A System of Mixed Equilibrium Problems, a General System of Variational Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for Nonexpansive Semigroup and Strictly Pseudocontractive Mappings

Abstract

We introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a general system of variational inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (2010) and many others.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻. Recall that a mapping 𝑇∶𝐶→𝐶 is nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.1) We denote the set of fixed points of 𝑇 by 𝐹(𝑇), that is 𝐹(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}. A mapping 𝑓∶𝐶→𝐶 is said to be an 𝛼-contraction if there exists a coefficient 𝛼∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.2) Let 𝐵∶𝐶→𝐻 be a mapping. Then 𝐵 is called:(1)monotone if ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥0,∀𝑥,𝑦∈𝐶;(1.3)(2)𝑑-strongly monotone if there exists a positive real number 𝑑 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝑑‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶,(1.4) for constant 𝑑>0, this implies that ‖𝐵𝑥−𝐵𝑦‖≥𝑑‖𝑥−𝑦‖,(1.5) that is, 𝐵 is 𝑑-expansive and when 𝑑=1, it is expansive;(3)𝐿-Lipschitz continuous if there exists a positive real number 𝐿 such that‖𝐵𝑥−𝐵𝑦‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶;(1.6)(4)𝑐-cocoercive [1, 2] if there exists a positive real number 𝑐 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝑐‖𝐵𝑥−𝐵𝑦‖²,∀𝑥,𝑦∈𝐶,(1.7) Clearly, every 𝑐-cocoercive map 𝐵 is (1/𝑐)-Lipschitz continuous;(5)relaxed 𝑐-cocoercive, if there exists a positive real number 𝑐 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥(−𝑐)‖𝐵𝑥−𝐵𝑦‖²,∀𝑥,𝑦∈𝐶;(1.8)(6)relaxed (𝑐,𝑑)-cocoercive, if there exists a positive real number 𝑐,𝑑 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥(−𝑐)‖𝐵𝑥−𝐵𝑦‖²+𝑑‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶,(1.9) for 𝑐=0, 𝐵 is 𝑑-strongly monotone. This class of mapping is more general than the class of strongly monotone mapping. It is easy to see that we have the following implication: 𝑑-strongly monotonicity implying relaxed (𝑐,𝑑)-cocoercivity,(7)𝑘-strictly pseudocontractive, if there exists a constant 𝑘∈[0,1) such that ‖𝐵𝑥−𝐵𝑦‖²≤‖𝑥−𝑦‖²+𝑘‖(𝐼−𝐵)𝑥−(𝐼−𝐵)𝑦‖²,∀𝑥,𝑦∈𝐶.(1.10)

Remark 1.1 (see [3, Remark  1.1 pages 135-136]). If 𝐵∶𝐶→𝐻 is a 𝐿_𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping with 𝑑>𝑐𝐿^2𝐵 and 0<𝜏<2(𝑑−𝑐𝐿^2𝐵)/𝐿^2𝐵, then 𝐼−𝜏𝐵 satisfies the following: ‖(𝐼−𝜏𝐵)𝑥−(𝐼−𝜏𝐵)𝑦‖≤(1−𝜏𝜉)‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶,(1.11) where 𝜉=(𝐿^2𝐵/2)[2(𝑑−𝑐𝐿^2𝐵)/𝐿^2𝐵−𝜏].
Similarly, if 𝐷∶𝐶→𝐻is𝐿_𝐷-Lipschitz continuous and relaxed (𝑐′,𝑑′)-cocoercive mapping with 𝑑′>𝑐′𝐿^2𝐷 and 0<𝛿<2(𝑑′−𝑐′𝐿^2𝐷)/𝐿^2𝐷, then the mapping 𝐼−𝛿𝐷 satisfies the following: ‖(𝐼−𝛿𝐷)𝑥−(𝐼−𝛿𝐷)𝑦‖≤1−𝛿𝜉^‖𝑥−𝑦‖,(1.12) where 𝜉′=(𝐿^2𝐷/2)[2(𝑑′−𝑐′𝐿^2𝐷)/𝐿^2𝐷−𝛿].

Let 𝐴 be a strongly positive linear bounded operator on 𝐻 if there is a constant 𝛾>0 with the property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻.(1.13) We recall optimization problem (for short, OP) as the following min_𝑥∈𝐹𝜇2⟨𝐴𝑥,𝑥⟩+21‖𝑥−𝑢‖²−ℎ(𝑥),(1.14) where 𝐹=∩^∞𝑛=1𝐶_𝑛,𝐶₁,𝐶₂,… are infinitely 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 𝑥∈𝐻). This kind of optimization problem has been studied extensively by many authors, see, for example, [4–7] when 𝐹=∩^∞𝑛=1𝐶_𝑛 and ℎ(𝑥)=⟨𝑥,𝑏⟩, where 𝑏 is a given point in 𝐻.

On the other hand, a family 𝒮={𝑆(𝑠)∶0≤𝑠<∞} of mappings of 𝐶 into itself is called a nonexpansive semigroup on 𝐶 if it satisfies the following conditions:(i)𝑆(0)𝑥=𝑥 for all 𝑥∈𝐶;(ii)𝑆(𝑠+𝑡)=𝑆(𝑠)𝑆(𝑡) for all 𝑠,𝑡≥0;(iii)‖𝑆(𝑠)𝑥−𝑆(𝑠)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 and 𝑠≥0;(iv)for all 𝑥∈𝐶,𝑠↦𝑆(𝑠)𝑥 is continuous.

We denote by 𝐹(𝒮) the set of all common fixed points of 𝒮={𝑆(𝑠)∶𝑠≥0}, that is, 𝐹(𝒮)=∩_𝑠≥0𝐹(𝑆(𝑠)). It is known that 𝐹(𝒮) is closed and convex.

Let 𝜙∶𝐶→ℝ be a real-valued function and let {Θ_𝑘∶𝐶×𝐶→ℝ,𝑘=1,2,…,𝑁} be a finite family of equilibrium functions, that is, Θ_𝑘(𝑢,𝑢)=0 for each 𝑢∈𝐶. The system of mixed equilibrium problems (for short, SMEP) for function (Θ₁,Θ₂,…,Θ_𝑁,𝜙) is to find 𝑧∈𝐶 such that ΘΘ₁(𝑧,𝑦)+𝜙(𝑦)−𝜙(𝑧)≥0,∀𝑦∈𝐶,₂Θ⋮(𝑧,𝑦)+𝜙(𝑦)−𝜙(𝑧)≥0,∀𝑦∈𝐶,_𝑁(𝑧,𝑦)+𝜙(𝑦)−𝜙(𝑧)≥0,∀𝑦∈𝐶.(1.15) The set of solutions of (1.15) is denoted by ∩^𝑁𝑘=1MEP(Θ_𝑘,𝜙), where MEP(Θ_𝑘,𝜙) is the set of solutions of the mixed equilibrium problem (for short, MEP), which is to find 𝑧∈𝐶 such that Θ_𝑘(𝑧,𝑦)+𝜙(𝑦)−𝜙(𝑧)≥0,∀𝑦∈𝐶.(1.16) In particular, if 𝜙≡0, and 𝑁=1, then the problem (1.15) reduces to the equilibrium problem (for short, EP), which is to find 𝑧∈𝐶 such that Θ(𝑧,𝑦)≥0,∀𝑦∈𝐶.(1.17) It is well known that the SMEP includes fixed point problem, optimization problem, variational inequality problem, and Nash equilibrium problem as its special cases (see [8–13] for more details).

For solving the solutions of a nonexpansive semigroup and the solutions of the system of mixed equilibrium problems were studied by many authors see [14–23] and reference therein. In 2010, Chang et al. [24] studied the following approximation method: Θ₁𝑢^𝑛(1),𝑥+𝜙(𝑥)−𝜙𝑢^𝑛(1)+1Θ𝑟₁𝐾^𝑢^𝑛(1)−𝐾^𝑥_𝑛,𝜂𝑥,𝑢^𝑛(1)≥0,∀𝑥∈𝐶,₂Θ⋮𝑢^𝑛(2),𝑥+𝜙(𝑥)−𝜙𝑢^𝑛(2)+𝑟1₂𝐾^𝑢^𝑛(2)−𝐾^𝑥_𝑛,𝜂𝑥,𝑢^𝑛(2)≥0,∀𝑥∈𝐶,_𝑁𝑥𝑢^𝑛(𝑁),𝑥+𝜙(𝑥)−𝜙𝑢^𝑛(𝑁)+𝑟1_𝑁𝐾^𝑢^𝑛(𝑁)−𝐾^𝑥_𝑛,𝜂𝑥,𝑢^𝑛(𝑁)≥0,∀𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑊_𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑢^𝑛(𝑁)𝑑𝑠,(1.18) where 𝑢𝑢^𝑛(1)_𝑛(𝑘)=𝐽^Θ1_𝑟1𝑥_𝑛,==𝐽𝐽^Θ𝑘_{𝑟𝑘Θ𝑘𝑟𝑘}𝑢^{𝑛(𝑘−1)}=𝐽^Θ𝑘_𝑟𝑘𝐽^Θ𝑘−1_𝑟𝑘−1𝑢^{𝑛(𝑘−2)}=𝐽^Θ𝑘_𝑟𝑘⋯𝐽^Θ2_𝑟2𝑢^𝑛(1),⋯𝐽^Θ2_𝑟2𝐽^Θ1_𝑟1𝑥_𝑛,𝑘=2,3,…,𝑁,(1.19)𝐽^Θ𝑘_𝑟𝑘∶𝐶→𝐶,𝑘=1,2,…,𝑁 is the mapping defined by (2.22) below, 𝑊_𝑛 is the mapping defined by (2.12), and 𝒮={𝑆(𝑠)∶0≤𝑠<∞} is a nonexpansive semigroup. They proved that {𝑥_𝑛} converges strongly to a fixed point of 𝐹(𝒮)∩𝐹(𝑊)∩(∩^𝑁𝑘=1MEP(Θ_𝑘,𝜙)) under control conditions on the parameters.

Let 𝐵,𝐷∶𝐶→𝐻 be two mappings. The general system of variational inequalities problem (see [25]) is to find (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that ⟨⟨𝛿𝜏𝐷𝐵𝑦𝑥^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐶,(1.20) where 𝜏 and 𝛿 are two positive real numbers. The set of solutions of the general system of variational inequalities problem is denoted by SVI(𝐶,𝐵,𝐷). In particular, if 𝐵=𝐷, then the problem (1.20) reduces to the following equation: ⟨⟨𝜏𝛿𝐵𝐵𝑥𝑦^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐶,(1.21) which is defined by Verma [26] (see also Verma [27]), and is called the new system of variational inequalities. Further, if we set 𝐷=0, then problem (1.20) reduces to the classical variational inequality is to find 𝑥^∗∈𝐶 such that ⟨𝐵𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶.(1.22) We denoted by VI(𝐶,𝐵) the set of solutions of the variational inequality problem. The variational inequality problem has been extensively studied in literature, see, for example, [28–31] and references therein. In order to find the solutions of the general system of variational inequality problem (1.20), Wangkeeree and Kamraksa [32] considered the following iterative algorithm: Θ𝑢_𝑛,𝑥+𝜙(𝑥)−𝜙𝑢_𝑛+1𝑧𝑟𝐾^𝑢_𝑛−𝐾^𝑥_𝑛,𝜂𝑥,𝑢_𝑛≥0,∀𝑥∈𝐶,_𝑛𝑥_𝑛+1=𝑃_𝐶𝑢_𝑛−𝛿𝐷𝑢_𝑛,=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑃_𝐶𝑧_𝑛−𝜏𝐵𝑧_𝑛,(1.23) where 𝐵,𝐷∶𝐶→𝐻 is a 𝐿_𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping and 𝐿_𝐷-Lipschitz continuous and relaxed (𝑐′,𝑑′)-cocoercive mapping, respectively. They proved that {𝑥_𝑛} converges strongly to a fixed point of 𝐹(𝑊_𝑛)∩MEP(Θ,𝜙)∩SVI(𝐶,𝐵,𝐷) which is a solution of general system of variational inequality (1.20). Very recently, Jaiboon and Kumam [33] studied a new general iterative method for finding a common element of the set of solution of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces, which solves some optimization problems.

Inspired and motivated by Chang et al. [24], Jaiboon and Kumam [33], Kumam and Jaiboon [34] and Wangkeeree and Kamraksa [32], the purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroup, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings. Consequently, we prove the strong convergence theorem in Hilbert spaces under control conditions on the parameters. Furthermore, we can apply our results for solving some optimization problems. Our results extend and improve the corresponding results in Chang et al. [24], Kumam and Jaiboon [34], Wangkeeree and Kamraksa [32], and many others.

2. Preliminaries

Let 𝐻 a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. We denote strong convergence (weak convergence) by notation →(⇀). In a real Hilbert space 𝐻, it is well known that ‖𝑥−‖𝑥𝑦‖+²𝑦=‖₂‖𝑥‖²−‖𝑦‖²−2⟨𝑥−𝑦,𝑦⟩,(2.1)‖𝑥+𝑦‖≤₂‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,(2.2)‖𝜆𝑥+(1−𝜆)𝑦‖₂≥‖𝑥‖²+2⟨𝑦,𝑥⟩,(2.3)=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖²(2.4) for all 𝑥,𝑦∈𝐻 and 𝜆∈ℝ.

Recall that for every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that ‖‖𝑥−𝑃_𝐶𝑥‖‖≤‖𝑥−𝑦‖,∀𝑦∈𝐶.(2.5)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is well known that 𝑃_𝐶 is a nonexpansive mapping of 𝐻 onto 𝐶 and satisfies ⟨𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦⟩≥‖‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖²(2.6) for every 𝑥,𝑦∈𝐻. Obviously, this immediately implies that ‖‖(𝑥−𝑦)−𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖²≤‖𝑥−𝑦‖²−‖‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖²,∀𝑥,𝑦∈𝐻.(2.7) Moreover, 𝑃_𝐶𝑥 is characterized by the following properties: 𝑃_𝐶𝑥∈𝐶 and ‖𝑥−𝑦‖⟨₂𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥⟩≤0,≥‖‖𝑥−𝑃_𝐶𝑥‖‖²+‖‖𝑦−𝑃_𝐶𝑥‖‖²(2.8) for all 𝑥∈𝐻,𝑦∈𝐶.

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

Lemma 2.1 (see [35]). Let 𝑉∶𝐶→𝐻 be a k-strict pseudo-contraction, then(1)the fixed point set 𝐹(𝑉) of 𝑉 is closed convex so that the projection 𝑃_𝐹(𝑉) is well defined;(2)define a mapping 𝑇∶𝐶→𝐻 by 𝑇𝑥=𝑡𝑥+(1−𝑡)𝑉𝑥,∀𝑥∈𝐶.(2.9) If 𝑡∈[𝑘,1), then 𝑇 is a nonexpansive mapping such that 𝐹(𝑉)=𝐹(𝑇).

A family of mappings {𝑉_𝑖∶𝐶→𝐻}^∞𝑖=1 is called a family of uniformly 𝑘-strict pseudo-contractions, if there exists a constant 𝑘∈[0,1) such that ‖‖𝑉_𝑖𝑥−𝑉_𝑖𝑦‖‖²≤‖𝑥−𝑦‖²+𝑘‖‖𝐼−𝑉_𝑖𝑥−𝐼−𝑉_𝑖𝑦‖‖²,∀𝑥,𝑦∈𝐶,∀𝑖≥1.(2.10) Let {𝑉_𝑖∶𝐶→𝐶}^∞𝑖=1 be a countable family of uniformly 𝑘-strict pseudo-contractions. Let {𝑇_𝑖∶𝐶→𝐶}^∞𝑖=1 be the sequence of nonexpansive mappings defined by (2.9), that is, 𝑇_𝑖𝑥=𝑡𝑥+(1−𝑡)𝑉_𝑖𝑥,∀𝑥∈𝐶,∀𝑖≥1,𝑡∈[𝑘,1).(2.11)

Let {𝑇_𝑖} be a sequence of nonexpansive mappings of 𝐶 into itself defined by (2.11) and let {𝜇_𝑖} be a sequence of nonnegative numbers in [0,1]. For each 𝑛≥1, define a mapping 𝑊_𝑛 of 𝐶 into itself as follows: 𝑈𝑈_𝑛,𝑛+1=𝐼,_𝑛,𝑛𝑈=𝜇_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜇_𝑛𝐼,_{𝑛,𝑛−1}𝑈=⋮𝜇_𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+1−𝜇_𝑛−1𝐼,_𝑛,𝑘𝑈=𝜇_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜇_𝑘𝐼,_{𝑛,𝑘−1}𝑈=⋮𝜇_𝑘−1𝑇_𝑘−1𝑈_𝑛,𝑘+1−𝜇_𝑘−1𝐼,_𝑛,2𝑊=𝜇₂𝑇₂𝑈_𝑛,3+1−𝜇₂𝐼,_𝑛=𝑈_𝑛,1=𝜇₁𝑇₁𝑈_𝑛,2+1−𝜇₁𝐼.(2.12) Such a mapping 𝑊_𝑛 is nonexpansive from 𝐶 to 𝐶 and it is called the 𝑊-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑛 and 𝜇₁,𝜇₂,…,𝜇_𝑛.

For each 𝑛,𝑘∈ℕ, let the mapping 𝑈_𝑛,𝑘 be defined by (2.12). Then we can have the following crucial conclusions concerning 𝑊_𝑛. You can find them in [36]. Now we only need the following similar version in Hilbert spaces.

Lemma 2.2 (see [36]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇₁,𝑇₂,… be nonexpansive mappings of 𝐶 into itself such that ∩^∞𝑛=1𝐹(𝑇_𝑛) is nonempty, let 𝜇₁,𝜇₂,… be real numbers such that 0≤𝜇_𝑛≤𝑏<1 for every 𝑛≥1. Then,(1)𝑊_𝑛 is nonexpansive and 𝐹(𝑊_𝑛)=∩^𝑛𝑖=1𝐹(𝑇_𝑖), for all 𝑛≥1;(2)for every 𝑥∈𝐶 and 𝑘∈ℕ, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists;(3)a mapping 𝑊∶𝐶→𝐶 defined by 𝑊𝑥∶=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥,∀𝑥∈𝐶(2.13) is a nonexpansive mapping satisfying 𝐹(𝑊)=∩^∞𝑖=1𝐹(𝑇_𝑖) and it is called the 𝑊-mapping generated by 𝑇₁,𝑇₂,… and 𝜇₁,𝜇₂,….

Lemma 2.3 (see [37]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, {𝑇_𝑖∶𝐶→𝐶} a countable family of nonexpansive mappings with ∩^∞𝑖=1𝐹(𝑇_𝑖)≠∅, {𝜇_𝑖} a real sequence such that 0<𝜇_𝑖≤𝑏<1,forall𝑖≥1. If 𝐷 is any bounded subset of 𝐶, then lim_𝑛→∞sup_𝑥∈𝐷‖‖𝑊𝑥−𝑊_𝑛𝑥‖‖=0.(2.14)

Lemma 2.4 (see [38]). Each Hilbert space 𝐻 satisfies Opial’s 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‖‖(2.15) holds for each 𝑦∈𝐻 with 𝑦≠𝑥.

Lemma 2.5 (see [39]). Assume 𝐴 is a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then, ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

For solving the system of mixed equilibrium problems (1.15), let us assume that function Θ_𝑘∶𝐻×𝐻→ℝ,𝑘=1,2,…,𝑁 satisfies the following conditions:(H1)Θ_𝑘 is monotone, that is, Θ_𝑘(𝑥,𝑦)+Θ_𝑘(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝐻;(H2) for each fixed 𝑦∈𝐻, 𝑥↦Θ_𝑘(𝑥,𝑦) is convex and upper semicontinuous;(H3) for each 𝑥∈𝐻,𝑦↦Θ_𝑘(𝑥,𝑦) is convex.

Let 𝜂∶𝐻×𝐻→𝐻 and 𝐵∶𝐻→𝐻 be two mappings. 𝐵 is said to be(1)monotone if ⟨𝐵𝑥−𝐵𝑦,𝜂(𝑥,𝑦)⟩≥0,∀𝑥,𝑦∈𝐻;(2.16)(2)𝑑-strongly monotone if there exists a positive real number 𝑑 such that ⟨𝐵𝑥−𝐵𝑦,𝜂(𝑥,𝑦)⟩≥𝑑‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐻;(2.17)(3)𝐿-Lipschitz continuous if there exists a constant 𝐿>0 such that ‖𝜂(𝑥,𝑦)‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐻.(2.18)

Let 𝐾∶𝐻→ℝ be a differentiable functional on 𝐻, which is called:(1)𝜂-convex [40] if 𝐾(𝑦)−𝐾(𝑥)≥𝐾^(𝑥),𝜂(𝑦,𝑥),∀𝑥,𝑦∈𝐻,(2.19) where 𝐾^(𝑥) is the Fréchet derivative of 𝐾 at 𝑥;(2)𝜂-strongly convex [41] if there exists a constant 𝜎>0 such that 𝐾(𝑦)−𝐾(𝑥)−𝐾^(𝑥),𝜂(𝑦,𝑥)≥𝜎2‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐻.(2.20)

In particular, if 𝜂(𝑥,𝑦)=𝑥−𝑦 for all 𝑥,𝑦∈𝐻, then 𝐾 is said to be strongly convex.

Lemma 2.6 (see [42]). Let 𝐻 be a real Hilbert space and let 𝜙 be a lower semicontinuous and convex functional from 𝐻 to ℝ. Let Θ be a bifunction from 𝐻×𝐻 to ℝ satisfying (H1)–(H3). Assume that(i)𝜂∶𝐻×𝐻→𝐻 is 𝜆-Lipschitz continuous with constant 𝜆>0 such that(a)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0,forall𝑥,𝑦∈𝐻, (b)𝜂(⋅,⋅) is affine in the first variable,(c)for each fixed 𝑥∈𝐻, 𝑦↦𝜂(𝑥,𝑦) is sequentially continuous from the weak topology to the weak topology;(ii)𝐾∶𝐻→ℝ is 𝜂-strongly convex with constant 𝜎>0 and its derivative 𝐾^ is sequentially continuous from the weak topology to the strong topology;(iii)for each 𝑥∈𝐻, there exist bounded subsets 𝐸_𝑥⊂𝐻 and 𝑧_𝑥∈𝐻 such that for any 𝑦∈𝐻⧵𝐸_𝑥, Θ𝑦,𝑧_𝑥+𝜙𝑧_𝑥−𝜙(𝑦)+1𝑟𝐾^(𝑦)−𝐾^(𝑥),𝜂𝑧_𝑥,𝑦<0.(2.21) For given 𝑟>0, let 𝐽^Θ𝑟∶𝐻→𝐻 be the mapping defined by 𝐽^Θ𝑟(𝑥)=𝑦∈𝐻∶Θ(𝑦,𝑧)+𝜙(𝑧)−𝜙(𝑦)+1𝑟𝐾^(𝑦)−𝐾^(𝑥),𝜂(𝑧,𝑦)≥0,∀𝑧∈𝐻(2.22)for all 𝑥∈𝐻. Then(1)𝐽^Θ𝑟 is single-valued.(2)𝐹(𝐽^Θ𝑟)=MEP(Θ,𝜙), where MEP(Θ,𝜙) is the set of solution of the mixed equilibrium problem, Θ(𝑥,𝑦)+𝜙(𝑦)−𝜙(𝑥)≥0,∀𝑦∈𝐻.(2.23)(3)MEP(Θ,𝜙) is closed and convex.

Lemma 2.7 (see [43]). Let {𝑥_𝑛} and {𝑣_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 𝑥_𝑛+1=(1−𝛽_𝑛)𝑣_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑣_𝑛+1−𝑣_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑣_𝑛−𝑥_𝑛‖=0.

Lemma 2.8 (see [44]). Assume {𝑥_𝑛} is a sequence of nonnegative real numbers such that 𝑥_𝑛+1≤1−𝑎_𝑛𝑥_𝑛+𝑏_𝑛,∀𝑛≥0,(2.24) where {𝑎_𝑛} is a sequence in (0,1) and {𝑏_𝑛} is a sequence in ℝ such that(1)∑^∞𝑛=1𝑎_𝑛=∞, (2)limsup_𝑛→∞(𝑏_𝑛/𝑎_𝑛)≤0 or ∑^∞𝑛=1|𝑏_𝑛|<∞.Then, lim_𝑛→∞𝑥_𝑛=0.

Lemma 2.9 (see [45]). 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. If 𝑧 is a solution to the minimization problem 𝑔(𝑧)=inf_𝑥∈𝐶𝑔(𝑥),(2.25) then 𝑔^(𝑥),𝑥−𝑧≥0,𝑥∈𝐶.(2.26) In particular, if 𝑧 solves problem 𝑂𝑃, then ⟨𝑢+[𝛾𝑓−(𝐼+𝜇𝐴)]𝑧,𝑥−𝑧⟩≤0.(2.27)

Lemma 2.10 (see [46]). Let 𝐶 be a nonempty bounded closed convex subset of a Hilbert space 𝐻 and let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on 𝐶, then for any ℎ≥0, lim_𝑡→∞sup_𝑥∈𝐶‖‖‖1𝑡^𝑡0𝑇(𝑠)𝑥𝑑𝑠−𝑇(ℎ)1𝑡^𝑡0𝑇(𝑠)𝑥𝑑𝑠‖‖‖=0.(2.28)

Lemma 2.11 (see [47]). Let C be a nonempty bounded closed convex subset of 𝐻, {𝑥_𝑛} a sequence in C, and 𝒮={𝑆(𝑠)∶0≤𝑠<∞} a nonexpansive semigroup on 𝐶. If the following conditions are satisfied:(i)𝑥_𝑛⇀𝑧;(ii)limsup_𝑠→∞limsup_𝑛→∞‖𝑆(𝑠)𝑥_𝑛−𝑥_𝑛‖=0, then 𝑧∈𝒮.

Lemma 2.12 (see [25]). For given 𝑥^∗,𝑦^∗∈𝐶 and (𝑥^∗,𝑦^∗) is a solution of the problem (1.20) if and only if 𝑥^∗ is a fixed point of the mapping 𝐺∶𝐶→𝐶 is defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶(𝑥−𝛿𝐷𝑥)−𝜏𝐵𝑃_𝐶(𝑥−𝛿𝐷𝑥),∀𝑥∈𝐻,(2.29) where 𝑦^∗=𝑃_𝐶(𝑥−𝛿𝐷𝑥), 𝛿 and 𝜏 are positive constants and 𝐵,𝐷∶𝐻→𝐻 are two mappings.

Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by SVI(𝐶,𝐵,𝐷).

Lemma 2.13 (see [32]). Let 𝐺∶𝐶→𝐶 be defined in Lemma 2.12. If 𝐵∶𝐻→𝐻 is a 𝐿_𝐵-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mapping and 𝐷∶𝐻→𝐻 is a 𝐿_𝐷-Lipschitz and relaxed (𝑐′,𝑑′)-cocoercive mapping where 𝜏≤2(𝑑−𝑐𝐿^2𝐵)/𝐿^2𝐵 and 𝛿≤2(𝑑′−𝑐′𝐿^2𝐷)/𝐿^2𝐷, then G is nonexpansive.

Lemma 2.14 (demiclosedness principle [48]). Assume that 𝑆 is a nonexpansive self-mapping of a nonempty closed convex subset 𝐶 of a real Hilbert space 𝐻. If 𝑆 has a fixed point, then 𝐼−𝑆 is demiclosed; that is, whenever {𝑥_𝑛} is a sequence in 𝐶 converging weakly to some 𝑥∈𝐶 (for short, 𝑥_𝑛⇀𝑥∈𝐶), and the sequence {(𝐼−𝑆)𝑥_𝑛} converges strongly to some 𝑦 (for short, (𝐼−𝑆)𝑥_𝑛→𝑦), it follows that (𝐼−𝑆)𝑥=𝑦. Here 𝐼 is the identity operator of 𝐻.

3. Main Results

In this section, we prove a strong convergence theorem of an iterative algorithm (3.1) for finding the solutions of a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 which 𝐶+𝐶⊂𝐶 and let 𝑓 be a contraction of 𝐶 into itself with 𝛼∈(0,1). Let 𝜙 be a lower semicontinuous and convex functional from 𝐻 to ℝ and let {Θ_𝑘∶𝐻×𝐻→ℝ,𝑘=1,2,…,𝑁} be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on 𝐶 and let {𝑡_𝑛} be a positive real divergent sequence. Let {𝑉_𝑖∶𝐶→𝐶}^∞𝑖=1 be a countable family of uniformly 𝑘-strict pseudo-contractions, let {𝑇_𝑖∶𝐶→𝐶}^∞𝑖=1 be the countable family of nonexpansive mappings defined by 𝑇_𝑖𝑥=𝑡𝑥+(1−𝑡)𝑉_𝑖𝑥,forall𝑥∈𝐶,forall𝑖≥1,𝑡∈[𝑘,1), let 𝑊_𝑛 be the 𝑊-mapping defined by (2.12), and let 𝑊 be a mapping defined by (2.13) with 𝐹(𝑊)≠∅. Let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and let 0<𝛾<(1+𝜇𝛾)/𝛼, 𝐵∶𝐻→𝐻 be a 𝐿_𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping with 𝑑>𝑐𝐿^2𝐵, and let 𝐷∶𝐻→𝐻 be a 𝐿_𝐷-Lipschitz continuous and relaxed (𝑐′,𝑑′)-cocoercive mapping with 𝑑′>𝑐′𝐿^2𝐷. Suppose that Ω∶=𝐹(𝒮)∩𝐹(𝑊)∩𝔉∩SVI(𝐶,𝐵,𝐷)≠∅, where 𝔉=(∩^𝑁𝑘=1MEP(Θ_𝑘,𝜙)). Let 𝜇>0, 𝛾>0 and 𝑟_𝑘>0,𝑘=1,2,…,𝑁, which are constants. For given 𝑥₁∈𝐻 arbitrarily and fixed 𝑢∈𝐻, suppose {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛} and {𝑢^𝑛(𝑘)},𝑘=1,2,…,𝑁 are the sequences generated iteratively by Θ₁𝑢^𝑛(1),𝑥+𝜙(𝑥)−𝜙𝑢^𝑛(1)+1Θ𝑟₁𝐾^𝑢^𝑛(1)−𝐾^𝑥_𝑛,𝜂𝑥,𝑢^𝑛(1)≥0,∀𝑥∈𝐻,₂Θ⋮𝑢^𝑛(2),𝑥+𝜙(𝑥)−𝜙𝑢^𝑛(2)+𝑟1₂𝐾^𝑢^𝑛(2)−𝐾^𝑥_𝑛,𝜂𝑥,𝑢^𝑛(2)≥0,∀𝑥∈𝐻,_𝑁𝑧𝑢^𝑛(𝑁),𝑥+𝜙(𝑥)−𝜙𝑢^𝑛(𝑁)+𝑟1_𝑁𝐾^𝑢^𝑛(𝑁)−𝐾^𝑥_𝑛,𝜂𝑥,𝑢^𝑛(𝑁)≥0,∀𝑥∈𝐻,_𝑛𝑦_𝑛=𝑃_𝐶𝑢^𝑛(𝑁)−𝛿𝐷𝑢^𝑛(𝑁),𝑥=_𝑛+1𝑃_𝐶𝑧_𝑛−𝜏𝐵𝑧_𝑛,=𝛼_𝑛𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠,(3.1) where 𝑢𝑢^𝑛(1)_𝑛(𝑘)=𝐽^Θ1_𝑟1𝑥_𝑛,==𝐽𝐽^Θ𝑘_{𝑟𝑘Θ𝑘𝑟𝑘}𝑢^{𝑛(𝑘−1)}=𝐽^Θ𝑘_𝑟𝑘𝐽^Θ𝑘−1_𝑟𝑘−1𝑢^{𝑛(𝑘−2)}=𝐽^Θ𝑘_𝑟𝑘⋯𝐽^Θ2_𝑟2𝑢^𝑛(1),⋯𝐽^Θ2_𝑟2𝐽^Θ1_𝑟1𝑥_𝑛,𝑘=2,3,…,𝑁,(3.2)𝐽^Θ𝑘_𝑟𝑘∶𝐻→𝐻,𝑘=1,2,…,𝑁 is the mapping defined by (2.22) and {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) for all 𝑛∈ℕ. Assume the following conditions are satisfied:(C1)𝜂∶𝐻×𝐻→𝐻 is 𝜆-Lipschitz continuous with constant 𝜆>0 such that(a)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0,forall𝑥,𝑦∈𝐻, (b)𝑥↦𝜂(𝑥,𝑦) is affine,(c)for each fixed 𝑦∈𝐻, 𝑦↦𝜂(𝑥,𝑦) is sequentially continuous from the weak topology to the weak topology;(C2)𝐾∶𝐻→ℝ is 𝜂-strongly convex with constant 𝜎>0 and its derivative 𝐾^ is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant 𝜈>0 such that 𝜎>𝜆𝜈;(C3) for each 𝑘∈{1,2,…,𝑁} and for all 𝑥∈𝐻, there exist bounded subsets 𝐸_𝑥⊂𝐻 and 𝑧_𝑥∈𝐻 such that for any 𝑦∈𝐻⧵𝐸_𝑥, Θ_𝑘𝑦,𝑧_𝑥+𝜙𝑧_𝑥−𝜙(𝑦)+1𝑟_𝑘𝐾^(𝑦)−𝐾^(𝑥),𝜂𝑧_𝑥,𝑦<0;(3.3)(C4)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞;(C5)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1; (C6)0<𝜏<2(𝑑−𝑐𝐿^2𝐵)/𝐿^2𝐵 and 0<𝛿<2(𝑑′−𝑐′𝐿^2𝐷)/𝐿^2𝐷.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Ω, which solves the following optimization problem (OP): min_𝑥∗∈Ω𝜇2⟨𝐴𝑥^∗,𝑥^∗⟩+12‖𝑥^∗−𝑢‖²−ℎ𝑥^∗,(3.4) and (𝑥^∗,𝑦^∗) is a solution of the general system of variational inequality problem (1.20) such that 𝑦^∗=𝑃_𝐶(𝑥^∗−𝛿𝐷𝑥^∗).

Proof. By the condition (C4) and (C5), we may assume, without loss of generality, that 𝛼_𝑛≤(1−𝛽_𝑛)(1+𝜇‖𝐴‖)⁻¹ for all 𝑛∈ℕ. Indeed, 𝐴 is a strongly positive bounded linear operator on 𝐻, we have ‖𝐴‖=sup||⟨𝐴𝑥,𝑥⟩||∶𝑥∈𝐻,‖𝑥‖=1.(3.5) Observe that 1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝑥,𝑥≥=11−−𝛽𝛽_𝑛𝑛−𝛼_𝑛−𝛼_𝑛𝜇⟨𝐴𝑥,𝑥⟩≥0,−𝛼_𝑛−𝛼_𝑛𝜇‖𝐴‖(3.6) so this shows that (1−𝛽_𝑛)𝐼−𝛼_𝑛(𝐼+𝜇𝐴) is positive. It follows that ‖‖1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)‖‖==ssuupp||1−𝛽1_𝑛−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝑥,𝑥||∶𝑥∈𝐻,‖𝑥‖=1≤1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇⟨𝐴𝑥,𝑥⟩∶𝑥∈𝐻,‖𝑥‖=1−𝛼_𝑛−𝛼_𝑛𝜇𝛾.(3.7) We shall divide the proofs into several steps.Step 1. We show that {𝑥_𝑛} is bounded.
Let 𝑥^∗∈Ω∶=𝐹(𝒮)∩𝐹(𝑊)∩(∩^𝑁𝑘=1MEP(Θ_𝑘,𝜙))∩SVI(𝐶,𝐵,𝐷). In fact, by the assumption that for each 𝑘∈{1,2,…,𝑁}, 𝐽^Θ𝑘_𝑟𝑘 is nonexpansive. Let 𝒜^𝑁∶=𝐽^Θ𝑁_𝑟𝑁⋯𝐽^Θ2_𝑟2𝐽^Θ1_𝑟1 and 𝒜⁰=𝐼. Then, we have 𝑥^∗=𝒜^𝑁𝑥^∗ and 𝑢^𝑛(𝑁)=𝒜^𝑁𝑥_𝑛. Since 𝑥^∗∈SVI(𝐶,𝐵,𝐷), then 𝑥^∗=𝑃_𝐶𝑃_𝐶𝑥^∗−𝛿𝐷𝑥^∗−𝜏𝐵𝑃_𝐶𝑥^∗−𝛿𝐷𝑥^∗=𝑃_𝐶𝑃_𝐶(𝐼−𝛿𝐷)𝒜^𝑁𝑥^∗−𝜏𝐵𝑃_𝐶(𝐼−𝛿𝐷)𝒜^𝑁𝑥^∗.(3.8) Putting 𝑦^∗=𝑃_𝐶(𝑥^∗−𝛿𝐷𝑥^∗)=𝑃_𝐶(𝐼−𝛿𝐷)𝒜^𝑁𝑥^∗, we have 𝑥^∗=𝑃_𝐶(𝑦^∗−𝜏𝐵𝑦^∗). Since 𝑥^∗=𝑆(𝑠)𝑥^∗,forall𝑠≥0 and 𝑥^∗=𝑊_𝑛𝑥^∗,forall𝑛≥1, therefore, we have 𝑥^∗=𝒜^𝑁𝑥^∗=𝑃_𝐶𝑦^∗−𝜏𝐵𝑦^∗=𝑊_𝑛𝑃_𝐶𝑦^∗−𝜏𝐵𝑦^∗=𝑆(𝑠)𝑊_𝑛𝑃_𝐶𝑦^∗−𝜏𝐵𝑦^∗.(3.9) Because 𝑃_𝐶 and 𝒜^𝑁 are nonexpansive mappings and from Remark 1.1, we have ‖‖𝑦_𝑛−𝑥^∗‖‖≤=‖‖‖‖𝑃(_𝐶𝐼−𝑧𝜏_𝑛𝐵−)𝜏𝑧𝐵_𝑛𝑧_𝑛−𝑃_𝐶𝑦^∗−𝜏𝐵𝑦^∗‖‖≤‖‖𝑧_𝑛−(𝐼−𝜏𝐵)𝑦^∗‖‖=‖‖𝑃_𝐶−𝑦^∗‖‖≤‖‖(𝐼−𝑢𝛿𝐷^𝑛(𝑁))−𝑢𝛿𝐷_𝑛(𝑁)𝑢^𝑛(𝑁)−𝑃^𝐶𝑥^∗−𝛿𝐷𝑥^∗‖‖≤‖‖𝑢_𝑛(𝑁)−(𝐼−𝛿𝐷)𝑥^∗‖‖=‖‖𝒜_𝑁−𝑥^∗‖‖≤‖‖𝑥_𝑛𝑥_𝑛−𝒜^𝑁𝑥^∗‖‖−𝑥^∗‖‖(3.10) which yields that ‖‖𝑥_𝑛+1−𝑥^∗‖‖=‖‖‖𝛼+_𝑛𝑢+1𝛼−_𝑛𝛽𝛾_𝑛𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝑡1_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠−𝑥^∗‖‖‖+‖1𝑢−‖𝛽+_𝑛𝛼_𝑛‖‖𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖≤𝛼_𝑛−𝛼_𝑛1+𝜇𝛾‖‖𝑥_𝑛−𝑥^∗‖‖+‖1𝑢−‖𝛽+_𝑛𝛼_𝑛‖‖𝛾𝑓𝑊_𝑛𝑥_𝑛−𝛾𝑓𝑥^∗‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖≤𝛼_𝑛−𝛼_𝑛1+𝜇𝛾‖‖𝑥_𝑛−𝑥^∗‖‖+‖1𝑢−‖𝛽+_𝑛𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖=𝛼_𝑛−𝛼_𝑛1+𝜇𝛾‖‖𝑥_𝑛−𝑥^∗‖‖=1−‖𝑢𝛼‖_𝑛+‖‖𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖+1−𝛼_𝑛1+𝜇𝛾+𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛1+𝜇𝛾−𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖1+𝜇𝛾−𝛾𝛼‖𝑢‖+‖‖𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖1+𝜇𝛾−𝛾𝛼.(3.11) It follows from (3.11) and induction that ‖‖𝑥_𝑛−𝑥^∗‖‖≤max‖‖𝑥₁−𝑝‖‖,‖𝑢‖+‖‖𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖1+𝜇𝛾−𝛾𝛼,𝑛≥1.(3.12) Hence, {𝑥_𝑛} is bounded, so are {𝑦_𝑛}, {𝑧_𝑛}, {𝑊_𝑛𝑥_𝑛}, {𝑓(𝑊_𝑛𝑥_𝑛)}, {𝑢^𝑛(𝑘)} for all 𝑘=1,2,…,𝑁 and {𝐾_𝑛𝑊_𝑛𝑦_𝑛}, where 𝐾_𝑛=(1/𝑡_𝑛)∫^𝑡𝑛₀𝑆(𝑠)𝑑𝑠.Step 2. We prove that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑢^(𝑁)𝑛+1−𝑢^𝑛(𝑁)‖=0.
Again, from Remark 1.1, we have the following estimates: ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖=≤‖‖‖‖𝑃_𝐶𝑧𝑧_𝑛+1𝑛+1−𝜏𝐵𝑧_𝑛+1−𝑃_𝐶𝑧_𝑛−𝜏𝐵𝑧_𝑛‖‖≤‖‖𝑧_𝑛+1−𝜏𝐵𝑧_𝑛+1−𝑧_𝑛−𝜏𝐵𝑧_𝑛‖‖=‖‖𝑃_𝐶−𝑧_𝑛‖‖≤‖‖𝑢𝑢^(𝑁)𝑛+1_(𝑁)𝑛+1−𝛿𝐷𝑢^(𝑁)𝑛+1−𝑃^𝐶𝑢^𝑛(𝑁)−𝛿𝐷𝑢^𝑛(𝑁)‖‖≤‖‖𝑢_(𝑁)𝑛+1−𝛿𝐷𝑢^(𝑁)𝑛+1−𝑢^𝑛(𝑁)−𝛿𝐷𝑢^𝑛(𝑁)‖‖=‖‖𝒜_𝑁−𝑢^𝑛(𝑁)‖‖≤‖‖𝑥_𝑛+1𝑥_𝑛+1−𝒜^𝑁𝑥_𝑛‖‖−𝑥_𝑛‖‖.(3.13) On the other hand, since 𝑇_𝑖 and 𝑈_𝑛,𝑖 are nonexpansive, we have ‖‖𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛‖‖≤=𝜇‖‖𝜇₁₁𝑇₁𝑈_𝑛+1,2𝑦_𝑛−𝜇₁𝑇₁𝑈_𝑛,2𝑦_𝑛‖‖=𝜇₁‖‖𝑈_𝑛+1,2𝑦_𝑛−𝑈_𝑛,2𝑦_𝑛‖‖≤𝜇₁‖‖𝜇₂𝑇₂𝑈_𝑛+1,3𝑦_𝑛−𝜇₂𝑇₂𝑈_𝑛,3𝑦_𝑛‖‖≤⋮𝜇₁𝜇₂‖‖𝑈_𝑛+1,3𝑦_𝑛−𝑈_𝑛,3𝑦_𝑛‖‖≤𝑀𝜇_21𝑛⋯𝜇_𝑛‖‖𝑈_{𝑛+1,𝑛+1}𝑦_𝑛−𝑈_𝑛,𝑛+1𝑦_𝑛‖‖_𝑖=1𝜇_𝑖,(3.14) where 𝑀₁≥0 is a constant such that ‖𝑈_{𝑛+1,𝑛+1}𝑦_𝑛−𝑈_𝑛,𝑛+1𝑦_𝑛‖≤𝑀₁ for all 𝑛≥0. It follows from (3.13) and (3.14) that we have ‖‖𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛𝑦_𝑛‖‖≤≤‖‖‖‖𝑊𝑦_𝑛+1𝑛+1𝑦_𝑛+1−𝑊_𝑛+1𝑦_𝑛‖‖+‖‖𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛‖‖−𝑦_𝑛‖‖+𝑀_1𝑛_𝑖=1𝜇_𝑖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝑀_1𝑛_𝑖=1𝜇_𝑖.(3.15) It follows that ‖‖𝐾_𝑛+1𝑊_𝑛+1𝑦_𝑛+1−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖=‖‖‖1𝑡_𝑛+1^𝑡𝑛+1₀𝑆(𝑠)𝑊_𝑛+1𝑦_𝑛+1𝑑𝑠−1𝑡_𝑛^𝑡𝑛₀≤1𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠‖‖‖𝑡_𝑛+1^𝑡𝑛+1₀+‖‖‖1‖‖𝑆(𝑠)𝑊_𝑛+1𝑦_𝑛+1−𝑆(𝑠)𝑊_𝑛𝑦_𝑛‖‖𝑑𝑠𝑡_𝑛+1^𝑡𝑛+1₀𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠−1𝑡_𝑛^𝑡𝑛₀≤‖‖𝑊𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠_𝑛+1‖‖‖𝑦_𝑛+1−𝑊_𝑛𝑦_𝑛‖‖+‖‖‖1𝑡_𝑛+1^𝑡𝑛+1_𝑡𝑛𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠+1𝑡_𝑛+1^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠−1𝑡_𝑛^𝑡𝑛₀≤‖‖𝑊𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠_𝑛+1‖‖‖𝑦_𝑛+1−𝑊_𝑛𝑦_𝑛‖‖+1𝑡_𝑛+1^𝑡𝑛+1_𝑡𝑛+||||1‖‖𝑆(𝑠)𝑊_𝑛𝑦_𝑛‖‖𝑑𝑠𝑡_𝑛+1−1𝑡_𝑛||||^𝑡_𝑛0≤‖‖𝑊‖‖𝑆(𝑠)𝑊_𝑛𝑦_𝑛‖‖𝑑𝑠_𝑛+1𝑦_𝑛+1−𝑊_𝑛𝑦_𝑛‖‖+21−𝑡^𝑛𝑡_𝑛+1𝑀²≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝑀_1𝑛_𝑖=1𝜇_𝑖+21−𝑡^𝑛𝑡_𝑛+1𝑀²,(3.16) where 𝑀₂=max{‖𝑆(𝑠)𝑊_𝑛𝑦_𝑛‖}.
Setting 𝑥_𝑛+1=(1−𝛽_𝑛)𝑣_𝑛+𝛽_𝑛𝑥_𝑛, for all 𝑛≥1, we have 𝑣_𝑛=𝑥^𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼^𝑛𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛1−𝛽_𝑛.(3.17) Then, we obtain 𝑣_𝑛+1−𝑣_𝑛=𝛼^𝑛+1𝑢+𝛾𝑓𝑊_𝑛+1𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝛼_𝑛+1(𝐼+𝜇𝐴)𝐾_𝑛+1𝑊_𝑛+1𝑦_𝑛+11−𝛽_𝑛+1−𝛼^𝑛𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛1−𝛽_𝑛=𝛼^𝑛+11−𝛽_𝑛+1𝑢+𝛾𝑓𝑊_𝑛+1𝑥_𝑛+1−𝛼^𝑛1−𝛽_𝑛𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛+𝐾_𝑛+1𝑊_𝑛+1𝑦_𝑛+1−𝐾_𝑛𝑊_𝑛𝑦_𝑛+𝛼^𝑛1−𝛽_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝛼^𝑛+11−𝛽_𝑛+1(𝐼+𝜇𝐴)𝐾_𝑛+1𝑊_𝑛+1𝑦_𝑛+1=𝛼^𝑛+11−𝛽_𝑛+1𝑢+𝛾𝑓𝑊_𝑛+1+𝛼_𝑛𝑥_𝑛+1−(𝐼+𝜇𝐴)𝐾_𝑛+1𝑊_𝑛+1𝑦_𝑛+11−𝛽_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑢−𝛾𝑓𝑊_𝑛𝑥_𝑛+𝐾_𝑛+1𝑊_𝑛+1𝑦_𝑛+1−𝐾_𝑛𝑊_𝑛𝑦_𝑛.(3.18) It follows from (3.16) and (3.18) that ‖‖𝑣_𝑛+1−𝑣_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝛼^𝑛+11−𝛽_𝑛+1‖𝑢‖+‖‖𝛾𝑓𝑊_𝑛+1+𝛼_𝑛𝑥_𝑛+1‖‖+‖‖(𝐼+𝜇𝐴)𝐾_𝑛+1𝑊_𝑛+1𝑦_𝑛+1‖‖1−𝛽_𝑛‖‖(𝐼+𝜇𝐴)𝐾_𝑛+𝑀_1𝑛𝑊_𝑛𝑦_𝑛‖‖+‖𝑢‖+‖‖𝛾𝑓𝑊_𝑛𝑥_𝑛‖‖_𝑖=1𝜇_𝑖+21−𝑡^𝑛𝑡_𝑛+1𝑀².(3.19) By the conditions (C4), (C5) and from 𝑡_𝑛∈(0,∞), 𝑡_𝑛→∞ and 0<𝜇_𝑖≤𝑏<1,forall𝑖≥1, we have limsup_𝑛→∞‖‖𝑣_𝑛+1−𝑣_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.20) Hence, by Lemma 2.7, we obtain lim_𝑛→∞‖‖𝑣_𝑛−𝑥_𝑛‖‖=0.(3.21) It follows that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞1−𝛽_𝑛‖‖𝑣_𝑛−𝑥_𝑛‖‖=0.(3.22) Applying (3.22) into (3.13), we obtain that lim_𝑛→∞‖‖𝑦_𝑛+1−𝑦_𝑛‖‖=lim_𝑛→∞‖‖𝑧_𝑛+1−𝑧_𝑛‖‖=lim_𝑛→∞‖‖𝑢^(𝑁)𝑛+1−𝑢^𝑛(𝑁)‖‖=0.(3.23)Step 3. We show that lim_𝑛→∞‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑦_𝑛‖=0, lim_𝑛→∞‖𝑦_𝑛−𝑆(𝑠)𝑦_𝑛‖=0, and lim_𝑛→∞‖𝑢^𝑛(𝑘+1)−𝑢^𝑛(𝑘)‖=0, where 𝐾_𝑛=(1/𝑡_𝑛)∫^𝑡𝑛₀𝑆(𝑠)𝑑𝑠.
Since 𝑥_𝑛+1=𝛼_𝑛(𝑢+𝛾𝑓(𝑊_𝑛𝑥_𝑛))+𝛽_𝑛𝑥_𝑛+((1−𝛽_𝑛)𝐼−𝛼_𝑛(𝐼+𝜇𝐴))𝐾_𝑛𝑊_𝑛𝑦_𝑛, we have ‖‖𝑥_𝑛−≤𝐾‖‖_𝑛𝑥𝑊_𝑛𝑛𝑦_𝑛‖‖=‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖+‖‖𝛼−_𝑛𝑥_𝑛+1‖‖=‖‖𝑥_𝑛𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝛼_𝑛𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛+𝛽_𝑛𝑥_𝑛−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖+𝛽_𝑛−𝑥_𝑛+1‖‖+𝛼_𝑛‖𝑢‖+‖‖𝛾𝑓𝑊_𝑛𝑥_𝑛‖‖+‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖‖‖𝑥_𝑛−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖,(3.24) that is ‖‖𝑥_𝑛−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖≤11−𝛽_𝑛‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+𝛼^𝑛1−𝛽_𝑛‖𝑢‖+‖‖𝛾𝑓𝑊_𝑛𝑥_𝑛‖‖+‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖.(3.25) By (C4), (C5), and (3.22) it follows that lim_𝑛→∞‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥_𝑛‖‖=0.(3.26)
Since 𝐽^Θ𝑁_𝑟𝑁∶𝐶→𝐶 is firmly nonexpansive, 𝑢^𝑛(𝑁)=𝒜^𝑁𝑥_𝑛, where 𝒜^𝑁∶=𝐽^Θ𝑁_𝑟𝑁⋯𝐽^Θ2_𝑟2𝐽^Θ1_𝑟1 and 𝑥^∗∈Ω, we have ‖‖𝑢^𝑛(𝑁)−𝑥^∗‖‖²=‖‖𝒜^𝑁𝑥_𝑛−𝒜^𝑁𝑥^∗‖‖²≤=𝑢𝒜^𝑁_𝑛(𝑁)𝑥_𝑛−𝒜^𝑁𝑥^∗,𝑥_𝑛−𝑥^∗=1−𝑥^∗,𝑥_𝑛−𝑥^∗2‖‖𝑢^𝑛(𝑁)−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢^𝑛(𝑁)‖‖²,(3.27) and hence ‖‖𝑢^𝑛(𝑁)−𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢^𝑛(𝑁)‖‖².(3.28) Observe that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖+𝛼1_𝑛−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²=‖‖1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗‖‖²+𝛼^2𝑛‖‖𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²++22𝛼𝛽_𝑛𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗≤1−𝛽_𝑛1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛼^2𝑛‖‖𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²++22𝛼𝛽_𝑛𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗=1−𝛽_𝑛1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾²‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖²+𝛽^2𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+21−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+𝑐_𝑛≤1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾²‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖²+𝛽^2𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝐾^𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐^𝑛=1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²−21−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛+𝛽^2𝑛‖‖𝐾^𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖²+𝛽^2𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛−𝛽^2𝑛‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐^𝑛=1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²−1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝐾^𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑦_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛,(3.29) where 𝑐_𝑛=+𝛼^2𝑛2‖‖𝛼𝑢_𝑛+𝛾𝑓(𝑊_𝑛𝑥_𝑛)−(𝐼+𝜇𝐴)𝑥^∗‖‖²+2𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗.(3.30) It follows from condition (C4) that lim_𝑛→∞𝑐_𝑛=0.(3.31) Putting (3.28) into (3.29) and using also (3.10), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑦_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑢^𝑛(𝑁)−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤+11−−𝛼𝛼^𝑛_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢^𝑛(𝑁)‖‖²−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛=1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²−1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑥_𝑛−𝑢^𝑛(𝑁)‖‖²+𝑐_𝑛≤‖‖𝑥_𝑛−𝑥^∗‖‖²−1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑥_𝑛−𝑢^𝑛(𝑁)‖‖²+𝑐_𝑛.(3.32) It follows that 1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑥_𝑛−𝑢^𝑛(𝑁)‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²+𝑐_𝑛≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖+𝑐_𝑛.(3.33) Therefore, by (3.22) and (3.31), we get lim_𝑛→∞‖‖𝑥_𝑛−𝑢^𝑛(𝑁)‖‖=0.(3.34) Since ‖‖𝑢^𝑛(𝑁)−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖≤‖‖𝑢^𝑛(𝑁)−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖,(3.35) and by (3.26) and (3.70), we have lim_𝑛→∞‖‖𝑢^𝑛(𝑁)−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖=0.(3.36) Since 𝐵 is a 𝐿_𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping on 𝐵 and 0<𝜏<2(𝑑−𝑐𝐿^2𝐵)/𝐿^2𝐵 for any 𝑥^∗∈Ω, we have ‖‖𝑦_𝑛−𝑥^∗‖‖²=‖‖𝑃_𝐶𝑧_𝑛−𝜏𝐵𝑧_𝑛−𝑃_𝐶𝑦^∗−𝜏𝐵𝑦^∗‖‖²≤‖‖𝑧_𝑛−𝑦^∗−𝜏𝐵𝑧_𝑛−𝐵𝑦^∗‖‖²=‖‖𝑧_𝑛−𝑦^∗‖‖²−2𝜏⟨𝑧_𝑛−𝑦^∗,𝐵𝑧_𝑛−𝐵𝑦^∗⟩+𝜏²‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−2𝜏−𝑐‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖²+𝑑‖‖𝑧_𝑛−𝑦^∗‖‖²+𝜏²‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖²=‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝜏𝑐‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖²−2𝜏𝑑‖‖𝑧_𝑛−𝑦^∗‖‖²+𝜏²‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝜏𝑐‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖²−2𝜏𝑑𝐿^2𝐵‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖²+𝜏²‖‖𝐵𝑦_𝑛−𝐵𝑝‖‖²=‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝜏𝑐+𝜏²−2𝜏𝑑𝐿^2𝐵‖‖𝐵𝑧^𝑛−𝐵𝑦^∗‖‖².(3.37) Similarly, since 𝐷 is a 𝐿_𝐷-Lipschitz continuous and relaxed (𝑐′,𝑑′)-cocoercive mapping on 𝐷 and 0<𝛿<2(𝑑′−𝑐′𝐿^2𝐷)/𝐿^2𝐷, we also have ‖‖𝑧_𝑛−𝑦^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛿𝑐^+𝛿²−2𝛿𝑑^𝐿^2𝐷‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖².(3.38) Substituting (3.37) into (3.29), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤×1−‖𝛼𝑥_𝑛𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾−𝑥^∗‖²+2𝜏𝑐+𝜏²−2𝜏𝑑𝐿^2𝐵‖‖𝐵𝑧^𝑛−𝐵𝑦^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛=1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾2𝜏𝑐+𝜏²−2𝜏𝑑𝐿^2𝐵‖‖𝐵𝑧^𝑛−𝐵𝑦^∗‖‖²+𝑐_𝑛≤‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝜏𝑐+𝜏²−2𝜏𝑑𝐿^2𝐵‖‖𝐵𝑧^𝑛−𝐵𝑦^∗‖‖²+𝑐_𝑛.(3.39) Again, substituting (3.38) into (3.29) and using also (3.10), we get ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑦_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑧_𝑛−𝑦^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤×1−‖‖𝛼𝑥_𝑛𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾−𝑥^∗‖‖²+2𝛿𝑐^+𝛿²−2𝛿𝑑^𝐿^2𝐷‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛=×1−2𝛼𝛿𝑐_𝑛−𝛼_𝑛𝜇𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾+𝛿²−2𝛿𝑑^𝐿^2𝐷‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²+𝑐_𝑛≤‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛿𝑐^+𝛿²−2𝛿𝑑^𝐿^2𝐷‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²+𝑐_𝑛.(3.40) Therefore, by (3.39) and (3.40), we have 2𝐿𝜏^2𝐵𝑑−2𝜏𝑐−𝜏²‖‖𝐵𝑧^𝑛−𝐵𝑦^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²+𝑐_𝑛2𝛿𝑑_≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖+𝑐_𝑛,𝐿^2𝐷−2𝛿𝑐^−𝛿²‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²+𝑐_𝑛≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖+𝑐_𝑛.(3.41) It follows from (3.22) and (3.31) that we obtain lilmim_{𝑛→∞𝑛→∞}‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖=0,(3.42)‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖=0.(3.43) From (2.6), we have ‖‖𝑧_𝑛−𝑦^∗‖‖²=‖‖𝑃_𝐶𝑢^𝑛(𝑁)−𝛿𝐷𝑢^𝑛(𝑁)−𝑃^𝐶𝑥^∗−𝛿𝐷𝑥^∗‖‖²=≤1𝑢^𝑛(𝑁)−𝛿𝐷𝑢^𝑛(𝑁)−𝑥^∗−𝛿𝐷𝑥^∗,𝑧_𝑛−𝑦^∗2‖‖𝑢^𝑛(𝑁)−𝛿𝐷𝑢^𝑛(𝑁)−𝑥^∗−𝛿𝐷𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑦^∗‖‖²−‖‖𝑢^𝑛(𝑁)−𝛿𝐷𝑢^𝑛(𝑁)−𝑥^∗−𝛿𝐷𝑥^∗−𝑧_𝑛−𝑦^∗‖‖²≤12‖‖𝑢^𝑛(𝑁)−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑦^∗‖‖²−‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗−𝛿𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²=12‖‖𝑢^𝑛(𝑁)−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑦^∗‖‖²−‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²≤1+2𝛿𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗,𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗−𝛿²‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²2‖‖𝑥^𝑛−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑦^∗‖‖²−‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²+2𝛿‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖−𝛿²‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖².(3.44) So, we obtain ‖‖𝑧_𝑛−𝑦^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²+2𝛿‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖−𝛿²‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖².(3.45) By (3.29), we get ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑦_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑧_𝑛−𝑦^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤×1−‖‖𝑥𝛼_𝑛𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾−𝑥^∗‖‖²−‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²++12−𝛿𝛼‖‖_𝑛𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖−𝛿²‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛=×1−−𝛼‖‖_𝑛−𝑢𝛼_{𝑛𝑛(𝑁)}𝜇𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²+2𝛿‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖−𝛿²‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²+𝑐^𝑛≤‖‖𝑥_𝑛−𝑥^∗‖‖²−1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²+−2𝛿𝛿₂‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²+𝑐_𝑛(3.46) which implies that 1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²+−2𝛿𝛿₂‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²+𝑐_𝑛≤+‖‖2𝑥𝛿_𝑛−‖‖𝑥𝑢_{𝑛+1𝑛(𝑁)}‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖−𝛿₂−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝐷𝑢^𝑛(𝑁)−𝐷𝑥^∗‖‖²+𝑐_𝑛.(3.47) From (3.22), (3.31), and (3.43), we have lim_𝑛→∞‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖=0.(3.48) Now, from (2.2) and (2.7), we observe that ‖‖𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗‖‖²=‖‖−𝑧_𝑛𝑃−_𝐶𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗𝑧_𝑛−𝜏𝐵𝑧_𝑛−𝑃_𝐶𝑦^∗−𝜏𝐵𝑦^∗+𝜏𝐵𝑧_𝑛−𝐵𝑦^∗‖‖²≤‖‖𝑧_𝑛−𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗−𝑃_𝐶𝑧_𝑛−𝜏𝐵𝑧_𝑛−𝑃_𝐶𝑦^∗−𝜏𝐵𝑦^∗‖‖²≤+‖‖2𝑧𝜏𝐵𝑧_𝑛−𝐵𝑦^∗,𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗_𝑛−𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗‖‖²−‖‖𝑃_𝐶𝑧_𝑛−𝜏𝐵𝑧_𝑛−𝑃_𝐶𝑦^∗−𝜏𝐵𝑦^∗‖‖²≤+‖‖2𝑧𝜏_𝑛‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖‖‖𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗‖‖−𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗‖‖²−‖‖𝐾_𝑛𝑊_𝑛𝑃_𝐶𝑧_𝑛−𝜏𝐵𝑧_𝑛−𝐾_𝑛𝑊_𝑛𝑃_𝐶𝑦^∗−𝜏𝐵𝑦^∗‖‖²=+‖‖2𝑧𝜏_𝑛‖‖𝐵𝑧_𝑛−𝐵𝑦^∗‖‖‖‖𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗‖‖−𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗‖‖²−‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝐾_𝑛𝑊_𝑛𝑥^∗‖‖²=+‖‖2𝜏𝑧‖‖_𝑛𝐵𝑧_𝑛−𝐵𝑦^∗‖‖‖‖𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗‖‖×‖‖𝑧−_𝑛𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗‖‖−‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖+2𝜏‖‖𝐵𝑧−_𝑛𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗‖‖+‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖≤‖‖𝑧_𝑛−𝐵𝑦^∗‖‖‖‖𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗‖‖×‖‖𝑧−_𝑛𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗−𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖+2𝜏‖‖𝐵𝑧−_𝑛𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗‖‖+‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖=‖‖𝑢_𝑛(𝑁)−𝐵𝑦^∗‖‖‖‖𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗‖‖×‖‖𝑧_𝑛−𝐾_𝑛𝑊_𝑛𝑦_𝑛+𝑥^∗−𝑦^∗−𝑢^𝑛(𝑁)−𝑧_𝑛−𝜏𝐵𝑧^𝑛−𝐵𝑦^∗‖‖+2𝜏‖‖𝐵𝑧−_𝑛𝜏𝐵𝑧_𝑛−𝑦^∗−𝜏𝐵𝑦^∗‖‖+‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖−𝐵𝑦^∗‖‖‖‖𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗‖‖.(3.49) It follows from (3.36), (3.42), and (3.48) that we have lim_𝑛→∞‖‖𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗‖‖=0,(3.50) since ‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑦_𝑛‖‖≤‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑢^𝑛(𝑁)‖‖+‖‖𝑢^𝑛(𝑁)−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖+‖‖𝑧_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗‖‖.(3.51) It follows from (3.36), (3.48) and (3.50), we get lim_𝑛→∞‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑦_𝑛‖‖=0,(3.52) and from (3.26), and (3.52) that we have lim_𝑛→∞‖‖𝑥_𝑛−𝑦_𝑛‖‖=0.(3.53) Since {𝑊_𝑛𝑦_𝑛} is a bounded sequence in 𝐶, from Lemma 2.10 for all 𝑠≥0, we have lim_𝑛→∞‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑆(𝑠)𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖=lim_𝑛→∞‖‖‖1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠−𝑆(𝑠)1𝑡_𝑛^𝑡𝑛₀=0,𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠‖‖‖(3.54) and since ‖‖𝑦_𝑛−𝑆(𝑠)𝑦_𝑛‖‖≤≤2‖‖‖‖𝑦_𝑛𝑦_𝑛−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖+‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑆(𝑠)𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖+‖‖𝑆(𝑠)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑆(𝑠)𝑦_𝑛‖‖−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖+‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑆(𝑠)𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖,(3.55) it follows from (3.52) and (3.54) that we get lim_𝑛→∞‖‖𝑦_𝑛−𝑆(𝑠)𝑦_𝑛‖‖=0.(3.56)
On the other hand, since 𝐽^Θ𝑘_𝑟𝑘∶𝐻→𝐻 is firmly nonexpansive, 𝒜^𝑘∶=𝐽^Θ𝑘_𝑟𝑘⋯𝐽^Θ2_𝑟2𝐽^Θ1_𝑟1,𝑘=1,2,…,𝑁 and 𝑥^∗∈Ω, we have ‖‖𝒜^𝑘+1𝑥_𝑛−𝑥^∗‖‖²=‖‖𝐽^Θ𝑘+1_𝑟𝑘+1𝒜^𝑘𝑥_𝑛−𝐽^Θ𝑘+1_𝑟𝑘+1𝑥^∗‖‖²≤=1𝐽^Θ𝑘+1_𝑟𝑘+1𝒜^𝑘𝑥_𝑛−𝑥^∗,𝒜^𝑘𝑥_𝑛−𝑥^∗2‖‖𝐽^{Θ𝑘+1𝑟𝑘+1}𝒜^𝑘𝑥_𝑛−𝑥^∗‖‖²+‖‖𝒜^𝑘𝑥_𝑛−𝑥^∗‖‖²−‖‖𝐽^Θ𝑘+1_𝑟𝑘+1𝒜^𝑘𝑥_𝑛−𝒜^𝑘𝑥_𝑛‖‖²,(3.57) and hence ‖‖𝒜^𝑘+1𝑥_𝑛−𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝒜^𝑘+1𝑥_𝑛−𝒜^𝑘𝑥_𝑛‖‖².(3.58) From (3.10), (3.29), and (3.58), for each 𝑘=1,2,…,𝑁−1, we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑦_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝒜^𝑘𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝒜^𝑘+1𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛≤+11−−𝛼𝛼^𝑛_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝒜^𝑘+1𝑥_𝑛−𝒜^𝑘𝑥_𝑛‖‖²−𝛼_𝑛𝜇𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑐_𝑛=1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²−1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝒜^𝑘+1𝑥_𝑛−𝒜^𝑘𝑥_𝑛‖‖²+𝑐_𝑛≤‖‖𝑥_𝑛−𝑥^∗‖‖²−1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝒜^𝑘+1𝑥_𝑛−𝒜^𝑘𝑥_𝑛‖‖²+𝑐_𝑛.(3.59) It follows that 1−𝛼_𝑛−𝛼_𝑛𝜇𝛾1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾‖‖𝒜^𝑘+1𝑥_𝑛−𝒜^𝑘𝑥_𝑛‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²+𝑐_𝑛≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖+𝑐_𝑛.(3.60) Therefore, by (3.22) and (3.31), we get lim_𝑛→∞‖‖𝒜^𝑘+1𝑥_𝑛−𝒜^𝑘𝑥_𝑛‖‖=0thatislim_𝑛→∞‖‖𝑢^𝑛(𝑘+1)−𝑢^𝑛(𝑘)‖‖=0.(3.61)Step 4. We prove that limsup_𝑛→∞⟨𝑢+[𝛾𝑓−(𝐼+𝜇𝐴)]𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0,(3.62) where 𝑥^∗ is a solution of the optimization problem: min_𝑥∈Ω𝜇2⟨𝐴𝑥^∗,𝑥^∗⟩+12‖𝑥^∗−𝑢‖²−ℎ𝑥^∗.(3.63) To show this inequality, we can choose a subsequence {𝑦_𝑛𝑖} of {𝑦_𝑛} such that lim_𝑖→∞𝑢+[𝛾𝑓−(𝐼+𝜇𝐴)]𝑥^∗,𝑦_𝑛𝑖−𝑥^∗=limsup_𝑛→∞⟨𝑢+[𝛾𝑓−(𝐼+𝜇𝐴)]𝑥^∗,𝑦_𝑛−𝑥^∗⟩.(3.64)
Since {𝑦_𝑛𝑖} is bounded, there exists a subsequence {𝑦_𝑛𝑖𝑗} of {𝑦_𝑛𝑖} which converges weakly to 𝑧∈𝐶. Without loss of generality, we can assume that 𝑦_𝑛𝑖⇀𝑧. From (3.53), we get 𝑥_𝑛𝑖⇀𝑧.
Next, we show that 𝑧∈Ω∶=𝐹(𝒮)∩𝐹(𝑊)∩𝔉∩SVI(𝐶,𝐵,𝐷), where 𝔉=(∩^𝑁𝑘=1MEP(Θ_𝑘,𝜙)).
(1) First, we prove that 𝑧∈𝐹(𝒮). Indeed, from Lemma 2.11 and (3.56), we get 𝑧∈𝐹(𝒮), that is, 𝑧=𝑆(𝑠)𝑧,forall𝑠≥0.
(2) Next, we show that 𝑧∈𝐹(𝑊)=∩^∞𝑛=1𝐹(𝑊_𝑛), where 𝐹(𝑊_𝑛)=∩^𝑛𝑖=1𝐹(𝑇_𝑖),forall𝑛≥1 and 𝐹(𝑊_𝑛+1)⊂𝐹(𝑊_𝑛). Assume that 𝑧∉𝐹(𝑊), then there exists a positive integer 𝑚 such that 𝑧∉𝐹(𝑇_𝑚) and so 𝑧∉∩^𝑚𝑖=1𝐹(𝑇_𝑖). Hence for any 𝑛≥𝑚,𝑧∉∩^𝑛𝑖=1𝐹(𝑇_𝑖)=𝐹(𝑊_𝑛), that is, 𝑧≠𝑊_𝑛𝑧. This together with 𝑧=𝑆(𝑠)𝑧,forall𝑠≥0, shows 𝑧=𝑆(𝑠)𝑧≠𝑆(𝑠)𝑊_𝑛𝑧,forall𝑠≥0; therefore, we have 𝑧≠𝐾_𝑛𝑊_𝑛𝑧,forall𝑛≥𝑚. It follows from the Opial’s condition and (3.52) that liminf_𝑖→∞‖‖𝑦_𝑛𝑖−𝑧‖‖<≤lliimmiinnf_𝑖→∞f_𝑖→∞‖‖𝑦_𝑛𝑖−𝐾_𝑛𝑖𝑊_𝑛𝑖𝑧‖‖≤liminf_𝑖→∞‖‖𝑦_𝑛𝑖−𝐾_𝑛𝑖𝑊_𝑛𝑖𝑦_𝑛𝑖‖‖+‖‖𝐾_𝑛𝑖𝑊_𝑛𝑖𝑦_𝑛𝑖−𝐾_𝑛𝑖𝑊_𝑛𝑖𝑧‖‖‖‖𝑦_𝑛𝑖−𝑧‖‖,(3.65) which is a contradiction. Thus, we get 𝑧∈𝐹(𝑊).
(3) Now, we prove that 𝑧∈𝔉. Since 𝒜^𝑘+1=𝐽^Θ𝑘+1_𝑟𝑘+1𝒜^𝑘,𝑘=1,2,…,𝑁−1, and 𝑢^𝑛(𝑘+1)=𝒜^𝑘+1𝑥_𝑛, we have Θ𝒜^𝑘+1𝑥_𝑛,𝑥+𝜙(𝑥)−𝜙𝒜^𝑘+1𝑥_𝑛+1𝑟_𝑘+1𝐾^𝒜^𝑘+1𝑥_𝑛−𝐾^𝒜^𝑘𝑥_𝑛,𝜂𝑥,𝒜^𝑘+1𝑥_𝑛∀𝑥∈≥𝐻0,.(3.66) It follows that 1𝑟_𝑘+1𝐾^𝒜^𝑘+1𝑥_𝑛𝑖−𝐾^𝒜^𝑘𝑥_𝑛𝑖,𝜂𝑥,𝒜^𝑘+1𝑥_𝑛𝑖≥−Θ𝒜^𝑘+1𝑥_𝑛𝑖,𝑥−𝜙(𝑥)+𝜙𝒜^𝑘+1𝑥_𝑛𝑖(3.67) for all 𝑥∈𝐻. From (3.61) and by conditions (C1)(c) and (C2), we get lim_{𝑛𝑖→∞}1𝑟_𝑘+1𝐾^𝒜^𝑘+1𝑥_𝑛𝑖−𝐾^𝒜^𝑘𝑥_𝑛𝑖,𝜂𝑥,𝒜^𝑘+1𝑥_𝑛𝑖=0.(3.68) By the assumption that 𝜙 is lower semicontinuous, then it is weakly lower semicontinuous and by the condition (H2) that 𝑥↦(−Θ_𝑖(𝑥,𝑦)) is lower semicontinuous, then it is weakly lower semicontinuous. Since 𝑦_𝑛𝑖⇀𝑧, it follows from (3.36), (3.52), and (3.61) that 𝑢^𝑛(𝑘)_𝑖⇀𝑧 for each 𝑘=1,2,…,𝑁−1. Taking the lower limit 𝑛_𝑖→∞ in (3.67), we have Θ_𝑘+1(𝑧,𝑥)+𝜙(𝑥)−𝜙(𝑧)≥0,∀𝑥∈𝐻,∀𝑘=0,1,2,…,𝑁−1.(3.69) Therefore, 𝑧∈∩^𝑁𝑘=1MEP(Θ_𝑘,𝜙).
(4) Next, we show that 𝑧∈SVI(𝐶,𝐵,𝐷). By (3.36) and (3.52), we have ‖‖𝑢^𝑛(𝑁)−𝑦_𝑛‖‖≤‖‖𝑢^𝑛(𝑁)−𝐾_𝑛𝑊_𝑛𝑦_𝑛‖‖+‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑦_𝑛‖‖⟶0as𝑛⟶∞.(3.70) By Lemma 2.13 that 𝐺 is a nonexpansive, we obtain ‖‖𝑦_𝑛−𝐺𝑦_𝑛‖‖=‖‖𝑃_𝐶=‖‖𝐺𝑢𝑃^𝐶_𝑛(𝑁)𝑢^𝑛(𝑁)−𝛿𝐷𝑢^𝑛(𝑁)−𝜏𝐵𝑃^𝐶𝑢^𝑛(𝑁)−𝛿𝐷𝑢^𝑛(𝑁)−𝐺𝑦^𝑛‖‖≤‖‖𝑢_𝑛(𝑁)−𝐺𝑦^𝑛‖‖−𝑦_𝑛‖‖.(3.71) Thus, lim_𝑛→∞‖‖𝑦_𝑛−𝐺𝑦_𝑛‖‖=0.(3.72) By Lemma 2.14, we obtain that 𝑧∈SVI(𝐶,𝐵,𝐷). Hence 𝑧∈Ω is proved.
Now, from Lemma 2.9, (3.64), and (3.53), we have limsup_𝑛→∞⟨𝑢+[𝛾𝑓−(𝐼+𝜇𝐴)]𝑥^∗,𝑥_𝑛−𝑥^∗⟩==lliimm_𝑖→∞sup_𝑛→∞⟨𝑢+[𝛾𝑓−(𝐼+𝜇𝐴)]𝑥^∗,𝑦_𝑛−𝑥^∗⟩=⟨𝑢+[𝛾𝑢𝑓+−[𝛾(𝐼𝑓+−𝜇(𝐼𝐴+)]𝜇𝑥𝐴_∗)]𝑥^∗,𝑦_𝑛𝑖−𝑥^∗,𝑧−𝑥^∗⟩≤0.(3.73) By (3.52), (3.53), and (3.73), we obtain limsup_𝑛→∞⟨𝑢+[𝛾𝑓−(𝐼+𝜇𝐴)]𝑥^∗,𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗⟩≤0.(3.74)Step 5. Finally, we show that 𝑥_𝑛→𝑥^∗. From (3.1), we obtain ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝛼_𝑛𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖²=‖‖+𝛼1_𝑛−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²=‖‖1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗‖‖²++22𝛼𝛼_𝑛𝑛1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗𝛽_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗+𝛼^2𝑛‖𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖²≤1−𝛽_𝑛−𝛼_𝑛1+𝜇𝛾‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²++22𝛼𝛼_𝑛𝑛1−𝛽_𝑛𝛾𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑓𝑊_𝑛𝑥_𝑛−𝑓𝑥^∗−2𝛼_2𝑛1−𝛽_𝑛𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗−2𝛼_2𝑛𝛾(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑓𝑊_𝑛𝑥_𝑛−𝑓𝑥^∗+2𝛼_𝑛(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗+𝛼_2𝑛𝛽_𝑛𝛾𝑥_𝑛−𝑥^∗,𝑓𝑊_𝑛𝑥_𝑛−𝑓𝑥^∗+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²≤1−𝛽_𝑛−𝛼_𝑛1+𝜇𝛾‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²++22𝛼𝛼_𝑛𝑛1−𝛽_𝑛𝛾‖‖𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑓𝑊_𝑛𝑥_𝑛−𝑓𝑥^∗‖‖−2𝛼_2𝑛1−𝛽_𝑛𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗−2𝛼_2𝑛𝛾‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑓𝑊_𝑛𝑥_𝑛−𝑓𝑥^∗‖‖+2𝛼_𝑛‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖+𝛼_2𝑛𝛽_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖‖‖𝑓𝑊_𝑛𝑥_𝑛−𝑓𝑥^∗‖‖+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²≤(1−𝛽_𝑛−𝛼_𝑛(1+𝜇𝛾))‖‖𝑥_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛1−𝛽_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖²+−22𝛼𝛼_𝑛2𝑛1−𝛽_𝑛𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗−2𝛼_2𝑛𝛾𝛼‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+2𝛼_𝑛‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖+𝛼_2𝑛𝛽_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²=1−𝛼_𝑛1+𝜇𝛾+2𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛼_𝑛−22𝛼1_𝑛−𝛽^𝑛𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗−2𝛼_𝑛𝛾𝛼‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+2𝛽_𝑛‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖+𝛼_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²=1−2𝛼_𝑛1+𝜇𝛾+𝛼^2𝑛1+𝜇𝛾²+2𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛼_𝑛2−21𝛼−_𝑛𝛽^𝑛𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗−2𝛼_𝑛𝛾𝛼‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+2𝛽_𝑛‖‖(𝐼+𝜇𝐴)𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖+𝛼_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²=1−2𝛼_𝑛.1+𝜇𝛾−𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛼_𝑛+22𝛽1_𝑛−𝛽^𝑛𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗+𝛼_𝑛𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗1+𝜇𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²−−22𝛾‖‖𝛼(𝐼‖‖+(𝐼𝜇+𝐴𝜇)𝐴𝐾)_𝑛𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑢+𝛾𝑓(𝑊_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖𝑥_𝑛)−(𝐼+𝜇𝐴)𝑥^∗‖‖².(3.75) Since {𝑥_𝑛}, {𝑓(𝑊_𝑛𝑥_𝑛)}, and {𝐾_𝑛𝑊_𝑛𝑦_𝑛} are bounded, there exist 𝑀>0 such that 1+𝜇𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²−−22𝛾‖‖𝛼(𝐼‖‖+(𝐼𝜇+𝐴𝜇)𝐴𝐾)_𝑛𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑢+𝛾𝑓𝑊_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗‖‖𝑥_𝑛−(𝐼+𝜇𝐴)𝑥^∗‖‖²≤𝑀(3.76) for all 𝑛≥0. It follows that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−𝛼_𝑛𝑎_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛼_𝑛𝑏_𝑛,(3.77) where 𝑎𝑏_𝑛=21+𝜇𝛾−𝛾𝛼_𝑛,=+221𝛽_𝑛−𝛽_𝑛𝐾_𝑛𝑊_𝑛𝑦_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗𝑥_𝑛−𝑥^∗,𝑢+𝛾𝑓𝑥^∗−(𝐼+𝜇𝐴)𝑥^∗+𝛼_𝑛𝑀.(3.78) Applying Lemma 2.8 to (3.77), we conclude that 𝑥_𝑛→𝑥^∗. This completes the proof.

Remark 3.2. For example, of the control conditions (C4)–(C6), we set 𝛼_𝑛=1/10𝑛,𝛽_𝑛=𝑛/(𝑛+1). We set 𝐵,𝐷 is a 1-Lipschitz continuous and relaxed (0,1)-cocoercive mapping, (i.e., 𝐿_𝐵=1=𝐿_𝐷 and 𝑐=0=𝑐^,𝑑=1=𝑑^).
Then, we can choose 𝜏∈(0,2) and 𝛿∈(0,2) which satisfies the condition (C6) in Theorem 3.1.

Corollary 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 which 𝐶+𝐶⊂𝐶 and let 𝑓 be a contraction of 𝐶 into itself with 𝛼∈(0,1). Let 𝜙 be a lower semicontinuous and convex functional from 𝐻 to ℝ and let Θ∶𝐻×𝐻→ℝ be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on 𝐶 and let {𝑡_𝑛} be a positive real divergent sequence. Let {𝑉_𝑖∶𝐶→𝐶}^∞𝑖=1 be a countable family of uniformly 𝑘- strict pseudo-contractions, let {𝑇_𝑖∶𝐶→𝐶}^∞𝑖=1 be the countable family of nonexpansive mappings defined by 𝑇_𝑖𝑥=𝑡𝑥+(1−𝑡)𝑉_𝑖𝑥,forall𝑥∈𝐶,forall𝑖≥1,𝑡∈[𝑘,1), let 𝑊_𝑛 be the 𝑊-mapping defined by (2.12), and let 𝑊 be a mapping defined by (2.13) with 𝐹(𝑊)≠∅. Let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and let 0<𝛾<(1+𝜇𝛾)/𝛼, 𝐵∶𝐻→𝐻 be a 𝐿_𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping with 𝑑>𝑐𝐿^2𝐵, and let 𝐷∶𝐻→𝐻 be a 𝐿_𝐷-Lipschitz continuous and relaxed (𝑐^,𝑑^)-cocoercive mapping with 𝑑^>𝑐^𝐿^2𝐷. Suppose that Ω∶=𝐹(𝒮)∩𝐹(𝑊)∩MEP(Θ,𝜙)∩SVI(𝐶,𝐵,𝐷)≠∅. Let 𝜇>0, 𝛾>0 and 𝑟>0, which are constants. For given 𝑥₁∈𝐻 arbitrarily and fixed 𝑢∈𝐻, suppose {𝑥_𝑛}, {𝑦_𝑛},{𝑧_𝑛}, and{𝑢_𝑛} are the sequences generated iteratively by Θ𝑢_𝑛,𝑥+𝜙(𝑥)−𝜙𝑢_𝑛+1𝑧𝑟𝐾^𝑢_𝑛−𝐾^𝑥_𝑛,𝜂𝑥,𝑢_𝑛≥0,∀𝑥∈𝐻,_𝑛𝑦_𝑛=𝑃_𝐶𝑢_𝑛−𝛿𝐷𝑢_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑧_𝑛−𝜏𝐵𝑧_𝑛,=𝛼_𝑛𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠,(3.79) where 𝑢_𝑛=𝐽^Θ𝑟𝑥_𝑛 such that 𝐽^Θ𝑟∶𝐻→𝐻 is the mapping defined by (2.22) and {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) for all 𝑛∈ℕ. If the functions 𝜂∶𝐻×𝐻→𝐻 and 𝐾∶𝐻→ℝ satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then {𝑥_𝑛} converges strongly to 𝑥^∗∈Ω, which solves the following optimization problem (OP): min_𝑥∗∈Ω𝜇2⟨𝐴𝑥^∗,𝑥^∗⟩+12‖𝑥^∗−𝑢‖²−ℎ𝑥^∗,(3.80) and (𝑥^∗,𝑦^∗) is a solution of the general system of variational inequality problem (1.20) such that 𝑦^∗=𝑃_𝐶(𝑥^∗−𝛿𝐷𝑥^∗).

Proof. Taking 𝑁=1 in Theorem 3.1. Hence, the conclusion follows. This completes the proof.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 which 𝐶+𝐶⊂𝐶 and let 𝑓 be a contraction of 𝐶 into itself with 𝛼∈(0,1). Let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on 𝐶 and let {𝑡_𝑛} be a positive real divergent sequence. Let {𝑉_𝑖∶𝐶→𝐶}^∞𝑖=1 be a countable family of uniformly 𝑘-strict pseudo-contractions, let {𝑇_𝑖∶𝐶→𝐶}^∞𝑖=1 be the countable family of nonexpansive mappings defined by 𝑇_𝑖𝑥=𝑡𝑥+(1−𝑡)𝑉_𝑖𝑥,forall𝑥∈𝐶,forall𝑖≥1,𝑡∈[𝑘,1), let 𝑊_𝑛 be the W-mapping defined by (2.12), and let 𝑊 be a mapping defined by (2.13) with 𝐹(𝑊)≠∅. Let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and let 0<𝛾<(1+𝜇𝛾)/𝛼, 𝐵∶𝐻→𝐻 be a 𝐿_𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping with 𝑑>𝑐𝐿^2𝐵, and let 𝐷∶𝐻→𝐻 be a 𝐿_𝐷-Lipschitz continuous and relaxed (𝑐^,𝑑^)-cocoercive mapping with 𝑑^>𝑐^𝐿^2𝐷. Suppose that Ω∶=𝐹(𝒮)∩𝐹(𝑊)∩SVI(𝐶,𝐵,𝐷)≠∅. Let 𝜇>0 and 𝛾>0, which are constants. For given 𝑥₁∈𝐻 arbitrarily and fixed 𝑢∈𝐻, suppose {𝑥_𝑛}, {𝑦_𝑛}, and{𝑧_𝑛} are the sequences generated iteratively by 𝑧𝑦_𝑛𝑛=𝑃_𝐶𝑥_𝑛−𝛿𝐷𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑧_𝑛−𝜏𝐵𝑧_𝑛,=𝛼_𝑛𝑢+𝛾𝑓𝑊_𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑦_𝑛𝑑𝑠,(3.81) where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) for all 𝑛∈ℕ. If the sequence {𝑥_𝑛} satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then {𝑥_𝑛} converges strongly to 𝑥^∗∈Ω, which solves the following optimization problem (OP): min_𝑥∗∈Ω𝜇2⟨𝐴𝑥^∗,𝑥^∗⟩+12‖𝑥^∗−𝑢‖²−ℎ𝑥^∗,(3.82) and (𝑥^∗,𝑦^∗) is a solution of the general system of variational inequality problem (1.20) such that 𝑦^∗=𝑃_𝐶(𝑥^∗−𝛿𝐷𝑥^∗).

Proof. Put Θ(𝑥,𝑦)≡𝜙(𝑥)≡0 for all 𝑥,𝑦∈𝐻 and 𝑟=1. Take 𝐾(𝑥)=‖𝑥‖²/2 and 𝜂(𝑦,𝑥)=𝑦−𝑥, for all 𝑥,𝑦∈𝐻. Then, we get 𝑢_𝑛=𝑃_𝐶𝑥_𝑛=𝑥_𝑛 in Corollary 3.3. Hence, the conclusion follows. This completes the proof.

Corollary 3.5. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑓 be a contraction of 𝐻 into itself with 𝛼∈(0,1). Let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on 𝐶 and let {𝑡_𝑛} be a positive real divergent sequence. Let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and let 0<𝛾<(1+𝜇𝛾)/𝛼, 𝐵∶𝐻→𝐻 be a 𝐿_𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping with 𝑑>𝑐𝐿^2𝐵. Suppose that Ω∶=𝐹(𝒮)∩𝐵⁻¹0≠∅. Let 𝜇>0 and 𝛾>0, which are constants. For given 𝑥₁∈𝐻 arbitrarily and fixed 𝑢∈𝐻, suppose the {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} are the sequences generated iteratively by 𝑧𝑦_𝑛𝑛=𝑥_𝑛−𝜏𝐵𝑥_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝜏𝐵𝑧_𝑛,=𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛(𝐼+𝜇𝐴)1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑦_𝑛𝑑𝑠,(3.83) where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) for all 𝑛∈ℕ. If the sequence {𝑥_𝑛} satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then {𝑥_𝑛} converges strongly to 𝑥^∗∈Ω.

Proof. Setting 𝜏=𝛿, 𝐶≡𝐻, 𝐷≡𝐵 and 𝑊_𝑛≡𝑃_𝐻≡𝐼 in Corollary 3.4, it follows from the proof of Theorem  4.1 in [25] that 𝐵⁻¹0=VI(𝐻,𝐵). Hence, the conclusion follows. This completes the proof.

Acknowledgments

The authors would like to thank the “Centre of Excellence in Mathematics” under the Commission on Higher Education, Ministry of Education, Thailand. Moreover, the authors are grateful to the reviewers for the careful reading of the paper and for the suggestions which improved the quality of this work.