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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 804642, 15 pages
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.
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.
Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of and be a mapping of into . We denote by the set of fixed points of and by the metric projection of onto . We also denote by the set of all real numbers.
Recall the following definitions.(i) is called monotone if (ii) is called -inverse-strongly monotone if there exists a positive constant such that (iii) is called -Lipschitz continuous if there exists a positive constant such that Clearly, every inverse strongly monotone mapping is Lipschitz continuous and monotone.
A mapping is said to be -strictly pseudocontractive if there exists a constant such that
It is known that if is a 0-strictly pseudocontractive mapping, then is a nonexpansive mapping. So the class of -strictly pseudocontractive mappings includes the class of nonexpansive mappings.
Let be a bifunction. The equilibrium problem for is to find that such that The set of solutions of problem (1.5) is denoted by .
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, [1–3]. In 1997, Combettes and Hirstoaga  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  introduced the following generalized mixed equilibrium problem of finding such that where is a nonlinear mapping, and is a function. The set of solutions of problem (1.7) is denoted by .
In the case of and , then problem (1.7) reduces to problem (1.5). In the case of , and , then problem (1.7) reduces to problem (1.6). In the case of , problem (1.7) reduces to the generalized equilibrium problem. In the case of , problem (1.7) reduces to the following mixed equilibrium problem of finding such that which was considered by Ceng and Yao . 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, )
Recently, S. Takahashi and W. Takahashi  introduced the following iteration process: and used this iteration process to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a generalized equilibrium problem in a Hilbert space.
In 2008, Bnouhachem et al.  introduced the following new extragradient iterative method. Let be a closed convex subset of a real Hilbert , be an -inverse strongly monotone mapping of into , and let be a nonexpansive mapping of into itself such that . Let the sequences , be given by where , , and satisfy some parameters controlling conditions. They proved that the sequence converges strongly to a common element of .
In 2010, Ceng et al.  introduced the following hybrid extragradient-like method. Let be a nonempty closed convex subset of a real Hilbert space , be a monotone, -Lipschitz continuous mapping, and let be a nonexpansive mapping such that . Let the sequences , , and be defined by Under the suitable conditions, they proved the sequences , , converge strongly to the same point .
In 2010, Inchan  introduced a new iterative scheme by the hybrid extragradient method in a Hilbert space as follows: ,, , and let where , , and satisfy some parameters controlling conditions. They proved that and 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  defined the mapping as follows: where are real numbers such that , , where is a -strictly pseudocontractive mapping of into itself and . It follows from  that is nonexpansive and . Nonexpansivity of each ensures the nonexpansivity of .
Motivated and inspired by the above work, in this paper, we introduced the following new iterative scheme by the extragradient-like method: , where and is a monotone, -Lipschitz continuous mapping, and is a -inverse strongly monotone mapping. Then under the suitable conditions, we derive some strong convergence results.
Let be a nonempty closed and convex subset of a Hilbert space , for any , and there exists a unique nearest point in , denoted by such that The projection operator is nonexpansive. Moreover, is characterized by the following properties: for every and ,
Suppose that is monotone and continuous. Then the solutions of the variational inequality can be characterized as solutions of the so-called Minty variational inequality:
In what follows, we shall make use of the following lemmas.
Lemma 2.1. Let be a real Hilbert space. Then for any , we have(i),
We denote by the normal cone for at a point , that is . In the following, we shall use the following Lemma.
Lemma 2.2 (see ). Let be a nonempty closed convex subset of a Banach space , and let be a monotone and hemicontinuous operator of into . Let be an operator defined as follows: Then is maximal monotone, and .
Lemma 2.3 (see ). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be nonexpansive mappings of into itself such that and be real numbers such that for every . Then for any and , the limit exists.
Using Lemma 2.3, define the mapping of into itself as follows: Such a mapping is called the modified mapping generalized by , and .
Lemma 2.4 (see ). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be nonexpansive mappings of into itself such that and be real numbers such that for every . Then is a nonexpansive mapping satisfying that .
Lemma 2.5 (see ). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be nonexpansive mappings of into itself such that and be real numbers such that for every . If is any bounded subset of , then
For solving the equilibrium problem, let us assume that satisfies the following conditions: for all , is monotone, that is, for all , for each is weakly upper semicontinuous, for each is convex and lower semicontinuous, for each and , there exists a bounded subset and such that for any , is a bounded set.
Lemma 2.6 (see ). Let be a closed subset of . Let be a lower semicontinuous and convex function, and be a bifunction from to satisfying (H1)–(H4). For and , define a mapping as follows: for all . Assume that either (A1) or (A2) holds. Then the following results hold: (1) for each , and is single valued,(2) is firmly nonexpansive, that is, for all ,(3), (4) is closed and convex.
3. Strong Convergence Theorems
Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from into satisfying (H1)–(H4) and be a lower semicontinuous and convex function with (A1) or (A2). Let be a monotone, -Lipschitz continuous mapping and be a -inverse-strongly monotone mapping. Let be a -strictly pseudocontractive mapping with and be a real sequence such that , for all . Assume that the control sequences , , , and satisfy the following conditions:(i),
(ii),(iii) and for all .
Then the sequence defined by (1.14) converges strongly to .
Proof. We divide the proof into several steps.
Step 1 ( is well defined). Indeed, for any . Put . Since , , and is -inverse-strongly monotone and , for any , we have It follows from (2.2) and (2.4) that In addition, we have and by (2.3), we obtain It follows from (3.2)–(3.4) that Setting . Therefore, from (1.14), (3.1), and (3.5), we get the following: So, and hence for all . It is easy to see that is closed and convex for all . This implies that and are well defined.Step 2 ( is a Cauchy sequence). It is easy to see that is closed and convex. From and , for any , we have So is bounded, and exists. So it follows from (3.1), (3.6), and the continuity of that , , and are bounded. By the construction of , we have and for any positive integer . So from (2.2), we have Letting in (3.8), we have , which implies that is a Cauchy sequence. So there exists such that .Step 3 (). From (3.8), we have Since , by (3.9) and condition (iii), we obtain that So Since from (3.11) and condition (i), we have For any , from (3.1) and (3.5), we obtain that Therefore, we have which implies that Combining the above inequality, (3.11) and conditions (i)–(iii), we have It follows from Lemma 2.6 that Therefore, By (3.5) and (3.19), we have It follows from (3.20) that Therefore, from (3.11), (3.17), (3.21), and conditions (i), (iii), which implies that . And from (3.1), (3.5), and (3.6), we have Thus it follows that Therefore, from (3.11), (3.24) and conditions (i)–(iii), we obtain and . Furthermore, we have which implies that . It follows from (3.22) and (3.25) that Note that , so by (3.26) and condition (i), we obtain that which implies that . Note that Therefore, by (3.13), (3.27), (3.28) and Lemma 2.5, we have Step 4 (). Since and is nonexpansive, by (3.29), we have So , that is,.
Next we show that . Indeed, from (H2) and (1.14), we get Put and . So . By (3.31), we have Let in (3.32), since is nonexpansive and is lower semicontinuous, by (3.22), condition (ii) and (H4), we have So, from (H1), (H4), and the above inequality, we obtain that is, Letting in the above inequality, we obtain for each , This implies that .
Finally, we show that . Define a mapping as Lemma 2.2. Let . Since and , we have . Since , we have and hence Since , and is Lipschitz continuous, by (3.25) and condition (ii), we deduce that . Since is maximal monotone, we have and so . Hence .Step 5 (). Put . Since , and the norm is lower semicontinuous, we have that is, . Hence , since is the unique element in that minimizes the distance from .
Thus, converges strongly to .
Remark 3.2. Theorem 3.1 mainly improves the results of Inchan . To be more precise, Theorem 3.1 improves and extends Theorem 3.1 of  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 relaxes to monotone, Lipschitz continuous.
Theorem 4.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from into satisfying (H1)–(H4) and be a lower semicontinuous and convex function with (A1) or (A2). Let be a monotone, -Lipschitz continuous mapping and be a -strictly pseudocontractive mapping. Let be a -strictly pseudocontractive mapping with and be a real sequence such that , for all . Let the sequence be generated , Assume that the control sequence , , and satisfy the following conditions:(i), (ii), . Then converges strongly to .
Proof. A -strictly pseudocontractive mapping is -inverse-strongly monotone. Then taking , for all in Theorem 3.1, we obtain the conclusion.
The authors are extremely grateful to the referees for their useful suggestions that improved the content of the paper.
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