Viscosity Approximation Methods for Equilibrium Problems, Variational Inequality Problems of Infinitely Strict Pseudocontractions in Hilbert Spaces

Abstract

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of the solutions of the equilibrium problem and the set of fixed points of infinitely strict pseudocontractive mappings. Strong convergence theorems are established in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

1. Introduction

Let 𝐻 be a real Hilbert space and let 𝐶 be a nonempty convex subset of 𝐻.

A mapping 𝑆 of 𝐶 is said to be a 𝜅-strict pseudocontraction if there exists a constant 𝜅∈[0,1) such that 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+𝜅(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦², (1.1) for all 𝑥,𝑦∈𝐶; see [1]. We denote the set of fixed points 𝑆 by 𝐹(𝑆) (i.e., 𝐹(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥}).

Note that the class of strict pseudocontraction strictly includes the class of nonexpansive mappings which are mappings 𝑆 on 𝐶 such that 𝑆𝑥−𝑆𝑦≤𝑥−𝑦, (1.2) for all 𝑥,𝑦∈𝐶. That is, 𝑆 is nonexpansive if and only if 𝑆 is a 0-strict pseudocontraction. Let Φ be a bifunction from 𝐶×𝐶 to ℝ, where ℝ is the set of real numbers. The equilibrium problem for Φ∶𝐶×𝐶→ℝ is to find 𝑥∈𝐶 such that Φ𝑥,𝑦≥0,∀𝑦∈𝐶. (1.3) The set of solutions of (1.3) is denoted by 𝐸𝑃(Φ). Given a mapping 𝐵∶𝐶→𝐻, let Φ(𝑥,𝑦)=⟨𝐵𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. Then the classical variational inequality problem is to find 𝑥∈𝐶 such that ⟨𝐵𝑥,𝑦−𝑥⟩≥0. We denote the solution of the variational inequality by VI(𝐶,𝐵); that is VI(𝐶,𝐵)=𝑥∈𝐶∶𝐵𝑥,𝑦−𝑥≥0. (1.4) Let 𝐴 be a strongly positive linear-bounded operator on 𝐻 if there is a constant 𝛾>0 with property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (1.5) A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space 𝐻: min_𝑥∈𝐸12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩, (1.6) where 𝐴 is a linear-bounded operator, 𝐸 is the fixed point set of a nonexpansive mapping 𝑆 on 𝐻, and 𝑏 is a given point in 𝐻. The problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see [1–11]. In particular, Combettes and Hirstoaga [4] proposed several methods for solving the equilibrium problem. On the other hand, Mann [6], Shimoji and Takahashi [8] considered iterative schemes for finding a fixed point of a nonexpansive mapping. Further, Acedo and Xu [12] projected new iterative methods for finding a fixed point of strict pseudocontractions.

In 2006, Marino and Xu [7] introduced the general iterative method: for 𝑥₁=𝑥∈𝐶, 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,𝑛≥1. (1.7) They proved that the sequence {𝑥_𝑛} of parameters satisfies appropriate condition and that the sequence {𝑥_𝑛} generated by (1.7) converges strongly to the unique solution of the variational inequality ⟨(𝛾𝑓−𝐴)𝑞,𝑝−𝑞⟩≤0,𝑝∈𝐹(𝑇). Recently, Liu [5] considered a general iterative method for equilibrium problems and strict pseudocontractions: Φ𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛽_𝑛𝑢_𝑛+1−𝛽_𝑛𝑆𝑢_𝑛,𝑥_𝑛+1=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜀_𝑛𝐴𝑢_𝑛,∀𝑛≥1, (1.8) where 𝑆 is a 𝑘-strict pseudocondition mapping and {𝜀_𝑛}, {𝛽_𝑛} are sequences in (0,1). They proved that under certain appropriate conditions over {𝜀_𝑛}, {𝛽_𝑛}, and {𝑟_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} both converge strongly to some 𝑞∈𝐹(𝑆)⋂𝐸𝑃(Φ), which solves some variational inequality problems. Tian [10] proposed a new general iterative algorithm: for nonexpansive mapping 𝑇∶𝐻→𝐻 with 𝐹(𝑇)≠𝜙, 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛,∀𝑛≥1, (1.9) where 𝐹 is a 𝑘-Lipschitzian and 𝜂-strong monotone operator. He obtained that the sequence 𝑥_𝑛 generated by (1.9) converges to a point 𝑞 in 𝐹(𝑇), which is the unique solution of the variational inequality ⟨(𝛾𝑓−𝐴)𝑞,𝑝−𝑞⟩≤0,𝑝∈𝐹(𝑇). Very recently, Wang [13] considered a general composite iterative method for infinite family strict pseudocontractions: for 𝑥₁=𝑥∈𝐶, 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑊_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑦_𝑛,∀𝑛≥1, (1.10) where 𝑊_𝑛 is a mapping defined by (2.5), 𝐹 is a 𝑘-Lipschitzian, and 𝜂-strongly monotone operator. With some appropriate condition, the sequence {𝑥_𝑛} generated by (1.10) converges strongly to a common element of the fixed point of an infinite family of 𝜆_𝑖-strictly pseudocontractive mapping, which is a unique solution of the variational inequality ⟨(𝛾𝑓−𝐴)𝑞,𝑝−𝑞⟩≤0,𝑝∈𝐹(𝑇). Kumam proposed many algorithms for the equilibrium and the fixed point problems with 𝑘-strict pseudoconditions; see [14–16]. In particular, in 2011, Kumam and Jaiboon [14] considered a system of mixed equilibrium problems, variational inequality problems, and strict pseudocontractive mappings: 𝑥₁∈𝐸,𝑢_𝑛∈𝐸,𝑣_𝑛𝑢∈𝐸,_𝑛=𝑇^𝜙1_𝑟^,𝜑𝑥_𝑛,𝑣_𝑛=𝑇^𝜙2_𝑠^,𝜑𝑥_𝑛,𝑧_𝑛=𝑃_𝐸𝑢_𝑛−𝜇_𝑛𝐶𝑢_𝑛,𝑦_𝑛=𝑃_𝐸𝑣_𝑛−𝜆_𝑛𝐶𝑣_𝑛,𝑘_𝑛=𝑎_𝑛𝑆_𝑘𝑥_𝑛+𝑏_𝑛𝑦_𝑛+𝑐_𝑛𝑧_𝑛,𝑥_𝑛+1=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑘_𝑛,∀𝑛≥1, (1.11) where 𝑆 is a 𝑘-strict pseudocondition mapping. They proved that under certain appropriate conditions over {𝜀_𝑛}, {𝛽_𝑛}, {𝑟_𝑛}, {𝑎_𝑛}, {𝑏_𝑛}, {𝑐_𝑛}, {𝜆_𝑛}, and {𝜇_𝑛}, the sequence {𝑥_𝑛} converges strongly to a point 𝑞∈Θ which is the unique solution of the variational inequality ⟨(𝐴−𝛾𝑓)𝑞,𝑥−𝑞⟩≥0. Inprasit [17] proposed a viscosity approximation methods to solving the generalized equilibrium and fixed point problems of finite family of nonexpansive mapping in Hilbert spaces.

In this paper, motivated by the above facts, we use the viscosity approximation method to find a common element of the set of solutions of the equilibrium problem VI(𝐶,𝐵) and the set of fixed points of a infinite family of strict pseudocontractions.

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 [18]). Assume that {𝛼_𝑛} is a sequence of nonnegative real numbers such that 𝛼_𝑛+1≤1−𝜌_𝑛𝛼_𝑛+𝜎_𝑛, (2.1) where {𝜌_𝑛} is a sequence in (0,1) and {𝜎_𝑛} is a sequence such that(i)∑^∞_𝑛=1𝜌_𝑛=∞, (ii)limsup_𝑛→∞(𝜎_𝑛/𝜌_𝑛)≤0 or ∑^∞_𝑛=1|𝜎_𝑛|<∞.
Then lim_𝑛→∞𝑎_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 we all know, 𝑦=𝑃_𝐶𝑥 if and only if there holds the relation: 𝑥−𝑦,𝑦−𝑧≥0∀𝑧∈𝐶. (2.3)

Lemma 2.3 (see [13]). Let 𝐴∶𝐻→𝐻 be an 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 [10]). Let 𝐻 be a Hilbert space and 𝑓∶𝐻→𝐻 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−𝛾𝑓𝑥−𝜇𝐴−𝛾𝑓𝑦≥𝜇𝜂−𝛾𝛼𝑥−𝑦²,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-concontractions. 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, we consider the mapping 𝑊_𝑛 defined by 𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝑡_𝑛𝑆^′_𝑛𝑈_𝑛,𝑛+1+1−𝑡_𝑛𝑈𝐼,_{𝑛,𝑛−1}=𝑡_𝑛−1𝑆^′_𝑛−1𝑈_𝑛,𝑛+1−𝑡_𝑛−1𝑈𝐼,⋯,_𝑛,𝑖=𝑡_𝑖𝑆^′_𝑖𝑈_𝑛,𝑖+1+1−𝑡_𝑖𝑈𝐼,⋯,_𝑛,2=𝑡₂𝑆^′₂𝑈_𝑛,3+1−𝑡₂𝑊𝐼,_𝑛=𝑈_𝑛,1=𝑡₁𝑆^′₁𝑈_𝑛,2+1−𝑡₁𝐼. (2.5)

Lemma 2.6 (see [8]). 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 [8]). 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∈𝐾𝑊𝑥−𝑊_𝑛𝑥=0. (2.7)

Lemma 2.8 (see [3]). 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)≠∅, and 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, let us assume that the bifunction Φ satisfies the following conditions:(A1)Φ(𝑥,𝑥)=0 for all 𝑥∈𝐶;(A2)Φ is monotone; that is Φ(𝑥,𝑦)+Φ(𝑦,𝑥)≤0 for any 𝑥,𝑦∈𝐶;(A3)for each 𝑥,𝑦,𝑧∈𝐶,limsup_𝑡→0Φ(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤Φ(𝑥,𝑦);(A4)Φ(𝑥,⋅) is convex and lower semicontinuous for each 𝑥∈𝐶.

We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.9 (see [2]). Let 𝐶 be a nonempty closed convex subset of 𝐻, let Φ be bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), and let 𝑟>0 and 𝑥∈𝐻. Then there exists 𝑧∈𝐶 such that Φ+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (2.8)

Lemma 2.10 (see [4]). Let 𝜙 be a bifunction from 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Then, for any 𝑟>0 and 𝑥∈𝐻, there exists 𝑧∈𝐶 such that Φ+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (2.9) Further, if 𝑇_𝑟𝑥={𝑧∈𝐶;Φ(𝑧,𝑦)+(1/𝑟)⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶}, then the following hold:(1)𝑇_𝑟 is single-valued;(2)𝑇_𝑟 is firmly nonexpansive;(3)𝐹(𝑇_𝑟)=𝐸𝑃(𝜙);(4)𝐸𝑃(𝜙) is closed and convex.

Lemma 2.11 (see [9]). 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 [11]). Let 𝐶,𝐻,𝐹, and 𝑇_𝑟𝑥 be as in Lemma 2.9. Then the following holds: 𝑇_𝑠𝑥−𝑇_𝑡𝑥²≤𝑇_𝑠𝑥−𝑇_𝑡𝑥,𝑇_𝑠𝑥−𝑥, (2.10) for all 𝑠,𝑡>0, and 𝑥∈𝐻.

Lemma 2.13 (see [13]). 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 Results

Now we start and prove our main result of this paper.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝜙 be a bifunction from 𝐶×𝐶→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=VI(𝐶,𝐵)∩^∞_𝑖=1𝐹(𝑆_𝑖)∩𝐸𝑃(𝜙)≠∅. Let 𝑓 be a contraction of 𝐻 into itself with 𝛼∈(0,1), and let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and 0<𝛾<𝛾/𝛾. Let 𝐵∶𝐶→𝐻 be an 𝜉-inverse strongly monotone mapping. Let 𝑊_𝑛 be the mapping generated by 𝑆^′_𝑖 and 𝑡_𝑖 as in (2.5). Let {𝑥_𝑛} be a sequence generated by the following algorithm: 𝜙𝑧_𝑛+1,𝑦𝜆_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑦≥0,_𝑛=𝑃_𝐶𝐼−𝜇_𝑛𝐵𝑧_𝑛,𝐾_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑊_𝑛𝑦_𝑛,𝑥_𝑛+1=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛, (3.1) where {𝜀_𝑛}, {𝛽_𝑛}, {𝛼_𝑛}, and {𝜆_𝑛} are sequences in (0,1). Assume that the control sequences satisfy the following restrictions:(i)lim_𝑛→∞𝜀_𝑛=0 and Σ^∞_𝑛=1𝜀_𝑛=∞;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1; (iii)0<lim_𝑛→∞(𝜆_𝑛/𝜆_𝑛+1)=1; (iv)lim_𝑛→∞|𝛼_𝑛+1−𝛼_𝑛|=0; (v)0<𝜇_𝑛≤2𝜉; (vi)lim_𝑛→∞𝛼_𝑛=𝑎.
Then {𝑥_𝑛} converges strongly to 𝑞∈Ω which is the unique solution of the variational inequality 𝐴−𝛾𝑓𝑞,𝑥−𝑞≥0,∀𝑥∈Ω, (3.2) or equivalent 𝑞=𝑃_Ω(𝐼−𝐴+𝛾𝑓)(𝑞), where 𝑃 is a metric projection mapping form 𝐻 onto Ω.

Proof. Since 𝜀_𝑛→0, as 𝑛→∞, we may assume, without loss of generality, that 𝜀_𝑛≤(1−𝛽_𝑛)‖𝐴‖⁻¹ for all 𝑛∈ℕ. By Lemma 2.3, 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 bounded linear operator on 𝐻, we have ‖𝐴‖=sup{|⟨𝐴𝑥,𝑥⟩|∶𝑥⊆𝐻,‖𝑥‖=1}. (3.3) Observe that 1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑥,𝑥=1−𝛽_𝑛‖𝑥‖²−𝜀_𝑛⟨⟩≥𝐴𝑥,𝑥1−𝛽_𝑛‖𝑥‖²−𝜀_𝑛‖𝐴‖≥0. (3.4) So this shows that (1−𝛽_𝑛)𝐼−𝜀_𝑛𝐴 is positive. It follows that 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. (3.5)
Step 1. We claim that the mapping 𝑃_Ω(𝐼−𝐴+𝛾𝑓) where Ω=⋂^∞_𝑖=1𝐹(𝑆_𝑖)⋂𝐸𝑃(Φ) has a unique fixed point. Let 𝑓 be a contraction of 𝐻 into itself with 𝛼∈(0,1). Then, we have 𝑃_𝐶𝐼−𝐴+𝛾𝑓(𝑥)−𝑃_𝐶𝑦≤(𝑥)−𝑦≤‖𝑓(𝑥)𝑦≤𝐼−𝐴+𝛾𝑓𝐼−𝐴+𝛾𝑓I−𝐴+𝛾𝑓𝐼−𝐴‖𝑥−𝑦+𝛾−𝑓1−𝛾=𝑥−𝑦+𝛾𝛼𝑥−𝑦1−𝛾−𝛾𝛼𝑥−𝑦, (3.6) for all 𝑥,𝑦∈𝐻. Since 0<1−(𝛾−𝛾𝛼)<1, it follows that 𝑃_Ω(𝐼−𝐴+𝛾𝑓) is a contraction of 𝐻 into itself. Therefore the Banach contraction mapping principle implies that there exists a unique element 𝑞∈𝐻 such that 𝑞=𝑃_Ω(𝐼−𝐴+𝛾𝑓)(𝑞).
Step 2. We shall show that (𝐼−𝜇_𝑛𝐵) is nonexpansive. Let 𝑥,𝑦∈𝐶. Since 𝐵 is 𝜉-inverse strongly monotone and 𝜆_𝑛<2𝛼 for all 𝑛∈ℕ, we obtain 𝐼−𝜇_𝑛𝐵𝑥−𝐼−𝜇_𝑛𝐵𝑦²=𝑥−𝑦−𝜇_𝑛𝐵𝑥−𝐵𝑦²=𝑥−𝑦²−2𝜇_𝑛𝑥−𝑦,𝐵𝑥−𝐵𝑦+𝜇²_𝑛𝐵𝑥−𝐵𝑦²≤𝑥−𝑦²−2𝜉𝜇_𝑛𝐵𝑥−𝐵𝑦²+𝜇²_𝑛𝐵𝑥−𝐵𝑦²=𝑥−𝑦²+𝜇_𝑛𝜇_𝑛−2𝜉𝐵𝑥−𝐵𝑦²≤𝑥−𝑦², (3.7) where 𝜇_𝑛≤2𝜉, for all 𝑛∈𝑁. So we have that the mapping (𝐼−𝜆_𝑛𝐴) is nonexpansive.
Step 3. We claim that {𝑥_𝑛} is bounded.
Let 𝑝∈Ω; from Lemma 2.10, we have 𝑝=𝑃_𝐶𝑝−𝜇_𝑛𝐵𝑝=𝑇_𝜆𝑛𝑧𝑝,_𝑛−𝑝=𝑇_𝜆𝑛𝑥_𝑛−𝑇_𝜆𝑛𝑝≤𝑥_𝑛−𝑝. (3.8) Note that 𝑦_𝑛−𝑝=𝑃_𝐶𝐼−𝜇_𝑛𝐵𝑧_𝑛=𝑃−𝑝_𝐶𝐼−𝜇_𝑛𝐵𝑧_𝑛−𝑃_𝐶𝐼−𝜇_𝑛𝐵𝑝≤𝑧_𝑛−𝜇_𝑛𝐵𝑧_𝑛−𝑝−𝜇_𝑛=𝐵𝑝𝐼−𝜇_𝑛𝐵𝑧_𝑛≤𝑧−𝑝_𝑛≤𝑥−𝑝_𝑛,𝐾−𝑝_𝑛−𝑝=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑊_𝑛𝑦_𝑛=𝛼−𝑝_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑊_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑊_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑦_𝑛≤𝑥−𝑝_𝑛−𝑝. (3.9) It follows that 𝑥_𝑛+1−𝑝=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛=𝜀−𝑝_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑝≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛−𝜀_𝑛𝛾𝐾_𝑛−𝑝≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛−𝜀_𝑛𝛾𝑥_𝑛≤−𝑝1−𝜀_𝑛𝛾𝑥_𝑛−𝑝+𝜀_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑝+𝜀_𝑛𝑝≤𝛾𝑓−𝐴𝑝1−𝜀_𝑛𝛾𝑥_𝑛−𝑝+𝜀_𝑛𝛾𝛼𝑥_𝑛−𝑝+𝜀_𝑛𝑝=𝛾𝑓−𝐴𝑝1−𝛾−𝛼𝛾𝜀_𝑛𝑥_𝑛−𝑝+𝛾−𝛼𝛾𝜀_𝑛𝛾𝑓𝑝−𝐴𝑝𝑥𝛾−𝛼𝛾≤max_𝑛,𝑝−𝑝𝛾𝑓−𝐴𝑝.𝛾−𝛼𝛾 (3.10) By simple induction, we have 𝑥_𝑛−𝑝≤max𝑥₁,𝑝−𝑝𝛾𝑓−𝐴𝑝𝛾−𝛼𝛾,∀𝑛∈ℕ. (3.11) Hence {𝑥_𝑛} is bounded. This implies that {𝐾_𝑛}, {𝑓(𝑥_𝑛)} are also bounded.
Step 4. Show that limsup_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Observing that 𝑧_𝑛=𝑇_𝜆𝑛𝑥_𝑛 and 𝑧_𝑛+1=𝑇_𝜆𝑛+1, we get 𝑧_𝑛+1−𝑧_𝑛=𝑇_𝜆𝑛+1𝑥_𝑛+1−𝑇_𝜆𝑛𝑥_𝑛=𝑇_𝜆𝑛+1𝑥_𝑛+1−𝑇_𝜆𝑛+1𝑥_𝑛+𝑇_𝜆𝑛+1𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑇_𝜆𝑛+1𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛. (3.12) By Lemma 2.10, we obtain Φ𝑇_𝜆𝑛𝑥_𝑛+1,𝑦𝜆_𝑛𝑦−𝑇_𝜆𝑛𝑥_𝑛,𝑇_𝜆𝑛𝑢_𝑛−𝑥_𝑛Φ𝑇≥0,∀𝑦∈𝐶,_𝜆𝑛+1𝑥_𝑛+1,𝑦𝜆_𝑛+1𝑦−𝑇_𝜆𝑛+1𝑥_𝑛,𝑇_𝜆𝑛+1𝑥_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.13) In particular, we have Φ𝑇_𝜆𝑛𝑥_𝑛,𝑇_𝜆𝑛+1𝑥_𝑛+1𝜆_𝑛𝑇_𝜆𝑛+1𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛,𝑇_𝜆𝑛𝑥_𝑛−𝑥_𝑛Φ𝑇≥0,_𝜆𝑛+1𝑥_𝑛,𝑇_𝜆𝑛𝑥_𝑛+1𝜆_𝑛+1𝑇_𝜆𝑛𝑥_𝑛−𝑇_𝜆𝑛+1𝑥_𝑛,𝑇_𝜆𝑛+1𝑥_𝑛−𝑥_𝑛≥0. (3.14) Summing up (3.14) and using (A₂), we obtain1𝜆_𝑛+1𝑇_𝜆𝑛𝑥_𝑛−𝑇_𝜆𝑛+1𝑥_𝑛,𝑇_𝜆𝑛+1𝑥_𝑛−𝑥_𝑛+1𝜆_𝑛𝑇_𝜆𝑛+1𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛,𝑇_𝜆𝑛𝑥_𝑛−𝑥_𝑛≥0, (3.15) for all 𝑦∈𝐶. It follows that 𝑇_𝜆𝑛𝑥_𝑛−𝑇_𝜆𝑛+1𝑥_𝑛,𝑇_𝜆𝑛+1𝑥_𝑛−𝑥_𝑛𝜆_𝑛+1−𝑇_𝜆𝑛𝑥_𝑛−𝑥_𝑛𝜆_𝑛≥0. (3.16) This implies 0≤𝑇_𝜆𝑛+1𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛,𝑇_𝜆𝑛𝑥_𝑛−𝑥_𝑛−𝜆_𝑛𝜆_𝑛+1𝑇_𝜆𝑛+1𝑥_𝑛−𝑥_𝑛=𝑇_𝜆𝑛+1𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛,𝑇_𝜆𝑛𝑥_𝑛−𝑇_𝜆𝑛+1𝑥_𝑛+𝜆1−^𝑛𝜆_𝑛+1𝑇_𝜆𝑛+1𝑥_𝑛−𝑥_𝑛. (3.17) It follows that 𝑇_𝜆𝑛+1𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛²≤𝜆1−^𝑛𝜆_𝑛+1𝑇_𝜆𝑛+1𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛𝑇_𝜆𝑛+1𝑥_𝑛+𝑥_𝑛. (3.18) Hence, we obtain 𝑇_𝜆𝑛+1𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛²≤𝜆1−^𝑛𝜆_𝑛+1𝐿, (3.19) where 𝐿=sup{‖𝑥_𝑛‖+‖𝑇_𝜆𝑛+1𝑥_𝑛‖∶𝑛∈ℕ}. By (3.12) and (3.19), we obtain 𝑧_𝑛+1−𝑧_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝑇_𝜆𝑛+1𝑢_𝑛−𝑇_𝜆𝑛𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆1−^𝑛𝜆_𝑛+1𝑦𝐿,_𝑛+1−𝑦_𝑛=𝑃_𝐶𝐼−𝜇_𝑛𝐵𝑧_𝑛+1−𝑃_𝐶𝐼−𝜇_𝑛𝐵𝑧_𝑛≤𝐼−𝜇_𝑛𝐵𝑧_𝑛+1−𝑧_𝑛≤𝑧_𝑛+1−𝑧_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆1−^𝑛𝜆_𝑛+1𝐿. (3.20) From (2.5), we have 𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛=𝑡₁𝑆^′₁𝑈_𝑛+1,2𝑦_𝑛−𝑡₁𝑆^′₁𝑈_𝑛,2𝑦_𝑛≤𝑡₁𝑈_𝑛+1,2𝑦_𝑛−𝑈_𝑛,2≤𝑡₁𝑡₂𝑆^′₂𝑈_𝑛+1,3𝑦_𝑛−𝑡₂𝑆^′₂𝑈_𝑛,3𝑦_𝑛≤𝑡₁𝑡₂𝑈_𝑛+1,3𝑦_𝑛−𝑈_𝑛,3𝑦_𝑛≤⋯≤𝑀₁^𝑛_𝑖=1𝑡_𝑖, (3.21) where 𝑀₁=sup_𝑛{‖𝑈_{𝑛+1,𝑛+1}𝑦_𝑛−𝑈_𝑛,𝑛+1𝑦_𝑛‖}.
Note that 𝐾_𝑛+1−𝐾_𝑛=𝛼_𝑛+1𝑥_𝑛+1+1−𝛼_𝑛+1𝑊_𝑛+1𝑦_𝑛+1−𝛼_𝑛𝑥_𝑛−1−𝛼_𝑛𝑊_𝑛𝑦_𝑛=𝛼_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1𝑥_𝑛+1−𝛼_𝑛+1𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛𝑦_𝑛−𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛+1𝑊_𝑛𝑦_𝑛−1−𝛼_𝑛𝑊_𝑛𝑦_𝑛=𝛼_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝑥_𝑛1−𝛼_𝑛+1𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛𝑦_𝑛+𝛼_𝑛−𝛼_𝑛+1𝑊_𝑛𝑦_𝑛≤𝛼_𝑛+1𝑥_𝑛+1−x_𝑛+𝛼_𝑛+1−𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛+1𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛𝑦_𝑛+𝛼_𝑛−𝛼_𝑛+1𝑊_𝑛𝑦_𝑛≤𝛼_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛+1𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛+1𝑦_𝑛+𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛+𝛼_𝑛−𝛼_𝑛+1𝑊_𝑛𝑦_𝑛≤𝛼_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛+1𝑥_𝑛+1−𝑥_𝑛+1−𝛼_𝑛+1𝜆1−^𝑛𝜆_𝑛+1𝐿+𝑀₁^𝑛_𝑖=1𝑡_𝑖+𝛼_𝑛−𝛼_𝑛+1𝑊_𝑛𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+2𝛼_𝑛+1−𝛼_𝑛𝑀^′₂+1−𝛼_𝑛+1𝑀₁^𝑛_𝑖=1𝑡_𝑖+𝜆1−^𝑛𝜆_𝑛+1𝐿, (3.22) where 𝑀^′₂=sup{‖𝑥_𝑛‖,‖𝑊_𝑛𝑦_𝑛‖}.
Suppose 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑙_𝑛, then 𝑙_𝑛=(𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛)/(1−𝛽_𝑛) =(𝜀_𝑛𝛾𝑓(𝑥_𝑛)+((1−𝛽_𝑛)𝐼−𝜀_𝑛𝐹)𝐾_𝑛)/(1−𝛽_𝑛).
Hence, we have 𝑙_𝑛+1−𝑙_𝑛=𝜀_𝑛+1𝛾𝑓𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝜀_𝑛𝐴𝐾_𝑛+11−𝛽_𝑛+1−𝜀_𝑛𝛾𝑓𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛1−𝛽_𝑛=𝜀_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝜀_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛+𝐾_𝑛+1−𝐾_𝑛+𝜀_𝑛1−𝛽_𝑛𝐴𝐾_𝑛−𝜀_𝑛+11−𝛽_𝑛+1𝐴𝐾_𝑛+1=𝜀_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝐾_𝑛+1+𝜀_𝑛1−𝛽_𝑛𝐴𝐾_𝑛−𝛾𝑓𝑥_𝑛+𝐾_𝑛+1−𝐾_𝑛. (3.23) Then 𝑙_𝑛+1−𝑙_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝜀_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1+𝐴𝐾_𝑛+1+𝜀_𝑛1−𝛽_𝑛𝐴𝐾_𝑛+𝛾𝑓𝑥_𝑛+𝐾_𝑛+1−𝐾_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝜀_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1+𝐴𝐾_𝑛+1+𝜀_𝑛1−𝛽_𝑛𝐴𝐾_𝑛+𝛾𝑓𝑥_𝑛+2𝛼_𝑛+1−𝛼_𝑛𝑀^′₂+1−𝛼_𝑛+1𝜆1−^𝑛𝜆_𝑛+1𝐿+𝑀₁^𝑛_𝑖=1𝑡_𝑖. (3.24) Combining with (i), (iii), and (iv), we have limsup_𝑛→∞𝑙_𝑛+1−𝑙_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.25) Hence, by Lemma 2.11, we obtain ‖𝑙_𝑛−𝑥_𝑛‖→0 as 𝑛→∞. It follows that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑙_𝑛−𝑥_𝑛=0. (3.26) We also know that 𝑥_𝑛+1−𝑥_𝑛=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑥_𝑛=𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑥_𝑛. (3.27) So lim_𝑛→∞𝐾_𝑛−𝑥_𝑛=0. (3.28)
Step 5. We claim that ‖𝑥_𝑛−𝑊𝑥_𝑛‖→0.
Observe that 𝑥_𝑛−𝑊_𝑛𝑦_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑦_𝑛+𝐾_𝑛−𝑊_𝑛𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑦_𝑛+𝛼_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛. (3.29) From (A1), (A3), and (3.28), using step 2, we have 1−𝛼_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝛽_𝑛𝐾_𝑛−𝜀_𝑛𝐴𝐾_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝐾_𝑛+𝛽_𝑛𝑥_𝑛−𝐾_𝑛. (3.30) This implies that 𝑥_𝑛−𝑊_𝑛𝑦_𝑛→0(as𝑛→∞). (3.31) Next we want to show lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.
Let 𝑝∈⋂^∞_𝑖=1𝐹(𝑆_𝑖)⋂𝐸𝑃(Φ); we have 𝑧_𝑛−𝑝²=𝑇_𝜆𝑛𝑥_𝑛−𝑇_𝜆𝑛𝑝²≤𝑇_𝜆𝑛𝑥_𝑛−𝑇_𝜆𝑛𝑝,𝑥_𝑛=𝑧−𝑝_𝑛−𝑝,𝑥_𝑛=1−𝑝2𝑧_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛−𝑧_𝑛². (3.32) Therefore 𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑧_𝑛². (3.33) Note that 𝐾_𝑛−𝑝=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑊_𝑛𝑦_𝑛−𝑝²=𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑊_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝑥_𝑛−𝑧_𝑛²=𝑥_𝑛−𝑝²−1−𝛼_𝑛𝑥_𝑛−𝑧_𝑛². (3.34) From (3.34), we have 𝑥_𝑛+1−𝑝²=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑝²=𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑝²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛−𝜀_𝑛𝛾𝐾_𝑛−𝑝²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛−𝜀_𝑛𝛾𝑥_𝑛−𝑝²−1−𝛼_𝑛𝑥_𝑛−𝑧_𝑛²=𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+1−𝜀_𝑛𝛾𝑥_𝑛−𝑝²−1−𝛽_𝑛−𝜀_𝑛𝛾1−𝛼_𝑛,𝑥_𝑛−𝑧_𝑛²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝑥_𝑛−𝑝²+1−𝛽_𝑛−𝜀_𝑛𝛾1−𝛼_𝑛𝑥_𝑛−𝑧_𝑛². (3.35) It follows that 1−𝛽_𝑛−𝜀_𝑛𝛾1−𝛼_𝑛𝑥_𝑛−𝑧_𝑛²≤𝜀_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²=𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝑥_𝑛−𝑝+𝑥_𝑛+1×𝑥−𝑝_𝑛−𝑥_𝑛+1. (3.36) From conditions (i), (vi) and (3.26), we have 𝑥_𝑛−𝑧_𝑛→0. (3.37) We also compute 𝑦_𝑛−𝑝²=𝑃_𝐶𝐼−𝜇_𝑛𝐵𝑧_𝑛−𝑃_𝐶𝐼−𝜇_𝑛𝐵𝑝≤𝑧_𝑛−𝜇_𝑛𝐵𝑧_𝑛−𝑝−𝜇_𝑛𝐵𝑝²=𝑧_𝑛−𝑝²−2𝜇_𝑛𝑧_𝑛−𝑝,𝐵𝑧_𝑛−𝐵𝑝+𝜇²_𝑛𝐵𝑧_𝑛−𝐵𝑝²≤𝑥_𝑛−𝑝²−𝜇_𝑛2𝜉−𝜇_𝑛𝐵𝑧_𝑛−𝐵𝑝²,𝐾_𝑛−𝑝²=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑊_𝑛𝑦_𝑛−𝑝²=𝛼_𝑛𝑥_𝑛−𝑝−1−𝛼_𝑛𝑊_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑦_𝑛−𝑝²=𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝜇_𝑛2𝜉−𝜇_𝑛𝐵𝑧_𝑛−𝐵𝑝²=𝑥_𝑛−𝑝²−𝜇_𝑛1−𝛼_𝑛2𝜉−𝜇_𝑛𝐵𝑧_𝑛−𝐵𝑝². (3.38)(3.39) So, from (3.39), we get 𝑥_𝑛+1−𝑝²=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑝²=𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑝²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛−𝜀_𝑛𝛾𝐾_𝑛−𝑝²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛−𝜀_𝑛𝛾×𝑥_𝑛−𝑝²−𝜇_𝑛1−𝛼_𝑛2𝜉−𝜇_𝑛𝐵𝑧_𝑛−𝐵𝑝²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝑥_𝑛−𝑝²−1−𝛽_𝑛−𝜀_𝑛𝛾𝜇_𝑛1−𝛼_𝑛2𝜉−𝜇_𝑛𝐵𝑧_𝑛−𝐵𝑝². (3.40) It follows that1−𝛽_𝑛−𝜀_𝑛𝛾𝜇_𝑛1−𝛼_𝑛2𝜉−𝜇_𝑛𝐵𝑧_𝑛−𝐵𝑝²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝. (3.41) So 𝐵𝑧_𝑛−𝐵𝑝→0. (3.42) On the other hand, we also know that 𝑦_𝑛−𝑝²=𝑃_𝐶𝐼−𝜇_𝑛𝐵𝑧_𝑛−𝑃_𝐶𝐼−𝜇_𝑛𝐵𝑝²≤𝐼−𝜇_𝑛𝐵𝑧_𝑛−𝐼−𝜇_𝑛𝐵𝑝,𝑦_𝑛=1−𝑝2𝐼−𝜇_𝑛𝐵𝑧_𝑛−𝐼−𝜇_𝑛𝐵𝑝²+𝑦_𝑛−𝑝²−𝐼−𝜇_𝑛𝐵𝑧_𝑛−𝐼−𝜇_𝑛𝐵𝑝−𝑦_𝑛−𝑝²≤12𝑥_𝑛−𝑝²+𝑦_𝑛−𝑝²−𝑧_𝑛−𝑦_𝑛²−𝜇_𝑛𝐵𝑧_𝑛−𝐵𝑝²+2𝜇_𝑛𝑧_𝑛−𝑦_𝑛,𝐵𝑧_𝑛−𝐵𝑝, (3.43) and hence 𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑧_𝑛−𝑦_𝑛²+2𝜇_𝑛𝑧_𝑛−𝑦_𝑛²𝐵𝑧_𝑛−𝐵𝑝. (3.44) So 𝐾_𝑛−𝑝²=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑊_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝑧_𝑛−𝑦_𝑛²+2𝜇_𝑛𝑧_𝑛−𝑦_𝑛𝐵𝑧_𝑛=𝑥−𝐵𝑝_𝑛−𝑝²−1−𝛼_𝑛𝑧_𝑛−𝑦_𝑛²+21−𝛼_𝑛𝜇_𝑛𝑧_𝑛−𝑦_𝑛𝐵𝑧_𝑛,𝑥−𝐵𝑝_𝑛+1−𝑝²=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑝²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛−𝜀_𝑛𝛾𝐾_𝑛−𝑝²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+𝑥_𝑛−𝑝²−1−𝛽_𝑛−𝜀_𝑛𝛾1−𝛼_𝑛𝑧_𝑛−𝑦_𝑛²+21−𝛼_𝑛𝜇_𝑛1−𝛽_𝑛−𝜀_𝑛𝛾𝑧_𝑛−𝑦_𝑛𝐵𝑧_𝑛−𝐵𝑝. (3.45)(3.46) Hence 1−𝛽_𝑛−𝜀_𝑛𝛾1−𝛼_𝑛𝑧_𝑛−𝑦_𝑛²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+21−𝛼_𝑛𝜇_𝑛1−𝛽_𝑛−𝜀_𝑛𝛾𝑧_𝑛−𝑦_𝑛𝐵𝑧_𝑛+𝑥−𝐵𝑝_𝑛−𝑝²−𝑥_𝑛+1−𝑝²≤𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝²+21−𝛼_𝑛𝜇_𝑛1−𝛽_𝑛−𝜀_𝑛𝛾𝑧_𝑛−𝑦_𝑛𝐵𝑧_𝑛+𝑥−𝐵𝑝_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝. (3.47) From (i), (3.42), and (3.26), we know that 𝑧_𝑛−𝑦_𝑛→0. (3.48) From (3.37) and (3.42), we can get 𝑥_𝑛−𝑦_𝑛→0. (3.49) On the other hand, we have 𝑥_𝑛−𝑊𝑥_𝑛≤𝑥_𝑛−𝑊_𝑛𝑦_𝑛+𝑊_𝑛𝑦_𝑛−𝑊𝑥_𝑛≤𝑥_𝑛−𝑊_𝑛𝑦_𝑛+sup𝑊_𝑛𝑦_𝑛−𝑊𝑥_𝑛. (3.50) By (3.30), (3.49), and using Lemma 2.7, we have 𝑥_𝑛−𝑊𝑥_𝑛→0. (3.51)
Step 6. We claim that limsup_𝑛→∞⟨(𝐴−𝛾𝑓)𝑞,𝑞−𝑥_𝑛⟩≤0, where 𝑞=𝑃_Ω(𝐼−𝐴+𝛾𝑓)(𝑞) is the unique solution of ⟨(𝐴−𝛾𝑓)𝑞,𝑥−𝑞⟩≥0, for all 𝑥∈Ω.
Indeed, take a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that limsup_𝑛→∞𝐴−𝛾𝑓𝑞,𝑞−𝑥_𝑛=limsup_𝑛→∞𝐴−𝛾𝑓𝑞,𝑞−𝑥_𝑛𝑗. (3.52) Since {𝑥_𝑛𝑗} is bounded, there exists a subsequence {𝑥_𝑛𝑗𝑘} of {𝑥_𝑛𝑗}, which converges weakly to 𝑝; without loss of generality, we can assume 𝑥_𝑛𝑗⇀𝑝 and 𝑊𝑥_𝑛𝑗⇀𝑝, we arrive at limsup_𝑛→∞𝐴−𝛾𝑓𝑞,𝑞−𝑥_𝑛𝑗=𝐴−𝛾𝑓𝑞,𝑞−𝑝≤0. (3.53)
Step 7. We show that 𝑥_𝑛→𝑞.
Since 𝐴−𝛾𝑓𝑞,𝑞−𝑥_𝑛+1=𝐴−𝛾𝑓𝑞,𝑥_𝑛−𝑥_𝑛+1+𝐴−𝛾𝑓𝑞,𝑞−𝑥_𝑛≤𝐴−𝛾𝑓𝑞⋅𝑥_𝑛−𝑥_𝑛+1+𝐴−𝛾𝑓𝑞,𝑞−𝑥_𝑛, (3.54) so limsup_𝑛→∞𝐴−𝛾𝑓𝑞,𝑞−𝑥_𝑛+1≤0. (3.55) Note that 𝑥_𝑛+1−𝑞²=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑞²=𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+𝛽_𝑛𝑥_𝑛−𝑞+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛−𝑞²≤1−𝛽_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴1−𝛽_𝑛𝐾_𝑛−𝑞+𝛽_𝑛𝑥_𝑛−𝑞²+2𝜀_𝑛𝐴−𝛾𝑓𝑞,𝑥_𝑛+1≤−𝑞1−𝛽_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴1−𝛽_𝑛𝐾_𝑛−𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+2𝜀_𝑛×𝑓𝑥𝛾𝛼_𝑛−𝑓𝑞,𝑥_𝑛+1−𝑞+2𝜀_𝑛𝛾𝑓𝑞−𝐴𝑞,𝑥_𝑛+1≤−𝑞1−𝛽_𝑛1−𝛽_𝑛𝐼−𝜀_𝑛𝐴1−𝛽_𝑛𝐾_𝑛−𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+2𝜀_𝑛×𝑥𝛾𝛼_𝑛−𝑞⋅𝑥_𝑛+1−𝑞+2𝜀_𝑛𝛾𝑓𝑞−𝐴𝑞,𝑥_𝑛+1≤−𝑞1−𝛽_𝑛𝐼−𝜀_𝑛𝐴²1−𝛽_𝑛𝐾_𝑛−𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝜀_𝑛×𝑥𝛾𝛼_𝑛−𝑞²+𝑥_𝑛+1−𝑞²+2𝜀_𝑛𝛾𝑓𝑞−𝐴𝑞,𝑥_𝑛+1≤−𝑞1−𝛽_𝑛−𝛾𝜀_𝑛²1−𝛽_𝑛+𝛽_𝑛+𝜀_𝑛𝛾𝛼𝑥_𝑛−𝑞²+𝜀_𝑛𝛾𝛼𝑥_𝑛+1−𝑞²+2𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞,𝑥_𝑛+1−𝑞, (3.56) which implies that 𝑥_𝑛+1−𝑞²≤21−𝛾−𝛼𝛾𝜀_𝑛1−𝛼𝛾𝜀_𝑛𝑥_𝑛−𝑞²+𝜀_𝑛1−𝛼𝛾𝜀_𝑛𝛾²𝜀²_𝑛1−𝛽_𝑛𝑥_𝑛−𝑞²+2𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞,𝑥_𝑛+1−𝑞. (3.57) Let 𝜎_𝑛=𝜀_𝑛1−𝛼𝛾𝜀_𝑛𝛾²𝜀²_𝑛1−𝛽_𝑛𝑥_𝑛−𝑞²+2𝜀_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞,𝑥_𝑛+1,𝜌−𝑞_𝑛=2𝛾−𝛼𝛾𝜀_𝑛1−𝛼𝛾𝜀_𝑛, (3.58) then we have limsup_𝑛→∞𝜎_𝑛𝜌_𝑛≤0. (3.59) Applying Lemma 2.2, we can conclude that {𝑥_𝑛} converges strongly to 𝑞 in norm. This completes the proof.

As direct consequences of Theorem 3.1, we obtain the following corollary.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹 be a bifunction from 𝐶×𝐶→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 an 𝛼-inverse strongly monotone mapping. Let 𝐹 be a strongly positive linear-bounded operator on 𝐻 with coefficient 𝛾>0 and 0<𝛾<𝛾/𝛼 and 𝜏<1. Let 𝑊_𝑛 be the mapping generated by 𝑆^′_𝑖 and 𝑡_𝑖, where 𝑆_𝑖∶𝐶→𝐶 is a nonexpansive mapping with a fixed point. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by the following algorithm: 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝐾≥0,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑊_𝑛𝑧_𝑛,𝑥_𝑛+1=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝐾_𝑛, (3.60) where{𝜀_𝑛}, {𝛽_𝑛}, {𝛼_𝑛}, and {𝜆_𝑛} are sequences in (0,1). Assume that the control sequences satisfy the following restrictions: (i)lim_𝑛→∞𝜀_𝑛=0 and Σ^∞_𝑛=1𝜀_𝑛=∞;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(iii)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1;(iv)lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=lim_𝑛→∞|𝛼_𝑛+1−𝛼_𝑛|=0;(v)0<𝑡_𝑛≤𝑏<1;(vi)𝜆_𝑛<2𝛼.
Then {𝑥_𝑛} converges strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤.

Acknowledgments

This research is supported by the Science Research Foundation Program in the Civil Aviation University of China (07kys08) and the Fundamental Research Funds for the Science of the Central Universities (program no. ZXH2012K001).