General Iterative Methods for Equilibrium Problems and Infinitely Many Strict Pseudocontractions in Hilbert Spaces

Abstract

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of fixed point of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by many others.

1. Introduction

Let 𝐻 be a real Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ, where ℝ is the set of real numbers.

The equilibrium problem for 𝐹∶𝐶×𝐶→ℝ is to find 𝑥∈𝐶 such that𝐹(𝑥,𝑦)≥0(1.1) for all 𝑦∈𝐶. The set of such solutions is denoted by EP(𝐹).

A mapping 𝑆 of 𝐶 is said to be a 𝜅-strict pseudocontraction if there exists a constant 𝜅∈[0,1) such that‖𝑆𝑥−𝑆𝑦‖²≤‖𝑥−𝑦‖²+𝜅‖(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦‖²(1.2) for all 𝑥,𝑦∈𝐶; see [1]. We denote the set of fixed points of 𝑆 by 𝐹(𝑆) (i.e., 𝐹(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥}).

Note that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings which are mapping 𝑆on𝐶 such that‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖(1.3) for all 𝑥,𝑦∈𝐶. That is, 𝑆 is nonexpansive if and only if 𝑆 is a 0-strict pseudocontraction.

Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem. Some methods have been proposed to solve the equilibrium problem (1.1); see, for instance, [2–4]. In particular, Combettes and Hirstoaga [5] proposed several methods for solving the equilibrium problem. On the other hand, Mann [6], Shimoji and Takahashi [7] considered iterative schemes for finding a fixed point of a nonexpansive mapping. Further, Acedo and Xu [8] projected new iterative methods for finding a fixed point of strict pseudocontractions.

In 2006, Marino and Xu [3] introduced the general iterative method and proved that the algorithm converged strongly. Recently, Liu [2] considered a general iterative method for equilibrium problems and strict pseudocontractions. Tian [9] proposed a new general iterative algorithm combining an 𝐿-Lipschitzian and 𝜂-strong monotone operator. Very recently, Wang [10] considered a general composite iterative method for infinite family strict pseudocontractions.

In this paper, motivated by the above facts, we introduce two iterative schemes and obtain strong convergence theorems for finding a common element of the set of fixed points of a infinite family of strict pseudocontractions and the set of solutions of the equilibrium problem (1.1).

2. Preliminaries

Throughout this paper, we always write ⇀ for weak convergence and → for strong convergence. We need some facts and tools in a real Hilbert space 𝐻 which are listed as below.

Lemma 2.1. Let 𝐻 be a real Hilbert space. There hold the following identities:(i)‖𝑥−𝑦‖²=‖𝑥‖²−‖𝑦‖²−2⟨𝑥−𝑦,𝑦⟩, ∀𝑥,𝑦∈𝐻;(ii)‖𝑡𝑥+(1−𝑡)𝑦‖²=𝑡‖𝑥‖²+(1−𝑡)‖𝑦‖²−𝑡(1−𝑡)‖𝑥−𝑦‖²,∀𝑡∈[0,1],∀𝑥,𝑦∈𝐻.

Lemma 2.2 (see [11]). Assume that {𝛼_𝑛} is a sequence of nonnegative real numbers such that 𝛼_𝑛+1≤1−𝛾_𝑛𝛼_𝑛+𝛿_𝑛,(2.1) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that(i)∑^∞𝑛=1𝛾_𝑛=∞;(ii)lim_𝑛→∞sup(𝛿_𝑛/𝛾_𝑛)≤0or∑^∞𝑛=1|𝛿_𝑛|<∞.Then, lim_𝑛→∞𝛼_𝑛=0.
Recall that given a nonempty closed convex subset 𝐶 of a real Hilbert space 𝐻, for any 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that ‖‖𝑥−𝑃_𝐶𝑥‖‖≤‖𝑥−𝑦‖(2.2) for all 𝑦∈𝐶. Such a 𝑃_𝐶 is called the metric (or the nearest point) projection of 𝐻 onto 𝐶. As known, 𝑦=𝑃_𝐶𝑥 if and only if there holds the relation: ⟨𝑥−𝑦,𝑦−𝑧⟩≥0∀𝑧∈𝐶.(2.3)

Lemma 2.3 (see [10]). Let 𝐴∶𝐻→𝐻 be a 𝐿-Lipschitzian and 𝜂-strongly monotone operator on a Hilbert space 𝐻 with 𝐿>0, 𝜂>0, 0<𝜇<2𝜂/𝐿², and 0<𝑡<1. Then, 𝑆=(𝐼−𝑡𝜇𝐴)∶𝐻→𝐻 is a contraction with contractive coefficient 1−𝑡𝜏 and 𝜏=(1/2)𝜇(2𝜂−𝜇𝐿²).

Lemma 2.4 (see [1]). Let 𝑆∶𝐶→𝐶 be a 𝜅-strict pseudocontraction. Define 𝑇∶𝐶→𝐶 by 𝑇𝑥=𝜆𝑥+(1−𝜆)𝑆𝑥 for each 𝑥∈𝐶. Then, as 𝜆∈[𝜅,1), 𝑇 is a nonexpansive mapping such that 𝐹(𝑇)=𝐹(𝑆).

Lemma 2.5 (see [9]). Let 𝐻 be a Hilbert space and 𝑓∶𝐻→𝐻 be a contraction with coefficient 0<𝛼<1, and 𝐴∶𝐻→𝐻 an 𝐿-Lipschitzian continuous operator and 𝜂-strongly monotone with 𝐿>0, 𝜂>0. Then for 0<𝛾<𝜇𝜂/𝛼: ⟨𝑥−𝑦,(𝜇𝐴−𝛾𝑓)𝑥−(𝜇𝐴−𝛾𝑓)𝑦⟩≥(𝜇𝜂−𝛾𝛼)‖𝑥−𝑦‖²,𝑥,𝑦∈𝐻.(2.4) That is, 𝜇𝐴−𝛾𝑓 is strongly monotone with coefficient 𝜇𝜂−𝛾𝛼.
Let {𝑆_𝑛} be a sequence of 𝜅_𝑛-strict pseudo-contractions. Define 𝑆^𝑛=𝜃_𝑛𝐼+(1−𝜃_𝑛)𝑆_𝑛,𝜃_𝑛∈[𝜅_𝑛,1). Then, by Lemma 2.4, 𝑆^𝑛 is nonexpansive. In this paper, consider the mapping 𝑊_𝑛 defined by 𝑈𝑈_𝑛,𝑛+1=𝐼,_𝑛,𝑛𝑈=𝑡_𝑛𝑆^𝑛𝑈_𝑛,𝑛+1+1−𝑡_𝑛𝐼,_{𝑛,𝑛−1}𝑈…,_𝑛,𝑖=𝑡_𝑛−1𝑆^𝑛−1𝑈_𝑛,𝑛+1−𝑡_𝑛−1𝐼,𝑈…,_𝑛,2=𝑡_𝑖𝑆^𝑖𝑈_𝑛,𝑖+1+1−𝑡_𝑖𝐼,𝑊=𝑡₂𝑆^2𝑈_𝑛,3+1−𝑡₂𝐼,_𝑛=𝑈_𝑛,1=𝑡₁𝑆^1𝑈_𝑛,2+1−𝑡₁𝐼,(2.5) where 𝑡₁,𝑡₂,… are real numbers such that 0≤𝑡_𝑛<1. Such a mapping 𝑊_𝑛 is called a 𝑊-mapping generated by 𝑆^1,𝑆^2,… and 𝑡₁,𝑡₂,…. It is easy to see 𝑊_𝑛 is nonexpansive.

Lemma 2.6 (see [7]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝐸, let 𝑆^1,𝑆^2,… be nonexpansive mappings of 𝐶 into itself such that ∩^∞𝑖=1𝐹(𝑆^𝑖)≠∅ and let 𝑡₁,𝑡₂,… be real numbers such that 0<𝑡_𝑖≤𝑏<1, for every 𝑖=1,2,…. Then, for any 𝑥∈𝐶 and 𝑘∈𝑁, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists.
Using Lemma 2.6, one can define the mapping 𝑊 of 𝐶 into itself as follows: 𝑊𝑥∶=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥,𝑥∈𝐶.(2.6)

Lemma 2.7 (see [7]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝐸. Let 𝑆^1,𝑆^2,… be nonexpansive mappings of 𝐶 into itself such that ∩^∞𝑖=1𝐹(𝑆^𝑖)≠∅ and let 𝑡₁,𝑡₂,… be real numbers such that 0<𝑡_𝑖≤𝑏<1,for all 𝑖≥1. If 𝐾 is any bounded subset of 𝐶, then lim_𝑛→∞sup_𝑥∈𝐾‖‖𝑊𝑥−𝑊_𝑛𝑥‖‖=0.(2.7)

Lemma 2.8 (see [12]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻,let{𝑆^𝑖∶𝐶→𝐶} be a family of infinite nonexpansive mappings with ∩^∞𝑖=1𝐹(𝑆^𝑖)≠∅, let 𝑡₁,𝑡₂,… be real numbers such that 0<𝑡_𝑖≤𝑏<1, for every 𝑖=1,2,…. Then 𝐹(𝑊)=∩^∞𝑖=1𝐹(𝑆^𝑖).
For solving the equilibrium problem, assume that the bifunction 𝐹 satisfies the following conditions:(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶;(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for any 𝑥,𝑦∈𝐶;(A3) for each 𝑥,𝑦,𝑧∈𝐶,limsup_𝑡→0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦);(A4)𝐹(𝑥,⋅) is convex and lower semicontinuous for each 𝑥∈𝐶.
Recall some lemmas which will be needed in the rest of this paper.

Lemma 2.9 (see [13]). Let 𝐶 be a nonempty closed convex subset of 𝐻, let 𝐹 be bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), and let 𝑟>0 and 𝑥∈𝐻. Then, there exists 𝑧∈𝐶 such that 𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶.(2.8)

Lemma 2.10 (see [5]). For 𝑟>0,𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=𝑧∈𝐶∣𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶(2.9) for all 𝑥∈𝐻. Then, the following statements hold:(i)𝑇_𝑟 is single-valued;(ii)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻,‖‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩;(2.10)(iii)𝐹(𝑇_𝑟)=𝐸𝑃(𝐹); (iv)𝐸𝑃(𝐹) is closed and convex.

Lemma 2.11 (see [14]). Let {𝑥_𝑛} and {𝑧_𝑛} be bounded sequences in a Banach space and let {𝛽_𝑛} be a sequence of real numbers such that 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 for all 𝑛=0,1,2,…. Suppose that 𝑥_𝑛+1=(1−𝛽_𝑛)𝑧_𝑛+𝛽_𝑛𝑥_𝑛 for all 𝑛=0,1,2,… and limsup_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖≤0. Then lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0.

Lemma 2.12 (see [4]). Let 𝐶,𝐻,𝐹, and 𝑇_𝑟𝑥 be as in Lemma 2.10. Then, the following holds: ‖‖𝑇_𝑠𝑥−𝑇_𝑡𝑥‖‖²≤𝑠−𝑡𝑠⟨𝑇^𝑠𝑥−𝑇_𝑡𝑥,𝑇_𝑠𝑥−𝑥⟩(2.11) for all 𝑠,𝑡>0 and 𝑥∈𝐻.

Lemma 2.13 (see [10]). Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻, and 𝑇∶𝐶→𝐶 a nonexpansive mapping with 𝐹(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦.

3. Main Result

Throughout the rest of this paper, we always assume that 𝑓 is a contraction of 𝐻 into itself with coefficient 𝛼∈(0,1), and 𝐴 is a 𝐿-Lipschitzian continuous operator and 𝜂-strongly monotone on 𝐻 with 𝐿>0, 𝜂>0. Assume that 0<𝜇<2𝜂/𝐿² and 0<𝛾<𝜇(𝜂−(𝜇𝐿²/2))/𝛼=𝜏/𝛼.

Define a mapping 𝑉_𝑛=𝛽_𝑛𝐼+(1−𝛽_𝑛)𝑊_𝑛𝑇_𝑟𝑛. Since both 𝑊_𝑛 and 𝑇_𝑟𝑛 are nonexpansive, it is easy to get 𝑉_𝑛 is also nonexpansive. Consider the following mapping 𝐺_𝑛 on 𝐻 defined by𝐺_𝑛𝑥=𝛼_𝑛𝛾𝑓(𝑥)+𝐼−𝛼_𝑛𝜇𝐴𝑉_𝑛𝑥,∀𝑥∈𝐻,𝑛∈𝑁,(3.1) where 𝛼_𝑛∈(0,1). By Lemmas 2.3 and 2.10, we have‖‖𝐺_𝑛𝑥−𝐺_𝑛𝑦‖‖≤≤𝛼𝛼_𝑛𝑛𝛾‖𝑓(𝑥)−𝑓(𝑦)‖+1−𝛼_𝑛𝜏‖‖𝑉_𝑛𝑥−𝑉_𝑛𝑦‖‖=1𝛾−𝛼𝛼‖_𝑛𝑥−𝑦‖+1−𝛼_𝑛𝜏‖𝑥−𝑦‖(𝜏−𝛾𝛼)‖𝑥−𝑦‖.(3.2) Since 0<1−𝛼_𝑛(𝜏−𝛾𝛼)<1, it follows that 𝐺_𝑛 is a contraction. Therefore, by the Banach contraction principle, 𝐺_𝑛 has a unique fixed pointed 𝑥^𝑓𝑛∈𝐻 such that𝑥^𝑓𝑛=𝛼_𝑛𝛾𝑓𝑥^𝑓𝑛+𝐼−𝛼^𝑛𝜇𝐴𝑉_𝑛𝑥^𝑓𝑛.(3.3)

For simplicity, we will write 𝑥_𝑛 for 𝑥^𝑓𝑛 provided no confusion occurs. Next we prove the sequences {𝑥_𝑛} converges strongly to a 𝑥^∗∈Ω=∩^∞𝑖=1𝐹(𝑆_𝑖)∩𝐸𝑃(𝐹) which solves the variational inequality:⟨(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑝−𝑥^∗⟩≤0,∀𝑝∈Ω.(3.4) Equivalently, 𝑥^∗=𝑃_Ω(𝐼−𝜇𝐴+𝛾𝑓)𝑥^∗.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐹 a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). Let 𝑆_𝑖∶𝐶→𝐶 be a family 𝜅_𝑖-strict pseudocontractions for some 0≤𝜅_𝑖<1. Assume the set Ω=∩^∞𝑖=1𝐹(𝑆_𝑖)∩𝐸𝑃(𝐹)≠∅. Let 𝑓 be a contraction of 𝐻 into itself with 𝛼∈(0,1) and let 𝐴 be a 𝐿-Lipschitzian continuous operator and 𝜂-strongly monotone with 𝐿>0,𝜂>0,0<𝜇<2𝜂/𝐿² and 0<𝛾<𝜇(𝜂−(𝜇𝐿²/2))/𝛼=𝜏/𝛼. For every 𝑛∈ℕ, let 𝑊_𝑛 be the mapping generated by 𝑆^𝑖 and 𝑡_𝑖 as in (2.5). Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by the following algorithm: 𝑦_𝑛𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛,𝑥_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑊_𝑛𝑢_𝑛,=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛.(3.5) If {𝛼_𝑛}, {𝛽_𝑛}, and {𝑟_𝑛} satisfy the following conditions:(i){𝛼_𝑛}⊂(0,1), lim_𝑛→∞𝛼_𝑛=0;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1; (iii){𝑟_𝑛}⊂(0,∞), liminf_𝑛→∞𝑟_𝑛>0.Then, {𝑥_𝑛} converges strongly to a point 𝑥^∗∈Ω, which solves the variational inequality (3.4).

Proof. The proof is divided into several steps.
Step 1. Show first that {𝑥_𝑛} is bounded.
Take any 𝑝∈Ω, by (3.5) and Lemma 2.3, we derive that ‖‖𝑥_𝑛−𝑝‖‖=≤𝛼‖‖𝛼_𝑛𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑝+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐴𝑝‖‖≤1𝛼−𝛾‖‖𝛼𝑥_𝑛𝑛−𝑝‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖+1−𝛼_𝑛𝜏‖‖𝑦_𝑛−𝑝‖‖(𝜏−𝛾𝛼)‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖.(3.6) It follows that ‖𝑥_𝑛−𝑝‖≤(‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖)/(𝜏−𝛾𝛼).
Hence, {𝑥_𝑛} is bounded, so are {𝑢_𝑛} and {𝑦_𝑛}. It follows from the Lipschitz continuity of 𝐴 that {𝐴𝑥_𝑛} and {𝐴𝑢_𝑛} are also bounded. From the nonexpansivity of 𝑓 and 𝑊_𝑛, it follows that {𝑓(𝑥_𝑛)} and {𝑊_𝑛𝑥_𝑛} are also bounded.Step 2. Show that lim_𝑛→∞‖‖𝑢_𝑛−𝑥_𝑛‖‖=0,lim_𝑛→∞‖‖𝑢_𝑛−𝑦_𝑛‖‖=0.(3.7) Notice that ‖‖𝑢_𝑛−𝑦_𝑛‖‖≤‖‖𝑢_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑦_𝑛‖‖=‖‖𝑢_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝜇𝐴𝑦_𝑛‖‖.(3.8)
By Lemma 2.10, we have ‖‖𝑢_𝑛−𝑝‖‖²=‖‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑝‖‖²≤⟨𝑥_𝑛−𝑝,𝑢_𝑛−𝑝⟩=12‖‖𝑢^𝑛−𝑝‖‖²+‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.9) It follows that ‖‖𝑢_𝑛−𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.10)
Thus, from Lemma 2.1 and (3.10), we get ‖‖𝑥_𝑛−𝑝‖‖²=‖‖𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑝+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐴𝑝‖‖²≤≤11−−𝛼𝛼_𝑛𝑛𝜏²‖‖𝑦_𝑛−𝑝‖‖²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑝)+𝛾𝑓(𝑝)−𝜇𝐴𝑝,𝑥_𝑛−𝑝≤1−𝛼_𝑛𝜏²‖‖𝑢_𝑛−𝑝‖‖²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑝)+𝛾𝑓(𝑝)−𝜇𝐴𝑝,𝑥_𝑛−𝑝=1−2𝛼𝜏_𝑛²‖‖𝑥^𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²+2𝛼^𝑛𝛾𝛼‖‖𝑥_𝑛−𝑝‖‖²+2𝛼_𝑛‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖‖‖𝑥_𝑛−𝑝‖‖≤‖‖𝑥(𝜏−𝛾𝛼)+𝛼_𝑛𝜏²‖‖𝑥^𝑛−𝑝‖‖²−1−𝛼_𝑛𝜏²‖‖𝑥_𝑛−𝑢_𝑛‖‖²+2𝛼_𝑛‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖‖‖𝑥_𝑛−𝑝_𝑛‖‖−𝑝‖‖²+𝛼_𝑛𝜏²‖‖𝑥_𝑛−𝑝‖‖²−1−𝛼_𝑛𝜏²‖‖𝑥_𝑛−𝑢_𝑛‖‖²+2𝛼_𝑛‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖‖‖𝑥_𝑛−𝑝‖‖.(3.11) It follows that 1−𝛼_𝑛𝜏²‖‖𝑥_𝑛−𝑢_𝑛‖‖²≤𝛼_𝑛𝜏²‖‖𝑥_𝑛−𝑝‖‖²+2𝛼_𝑛‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖‖‖𝑥_𝑛−𝑝‖‖.(3.12) Since 𝛼_𝑛→0, we have lim_𝑛→∞‖‖𝑢_𝑛−𝑥_𝑛‖‖=0.(3.13) From (3.8), it is easy to get lim_𝑛→∞‖‖𝑢_𝑛−𝑦_𝑛‖‖=0.(3.14)Step 3. Show that ‖‖𝑢lim_𝑛→∞‖‖𝑢_𝑛−𝑊𝑢_𝑛‖‖=0,(3.15)_𝑛−𝑊_𝑛𝑢_𝑛‖‖≤‖‖𝑢_𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑊_𝑛𝑢_𝑛‖‖=‖‖𝑢_𝑛−𝑦_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖+‖‖𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖.(3.16) This implies that 1−𝛽_𝑛‖‖𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖≤‖‖𝑢_𝑛−𝑦_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖.(3.17) From condition (ii), (3.13), and (3.14), we have ‖‖𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖⟶0.(3.18) Notice that ‖‖𝑢_𝑛−𝑊𝑢_𝑛‖‖≤‖‖𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖+‖‖𝑊_𝑛𝑢_𝑛−𝑊𝑢_𝑛‖‖.(3.19) By Lemma 2.7 and (3.18), we get (3.15).
Since {𝑢_𝑛} is bounded, so there exists a subsequence {𝑢_𝑛𝑗} which converges weakly to 𝑥^∗.Step 4. Show that 𝑥^∗∈Ω.
Since 𝐶 is closed and convex, 𝐶 is weakly closed. So, we have 𝑥^∗∈𝐶.
From (3.15), we obtain 𝑊𝑢_𝑛𝑗⇀𝑥^∗. From Lemmas 2.8, 2.4, and 2.13, we have 𝑥^∗∈𝐹(𝑊)=∩^∞𝑖=1𝐹(𝑆^𝑖)=∩^∞𝑖=1𝐹(𝑆_𝑖).
By 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, for all 𝑛≥1, we have 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶.(3.20) It follows from (A2) that 1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥𝐹𝑦,𝑢_𝑛,∀𝑦∈𝐶.(3.21) Hence, we get 1𝑟_𝑛𝑗𝑦−𝑢^𝑛_𝑗,𝑢_𝑛𝑗−𝑥_𝑛𝑗≥𝐹𝑦,𝑢^𝑛_𝑗,∀𝑦∈𝐶.(3.22) It follows from condition (iii), (3.13), and (A4) that 0≥𝐹𝑦,𝑥^∗,∀y∈𝐶.(3.23) For 𝑠 with 0<𝑠≤1 and 𝑦∈𝐶, let 𝑦_𝑠=𝑠𝑦+(1−𝑠)𝑥^∗. Since 𝑦∈𝐶 and 𝑥^∗∈𝐶, we obtain 𝑦_𝑠∈𝐶 and hence 𝐹(𝑦_𝑠,𝑥^∗)≤0. So, we have 0=𝑓𝑦_𝑠,𝑦_𝑠≤𝑠𝐹𝑦_𝑠,𝑦+(1−𝑠)𝐹𝑦_𝑠,𝑥^∗≤𝑠𝐹𝑦_𝑠,𝑦.(3.24) Dividing by 𝑠, we get 𝐹𝑦_𝑠,y≥0,∀𝑦∈𝐶.(3.25) Letting 𝑠→0 and from (A3), we get 𝐹𝑥^∗,𝑦≥0(3.26) for all 𝑦∈𝐶and𝑥^∗∈EP(𝐹).Hence𝑥^∗∈Ω.Step 5. Show that 𝑥_𝑛→𝑥^∗,where𝑥^∗=𝑃_Ω(𝐼−𝜇𝐴+𝛾𝑓)𝑥^∗: 𝑥_𝑛−𝑥^∗=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑥^∗+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐴𝑥^∗.(3.27) Hence, we obtain ‖‖𝑥_𝑛−𝑥^∗‖‖²=≤𝛼𝛼_𝑛𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑥^∗,𝑥_𝑛−𝑥^∗+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐴𝑥^∗,𝑥_𝑛−𝑥^∗;𝛾𝑓𝑥_𝑛−𝜇𝐴𝑥^∗,𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝜏‖‖𝑥_𝑛−𝑥^∗‖‖².(3.28) It follows that ‖‖𝑥_𝑛−𝑥^∗‖‖²≤1𝜏𝛾𝑓𝑥^𝑛−𝜇𝐴𝑥^∗=1,𝑥_𝑛−𝑥^∗𝜏𝛾𝑓𝑥^𝑛−𝑓𝑥^∗,𝑥_𝑛≤1−𝑥^∗+𝛾𝑓𝑥^∗−𝜇𝐴𝑥^∗,𝑥_𝑛−𝑥^∗𝜏𝛾𝛼‖‖𝑥^𝑛−𝑥^∗‖‖²+𝛾𝑓𝑥^∗−𝜇𝐴𝑥^∗,𝑥_𝑛−𝑥^∗.(3.29) This implies that ‖‖𝑥_𝑛−𝑥^∗‖‖²≤𝛾𝑓𝑥^∗−𝜇𝐴𝑥^∗,𝑥_𝑛−𝑥^∗𝜏−𝛾𝛼.(3.30) In particular, ‖‖𝑥_𝑛𝑗−𝑥^∗‖‖²≤𝛾𝑓𝑥^∗−𝜇𝐴𝑥^∗,𝑥_𝑛𝑗−𝑥^∗𝜏−𝛾𝛼.(3.31)
Since 𝑥_𝑛𝑗⇀𝑥^∗, it follows from (3.31) that 𝑥_𝑛𝑗→𝑥^∗ as 𝑗→∞. Next, we show that 𝑥^∗ solves the variational inequality (3.4).
By the iterative algorithm (3.5), we have 𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐴𝑉_𝑛𝑥_𝑛.(3.32) Therefore, we have 𝜇𝛼_𝑛𝐴𝑥_𝑛−𝛼_𝑛𝛾𝑓𝑥_𝑛=𝜇𝛼_𝑛𝐴𝑥_𝑛−𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐴𝑉_𝑛𝑥_𝑛,(3.33) that is, (𝜇𝐴−𝛾𝑓)𝑥_𝑛=−1𝛼_𝑛𝐼−𝑉_𝑛𝑥_𝑛−𝜇𝛼_𝑛𝐴𝑥_𝑛−𝐴𝑉_𝑛𝑥_𝑛.(3.34) Hence, for 𝑝∈Ω, ⟨(𝜇𝐴−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑝⟩=−1𝛼_𝑛𝐼−𝑉_𝑛𝑥_𝑛=−1−𝜇𝛼_𝑛𝐴𝑥_𝑛−𝐴𝑉_𝑛𝑥_𝑛,𝑥_𝑛−𝑝𝛼_𝑛𝐼−𝑉_𝑛𝑥_𝑛≤𝜇⟨𝐴𝑥_𝑛−𝐼−𝑉_𝑛𝑝,𝑥_𝑛−𝑝+𝜇⟨𝐴𝑥_𝑛−𝐴𝑉_𝑛𝑥_𝑛,𝑥_𝑛−𝑝⟩−𝐴𝑉_𝑛𝑥_𝑛,𝑥_𝑛−𝑝⟩.(3.35)
Since 𝐼−𝑉_𝑛 is monotone (i.e., ⟨𝑥−𝑦,(𝐼−𝑉_𝑛)𝑥−(𝐼−𝑉_𝑛)𝑦⟩≥0, for all 𝑥,𝑦∈𝐻). This is due to the nonexpansivity of 𝑉_𝑛.
Now replacing 𝑛 in (3.35) with 𝑛_𝑗 and letting 𝑗→∞, we obtain ⟨(𝜇𝐴−𝛾𝑓)𝑥^∗,𝑥^∗−𝑝⟩=lim_𝑗→∞(𝜇𝐴−𝛾𝑓)𝑥^𝑛_𝑗,𝑥_𝑛𝑗−𝑝≤lim_𝑗→∞𝜇𝐴𝑥_𝑛𝑗−𝐴𝑉_𝑛𝑥_𝑛𝑗,𝑥_𝑛𝑗−𝑝=0.(3.36)
That is, 𝑥^∗∈Ω is a solution of (3.4). To show that the sequence {𝑥_𝑛} converges strongly to 𝑥^∗, we assume that 𝑥_𝑛𝑘→𝑥̂. By the same processing as the proof above, we derive 𝑥̂∈Ω. Moreover, it follows from the inequality (3.36) that ⟨(𝜇𝐴−𝛾𝑓)𝑥^∗,𝑥^∗−𝑥̂⟩≤0.(3.37) Interchanging 𝑥^∗ and 𝑥̂, we get ⟨(𝜇𝐴−𝛾𝑓)𝑥̂,𝑥̂−𝑥^∗⟩≤0.(3.38) By Lemma 2.5, adding up (3.37) and (3.38) yields (𝜇𝜂−𝛾𝛼)‖𝑥^∗−𝑥̂‖²≤⟨(𝜇𝐴−𝛾𝑓)𝑥^∗−(𝜇𝐴−𝛾𝑓)𝑥̂,𝑥^∗−𝑥̂⟩≤0.(3.39) Hence 𝑥^∗=𝑥̂ and, therefore, 𝑥_𝑛→𝑥^∗ as 𝑛→∞, ⟨(𝐼−𝜇𝐴+𝛾𝑓)𝑥^∗−𝑥^∗,𝑥^∗−𝑝⟩≥0,∀𝑝∈Ω.(3.40) This is equivalent to the fixed point equation: 𝑃_Ω(𝐼−𝜇𝐴+𝛾𝑓)𝑥^∗=𝑥^∗.(3.41)

Theorem 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐹 a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). Let 𝑆_𝑖∶𝐶→𝐶 be a family 𝜅_𝑖-strict pseudocontractions for some 0≤𝜅_𝑖<1. Assume the set Ω=∩^∞𝑖=1𝐹(𝑆_𝑖)∩𝐸𝑃(𝐹)≠∅. Let 𝑓 be a contraction of 𝐻 into itself with 𝛼∈(0,1) and let 𝐴 be a 𝐿-Lipschitzian continuous operator and 𝜂-strongly monotone with 𝐿>0, 𝜂>0,0<𝜇<2𝜂/𝐿², and 0<𝛾<𝜇(𝜂−(𝜇𝐿²/2))/𝛼=𝜏/𝛼. For every 𝑛∈ℕ, let 𝑊_𝑛 be the mapping generated by 𝑆^𝑖 and 0<𝑡_𝑖≤𝑏<1. Given 𝑥₁∈𝐻, let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by the following algorithm: 𝑦𝑢_𝑛𝑛=𝑇_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑊_𝑛𝑢_𝑛,=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛.(3.42) If {𝛼_𝑛}, {𝛽_𝑛} and {𝑟_𝑛} satisfy the following conditions:(i){𝛼_𝑛}⊂(0,1), lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1; (iii){𝑟_𝑛}⊂(0,∞), liminf_𝑛→∞𝑟_𝑛>0 and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Ω, which solves the variational inequality (3.4).

Proof. The proof is divided into several steps.
Step 1. Show first that {𝑥_𝑛} is bounded.
Taking any 𝑝∈Ω, we have ‖‖𝑥_𝑛+1−𝑝‖‖=≤‖‖𝛼𝛼_𝑛𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑝+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐴𝑝‖‖≤𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑝)‖‖+‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖+1−𝛼_𝑛𝜏‖‖𝑦_𝑛−𝑝‖‖=1𝛼−𝛾𝛼‖‖_𝑛𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖+1−𝛼_𝑛𝜏‖‖𝑦_𝑛−𝑝‖‖(𝜏−𝛼𝛾)‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛(𝜏−𝛼𝛾)‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖≤max‖‖𝑥_𝑛𝜏−𝛼𝛾−𝑝‖‖,‖𝛾𝑓(𝑝)−𝜇𝐴𝑝‖𝜏−𝛼𝛾.(3.43) By induction, we obtain ‖𝑥_𝑛−𝑝‖≤max{‖𝑥₁−𝑝‖,‖𝛾𝑓(𝑝)−𝜇𝐴𝑝)‖/(𝜏−𝛼𝛾)},𝑛≥1.Hence, {𝑥_𝑛} is bounded, so are {𝑢_𝑛} and {𝑦_𝑛}. It follows from the Lipschitz continuity of 𝐴 that {𝐴𝑥_𝑛} and {𝐴𝑢_𝑛} are also bounded. From the nonexpansivity of 𝑓 and 𝑊_𝑛, it follows that {𝑓(𝑥_𝑛)} and {𝑊_𝑛𝑥_𝑛} are also bounded.Step 2. Show that ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖⟶0.(3.44)
Observe that ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖=≤‖‖‖‖𝑇𝑇_{𝑟𝑛+1𝑟𝑛+1}𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛‖‖≤‖‖𝑥_𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛+1𝑥_𝑛‖‖+‖‖𝑇_𝑟𝑛+1𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖‖−𝑥_𝑛‖‖+‖‖𝑇_𝑟𝑛+1𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖‖,(3.45) and from (2.5), we have ‖‖𝑊_𝑛+1𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖=≤𝑡‖‖₁𝑡₁𝑆^1𝑈_𝑛+1,2𝑢_𝑛−𝑡₁𝑆^1𝑈_𝑛,2𝑢_𝑛‖‖=𝑡₁‖‖𝑈_𝑛+1,2𝑢_𝑛−𝑈_𝑛,2𝑢_𝑛‖‖≤𝑡₁‖‖𝑡₂𝑆^2𝑈_𝑛+1,3𝑢_𝑛−𝑡₂𝑆^2𝑈_𝑛,3𝑢_𝑛‖‖≤≤⋯_𝑛𝑡₂‖‖𝑈_𝑛+1,3𝑢_𝑛−𝑈_𝑛,3𝑢_𝑛‖‖_𝑖=1≤𝑀𝑡_𝑖1𝑛‖‖𝑈_{𝑛+1,𝑛+1}𝑢_𝑛−𝑈_𝑛,𝑛+1𝑢_𝑛‖‖_𝑖=1𝑡_𝑖,(3.46) where 𝑀₁=sup_𝑛{‖𝑈_{𝑛+1,𝑛+1}𝑢_𝑛−𝑈_𝑛,𝑛+1𝑢_𝑛‖}.
Suppose 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑧_𝑛, then 𝑧_𝑛=(𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛)/(1−𝛽_𝑛)=(𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝜇𝛼_𝑛𝐴)𝑦_𝑛−𝛽_𝑛𝑥_𝑛)/(1−𝛽_𝑛).
Hence, we have 𝑧_𝑛+1−𝑧_𝑛=𝛼^𝑛+1𝛾𝑓𝑥_𝑛+1+𝐼−𝜇𝛼_𝑛+1𝐴𝑦_𝑛+1−𝛽_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝛼^𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼^𝑛+1𝛾𝑓𝑥_𝑛+1−𝜇𝐴𝑦_𝑛+11−𝛽_𝑛+1+𝑦^𝑛+1−𝛽_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝛼^𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑦_𝑛1−𝛽_𝑛−𝑦^𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼^𝑛+1𝛾𝑓𝑥_𝑛+1−𝜇𝐴𝑦_𝑛+11−𝛽_𝑛+1+𝛽^𝑛+1𝑥_𝑛+1+1−𝛽_𝑛+1𝑊_𝑛+1𝑢_𝑛+1−𝛽_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝛼^𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑦_𝑛1−𝛽_𝑛−𝛽^𝑛𝑥_𝑛+1−𝛽_𝑛𝑊_𝑛𝑢_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛≤𝛼^𝑛+1𝛾𝑓𝑥_𝑛+1−𝜇𝐴𝑦_𝑛+11−𝛽_𝑛+1−𝛼^𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑦_𝑛1−𝛽_𝑛+𝑊_𝑛+1𝑢_𝑛+1−𝑊_𝑛𝑢_𝑛.(3.47)
It follows from (3.45), (3.46), and the above result that ‖‖𝑧≤_𝑛+1−𝛼𝑧_𝑛𝑛+1‖‖1−𝛽_𝑛+1‖‖𝛾𝑓𝑥_𝑛+1‖‖+‖‖𝜇𝐴𝑦_𝑛+1‖‖+𝛼^𝑛1−𝛽_𝑛‖‖𝛾𝑓𝑥_𝑛‖‖+‖‖𝜇𝐴𝑦_𝑛≤𝛼_𝑛+1‖‖+‖‖𝑊_𝑛+1𝑢_𝑛+1−𝑊_𝑛𝑢_𝑛‖‖1−𝛽_𝑛+1+𝛼^𝑛1−𝛽_𝑛𝑀²≤𝛼_𝑛+1+‖‖𝑊_𝑛+1𝑢_𝑛+1−𝑊_𝑛+1𝑢_𝑛‖‖+‖‖𝑊_𝑛+1𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖1−𝛽_𝑛+1+𝛼^𝑛1−𝛽_𝑛𝑀²≤‖‖𝑥_𝑛+1+‖‖𝑢_𝑛+1−𝑢_𝑛‖‖+‖‖𝑊_𝑛+1𝑢_𝑛−𝑊_𝑛𝑢_𝑛‖‖−𝑥_𝑛‖‖+‖‖𝑇_𝑟𝑛+1𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖‖+𝛼^𝑛+11−𝛽_𝑛+1+𝛼^𝑛1−𝛽_𝑛𝑀²+𝑀_1𝑛_𝑖=1𝑡_𝑖,(3.48) where 𝑀₂=sup_𝑛{‖𝛾𝑓(𝑥_𝑛)‖+‖𝜇𝐴𝑦_𝑛‖}. Hence, we get ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤‖‖𝑇_𝑟𝑛+1𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖‖+𝛼^𝑛+11−𝛽_𝑛+1+𝛼^𝑛1−𝛽_𝑛𝑀²+𝑀_1𝑛_𝑖=1𝑡_𝑖.(3.49) From condition (i), (iii), 0<𝑡_𝑛≤𝑏<1, and Lemma 2.12, we obtain limsup_𝑛→∞‖‖𝑧_𝑛+1−𝑧_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.50) By Lemma 2.11,we have lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0. Thus, lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞1−𝛽_𝑛‖‖𝑧_𝑛−𝑥_𝑛‖‖=0.(3.51)
By Lemma 2.12, (3.45) and (3.44), we obtain ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖⟶0.(3.52)Step 3. Show that ‖‖𝑥_𝑛−𝑊𝑥_𝑛‖‖⟶0.(3.53) Observe that ‖‖‖‖𝑥𝑥_𝑛𝑛−𝑊_𝑛𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑊_𝑛𝑢_𝑛‖‖+‖‖𝑊_𝑛𝑢_𝑛−𝑊_𝑛𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑊_𝑛𝑢_𝑛‖‖+‖‖𝑢_𝑛−𝑥_𝑛‖‖,−𝑊_𝑛𝑢_𝑛‖‖+≤‖‖‖‖𝑥𝑥_𝑛𝑛+1−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑊_𝑛𝑢_𝑛‖‖=‖‖𝑥_𝑛−𝑥_𝑛+1‖‖−𝑦_𝑛‖‖+𝛽_𝑛‖‖𝑢_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑊_𝑛𝑢_𝑛‖‖.(3.54) From condition (i) and (3.5),we can obtain 1−𝛽_𝑛‖‖𝑥_𝑛−𝑊_𝑛𝑢≤_𝑛‖‖‖‖≤𝑥_𝑛‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑦_𝑛‖‖+𝛽_𝑛‖‖𝑢_𝑛−𝑥_𝑛‖‖−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝜇𝐴𝑦_𝑛‖‖+𝛽_𝑛‖‖𝑢_𝑛−𝑥_𝑛‖‖.(3.55)
By Lemma 2.10, we get ‖‖𝑢_𝑛−𝑝‖‖²=‖‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑝‖‖²≤𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑝,𝑥_𝑛−𝑝=12‖‖𝑢^𝑛−𝑝‖‖²+‖‖𝑥_𝑛−𝑝‖‖²+‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.56) This implies that ‖‖𝑢_𝑛−𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.57) By nonexpansivity of 𝑊_𝑛, we have ‖‖𝑦_𝑛−𝑝‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛽_𝑛‖‖𝑢_𝑛−𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²−1−𝛽_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.58) It follows from (3.42) that ‖‖𝑥_𝑛+1−𝑝‖‖²=‖‖𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑝+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐴𝑝+𝛼_𝑛(𝑝−𝜇𝐴𝑝)‖‖²≤𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝑝‖‖²+1−𝛼_𝑛𝜏‖‖𝑦_𝑛−𝑝‖‖²+𝛼_𝑛‖𝑝−𝜇𝐴𝑝‖²≤𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝑝‖‖²+1−𝛼_𝑛𝜏‖‖𝑥_𝑛−𝑝‖‖²−1−𝛽_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²+𝛼^𝑛‖𝑝−𝜇𝐴𝑝‖²≤𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝑝‖‖²+‖‖𝑥_𝑛−𝑝‖‖²−1−𝛽_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²+𝛼_𝑛‖𝑝−𝜇𝐴𝑝‖².(3.59) This implies that 1−𝛽_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²≤𝛼_𝑛‖‖𝛾𝑓𝑥^𝑛−𝑝‖‖²+‖𝑝−𝜇𝐴𝑝‖²+‖‖𝑥^𝑛−𝑝‖‖²−‖‖𝑥_𝑛+1−𝑝‖‖²≤𝛼_𝑛‖‖𝛾𝑓𝑥^𝑛−𝑝‖‖²+‖𝑝−𝜇𝐴𝑝‖²+‖‖𝑥^𝑛−𝑝‖‖+‖‖𝑥_𝑛+1−𝑝‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.60) From condition (i), (ii), and (3.44), we have ‖‖𝑥_𝑛−𝑢_𝑛‖‖⟶0.(3.61) Further we have ‖𝑥_𝑛−𝑊_𝑛𝑢_𝑛‖→0. Thus we get ‖‖𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖‖⟶0.(3.62)
On the other hand, we have ‖‖𝑥_𝑛−𝑊𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖‖+‖‖𝑊_𝑛𝑥_𝑛−𝑊𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖‖+sup_{𝑥𝑛∈𝐶}‖‖𝑊_𝑛𝑥_𝑛−𝑊𝑥_𝑛‖‖.(3.63) Combining (3.62), the last inequality, and Lemma 2.7, we obtain (3.53).Step 4. Show that limsup_𝑛→∞⟨(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0,(3.64) where 𝑥^∗=𝑃_Ω(𝐼−𝜇𝐴+𝛾𝑓)𝑥^∗ is a unique solution of the variational inequality (3.4). Indeed, take a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that limsup_𝑛→∞⟨(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛−𝑥^∗⟩=lim_𝑗→∞(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛𝑗−𝑥^∗.(3.65)
Since {𝑥_𝑛𝑗} is bounded, there exists a subsequence {𝑥_𝑛𝑗𝑘} of {𝑥_𝑛𝑗} which converges weakly to 𝑞. Without loss of generality, we can assume 𝑥_𝑛𝑗⇀𝑞. From (3.53), we obtain 𝑊𝑥_𝑛𝑗⇀𝑞.
By the same argument as in the proof of Theorem 3.1, we have 𝑞∈Ω. Since 𝑥^∗=𝑃_Ω(𝐼−𝜇𝐴+𝛾𝑓)𝑥^∗, it follows that limsup_𝑛→∞⟨(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛−𝑥^∗⟩=lim_𝑗→∞(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛𝑗−𝑥^∗=⟨(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑞−𝑥^∗⟩≤0.(3.66)Step 5. Show that 𝑥_𝑛⟶𝑥^∗.(3.67) Since (𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗≤=‖((𝛾𝛾𝑓𝑓−−𝜇𝜇𝐴𝐴))𝑥𝑥^∗_∗,𝑥_𝑛+1−𝑥_𝑛+⟨(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛−𝑥^∗⟩‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+⟨(𝛾f−𝜇𝐴)𝑥^∗,𝑥_𝑛−𝑥^∗⟩.(3.68) It follows from (3.44) and (3.66) that ‖‖li𝑥msup_𝑛→∞(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗≤0._𝑛+1−𝑥^∗‖‖²=‖‖𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛−𝑥^∗‖‖²=‖‖𝐼−𝜇𝛼_𝑛𝐴𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐴𝑥^∗+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑥^∗‖‖²≤≤‖‖1𝐼−−𝛼_𝑛𝜇𝛼_𝑛𝐴𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐴𝑥^∗‖‖²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜏²‖‖𝑦_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓𝑥^∗,𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗𝜏²‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛼_𝑛𝛼𝛾‖‖𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛+1−𝑥^∗‖‖²+2𝛼^𝑛(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗.(3.69) This implies that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−𝛼^𝑛𝜏²+𝛼_𝑛𝛼𝛾1−𝛼_𝑛𝛼𝛾‖‖𝑥^𝑛−𝑥^∗‖‖²+2𝛼^𝑛1−𝛼_𝑛𝛼𝛾(𝛾𝑓−𝜇𝐴)𝑥^∗≤1−2𝛼_𝑛,𝑥_𝑛+1−𝑥^∗(𝜏−𝛼𝛾)1−𝛼_𝑛𝛼𝛾‖‖𝑥^𝑛−𝑥^∗‖‖²+2𝛼^𝑛1−𝛼_𝑛𝛼𝛾(𝛾𝑓−𝜇𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛼^𝑛𝜏²1−𝛼_𝑛𝛼𝛾𝑀³,(3.70) where 𝑀₃=sup_𝑛‖𝑥_𝑛−𝑥^∗‖², 𝑛≥1. It is easily to see that 𝛾_𝑛=2𝛼_𝑛(𝜏−𝛼𝛾)/(1−𝛼_𝑛𝛼𝛾). Hence, by Lemma 2.2, the sequence {𝑥_𝑛} converges strongly to 𝑥^∗.

Remark 3.3. If 𝐹≡0, then Theorem 3.2 reduces to Theorem 3.1 of Wang [10].

Acknowledgments

The authors would like to thank the referee for valuable suggestions to improve the manuscript NSFC Tianyuan Youth Foundation of Mathematics of China (no. 11126136), and the Fundamental Research Funds for the Central Universities (GRANT: ZXH2011C002).