑥_𝑛−𝐽^𝐹1_𝑟1,𝑛+1𝑥_𝑛. (78) Therefore, from (76), we conclude (75).
Second, we estimate ‖𝑢_𝑛+1−𝑢_𝑛‖. From 𝑢_𝑛+1=ℑ^𝑀_𝑛+1𝑥_𝑛+1 and 𝑢_𝑛=ℑ^𝑀_𝑛𝑥_𝑛=𝐽^𝐹𝑀_{𝑟𝑀,𝑛}ℑ_𝑛^𝑀−1𝑥_𝑛, we obtain 𝐹_𝑀𝑢_𝑛+1+1,𝑦𝑟_𝑀,𝑛+1𝑦−𝑢_𝑛+1,𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1𝐹≥0,∀𝑦∈𝐶,_𝑀𝑢_𝑛+1,𝑦𝑟_𝑀,𝑛𝑦−𝑢_𝑛,𝑢_𝑛−ℑ_𝑛^𝑀−1𝑥_𝑛≥0,∀𝑦∈𝐶. (79)(80) Taking 𝑦=𝑢_𝑛 in (79) and 𝑦=𝑢_𝑛+1 in (80), we have 𝐹_𝑀𝑢_𝑛+1,𝑢_𝑛+1𝑟_𝑀,𝑛+1𝑢_𝑛−𝑢_𝑛+1,𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1𝐹≥0,_𝑀𝑢_𝑛,𝑢_𝑛+1+1𝑟_𝑀,𝑛𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−ℑ_𝑛^𝑀−1𝑥_𝑛≥0. (81) So, from (A2), one has 𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−ℑ_𝑛^𝑀−1𝑥_𝑛𝑟_𝑀,𝑛−𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1𝑟_𝑀,𝑛+1≥0. (82) Furthermore, 𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−ℑ_𝑛^𝑀−1𝑥_𝑛−𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1+𝑟1−^𝑀,𝑛𝑟_𝑀,𝑛+1𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1≥0. (83) Since lim_𝑛→∞𝑟_𝑘,𝑛>0, we assume that there exists a real number such that 𝑟_𝑘,𝑛>𝑎>0 for all 𝑛∈𝑁. Thus, we obtain 𝑢_𝑛+1−𝑢_𝑛≤ℑ_𝑛^𝑀−1𝑥_𝑛−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1+𝑟1−^𝑀,𝑛𝑟_𝑀,𝑛+1𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1≤ℑ_𝑛^𝑀−1𝑥_𝑛−ℑ^𝑀−1_𝑛+1𝑥_𝑛+ℑ^𝑀−1_𝑛+1𝑥_𝑛−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1+1𝑎𝑟_𝑀,𝑛+1−𝑟_𝑀,𝑛𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1≤ℑ_𝑛^𝑀−1𝑥_𝑛−ℑ^𝑀−1_𝑛+1𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1+1𝑎𝑟_𝑀,𝑛+1−𝑟_𝑀,𝑛𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1. (84) Third, we estimate ‖𝑆_𝑛+1𝑢_𝑛−𝑆_𝑛𝑢_𝑛‖. It follows from (37) that 𝑆_𝑛+1𝑢_𝑛−𝑆_𝑛𝑢_𝑛²=𝑈_𝑛+1,1𝑢_𝑛−𝑈_𝑛,1𝑢_𝑛²=𝛼¹₁𝑇₁𝑈_𝑛+1,2𝑢_𝑛+𝛼¹₂𝑈_𝑛+1,2𝑢_𝑛+𝛼¹₃𝑢_𝑛−𝛼¹₁𝑇₁𝑈_𝑛,2𝑢_𝑛+𝛼¹₂𝑈_𝑛,2𝑢_𝑛+𝛼¹₃𝑢_𝑛²=𝛼¹₁𝑇₁𝑈_𝑛+1,2−𝑇₁𝑈_𝑛,2𝑢_𝑛+𝛼¹₂𝑈_𝑛+1,2−𝑈_𝑛,2𝑢_𝑛²≤𝛼¹₁𝑇₁𝑈_𝑛+1,2−𝑇₁𝑈_𝑛,2𝑢_𝑛²+𝛼¹₂𝑈_𝑛+1,2−𝑈_𝑛,2𝑢_𝑛²−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛+1,2−𝑇₁𝑈_𝑛,2𝑢_𝑛−𝑈_𝑛+1,2−𝑈_𝑛,2𝑢_𝑛²≤𝛼¹₁𝑈_𝑛+1,2𝑢_𝑛−𝑈_𝑛,2𝑢_𝑛²+𝜅𝐼−𝑇₁𝑈_𝑛+1,2𝑢_𝑛−𝐼−𝑇₁𝑈_𝑛,2𝑢_𝑛²+𝛼²₁𝑈_𝑛+1,2𝑢_𝑛−𝑈_𝑛,2𝑢_𝑛²−𝛼¹₁𝛼¹₂𝐼−𝑇₁𝑈_𝑛+1,2𝑢_𝑛−𝐼−𝑇₁𝑈_𝑛,2𝑢_𝑛²≤1−𝛼³₁𝑈_𝑛+1,2𝑢_𝑛−𝑈_𝑛,2𝑢_𝑛²⋮≤^𝑛_𝑖=11−𝛼^𝑖₃𝑈_{𝑛+1,𝑛+1}𝑢_𝑛−𝑈_𝑛,𝑛+1𝑢_𝑛², (85) which means that 𝑆_𝑛+1𝑢_𝑛−𝑆_𝑛𝑢_𝑛≤𝐿₁^𝑛_𝑖=11−𝛼^𝑖₃, (86) where 𝐿₁≥0 is a constant such that ‖𝑈_{𝑛+1,𝑛+1}𝑢_𝑛−𝑈_𝑛,𝑛+1𝑢_𝑛‖≤𝐿₁, for all 𝑛∈𝑁.
Next, we estimate ‖𝑦_𝑛+1−𝑦_𝑛‖. Substituting (84) and (86) into (74), one has 𝑧_𝑛+1−𝑧_𝑛≤𝑥_𝑛−𝑥_𝑛+1+ℑ_𝑛^𝑀−1𝑥_𝑛−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1𝑎𝑟_𝑀,𝑛+1−𝑟_𝑀,𝑛𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1+𝐿₁^𝑛_𝑖=11−𝛼^𝑖₃+𝛼_𝑛−𝛼_𝑛+1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛+2𝑡𝑑𝑠_𝑛+1−𝑡_𝑛𝑡_𝑛+1𝑢_𝑛−𝑝. (87) From (61), we have 𝑦_𝑛+1−𝑦_𝑛=𝜀_𝑛+1𝛾𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝜀_𝑛+1−𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛+𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑧_𝑛+1−𝛽_𝑛+1𝐼−𝜀_𝑛+1𝐴𝑧_𝑛+1−𝑧_𝑛≤𝜀_𝑛+1𝛾𝜂𝑥_𝑛+1−𝑥_𝑛+𝜀_𝑛+1−𝜀_𝑛×𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛+𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑧_𝑛+1−𝛽_𝑛+1−𝜀_𝑛+1𝛾𝑧_𝑛+1−𝑧_𝑛. (88) Substitution (87) into (88) yields that 𝑦_𝑛+1−𝑦_𝑛≤1−𝜀_𝑛+1𝛾−𝛾𝜂𝑥_𝑛+1−𝑥_𝑛+𝐿₂𝜀_𝑛+1−𝜀_𝑛+𝛽_𝑛+1−𝛽_𝑛+2𝑡_𝑛+1−𝑡_𝑛𝑡_𝑛+1+𝑟_𝑀,𝑛+1−𝑟_𝑀,𝑛+𝛼_𝑛−𝛼_𝑛+1+ℑ_𝑛^𝑀−1𝑥_𝑛−ℑ^𝑀−1_𝑛+1𝑥_𝑛+𝐿₁^𝑛_𝑖=11−𝛼^𝑖₃, (89) where 𝐿₂ is an appropriate constant such that 𝐿₂=maxsup_𝑛≥1^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠,sup_𝑛≥11𝑎𝑢_𝑛+1−ℑ^𝑀−1_𝑛+1𝑥_𝑛+1,sup_𝑛≥1𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛,sup_𝑛≥1𝑥_𝑛−𝑧_𝑛,sup_𝑛≥1𝑢_𝑛−𝑝. (90) It follows from (89) that 𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝐿₂𝜀_𝑛+1−𝜀_𝑛+𝛽_𝑛+1−𝛽_𝑛+2𝑡_𝑛+1−𝑡_𝑛𝑡_𝑛+1+𝑟_𝑀,𝑛+1−𝑟_𝑀,𝑛+𝛼_𝑛−𝛼_𝑛+1+ℑ_𝑛^𝑀−1𝑥_𝑛−ℑ^𝑀−1_𝑛+1𝑥_𝑛+𝐿₁^𝑛_𝑖=11−𝛼^𝑖₃. (91) Consequently, from (75) and the conditions in Theorem 16, we obtain lim_𝑛→∞sup𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (92) Hence, by Lemma 9, one has lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (93) Since 𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑦_𝑛, this shows that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛾_𝑛𝑦_𝑛−𝑥_𝑛=0. (94)
Step 3. We claim that lim_𝑛→∞‖(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑢_𝑛‖=0.
Observing 𝑦_𝑛=𝜀_𝑛𝛾𝑓(𝑥_𝑛)+𝛽_𝑛𝑥_𝑛+((1−𝛽_𝑛)𝐼−𝜀_𝑛𝐴)𝑧_𝑛, we obtain 𝑥_𝑛−𝑧_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑧_𝑛≤𝑥_𝑛−𝑦_𝑛+𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛+𝛽_𝑛𝑥_𝑛−𝑧_𝑛, (95) which means that 𝑥_𝑛−𝑧_𝑛≤11−𝛽_𝑛𝑥_𝑛−𝑦_𝑛+𝜀_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧_𝑛. (96) This together with the conditions (i) and (ii) imply that lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=0. (97) From (93) and (97), one has lim_𝑛→∞𝑦_𝑛−𝑧_𝑛≤lim_𝑛→∞𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑧_𝑛=0. (98)
For 𝑝∈Θ, we see that 𝑧_𝑛−𝑝²=𝑃_𝐶1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑝²=𝑃_𝐶1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑃_𝐶1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²≤1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²=1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²−2𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠_𝑁𝑖=1,𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠+𝛼²_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²≤𝑥_𝑛−𝑝²−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖2𝑣_𝑖𝜔²_𝑖−2𝜇_𝑖−𝛼_𝑛×𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠². (99) It follows from (42) that𝑦_𝑛−𝑝²=𝜀_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝²=1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝜀_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²=1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛≤−𝐴𝑝1−𝛽_𝑛−𝜀_𝑛𝛾𝑧_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛=−𝐴𝑝1−𝛽_𝑛−𝜀_𝑛𝛾²𝑧_𝑛−𝑝²+𝛽²_𝑛𝑥_𝑛−𝑝²+2𝛽_𝑛1−𝛽_𝑛−𝜀_𝑛𝛾×𝑧_𝑛−𝑝𝑥_𝑛−𝑝+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+2𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛≤−𝐴𝑝1−𝛽_𝑛−𝜀_𝑛𝛾²𝑧_𝑛−𝑝²+𝛽²_𝑛𝑥_𝑛−𝑝²+𝛽_𝑛1−𝛽_𝑛−𝜀_𝑛𝛾×𝑧_𝑛−𝑝²+𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+2𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛=−𝐴𝑝1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾𝑧_𝑛−𝑝²+1−𝜀_𝑛𝛾𝛽_𝑛𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+2𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝. (100) Substituting (99) into (100) yields that 𝑦_𝑛−𝑝²≤1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×𝑥_𝑛−𝑝²−𝑎𝛼_𝑛2𝑣₁𝜔²₁−2𝜇₁+𝛼_𝑛×𝐴₁1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²−(1−𝑎)𝛼_𝑛2𝑣₂𝜔²₂−2𝜇₂+𝛼_𝑛×𝐴₂1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²+1−𝜀_𝑛𝛾𝛽_𝑛𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛=−𝐴𝑝1−𝜀_𝑛𝛾²𝑥_𝑛−𝑝²+1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖2𝑣_𝑖𝜔²_𝑖−2𝜇_𝑖−𝛼_𝑛×𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²≤𝑥_𝑛−𝑝²+1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖2𝑣_𝑖𝜔²_𝑖−2𝜇_𝑖−𝛼_𝑛×𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝+𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝². (101) Furthermore, 𝑥_𝑛+1−𝑝²=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑦_𝑛−𝑝²≤𝛾_𝑛𝑥_𝑛−𝑝²+1−𝛾_𝑛𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝²+1−𝛾_𝑛1−𝜀_𝑛𝛾×1−𝛽_𝑛−𝜀_𝑛𝛾×−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖2𝑣_𝑖𝜔²_𝑖−2𝜇_𝑖−𝛼_𝑛𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²+1−𝛾_𝑛𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+21−𝛾_𝑛𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+21−𝛾_𝑛𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝. (102) It follows that 1−𝑒₁1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×^𝑁_𝑖=1𝛿_𝑖2𝑔₁𝑣_𝑖𝜔²_𝑖−2𝑔₂𝜇_𝑖−𝑔²₂×𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²≤1−𝛾_𝑛1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖2𝑣_𝑖𝜔²_𝑖−2𝜇_𝑖−𝛼_𝑛×𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+1−𝛾_𝑛𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+21−𝛾_𝑛𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝+21−𝛾_𝑛𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝. (103) From (94) and the condition (i), for 𝑖=1,2,…,𝑁, we have lim_𝑛→∞𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠=0. (104) Then, for 𝛿_𝑖∈(0,1) and ∑^𝑁_𝑖=1𝛿_𝑖=1, lim_𝑛→∞^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠=0. (105) On the other hand, one has𝑧_𝑛−𝑝²=𝑃_𝐶1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑝²≤1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠,𝑧_𝑛=1−𝑝21𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²+𝑧_𝑛−𝑝²−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠−𝑧_𝑛−𝑝²≤12𝑢_𝑛−𝑝²+𝑧_𝑛−𝑝²−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²≤12𝑥_𝑛−𝑝²+𝑧_𝑛−𝑝²−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛²−𝛼²_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠²+2𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛,1𝑝𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛, (106)which means that 𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝²−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛²+2𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛×1𝑝𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛. (107) It follows that 𝑦_𝑛−𝑝²≤1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾𝑧_𝑛−𝑝²+1−𝜀_𝑛𝛾𝛽_𝑛𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛≤−𝐴𝑝1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×𝑥_𝑛−𝑝²−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛²+2𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛×𝑆𝑝𝑑𝑠_𝑛𝑢_𝑛−𝑧_𝑛+1−𝜀_𝑛𝛾𝛽_𝑛𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛=−𝐴𝑝1−𝜀_𝑛𝛾²𝑥_𝑛−𝑝²−1−𝜀_𝑛𝛾×1−𝛽_𝑛−𝜀_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛²+2𝛼_𝑛1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑝𝑑𝑠𝑆_𝑛𝑢_𝑛−𝑧_𝑛+𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛≤𝑥−𝐴𝑝_𝑛−𝑝²−1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛²+2𝛼_𝑛1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛×𝑆𝑝𝑑𝑠_𝑛𝑢_𝑛−𝑧_𝑛+𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+2𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝. (108) Therefore, from (108) and (102), one has 𝑥_𝑛+1−𝑝²≤𝛾_𝑛𝑥_𝑛−𝑝²+1−𝛾_𝑛𝑦_𝑛−𝑝²≤𝛾_𝑛𝑥_𝑛−𝑝²+1−𝛾_𝑛×𝑥_𝑛−𝑝²−1−𝜀_𝑛𝛾×1−𝛽_𝑛−𝜀_𝑛𝛾×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛²+2𝛼_𝑛1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛×𝑆𝑝𝑑𝑠_𝑛𝑢_𝑛−𝑧_𝑛+𝜀²_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛=𝑥−𝐴𝑝_𝑛−𝑝²−1−𝛾_𝑛1−𝜀_𝑛𝛾×1−𝛽_𝑛−𝜀_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛²+2𝛼_𝑛1−𝛾_𝑛1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛×𝑆𝑝𝑑𝑠_𝑛𝑢_𝑛−𝑧_𝑛+1−𝛾_𝑛𝜀²_𝑛×𝑟𝑓𝑥_𝑛−𝐴𝑝²+21−𝛾_𝑛𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+21−𝛾_𝑛𝛽_𝑛𝜀_𝑛×𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝. (109) Then, 1−𝛾_𝑛1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝛼_𝑛1−𝛾_𝑛1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛×𝑆𝑝𝑑𝑠_𝑛𝑢_𝑛−𝑧_𝑛+1−𝛾_𝑛𝜀²_𝑛×𝑟𝑓𝑥_𝑛−𝐴𝑝²+21−𝛾_𝑛𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝+21−𝛾_𝑛𝛽_𝑛𝜀_𝑛×𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝. (110) From (94), (105), and condition (i), one has lim_𝑛→∞1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑧_𝑛=0. (111)
Let 𝑝∈Θ and 𝑘∈{1,2,…,𝑀}. Since 𝐽^𝐹𝑘_{𝑟𝑘,𝑛} is firmly nonexpansive, we obtain ℑ^𝑘_𝑛𝑥_𝑛−𝑝²=𝐽^𝐹𝑘_{𝑟𝑘,𝑛}ℑ_𝑛^𝑘−1𝑥_𝑛−𝐽^𝐹𝑘_{𝑟𝑘,𝑛}𝑝²=⟨𝐽^𝐹𝑘_{𝑟𝑘,𝑛}ℑ_𝑛^𝑘−1𝑥_𝑛−𝑝,ℑ_𝑛^𝑘−1𝑥_𝑛=1−𝑝⟩2ℑ^𝑘_𝑛𝑥_𝑛−𝑝²+ℑ_𝑛^𝑘−1𝑥_𝑛−𝑝²−ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛². (112) It follows that ℑ^𝑘_𝑛𝑥_𝑛−𝑝²≤𝑥_𝑛−𝑝²−ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛². (113) Consequently, from (108), one has 𝑦_𝑛−𝑝²≤1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾𝑧_𝑛−𝑝²+1−𝜀_𝑛𝛾𝛽_𝑛𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+2𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛≤−𝐴𝑝1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×𝑢_𝑛−𝑝²+1−𝜀_𝑛𝛾𝛽_𝑛𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+2𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛=−𝐴𝑝1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×ℑ^𝑘_𝑛𝑥_𝑛−𝑝²+1−𝜀_𝑛𝛾𝛽_𝑛×𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛≤−𝐴𝑝1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×𝑥_𝑛−𝑝²−ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛²+1−𝜀_𝑛𝛾𝛽_𝑛𝑥_𝑛−𝑝²+𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛≤𝑥−𝐴𝑝_𝑛−𝑝²−1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛²+𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝. (114) Then, 𝑥_𝑛+1−𝑝²=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑦_𝑛−𝑝²≤𝛾_𝑛𝑥_𝑛−𝑝²+1−𝛾_𝑛𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝²+1−𝛾_𝑛×−1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛²+𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥−𝑝𝑟𝑓_𝑛−𝐴𝑝+2𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝. (115) That is, 1−𝛾_𝑛−1−𝜀_𝑛𝛾1−𝛽_𝑛−𝜀_𝑛𝛾×ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+1−𝛾_𝑛𝜀²_𝑛𝑟𝑓(𝑥_𝑛)−𝐴𝑝²+21−𝛾_𝑛𝜀_𝑛×1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝+21−𝛾_𝑛𝛽_𝑛𝜀_𝑛𝑥_𝑛−𝑝,𝑟𝑓𝑥_𝑛−𝐴𝑝. (116) By condition (i) and (94), for 𝑘∈{1,2,…,𝑀}, we obtain lim_𝑛→∞ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛=0. (117) Therefore, we have 𝑢_𝑛−𝑥_𝑛=ℑ^𝑘_𝑛𝑥_𝑛−ℑ⁰_𝑛𝑥_𝑛≤ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛+ℑ_𝑛^𝑘−1𝑥_𝑛−ℑ_𝑛^𝑘−2𝑥_𝑛+⋯+ℑ¹_𝑛𝑥_𝑛−ℑ⁰_𝑛𝑥_𝑛. (118) From (117), one has lim_𝑛→∞𝑢_𝑛−𝑥_𝑛=0. (119)Notice that 𝑢_𝑛−𝑦_𝑛≤𝑢_𝑛−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛, (120) Applying (119) and (93), we have lim_𝑛→∞𝑢_𝑛−𝑦_𝑛=0. (121) Since 𝑢_𝑛−𝑧_𝑛≤𝑢_𝑛−𝑦_𝑛+𝑦_𝑛−𝑧_𝑛, (122) this together with (94) yields that lim_𝑛→∞𝑢_𝑛−𝑧_𝑛=0. (123) Consequently, we obtain lim_𝑛→∞1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝑢_𝑛=0. (124)
Step 4. Letting 𝑧=𝑃_Θ(𝐼−𝐴+𝑟𝑓)𝑧, we show lim_𝑛→∞sup𝑟𝑓−𝐴𝑧,𝑥_𝑛−𝑧≤0. (125) We know that 𝑃_Θ(𝐼−𝐴+𝑟𝑓) is a contraction. Indeed, for any 𝑥,𝑦∈𝐻, we have 𝑃_Θ𝐼−𝐴+𝑟𝑓𝑥−𝑃_Θ𝑦≤𝑦≤𝐼−𝐴+𝑟𝑓𝐼−𝐴+𝑟𝑓𝑥−𝐼−𝐴+𝑟𝑓1−𝛾−𝑟𝜂𝑥−𝑦, (126) and hence 𝑃_Θ(𝐼−𝐴+𝑟𝑓) is a contraction due to (1−(𝛾−𝑟𝜂))∈(0,1). Thus, Banach’s Contraction Mapping Principle guarantees that 𝑃_Θ(𝐼−𝐴+𝑟𝑓) has a unique fixed point, which implies 𝑧=𝑃_Θ(𝐼−𝐴+𝑟𝑓)𝑧.
We claim that 𝑧∈𝐹(Γ). Since {𝑢_𝑛𝑖}⊂{𝑢_𝑛} is bounded in 𝐶, without loss of generality, we can assume that {𝑢_𝑛𝑖}⇀𝑧. Since 𝐶 is closed and convex, 𝐶 is weakly closed. Thus we have 𝑧∈𝐶. For 0≤𝑠<∞, notice that 𝑢_𝑛𝑖−𝑇(ℎ)𝑢_𝑛𝑖≤𝑢_𝑛𝑖−1𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖+1𝑑𝑠𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖(ℎ)1𝑑𝑠−𝑇𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖+𝑇(ℎ)1𝑑𝑠𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖𝑑𝑠−𝑇(ℎ)𝑢_𝑛𝑖≤2𝑢_𝑛𝑖−1𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖+1𝑑𝑠𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖(ℎ)1𝑑𝑠−𝑇𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖𝑑𝑠. (127) It follows from (124) and Lemma 4 that lim_𝑛→∞𝑢_𝑛𝑖−𝑇(ℎ)𝑢_𝑛𝑖=0. (128) Thus, (128) and Lemma 5 assert that 𝑧∈𝐹(Γ). Since {𝑥_𝑛𝑖}⊂{𝑥_𝑛} is bounded in 𝐶, without loss of generality, we can assume that {𝑥_𝑛𝑖}⇀𝜔. It follows from (94) that 𝑧_𝑛𝑖⇀𝜔. Since 𝐶 is closed and convex, 𝐶 is weakly closed. Thus we have 𝜔∈𝐶.
Let us show 𝜔∈𝐹(𝑆). For the sake of contradiction, suppose that 𝜔∉𝐹(𝑆), that is, 𝑆𝜔≠𝜔. Since 𝑧∈𝐹(Γ), by our assumption, we have 𝑇_𝑖𝜔∈𝐹(Γ) and then 𝑆_𝑛𝜔∈𝐹(Γ). Hence (1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝜔𝑑𝑠=𝑆_𝑛𝜔. Therefore, by (124) and Opial condition, we have lim_𝑛→∞inf𝑢_𝑛𝑖−𝜔<lim_𝑛→∞inf𝑢_𝑛𝑖−𝑆𝜔≤lim_𝑛→∞inf𝑢_𝑛𝑖−1𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖+1𝑑𝑠𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖−1𝑑𝑠𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖+𝑆𝜔𝑑𝑠_𝑛𝑖𝜔−𝑆𝜔≤lim_𝑛→∞inf𝑢_𝑛𝑖−𝜔, (129) which derives a contradiction. Thus, we obtain 𝜔∈𝐹(𝑆)=∩^∞_𝑖=1𝐹(𝑇_𝑖).
Next, we claim that 𝜔∈∩^𝑀_𝑖=1SEP(𝐹_𝑖). Since 𝑢_𝑛=ℑ^𝑘_𝑛𝑥_𝑛 for 𝑘=1,2,…,𝑀, we obtain 𝐹_𝑘ℑ^𝑘_𝑛𝑥_𝑛+1,𝑦𝑟_𝑛𝑦−ℑ^𝑘_𝑛𝑥_𝑛,ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛≥0,∀𝑦∈𝐶. (130)From (A2), one has 1𝑟_𝑛𝑦−ℑ^𝑘_𝑛𝑥_𝑛,ℑ^𝑘_𝑛𝑥_𝑛−ℑ_𝑛^𝑘−1𝑥_𝑛≥𝐹𝑦,ℑ^𝑘_𝑛𝑥_𝑛. (131) Replacing 𝑛 by 𝑛_𝑖, we have 𝑦−ℑ^𝑘_𝑛𝑖𝑥_𝑛𝑖,1𝑟_𝑛𝑖ℑ^𝑘_𝑛𝑖𝑥_𝑛𝑖−ℑ_𝑛^𝑘−1_𝑖𝑥_𝑛𝑖≥𝐹_𝑘𝑦,ℑ^𝑘_𝑛𝑖𝑥_𝑛𝑖. (132) It follows from (1/𝑟_𝑛𝑖)(ℑ^𝑘_𝑛𝑖𝑥_𝑛𝑖−ℑ_𝑛^𝑘−1_𝑖𝑥_𝑛𝑖)→0 and ℑ^𝑘_𝑛𝑖𝑥_𝑛𝑖⇀𝜔 that 𝐹_𝑘𝑦,𝜔≤0,𝑦∈𝐶, (133) for 𝑘=1,2,…,𝑀.
Put 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝜔 for all 𝑡∈(0,1] and 𝑦∈𝐶. Then, we have 𝑧_𝑡∈𝐶 and then 𝐹(𝑧_𝑡,𝜔)≤0. Hence, from (A1) and (A4), we have 0=𝐹_𝑘𝑧_𝑡,𝑧_𝑡≤𝑡𝐹_𝑘𝑧_𝑡,𝑦+(1−𝑡)𝐹_𝑘𝑧_𝑡,𝑦≤𝑡𝐹_𝑘𝑧_𝑡,𝑦, (134) which means 𝐹_𝑘(𝑧_𝑡,𝑦)≥0. From (A3), we obtain 𝐹_𝑘(𝜔,𝑦)≥0 for 𝑦∈𝐶 and then 𝜔∈SEP(𝐹_𝑘) for 𝑘=1,2,…,𝑀, that is, 𝜔∈∩^𝑀_𝑖=1SEP(𝐹_𝑘).
Finally, we claim that 𝜔∈∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖).
We define the maximal monotone operator 𝑄𝑞₁=^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑞₁+𝑁_𝐶𝑞₁,𝜔₁∈𝐶,∅,𝜔₁∉𝐶. (135)
Since 𝐴_𝑖 is relaxed (𝜇_𝑖,𝑣_𝑖)-cocoercive for 𝑖=1,2, we have ^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑥−^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖=𝑦,𝑥−𝑦_𝑁𝑖=1𝛿_𝑖𝐴_𝑖𝑥−𝐴_𝑖≥𝑦,𝑥−𝑦_𝑁𝑖=1𝛿_𝑖−𝜇_𝑖𝐴_𝑖𝑥−𝐴_𝑖𝑦²+𝑣_𝑖𝑥−𝑦²≥^𝑁_𝑖=1𝛿_𝑖𝑣_𝑖−𝜇_𝑖𝜔^𝑖₁𝑥−𝑦²≥0, (136) which yields that ∑^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖 is monotone. Thus, 𝑄 is maximal monotone. Let (𝑞₁,𝑞₂)∈𝐺(𝑄). Since 𝑞₂−(∑^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖)𝑞₁∈𝑁_𝐶𝑞₁ and 𝑧_𝑛∈𝐶, we have 𝑞₁−𝑧_𝑛,𝑞₂−^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑞₁≥0. (137) On the other hand, it follows from 𝑧_𝑛=𝑃_𝐶((1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛(∑^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖)(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠) that 𝑞₁−𝑧_𝑛,𝑧_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠≥0, (138) and hence 𝑞₁−𝑧_𝑛,𝑧_𝑛−1/𝑡_𝑛∫^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠𝛼_𝑛+^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑆_𝑛𝑢_𝑛𝑑𝑠≥0. (139) It follows that 𝑞₁−𝑧_𝑛𝑖,𝑞₂≥𝑞₁−𝑧_𝑛𝑖,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑞₁≥𝑞₁−𝑧_𝑛𝑖,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑞₁−𝑞₁−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑆_𝑛𝑖𝑢_𝑛𝑖𝛼_𝑛𝑖+^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑆_𝑛𝑖𝑢_𝑛𝑖=𝑞₁−𝑧_𝑛𝑖,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×𝑞₁−1𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖−𝑧𝑑𝑠_𝑛𝑖−1/𝑡_𝑛𝑖∫^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖𝑑𝑠𝛼_𝑛𝑖=𝑞₁−𝑧_𝑛𝑖,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑞₁−𝑧_𝑛𝑖+𝑞₁−𝑧_𝑛𝑖,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑧_𝑛𝑖−𝑆_𝑛𝑖𝑢_𝑛𝑖−𝑞₁−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑆_𝑛𝑖𝑢_𝑛𝑖𝛼_𝑛𝑖≥𝑞₁−𝑧_𝑛𝑖,^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖×𝑧_𝑛𝑖−1𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖−𝑞𝑑𝑠₁−𝑧_𝑛𝑖,𝑧_𝑛𝑖−1/𝑡_𝑛𝑖∫^𝑡𝑛𝑖₀𝑇(𝑠)𝑆_𝑛𝑖𝑢_𝑛𝑖𝑑𝑠𝛼_𝑛𝑖, (140) which implies that 𝑞₁−𝜔,𝑞₂≥0. (141) Since 𝑄 is maximal monotone, we obtain that 𝜔∈𝑄⁻¹0. From Lemma 8, we obtain 𝜔∈VI(𝐶,∑^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖), that is, 𝜔∈(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖)). Thus, 𝜔∈Θ.
Since 𝑧=𝑃_Θ(𝐼−𝐴+𝑟𝑓)𝑧, one has lim_𝑛→∞sup𝑟𝑓−𝐴𝑧,𝑥_𝑛−𝑧=lim_𝑖→∞𝑟𝑓−𝐴𝑧,𝑥_𝑛𝑖=−𝑧𝑟𝑓−𝐴𝑧,𝜔−𝑧≤0. (142) Furthermore, 𝑟𝑓−𝐴𝑧,𝑦_𝑛−𝑧=𝑟𝑓−𝐴𝑧,𝑦_𝑛−𝑥_𝑛+𝑟𝑓−𝐴𝑧,𝑥_𝑛−𝑧. (143) From (93) and (142), we have lim_𝑛→∞sup𝑟𝑓−𝐴𝑧,𝑦_𝑛−𝑧≤0. (144)
Step 5. Finally, we show that 𝑥_𝑛 converges strongly to 𝑧=𝑃_Θ(𝐼−𝐴+𝑟𝑓)𝑧. Indeed, from (61) and (70), we obtain 𝑦_𝑛−𝑧²=𝜀_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑧²=1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑧+𝜀_𝑛𝑟𝑓𝑥_𝑛−𝐴𝑧²≤1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑝²+2𝜀_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑧+𝜀_𝑛𝑟𝑓𝑥_𝑛,𝑥−𝐴𝑧𝑟𝑓_𝑛=−𝐴𝑝1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑧²+2𝜀_𝑛⟨𝑦_𝑛−𝑧,𝑟𝑓𝑥_𝑛≤−𝐴𝑧1−𝛽_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴1−𝛽_𝑛𝑧_𝑛−𝑧²+𝛽_𝑛𝑥_𝑛−𝑧²+2𝑟𝜀_𝑛𝑦_𝑛−𝑧,𝑓𝑥_𝑛−𝑓(𝑧)+2𝜀_𝑛𝑦_𝑛(𝑧)≤−𝑧,𝑓−𝐴𝑧1−𝛽_𝑛−𝜀_𝑛𝛾²1−𝛽_𝑛𝑥_𝑛−𝑧²+𝛽_𝑛𝑥_𝑛−𝑧²+𝑟𝜂𝜀_𝑛𝑥_𝑛−𝑧²+𝑦_𝑛−𝑧²+2𝜀_𝑛𝑦_𝑛(𝑧)=2−𝑧,𝑓−𝐴𝑧1−+𝜀𝛾−𝑟𝜂^𝑛𝛾²1−𝛽_𝑛𝑥_𝑛−𝑧²+𝑟𝜂𝜀_𝑛𝑦_𝑛−𝑧²+2𝜀_𝑛𝑦_𝑛−𝑧,𝑓(𝑧)−𝐴𝑧, (145) which implies that 𝑦_𝑛−𝑧²≤21−𝛾−𝑟𝜂𝜀_𝑛1−𝑟𝜂𝜀_𝑛𝑥_𝑛−𝑧²+𝜀_𝑛1−𝑟𝜂𝜀_𝑛×𝛾²𝜀_𝑛1−𝛽_𝑛𝑥_𝑛−𝑧²+2𝑦_𝑛−𝑧,𝑓(𝑧)−𝐴𝑧. (146) It follows from (146) that 𝑥_𝑛+1−𝑧²=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑦_𝑛−𝑝²≤𝛾_𝑛𝑥_𝑛−𝑧²+1−𝛾_𝑛𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+1−𝛾_𝑛×21−𝛾−𝑟𝜂𝜀_𝑛1−𝑟𝜂𝜀_𝑛𝑥_𝑛−𝑧²+𝜀_𝑛1−𝑟𝜂𝜀_𝑛×𝛾²𝜀_𝑛1−𝛽_𝑛𝑥_𝑛−𝑧²+2𝑦_𝑛(𝑧)=2−𝑧,𝑓−𝐴𝑧1−1−𝛾_𝑛𝛾−𝑟𝜂𝜀_𝑛1−𝑟𝜂𝜀_𝑛𝑥_𝑛−𝑧²+1−𝛾_𝑛𝜀_𝑛1−𝑟𝜂𝜀_𝑛×𝛾²𝜀_𝑛1−𝛽_𝑛𝑥_𝑛−𝑧²+2𝑦_𝑛−𝑧,𝑓(𝑧)−𝐴𝑧. (147) From condition (i) and (142), we know that ^∞_𝑖=121−𝛾_𝑛𝛾−𝑟𝜂𝜀_𝑛1−𝑟𝜂𝜀_𝑛=∞,lim_𝑛→∞sup1−𝛾_𝑛𝜀_𝑛1−𝑟𝜂𝜀_𝑛𝛾²𝜀_𝑛1−𝛽_𝑛𝑥_𝑛−𝑧²+2𝑦_𝑛−𝑧,𝑓(𝑧)−𝐴𝑧≤0. (148) we can conclude from Lemma 10 that 𝑥_𝑛→𝑧 as 𝑛→∞. This completes the proof of Theorem 16.

Theorem 17. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝐹_𝑘,𝑘∈{1,2,…,𝑀} be bifunction from 𝐶×𝐶→𝑅 satisfying (A1)–(A4). Let {𝑇_𝑖}^∞_𝑖=1 be 𝑘_𝑖-strict pseudocontractive mappings of 𝐶 into self with 𝜅=sup_𝑖𝑘_𝑖 and let 𝜌_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂)∈𝐼×𝐼×𝐼, where 𝐼=[0,1], 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₂=1, 𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂∈(𝜅,1) for all 𝑗=1,2,…. For every 𝑛∈𝑁, let 𝑆_𝑛 and 𝑆 be 𝑆-mapping generated by 𝑇_𝑛,𝑇_𝑛−1,…,𝑇₁ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₁ and 𝑇_𝑛,𝑇_𝑛−1,…, and 𝜌_𝑛,𝜌_𝑛−1,…, respectively. Let 𝐵∶𝐶→𝐻 be 𝜔-Lipschitz continuous and relaxed (𝜇,𝑣)-cocoercive mappings with 𝑣−𝜇𝜔²>0, let 𝑓 be a contraction of 𝐻 into itself with 𝜂∈(0,1), and let 𝐴 be a strongly positive linear bounded self-adjoint operator with the coefficients 𝛾>0 and 0<𝑟<𝛾/𝜂. Assume that Θ∶=^∞_𝑖=1𝐹𝑇_𝑖∩^𝑀_𝑘=1SEP𝐹_𝑘∩VI(𝐶,𝐵). (149) Let {𝑥_𝑛} be a sequence generated by 𝑥₁∈𝐶 and 𝑢_𝑛=𝐽^𝐹𝑀_{𝑟𝑀,𝑛}𝐽^𝐹𝑀−1_{𝑟𝑀−1,𝑛}⋯𝐽^𝐹2_𝑟2,𝑛𝐽^𝐹1_𝑟1,𝑛𝑥_𝑛,𝑧_𝑛=𝑃_𝐶𝑆_𝑛𝑢_𝑛−𝛼_𝑛𝐵𝑆_𝑛𝑢_𝑛,𝑦_𝑛=𝜀_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,∀𝑛∈𝑁, (150) where {𝑆_𝑛∶𝐶→𝐶} is the sequence defined by (37). If {𝜀_𝑛}, {𝛽_𝑛} are two sequences in (0,1) and {𝛾_𝑛}⊂[𝑐₁,𝑐₂]⊂(0,1) and {𝑟_𝑘,𝑛}, for 𝑘∈{1,2,…,𝑀} is a real sequence in (0,∞) satisfing the following conditions:(i)lim_𝑛→∞𝜀_𝑛=0, ∑^∞_𝑖=1𝜀_𝑛=∞, (ii)0<lim_𝑛→∞inf𝛽_𝑛≤lim_𝑛→∞sup𝛽_𝑛<1 and lim_𝑛→∞|𝛽_𝑛+1−𝛽_𝑛|=0,(iii)lim_𝑛→∞inf𝑟_𝑘,𝑛>0 and lim_𝑛→∞|𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛|=0, for 𝑘∈{1,2,…,𝑀},(iv){𝛼_𝑛}⊂[𝑔₁,𝑔₂]⊂(0,(2(𝑣−𝜇𝜔²))/𝜔²) and lim_𝑛→∞|𝛼_𝑛+1−𝛼_𝑛|=0.
Then {𝑥_𝑛} converges strongly to 𝑧∈Θ, where 𝑧 is the unique solution of variational inequality lim_𝑛→∞sup𝑟𝑓−𝐴𝑧,𝑝−𝑧≤0,∀𝑝∈Θ, (151) which is the optimality condition for the minimization problem min_𝑧∈Θ12⟨𝐴𝑧,𝑧⟩−ℎ(𝑧), (152) where ℎ is a potential function for 𝑟𝑓(i.e., ℎ^′(𝑧)=𝑟𝑓(𝑧) for 𝑧∈𝐻).

Proof. By Theorem 16, for 𝑖=1,2,…,𝑁, letting 𝐴_𝑖=𝐵, we can obtain Theorem 17.

Theorem 18. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝐹_𝑘,𝑘∈{1,2,…,𝑀} be a bifunction from 𝐶×𝐶→𝑅 satisfying (A1)–(A4). Let {𝑇_𝑖}^∞_𝑖=1 be 𝑘_𝑖-strict pseudo-contractive mappings of 𝐶 into self with 𝜅=sup_𝑖𝑘_𝑖 and let 𝜌_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂)∈𝐼×𝐼×𝐼, where 𝐼=[0,1], 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₂=1, 𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₂∈(𝜅,1) for all 𝑗=1,2,…. For every 𝑛∈𝑁, let 𝑆_𝑛 and 𝑆 be 𝑆-mapping generated by 𝑇_𝑛,𝑇_𝑛−1,…,𝑇₁ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₁ and 𝑇_𝑛,𝑇_𝑛−1,…, and 𝜌_𝑛,𝜌_𝑛−1,⋯, respectively. Let 𝐴_𝑖∶𝐶→𝐻 be 𝜔_𝑖-Lipschitz continuous and relaxed (𝜇_𝑖,𝑣_𝑖)-cocoercive mappings with 𝑣_𝑖−𝜇_𝑖𝜔²_𝑖>0, for 𝑖=1,2,…,𝑁, let 𝑓 be a contraction of 𝐻 into itself with 𝜂∈(0,1), and let 𝐴 be is a strongly positive linear bounded self-adjoint operator with the coefficients 𝛾>0 and 0<𝑟<𝛾/𝜂. Assume that Θ∶=^∞_𝑖=1𝐹𝑇_𝑖∩EP(𝐹)∩^𝑁_𝑖=1VI𝐶,𝐴_𝑖. (153) Let {𝑥_𝑛} be a sequence generated by 𝑥₁∈𝐶 and 𝐹𝑢_𝑛+1,𝑦𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑧≥0,_𝑛=𝑃_𝐶𝑆_𝑛𝑢_𝑛−𝛼_𝑛^𝑁_𝑖=1𝛿_𝑖𝐴_𝑖𝑆_𝑛𝑢_𝑛,𝑦_𝑛=𝜀_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,∀𝑛∈𝑁, (154) where {𝑆_𝑛∶𝐶→𝐶} is the sequence defined by (37) and 𝛿_𝑖∈(0,1), ∑^𝑁_𝑖=1𝛿_𝑖=1. If {𝜀_𝑛}, {𝛽_𝑛} are two sequences in (0,1) and {𝛾_𝑛}⊂[𝑐₁,𝑐₂]⊂(0,1) and {𝑟_𝑘,𝑛}, for 𝑘∈{1,2,…,𝑀} is a real sequence in (0,∞) satisfing the following conditions:(i)lim_𝑛→∞𝜀_𝑛=0, ∑^∞_𝑖=1𝜀_𝑛=∞, (ii)0<lim_𝑛→∞inf𝛽_𝑛≤lim_𝑛→∞sup𝛽_𝑛<1 and lim_𝑛→∞|𝛽_𝑛+1−𝛽_𝑛|=0,(iii)lim_𝑛→∞inf𝑟_𝑛>0 and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0,(iv){𝛼_𝑛}⊂[𝑔₁,𝑔₂]⊂(0,(2∑^𝑁_𝑖=1𝛿_𝑖(𝑣_𝑖−𝜇_𝑖𝜔²_𝑖))/(∑^𝑁_𝑖=1𝛿_𝑖𝜔²_𝑖)) and lim_𝑛→∞|𝛼_𝑛+1−𝛼_𝑛|=0.
Then {𝑥_𝑛} converges strongly to 𝑧∈Θ, where 𝑧 is the unique solution of variational inequality lim_𝑛→∞sup𝑟𝑓−𝐴𝑧,𝑝−𝑧≤0,∀𝑝∈Θ, (155) which is the optimality condition for the minimization problem min_𝑧∈Θ12⟨𝐴𝑧,𝑧⟩−ℎ(𝑧), (156) where ℎ is a potential function for 𝑟𝑓(i.e., ℎ^′(𝑧)=𝑟𝑓(𝑧) for 𝑧∈𝐻).

Proof. By Theorem 16, letting 𝑀=1 for all 𝑛≥1, we can obtain Theorem 19.

Theorem 19. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝐹_𝑘,𝑘∈{1,2,…,𝑀} be a bifunction from 𝐶×𝐶→𝑅 satisfying (A1)–(A4). Let 𝐵∶𝐶→𝐻 be 𝜔-Lipschitz continuous and relaxed (𝜇,𝑣)-cocoercive mappings with 𝑣−𝜇𝜔²>0, and let 𝑓 be a contraction of 𝐻 into itself with 𝜂∈(0,1), and let 𝐴 be is a strongly positive linear bounded self-adjoint operator with the coefficients 𝛾>0 and 0<𝑟<𝛾/𝜂. Assume that Θ∶=^𝑀_𝑘=1SEP𝐹_𝑘∩VI(𝐶,𝐵). (157) Let {𝑥_𝑛} be a sequence generated by 𝑥₁∈𝐶 and 𝑢_𝑛=𝐽^𝐹𝑀_{𝑟𝑀,𝑛}𝐽^𝐹𝑀−1_{𝑟𝑀−1,𝑛}⋯𝐽^𝐹2_𝑟2,𝑛𝐽^𝐹1_𝑟1,𝑛𝑥_𝑛,𝑧_𝑛=𝑃_𝐶𝑢_𝑛−𝛼_𝑛𝐵𝑢_𝑛,𝑦_𝑛=𝜀_𝑛𝑟𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,∀𝑛∈𝑁. (158) If {𝜀_𝑛}, {𝛽_𝑛} are two sequences in (0,1) and {𝛾_𝑛}⊂[𝑐₁,𝑐₂]⊂(0,1) and {𝑟_𝑘,𝑛}, for 𝑘∈{1,2,…,𝑀} is a real sequence in (0,∞) satisfing the following conditions:(i)lim_𝑛→∞𝜀_𝑛=0, ∑^∞_𝑖=1𝜀_𝑛=∞, (ii)0<lim_𝑛→∞inf𝛽_𝑛≤lim_𝑛→∞sup𝛽_𝑛<1 and lim_𝑛→∞|𝛽_𝑛+1−𝛽_𝑛|=0,(iii)lim_𝑛→∞inf𝑟_𝑘,𝑛>0 and lim_𝑛→∞|𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛|=0, for 𝑘∈{1,2,…,𝑀}.(iv){𝛼_𝑛}⊂[𝑔₁,𝑔₂]⊂(0,(2(𝑣−𝜇𝜔²))/𝜔²) and lim_𝑛→∞|𝛼_𝑛+1−𝛼_𝑛|=0.Then {𝑥_𝑛} converges strongly to 𝑧∈Θ, where 𝑧 is the unique solution of variational inequality lim_𝑛→∞sup𝑟𝑓−𝐴𝑧,𝑝−𝑧≤0,∀𝑝∈Θ, (159) which is the optimality condition for the minimization problem min_𝑧∈Θ12⟨𝐴𝑧,𝑧⟩−ℎ(𝑧), (160) where ℎ is a potential function for 𝑟𝑓(i.e., ℎ^′(𝑧)=𝑟𝑓(𝑧) for 𝑧∈𝐻).

Proof. By Theorem 17, letting 𝑇_𝑛=𝐼 for all 𝑛≤1, we can obtain Theorem 19.

Acknowledgments

The authors would like to thank the anonymous referees and the editor for their constructive comments and suggestions, which greatly improved this paper. This project is supported by the Natural Science Foundation of China (Grant nos. 11171180, 11171193, 11126233, and 10901096) Shandong Provincial Natural Science Foundation (Grant no. ZR2011AM016), and the Project of Science and Technology Program of Weifang (Grant no. 20121103).