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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 804642, 15 pages
http://dx.doi.org/10.1155/2012/804642
Research Article

A Hybrid Extragradient-Like Method for Variational Inequalities, Equilibrium Problems, and an Infinitely Family of Strictly Pseudocontractive Mappings

1Department of Mathematics, Shaoxing University, Shaoxing 312000, China
2Mathematical College, Sichuan University, Sichuan, Chengdu 610064, China
3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received 31 December 2011; Accepted 15 January 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Yaqin Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan (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 and 𝐴 be a mapping of 𝐶 into 𝐻. We denote by 𝐹(𝐴) the set of fixed points of 𝐴 and by 𝑃𝐶 the metric projection of 𝐻 onto 𝐶. We also denote by 𝑅 the set of all real numbers.

Recall the following definitions.(i)𝐴 is called monotone if 𝐴𝑥𝐴𝑦,𝑥𝑦0,𝑥,𝑦𝐶.(1.1)(ii)𝐴 is called 𝛼-inverse-strongly monotone if there exists a positive constant 𝛼 such that 𝐴𝑥𝐴𝑦,𝑥𝑦𝛼𝐴𝑥𝐴𝑦2,𝑥,𝑦𝐶.(1.2)(iii)𝐴 is called 𝑘-Lipschitz continuous if there exists a positive constant 𝑘 such that𝐴𝑥𝐴𝑦𝑘𝑥𝑦,𝑥,𝑦𝐶.(1.3) Clearly, every inverse strongly monotone mapping is Lipschitz continuous and monotone.

A mapping 𝑇𝐶𝐶 is said to be 𝜉-strictly pseudocontractive if there exists a constant 𝜉[0,1) such that𝑇𝑥𝑇𝑦2𝑥𝑦2+𝜉(𝐼𝑇)𝑥(𝐼𝑇)𝑦2,𝑥,𝑦𝐶.(1.4)

It is known that if 𝑇 is a 0-strictly pseudocontractive mapping, then 𝑇 is a nonexpansive mapping. So the class of 𝜉-strictly pseudocontractive mappings includes the class of nonexpansive mappings.

Let Θ𝐶×𝐶𝑅 be a bifunction. The equilibrium problem for Θ is to find that 𝑥𝐶 such thatΘ(𝑥,𝑦)0,𝑦𝐶.(1.5) The set of solutions of problem (1.5) is denoted by EP.

Given a mapping 𝐴𝐶𝐻, let Θ(𝑥,𝑦)=𝐴𝑥,𝑦𝑥 for all 𝑥,𝑦𝐶. Then problem (1.5) reduces to the following classical variational inequality problem of finding 𝑥𝐶 such that𝐴𝑥,𝑦𝑥0,𝑦𝐶.(1.6) The set of solutions of problem (1.6) is denoted by VI(𝐶,𝐴).

Numerous problems in physics, optimization, saddle point problems, complementarity problems, mechanics, and economics reduce to find a solution of problem (1.5). Many methods have been proposed to solve problem (1.5); see, for instant, [13]. In 1997, Combettes and Hirstoaga [4] introduced an iterative scheme of finding the best approximation to initial data when EP is nonempty and proved a strong convergence theorem.

Recently, Peng and Yao [5] introduced the following generalized mixed equilibrium problem of finding 𝑥𝐶 such that Θ(𝑥,𝑦)+𝜑(𝑦)𝜑(𝑥)+𝐵𝑥,𝑦𝑥0,𝑦𝐶,(1.7) where 𝐵𝐶𝐻 is a nonlinear mapping, and 𝜑𝐶𝑅 is a function. The set of solutions of problem (1.7) is denoted by 𝐺MEP.

In the case of 𝐵=0 and 𝜑=0, then problem (1.7) reduces to problem (1.5). In the case of Θ=0,𝜑=0, and 𝐵=𝐴, then problem (1.7) reduces to problem (1.6). In the case of 𝜑=0, problem (1.7) reduces to the generalized equilibrium problem. In the case of 𝐵=0, problem (1.7) reduces to the following mixed equilibrium problem of finding 𝑥𝐶 such that Θ(𝑥,𝑦)+𝜑(𝑦)𝜑(𝑥)0,𝑦𝐶,(1.8) which was considered by Ceng and Yao [6]. The set of sulutions of this problem is denoted by MEP.

The problem (1.7) is very general in the sense that it includes, as special cases, some optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, economics, and others (see, for instant, [6])

Recently, S. Takahashi and W. Takahashi [7] introduced the following iteration process:𝑥1Θ𝑢=𝑢𝐶,𝑛,𝑦+𝐴𝑥𝑛,𝑦𝑢𝑛1+𝑟𝑦𝑢𝑛,𝑢𝑛𝑥𝑛𝑥0,𝑦𝐶,𝑛+1=𝛽𝑛𝑥𝑛+1𝛽𝑛𝑆𝛼𝑛𝑢+1𝛼𝑛𝑢𝑛,𝑛1,(1.9) and used this iteration process to find a common element of the set of fixed points of a nonexpansive mapping 𝑆 and the set of solutions of a generalized equilibrium problem in a Hilbert space.

In 2008, Bnouhachem et al. [8] introduced the following new extragradient iterative method. Let 𝐶 be a closed convex subset of a real Hilbert 𝐻, 𝐴 be an 𝛼-inverse strongly monotone mapping of 𝐶 into 𝐻, and let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑆)VI(𝐶,𝐴). Let the sequences {𝑥𝑛}, {𝑦𝑛} be given by𝑥1𝑦,𝑢𝐶chosenarbitrary,𝑛=𝑃𝐶𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑥𝑛+1=𝛽𝑛𝑥𝑛+1𝛽𝑛𝑆𝛼𝑛𝑢+1𝛼𝑛𝑃𝐶𝑥𝑛𝜆𝑛𝐴𝑦𝑛,𝑛1,(1.10) where {𝛼𝑛}, {𝛽𝑛}, and {𝜆𝑛}(0,1) satisfy some parameters controlling conditions. They proved that the sequence {𝑥𝑛} converges strongly to a common element of 𝐹(𝑆)VI(𝐶,𝐴).

In 2010, Ceng et al. [9] introduced the following hybrid extragradient-like method. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, 𝐴𝐶𝐻 be a monotone, 𝑘-Lipschitz continuous mapping, and let 𝑆𝐶𝐶 be a nonexpansive mapping such that 𝐹(𝑆)VI(𝐶,𝐴). Let the sequences {𝑥𝑛}, {𝑦𝑛}, and {𝑧𝑛} be defined by𝑥0𝑦𝐶,𝑛=1𝛾𝑛𝑥𝑛+𝛾𝑛𝑃𝐶𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑧𝑛=1𝛼𝑛𝛽𝑛𝑥𝑛+𝛼𝑛𝑦𝑛+𝛽𝑛𝑆𝑃𝐶𝑥𝑛𝜆𝑛𝐴𝑦𝑛,𝐶𝑛=𝑧𝑧𝐶𝑛𝑧2𝑥𝑛𝑧2+33𝛾𝑛+𝛼𝑛𝑏2𝐴𝑥𝑛2,𝑄𝑛=𝑧𝐶,𝑥𝑛𝑧,𝑥0𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝑄𝑛𝑥0,𝑛0.(1.11) Under the suitable conditions, they proved the sequences {𝑥𝑛}, {𝑦𝑛}, {𝑧𝑛} converge strongly to the same point 𝑃𝐹(𝑆)VI(𝐶,𝐴)𝑥0.

In 2010, Inchan [10] introduced a new iterative scheme by the hybrid extragradient method in a Hilbert space 𝐻 as follows: 𝑥0𝐻,𝐶1=𝐶𝐻, 𝑥1=𝑃𝐶𝑥0, and let𝑢𝑛𝑢𝐶,Θ𝑛,𝑦+𝐵𝑥𝑛,𝑦𝑢𝑛1+𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛𝑦0,𝑦𝐶,𝑛=𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛,𝑧𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑆𝛽𝑛𝑥𝑛+1𝛽𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑦𝑛,𝐶𝑛+1=𝑧𝐶𝑛𝑧𝑛𝑥𝑧𝑛,𝑥𝑧𝑛+1=𝑃𝐶𝑛+1𝑥0,𝑛0,(1.12) where {𝛼𝑛}, {𝛽𝑛}, and {𝜆𝑛}(0,1) satisfy some parameters controlling conditions. They proved that {𝑥𝑛} and {𝑢𝑛} strongly converge to the same common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality for nonexpansive mappings.

Very recently, Wang [11] defined the mapping 𝑊𝑛 as follows: 𝑈𝑛,𝑛+1𝑈=𝐼,𝑛,𝑛=𝛾𝑛𝑇𝑛𝑈𝑛,𝑛+1+1𝛾𝑛𝑈𝐼,𝑛,𝑛1=𝛾𝑛1𝑇𝑛1𝑈𝑛,𝑛+1𝛾𝑛1𝑈𝐼,𝑛,𝑘=𝛾𝑘𝑇𝑘𝑈𝑛,𝑘+1+1𝛾𝑘𝑈𝐼,𝑛,𝑘1=𝛾𝑘1𝑇𝑘1𝑈𝑛,𝑘+1𝛾𝑘1𝑈𝐼,𝑛,2=𝛾2𝑇2𝑈𝑛,3+1𝛾2𝑊𝐼,𝑛=𝑈𝑛,1=𝛾1𝑇1𝑈𝑛,2+1𝛾1𝐼,(1.13) where 𝛾1,𝛾2, are real numbers such that 0𝛾𝑛1, 𝑇𝑖=𝜃𝑖𝐼+(1𝜃𝑖)𝑇𝑖, where 𝑇𝑖 is a 𝜇𝑖-strictly pseudocontractive mapping of 𝐶 into itself and 𝜃𝑖[𝜇𝑖,1). It follows from [12] that 𝑇𝑖 is nonexpansive and 𝐹(𝑇𝑖)=𝐹(𝑇𝑖). Nonexpansivity of each 𝑇𝑖 ensures the nonexpansivity of 𝑊𝑛.

Motivated and inspired by the above work, in this paper, we introduced the following new iterative scheme by the extragradient-like method: 𝐶1=𝐶𝐻,𝑥1=𝑃𝐶𝑥0,𝑢𝑛𝑢𝐶suchthatΘ𝑛,𝑦+𝐵𝑥𝑛,𝑦𝑢𝑛𝑢+𝜑(𝑦)𝜑𝑛+1𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛𝑦0,𝑦𝐶,𝑛=1𝑠𝑛𝑢𝑛+𝑠𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛,𝑧𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑊𝑛𝛽𝑛𝑥𝑛+1𝛽𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑦𝑛,𝐶𝑛+1=𝑧𝐶𝑛𝑧𝑛𝑧2𝑥𝑛𝑧2+31𝑠𝑛𝑏2𝐴𝑢𝑛2,𝑥𝑛+1=𝑃𝐶𝑛+1𝑥0,𝑛0,(1.14) where 𝐴,𝐵𝐶𝐻 and 𝐴 is a monotone, 𝑘-Lipschitz continuous mapping, and 𝐵 is a 𝛽-inverse strongly monotone mapping. Then under the suitable conditions, we derive some strong convergence results.

2. Preliminaries

Let 𝐶 be a nonempty closed and convex subset of a Hilbert space 𝐻, for any 𝑥𝐻, and there exists a unique nearest point in 𝐶, denoted by 𝑃𝐶𝑥 such that 𝑥𝑃𝐶𝑥𝑥𝑦,𝑦𝐶.(2.1) The projection operator 𝑃𝐶𝐻𝐶 is nonexpansive. Moreover, 𝑃𝐶𝑥 is characterized by the following properties: for every 𝑥𝐻 and 𝑦𝐶, 𝑥𝑦2𝑥𝑃𝐶𝑥2+𝑦𝑃𝐶𝑥2,(2.2)𝑥𝑃𝐶𝑥,𝑦𝑃𝐶𝑥0.(2.3)

Suppose that 𝐴 is monotone and continuous. Then the solutions of the variational inequality VI(𝐶,𝐴) can be characterized as solutions of the so-called Minty variational inequality: 𝑥VI(𝐶,𝐴)𝐴𝑥,𝑥𝑥0,𝑥𝐶.(2.4)

In what follows, we shall make use of the following lemmas.

Lemma 2.1. Let 𝐻 be a real Hilbert space. Then for any 𝑥,𝑦𝐻, we have(i)𝑥±𝑦2=𝑥2±2𝑥,𝑦+𝑦2, (ii)𝑡𝑥+(1𝑡)𝑦2=𝑡𝑥2+(1𝑡)𝑦2𝑡(1𝑡)𝑥𝑦2,forall𝑡[0,1].
We denote by 𝑁𝐶(𝑣) the normal cone for 𝐶 at a point 𝑣𝐶, that is 𝑁𝐶(𝑣)={𝑥𝐸𝑣𝑦,𝑥0forall𝑦𝐶}. In the following, we shall use the following Lemma.

Lemma 2.2 (see [13]). Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸, and let 𝐴 be a monotone and hemicontinuous operator of 𝐶 into 𝐸. Let 𝑇𝐸×𝐸 be an operator defined as follows: 𝑇𝑣=𝐴𝑣+𝑁𝐶(𝑣),𝑣𝐶,,𝑣𝐶.(2.5) Then 𝑇 is maximal monotone, and 𝑇10=VI(𝐶,𝐴).

Lemma 2.3 (see [14]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝐸. Let 𝑇1,𝑇2, be nonexpansive mappings of 𝐶 into itself such that 𝑖=1𝐹(𝑇𝑖) and 𝛾1,𝛾2, be real numbers such that 0<𝛾𝑖𝑏<1 for every 𝑖=1,2,. Then for any 𝑥𝐶 and 𝑘𝑁, the limit lim𝑛𝑈𝑛,𝑘 exists.
Using Lemma 2.3, define the mapping 𝑊 of 𝐶 into itself as follows: 𝑊𝑥=lim𝑛𝑊𝑛𝑥=lim𝑛𝑈𝑛,1𝑥,𝑥𝐶.(2.6) Such a mapping 𝑊 is called the modified 𝑊 mapping generalized by 𝑇1,𝑇2,,𝛾1,𝛾2,, and 𝜃1,𝜃2,.

Lemma 2.4 (see [14]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝐸. Let 𝑇1,𝑇2, be nonexpansive mappings of 𝐶 into itself such that 𝑖=1𝐹(𝑇𝑖) and 𝛾1,𝛾2, be real numbers such that 0<𝛾𝑖𝑏<1 for every 𝑖=1,2,. Then 𝑊 is a nonexpansive mapping satisfying that 𝐹(𝑊)=𝑖=1𝐹(𝑇𝑖).

Lemma 2.5 (see [15]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝐸. Let 𝑇1,𝑇2, be nonexpansive mappings of 𝐶 into itself such that 𝑖=1𝐹(𝑇𝑖) and 𝛾1,𝛾2, be real numbers such that 0<𝛾𝑖𝑏<1 for every 𝑖=1,2,. If 𝐾 is any bounded subset of 𝐶, then lim𝑛sup𝑥𝐾𝑊𝑥𝑊𝑛𝑥=0.(2.7)
For solving the equilibrium problem, let us assume that Θ satisfies the following conditions:(H1)Θ(𝑥,𝑥)=0 for all 𝑥𝐶,(H2)Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)0 for all 𝑥,𝑦𝐶,(H3) for each 𝑦𝐶,𝑥Θ(𝑥,𝑦) is weakly upper semicontinuous,(H4) for each 𝑥𝐶,𝑦Θ(𝑥,𝑦) is convex and lower semicontinuous,(A1) for each 𝑥𝐻 and 𝑟>0, there exists a bounded subset 𝐷𝑥𝐶 and 𝑦𝑥𝐶 such that for any 𝑧𝐶𝐷𝑥, Θ𝑧,𝑦𝑥𝑦+𝜑𝑥1𝜑(𝑧)+𝑟𝑦𝑥𝑧,𝑧𝑥<0,(2.8)(A2)𝐶 is a bounded set.

Lemma 2.6 (see [6]). Let 𝐶 be a closed subset of 𝐻. Let 𝜑𝐶𝑅 be a lower semicontinuous and convex function, and Θ be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (H1)–(H4). For 𝑟>0 and 𝑥𝐻, define a mapping 𝑇𝑟𝐻𝐶 as follows: 𝑇𝑟1(𝑥)=𝑧𝐶Θ(𝑧,𝑦)+𝜑(𝑦)𝜑(𝑧)+𝑟𝑦𝑧,𝑧𝑥0,𝑦𝐶(2.9) for all 𝑥𝐻. Assume that either (A1) or (A2) holds. Then the following results hold: (1)𝑇𝑟(𝑥) for each 𝑥𝐻, and 𝑇𝑟 is single valued,(2)𝑇𝑟 is firmly nonexpansive, that is, for all 𝑥,𝑦𝐻,𝑇𝑟𝑥𝑇𝑟𝑦2𝑇𝑟𝑥𝑇𝑟𝑦,𝑥𝑦,(3)𝐹(𝑇𝑟)=MEP, (4)MEP is closed and convex.

3. Strong Convergence Theorems

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ be a bifunction from 𝐶×𝐶 into 𝑅 satisfying (H1)–(H4) and 𝜑𝐶𝑅 be a lower semicontinuous and convex function with (A1) or (A2). Let 𝐴𝐶𝐻 be a monotone, 𝑘-Lipschitz continuous mapping and 𝐵𝐶𝐻 be a 𝛽-inverse-strongly monotone mapping. Let 𝑇𝑖𝐶𝐶 be a 𝜇𝑖-strictly pseudocontractive mapping with 𝐹=𝑖=1𝐹(𝑇𝑖)VI(𝐶,𝐴)𝐺MEP and {𝛾𝑖} be a real sequence such that 0<𝛾𝑖𝑏<1, for all 𝑖1. Assume that the control sequences {𝛼𝑛}, {𝛽𝑛}, {𝑠𝑛}[0,1], {𝑟𝑛}(0,2𝛽),and {𝜆𝑛}(0,1/2𝑘) satisfy the following conditions:(i)limsup𝑛𝛼𝑛<1,limsup𝑛𝛽𝑛<1, (ii)0<𝑎𝜆𝑛𝑏<1/2𝑘,0<𝑑𝑟𝑛𝑒<2𝛽,(iii)𝑠𝑛1(𝑛) and 𝑠𝑛>3/4 for all 𝑛0.
Then the sequence {𝑥𝑛} defined by (1.14) converges strongly to 𝑃𝐹𝑥0.

Proof. We divide the proof into several steps.
Step 1 ({𝑥𝑛} is well defined). Indeed, for any 𝑞𝐹. Put 𝑡𝑛=𝑃𝐶(𝑢𝑛𝜆𝑛𝐴𝑦𝑛). Since 𝑢𝑛=𝑇𝑟𝑛(𝑥𝑛𝑟𝑛𝐵𝑥𝑛), 𝑞=𝑇𝑟𝑛(𝑞𝑟𝑛𝐵𝑞), and 𝐵 is 𝛽-inverse-strongly monotone and 𝑟𝑛[0,2𝛽], for any 𝑛0, we have 𝑢𝑛𝑞2=𝑇𝑟𝑛𝑥𝑛𝑟𝑛𝐵𝑥𝑛𝑇𝑟𝑛𝑞𝑟𝑛𝐵𝑞2𝑥𝑛𝑟𝑛𝐵𝑥𝑛𝑞𝑟𝑛𝐵𝑞2𝑥𝑛𝑞22𝑟𝑛𝐵𝑥𝑛𝐵𝑞,𝑥𝑛𝑞+𝑟2𝑛𝐵𝑥𝑛𝐵𝑞2𝑥𝑛𝑞2+𝑟𝑛𝑟𝑛2𝛽𝐵𝑥𝑛𝐵𝑞2𝑥𝑛𝑞2.(3.1) It follows from (2.2) and (2.4) that 𝑡𝑛𝑞2𝑢𝑛𝜆𝑛𝐴𝑦𝑛𝑞2𝑢𝑛𝜆𝑛𝐴𝑦𝑛𝑡𝑛2=𝑢𝑛𝑞2𝑢𝑛𝑡𝑛22𝜆𝑛𝐴𝑦𝑛,𝑡𝑛=𝑢𝑞𝑛𝑞2𝑢𝑛𝑦𝑛2𝑦𝑛𝑡𝑛22𝜆𝑛𝐴𝑦𝑛,𝑦𝑛𝑞2𝑢𝑛𝑦𝑛,𝑦𝑛𝑡𝑛+2𝜆𝑛𝐴𝑦𝑛,𝑦𝑛𝑡𝑛𝑢𝑛𝑞2𝑢𝑛𝑦𝑛2𝑦𝑛𝑡𝑛2+2𝑢𝑛𝑦𝑛𝜆𝑛𝐴𝑦𝑛,𝑡𝑛𝑦𝑛.(3.2) In addition, we have 𝑢𝑛𝑦𝑛𝜆𝑛𝐴𝑦𝑛,𝑡𝑛𝑦𝑛=𝑢𝑛𝑦𝑛𝜆𝑛𝐴𝑢𝑛,𝑡𝑛𝑦𝑛+𝜆𝑛𝐴𝑢𝑛𝐴𝑦𝑛,𝑡𝑛𝑦𝑛𝑠𝑛𝑢𝑛𝜆𝑛𝐴𝑢𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛,𝑡𝑛𝑦𝑛+𝜆𝑛𝑠𝑛1𝐴𝑢𝑛,𝑡𝑛𝑦𝑛+𝜆𝑛𝑘𝑢𝑛𝑦𝑛𝑡𝑛𝑦𝑛,(3.3) and by (2.3), we obtain 𝑢𝑛𝜆𝑛𝐴𝑢𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛,𝑡𝑛𝑦𝑛=𝑢𝑛𝜆𝑛𝐴𝑢𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛,1𝑠𝑛𝑡𝑛𝑢𝑛+𝑠𝑛𝑡𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛1𝑠𝑛𝜆𝑛𝐴𝑢𝑛𝑡𝑛𝑦𝑛+𝑦𝑛𝑢𝑛+𝑠𝑛𝑢𝑛𝜆𝑛𝐴𝑢𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛,𝑡𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛1𝑠𝑛𝜆𝑛𝐴𝑢𝑛𝑡𝑛𝑦𝑛+𝑦𝑛𝑢𝑛.(3.4) It follows from (3.2)–(3.4) that 𝑡𝑛𝑞2𝑢𝑛𝑞2𝑢𝑛𝑦𝑛2𝑦𝑛𝑡𝑛2+2𝑠𝑛1𝑠𝑛𝜆𝑛𝐴𝑢𝑛𝑡𝑛𝑦𝑛+𝑦𝑛𝑢𝑛+2𝜆𝑛1𝑠𝑛𝐴𝑢𝑛𝑡𝑛𝑦𝑛+2𝜆𝑛𝑘𝑢𝑛𝑦𝑛𝑡𝑛𝑦𝑛𝑢𝑛𝑞2𝑢𝑛𝑦𝑛2𝑦𝑛𝑡𝑛2+𝑠𝑛1𝑠𝑛2𝑏2𝐴𝑢𝑛2+𝑡𝑛𝑦𝑛2+𝑦𝑛𝑢𝑛2+1𝑠𝑛𝑏2𝐴𝑢𝑛2+𝑡𝑛𝑦𝑛2+𝑘𝑏𝑡𝑛𝑦𝑛2+𝑦𝑛𝑢𝑛2=𝑢𝑛𝑞2𝑠𝑛𝑏𝑘𝑢𝑛𝑦𝑛22𝑠𝑛1𝑘𝑏𝑡𝑛𝑦𝑛2+31𝑠𝑛𝑏2𝐴𝑢𝑛2𝑢𝑛𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2.(3.5) Setting 𝑤𝑛=𝛽𝑛𝑥𝑛+(1𝛽𝑛)𝑡𝑛. Therefore, from (1.14), (3.1), and (3.5), we get the following: 𝑧𝑛𝑞2𝛼𝑛𝑥𝑛𝑞2+1𝛼𝑛𝑊𝑛𝑤𝑛𝑞2𝛼𝑛𝑥𝑛𝑞2+1𝛼𝑛𝑤𝑛𝑞2𝛼𝑛𝑥𝑛𝑞2+1𝛼𝑛𝛽𝑛𝑥𝑛𝑞2+1𝛼𝑛1𝛽𝑛𝑡𝑛𝑞2𝛼𝑛+1𝛼𝑛𝛽𝑛+1𝛼𝑛1𝛽𝑛𝑥𝑛𝑞2+31𝛼𝑛1𝛽𝑛1𝑠𝑛𝑏2𝐴𝑢𝑛2𝑥𝑛𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2.(3.6) So, 𝑞𝐶𝑛 and hence 𝐹𝐶𝑛 for all 𝑛1. It is easy to see that 𝐶𝑛 is closed and convex for all 𝑛1. This implies that {𝑥𝑛} and {𝑢𝑛} are well defined.Step 2 ({𝑥𝑛} is a Cauchy sequence). It is easy to see that 𝐹 is closed and convex. From 𝑥𝑛+1=𝑃𝐶𝑛+1𝑥0𝐶𝑛+1𝐶𝑛 and 𝑥𝑛=𝑃𝐶𝑛𝑥0, for any 𝑞𝐹, we have 𝑥𝑛𝑥0𝑥𝑛+1𝑥0𝑞𝑥0.(3.7) So {𝑥𝑛} is bounded, and lim𝑛𝑥𝑛𝑥0 exists. So it follows from (3.1), (3.6), and the continuity of 𝐴 that {𝑢𝑛}, {𝑧𝑛}, and {𝐴𝑢𝑛} are bounded. By the construction of 𝐶𝑛, we have 𝐶𝑚𝐶𝑛 and 𝑥𝑚=𝑃𝐶𝑚𝑥0𝐶𝑛 for any positive integer 𝑚𝑛. So from (2.2), we have 𝑥𝑚𝑥𝑛2𝑥𝑚𝑥02𝑥𝑛𝑥02.(3.8) Letting 𝑚,𝑛 in (3.8), we have 𝑥𝑚𝑥𝑛0, which implies that {𝑥𝑛} is a Cauchy sequence. So there exists 𝑧𝐶 such that 𝑥𝑛𝑧(𝑛).Step 3 (lim𝑛𝑤𝑛𝑊𝑤𝑛=0). From (3.8), we have lim𝑛𝑥𝑛𝑥𝑛+1=0.(3.9) Since 𝑥𝑛+1𝐶𝑛+1, by (3.9) and condition (iii), we obtain that 𝑧𝑛𝑥𝑛+1𝑥𝑛𝑥𝑛+1+31𝑠𝑛𝑏2𝐴𝑢𝑛20(𝑛).(3.10) So lim𝑛𝑧𝑛𝑥𝑛=0.(3.11) Since 𝑧𝑛𝑥𝑛=1𝛼𝑛𝑥𝑛𝑊𝑛𝑤𝑛,(3.12) from (3.11) and condition (i), we have lim𝑛𝑥𝑛𝑊𝑛𝑤𝑛=0.(3.13) For any 𝑞𝐹, from (3.1) and (3.5), we obtain that 𝑤𝑛𝑞2𝛽𝑛𝑥𝑛𝑞2+1𝛽𝑛𝑡𝑛𝑞2𝛽𝑛𝑥𝑛𝑞2+1𝛽𝑛𝑢𝑛𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2𝑥𝑛𝑞21𝛽𝑛𝑟𝑛2𝛽𝑟𝑛𝐵𝑥𝑛𝐵𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2.(3.14) Therefore, we have 𝑧𝑛𝑞2𝛼𝑛𝑥𝑛𝑞2+1𝛼𝑛𝑤𝑛𝑞2𝑥𝑛𝑞21𝛼𝑛1𝛽𝑛𝑟𝑛2𝛽𝑟𝑛𝐵𝑥𝑛𝐵𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2,(3.15) which implies that 1𝛼𝑛1𝛽𝑛𝑟𝑛2𝛽𝑟𝑛𝐵𝑥𝑛𝐵𝑞2𝑥𝑛𝑞2𝑧𝑛𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2𝑧𝑛𝑥𝑛𝑧𝑛+𝑥𝑞𝑛𝑞+31𝑠𝑛𝑏2𝐴𝑢𝑛2.(3.16) Combining the above inequality, (3.11) and conditions (i)–(iii), we have lim𝑛𝐵𝑥𝑛𝐵𝑞=0.(3.17) It follows from Lemma 2.6 that 𝑢𝑛𝑞2=𝑇𝑟𝑛𝑥𝑛𝑟𝑛𝐵𝑥𝑛𝑇𝑟𝑛𝑞𝑟𝑛𝐵𝑞2𝑥𝑛𝑟𝑛𝐵𝑥𝑛𝑞𝑟𝑛𝐵𝑞,𝑢𝑛=1𝑞2𝑥𝑛𝑟𝑛𝐵𝑥𝑛𝑞𝑟𝑛𝐵𝑞2+𝑢𝑛𝑞2𝑥𝑛𝑟𝑛𝐵𝑥𝑛𝑞𝑟𝑛𝑢𝐵𝑞𝑛𝑞212𝑥𝑛𝑞2+𝑢𝑛𝑞2𝑥𝑛𝑢𝑛2+2𝑟𝑛𝑥𝑛𝑢𝑛,𝐵𝑥𝑛𝐵𝑞𝑟2𝑛𝐵𝑥𝑛𝐵𝑞2.(3.18) Therefore, 𝑢𝑛𝑞2𝑥𝑛𝑞2𝑥𝑛𝑢𝑛2+2𝑟𝑛𝑥𝑛𝑢𝑛,𝐵𝑥𝑛𝐵𝑞𝑟2𝑛𝐵𝑥𝑛𝐵𝑞2.(3.19) By (3.5) and (3.19), we have 𝑧𝑛𝑞2𝛼𝑛𝑥𝑛𝑞2+1𝛼𝑛𝑤𝑛𝑞2𝛼𝑛𝑥𝑛𝑞2+1𝛼𝑛𝛽𝑛𝑥𝑛𝑞2+1𝛽𝑛𝑡𝑛𝑞2𝛼𝑛𝑥𝑛𝑞2+1𝛼𝑛𝛽𝑛𝑥𝑛𝑞2+1𝛽𝑛𝑢𝑛𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2𝑥𝑛𝑞21𝛼𝑛1𝛽𝑛𝑥𝑛𝑢𝑛2+2𝑟𝑛1𝛼𝑛1𝛽𝑛𝑥𝑛𝑢𝑛,𝐵𝑥𝑛𝐵𝑞1𝛼𝑛1𝛽𝑛𝑟2𝑛𝐵𝑥𝑛𝐵𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2.(3.20) It follows from (3.20) that 1𝛼𝑛1𝛽𝑛𝑥𝑛𝑢𝑛2𝑥𝑛𝑞2𝑧𝑛𝑞2+2𝑟𝑛1𝛼𝑛1𝛽𝑛𝑥𝑛𝑢𝑛,𝐵𝑥𝑛𝐵𝑞1𝛼𝑛1𝛽𝑛𝑟2𝑛𝐵𝑥𝑛𝐵𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2𝑧𝑛𝑥𝑛𝑧𝑛+𝑥𝑞𝑛𝑥𝑞+2𝑒𝑛𝑢𝑛𝐵𝑥𝑛𝐵𝑞+𝑒2𝐵𝑥𝑛𝐵𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2.(3.21) Therefore, from (3.11), (3.17), (3.21), and conditions (i), (iii), lim𝑛𝑥𝑛𝑢𝑛=0,(3.22) which implies that 𝑢𝑛𝑧(𝑛). And from (3.1), (3.5), and (3.6), we have 𝑧𝑛𝑞2𝛼𝑛𝑥𝑛𝑞2+1𝛼𝑛𝛽𝑛𝑥𝑛𝑞2+1𝛼𝑛1𝛽𝑛𝑡𝑛𝑞2𝑥𝑛𝑞21𝛼𝑛1𝛽𝑛𝑠𝑛𝑢𝑏𝑘𝑛𝑦𝑛21𝛼𝑛1𝛽𝑛2𝑠𝑛𝑡1𝑘𝑏𝑛𝑦𝑛2+31𝑠𝑛𝑏2𝐴𝑢𝑛2.(3.23) Thus it follows that 1𝛼𝑛1𝛽𝑛𝑠𝑛𝑢𝑏𝑘𝑛𝑦𝑛2+1𝛼𝑛1𝛽𝑛2𝑠𝑛𝑡1𝑘𝑏𝑛𝑦𝑛2𝑥𝑛𝑞2𝑧𝑛𝑞2+31𝑠𝑛𝑏2𝐴𝑢𝑛2𝑧𝑛𝑥𝑛𝑧𝑛+𝑥𝑞𝑛𝑞+31𝑠𝑛𝑏2𝐴𝑢𝑛2.(3.24) Therefore, from (3.11), (3.24) and conditions (i)–(iii), we obtain lim𝑛𝑢𝑛𝑦𝑛=0 and lim𝑛𝑡𝑛𝑦𝑛=0. Furthermore, we have lim𝑛𝑡𝑛𝑢𝑛=0,(3.25) which implies that 𝑡𝑛𝑧(𝑛). It follows from (3.22) and (3.25) that lim𝑛𝑡𝑛𝑥𝑛=0.(3.26) Note that 𝑤𝑛𝑥𝑛=(1𝛽𝑛)(𝑡𝑛𝑥𝑛), so by (3.26) and condition (i), we obtain that lim𝑛𝑤𝑛𝑥𝑛=0,(3.27) which implies that 𝑤𝑛𝑧(𝑛). Note that 𝑤𝑛𝑊𝑤𝑛𝑤𝑛𝑥𝑛+𝑥𝑛𝑊𝑛𝑤𝑛+𝑊𝑛𝑤𝑛𝑊𝑤𝑛.(3.28) Therefore, by (3.13), (3.27), (3.28) and Lemma 2.5, we have lim𝑛𝑤𝑛𝑊𝑤𝑛=0.(3.29)Step 4 (𝑧𝐹). Since 𝑤𝑛𝑧(𝑛) and 𝑊 is nonexpansive, by (3.29), we have 𝑧𝑊𝑧𝑧𝑤𝑛+𝑤𝑛𝑊𝑤𝑛+𝑊𝑤𝑛𝑊𝑧0(𝑛).(3.30) So 𝑧=𝑊𝑧, that is,𝑧𝐹(𝑊)=𝑖=1𝐹(𝑇𝑖).
Next we show that 𝑧𝐺MEP. Indeed, from (H2) and (1.14), we get 𝐵𝑥𝑛,𝑦𝑢𝑛𝑢+𝜑(𝑦)𝜑𝑛+1𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛Θ𝑦,𝑢𝑛.(3.31) Put 𝑦𝑡=𝑡𝑦+(1𝑡)𝑧,𝑡(0,1] and 𝑦𝐶. So 𝑦𝑡𝐶. By (3.31), we have 𝐵𝑦𝑡,𝑦𝑡𝑢𝑛𝐵𝑦𝑡,𝑦𝑡𝑢𝑛𝐵𝑥𝑛,𝑦𝑡𝑢𝑛𝑦𝜑𝑡𝑢+𝜑𝑛𝑦𝑡𝑢𝑛,𝑢𝑛𝑥𝑛𝑟𝑛𝑦+Θ𝑡,𝑢𝑛=𝐵𝑦𝑡𝐵𝑢𝑛,𝑦𝑡𝑢𝑛+𝐵𝑢𝑛𝐵𝑥𝑛,𝑦𝑡𝑢𝑛𝑦𝜑𝑡𝑢+𝜑𝑛𝑦𝑡𝑢𝑛,𝑢𝑛𝑥𝑛𝑟𝑛𝑦+Θ𝑡,𝑢𝑛𝐵𝑢𝑛𝐵𝑥𝑛𝑦𝑡𝑢𝑛𝑦𝜑𝑡𝑢+𝜑𝑛𝑦𝑡𝑢𝑛𝑢𝑛𝑥𝑛𝑟𝑛𝑦+Θ𝑡,𝑢𝑛.(3.32) Let 𝑛 in (3.32), since 𝐵 is nonexpansive and 𝜑 is lower semicontinuous, by (3.22), condition (ii) and (H4), we have 𝐵𝑦𝑡,𝑦𝑡𝑦𝑧𝜑𝑡𝑦+𝜑(𝑧)+Θ𝑡,𝑧.(3.33) So, from (H1), (H4), and the above inequality, we obtain 𝑦0=Θ𝑡,𝑦𝑡𝑦+𝜑𝑡𝑦𝜑𝑡𝑦𝑡Θ𝑡𝑦,𝑦+(1𝑡)Θ𝑡𝑦,𝑧+𝑡𝜑(𝑦)+(1𝑡)𝜑(𝑧)𝜑𝑡Θ𝑦=𝑡𝑡𝑦,𝑦+𝜑(𝑦)𝜑𝑡+Θ𝑦(1𝑡)𝑡𝑦,𝑧+𝜑(𝑧)𝜑𝑡Θ𝑦𝑡𝑡𝑦,𝑦+𝜑(𝑦)𝜑𝑡+(1𝑡)𝑡𝐵𝑦𝑡,𝑦𝑧,(3.34) that is, Θ𝑦𝑡𝑦,𝑦+𝜑(𝑦)𝜑𝑡+(1𝑡)𝐵𝑦𝑡,𝑦𝑧0.(3.35) Letting 𝑡0 in the above inequality, we obtain for each 𝑦𝐶, Θ(𝑧,𝑦)+𝜑(𝑦)𝜑(𝑧)+𝐵𝑧,𝑦𝑧0.(3.36) This implies that 𝑧𝐺MEP.
Finally, we show that 𝑧VI(𝐶,𝐴). Define a mapping 𝑇 as Lemma 2.2. Let (𝑣,𝑢)𝐺(𝑇). Since 𝑢𝐴𝑣𝑁𝐶𝑣 and 𝑡𝑛𝐶, we have 𝑣𝑡𝑛,𝑢𝐴𝑣0. Since 𝑡𝑛=𝑃𝐶(𝑢𝑛𝜆𝑛𝐴𝑦𝑛), we have 𝑣𝑡𝑛,𝑡𝑛𝑢𝑛𝜆𝑛𝐴𝑦𝑛0,(3.37) and hence 𝑣𝑡𝑛,𝑢𝑣𝑡𝑛,𝐴𝑣𝑣𝑡𝑛,𝐴𝑣𝑣𝑡𝑛,𝑡𝑛𝑢𝑛𝜆𝑛+𝐴𝑦𝑛=𝑣𝑡𝑛,𝐴𝑣𝐴𝑦𝑛𝑡𝑛𝑢𝑛𝜆𝑛=𝑣𝑡𝑛,𝐴𝑣𝐴𝑡𝑛+𝑣𝑡𝑛,𝐴𝑡𝑛𝐴𝑦𝑛𝑣𝑡𝑛,𝑡𝑛𝑢𝑛𝜆𝑛𝑣𝑡𝑛𝐴𝑡𝑛𝐴𝑦𝑛𝑣𝑡𝑛𝑡𝑛𝑢𝑛𝜆𝑛.(3.38) Since lim𝑛𝑡𝑛𝑦𝑛=0, and 𝐴 is Lipschitz continuous, by (3.25) and condition (ii), we deduce that 𝑣𝑧,𝑢0. Since 𝑇 is maximal monotone, we have 𝑧𝑇10 and so 𝑧VI(𝐶,𝐴). Hence 𝑧𝐹.
Step 5 (𝑧=𝑃𝐹𝑥0). Put 𝑧=𝑃𝐹𝑥0. Since 𝑥𝑛=𝑃𝐶𝑛𝑥0, 𝑧𝐹 and the norm is lower semicontinuous, we have 𝑧𝑥0𝑧𝑥0liminf𝑛𝑥𝑛𝑥0=lim𝑛𝑥𝑛𝑥0𝑧𝑥0,(3.39) that is, 𝑧𝑥0=𝑧𝑥0. Hence 𝑧=𝑧=𝑃𝐹𝑥0, since 𝑧 is the unique element in 𝐹 that minimizes the distance from 𝑥0.
Thus, {𝑥𝑛} converges strongly to 𝑃𝐹𝑥0.

Remark 3.2. Theorem 3.1 mainly improves the results of Inchan [10]. To be more precise, Theorem 3.1 improves and extends Theorem 3.1 of [10] from the following several aspects:(i)from a single nonexpansive mapping to an infinite family of strictly pseudocontractive mappings,(ii)from generalized equilibrium problems to generalized mixed equilibrium problems,(iii)from hybrid extragradient methods to hybrid extragradient-like methods,(iv)the condition of 𝐴 relaxes to monotone, Lipschitz continuous.

4. Application

Theorem 4.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ be a bifunction from 𝐶×𝐶 into 𝑅 satisfying (H1)–(H4) and 𝜑𝐶𝑅 be a lower semicontinuous and convex function with (A1) or (A2). Let 𝐴𝐶𝐻 be a monotone, 𝑘-Lipschitz continuous mapping and 𝑇𝐶𝐶 be a 𝜉-strictly pseudocontractive mapping. Let 𝑇𝑖𝐶𝐶 be a 𝜇𝑖-strictly pseudocontractive mapping with 𝐹=𝑖=1𝐹(𝑇𝑖)VI(𝐶,𝐴)𝐺MEP and {𝛾𝑖} be a real sequence such that 0<𝛾𝑖𝑏<1, for all 𝑖1. Let the sequence {𝑥𝑛} be generated 𝐶1=𝐶𝐻,𝑥1=𝑃𝐶𝑥0, 𝑢𝑛𝑢𝐶suchthatΘ𝑛,𝑦+(𝐼𝑇)𝑥𝑛,𝑦𝑢𝑛𝑢+𝜑(𝑦)𝜑𝑛+1𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛𝑦0,𝑦𝐶,𝑛=𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛,𝑧𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑊𝑛𝛽𝑛𝑥𝑛+1𝛽𝑛𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑦𝑛,𝐶𝑛+1=𝑧𝐶𝑛𝑧𝑛𝑥𝑧𝑛,𝑥𝑧𝑛+1=𝑃𝐶𝑛+1𝑥0,𝑛0.(4.1) Assume that the control sequence {𝛼𝑛}, {𝛽𝑛}[0,1], {𝑟𝑛}(0,1𝜉) and {𝜆𝑛}(0,1/2𝑘) satisfy the following conditions:(i)limsup𝑛𝛼𝑛<1,limsup𝑛𝛽𝑛<1, (ii)0<𝑎𝜆𝑛𝑏<1/2𝑘, 0<𝑑𝑟𝑛𝑒<1𝜉. Then {𝑥𝑛} converges strongly to 𝑃𝐹𝑥0.

Proof. A 𝜉-strictly pseudocontractive mapping is (1𝜉)/2-inverse-strongly monotone. Then taking 𝑠𝑛=1, for all 𝑛1 in Theorem 3.1, we obtain the conclusion.

Acknowledgment

The authors are extremely grateful to the referees for their useful suggestions that improved the content of the paper.

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