ISRN Applied MathematicsVolume 2012 (2012), Article ID 482869, 22 pagesdoi:10.5402/2012/482869

Research Article

Hybrid Projection Algorithm for a New General System of Variational Inequalities in Hilbert Spaces

S. Imnang1,2

¹Department of Mathematics and Statistics, Faculty of Science, Thaksin University, Phatthalung Campus, Phatthalung 93110, Thailand²Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 5 November 2012; Accepted 18 December 2012

Academic Editors: I. K. Argyros, H. Y. Chung, and Y.-G. Zhao

Copyright © 2012 S. Imnang. 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.

A new general system of variational inequalities in a real Hilbert space is introduced and studied. The solution of this system is shown to be a fixed point of a nonexpansive mapping. We also introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Several strong convergence theorems of the proposed hybrid projection algorithm are established by using the demiclosedness principle. Our results extend and improve recent results announced by many others.

1. Introduction

Let be a real Hilbert space with inner product and be a nonempty closed convex subset of . Recall that is nonexpansive if , for all . The fixed point set of is denoted by .

Let be a nonlinear mapping. Then is called (i)-strongly monotone, if there exists a positive real number such that (ii)-Lipschitz continuous (or Lipschitzian), if there exists a constant such that (iii)relaxed -cocoercive, if there exist two constants such that for , is -strongly monotone. This class of mappings is more general than the class of strongly monotone mappings.

Next, we consider the following variational inequality problem of finding such that The set of solutions of the variational inequality (1.4) is denoted by . Variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, free, moving, equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. The variational inequality problem has been extensively studied in the literature, see, Piri [1], Qin et al. [2], Shehu [3], Wangkeeree and Preechasilp [4], Yao et al. [5], Yao et al. [6], and the references therein.

For solving the variational inequality problem in the finite-dimensional Euclidean space under the assumption that a set is closed and convex, a mapping of into is monotone and -Lipschitz-continuous and is nonempty, Korpelevič [7] introduced the following called extragradient method: for every , where and is the projection of onto . He showed that the sequences and generated by this iterative process converge to the same point . Later on, the idea of Korpelevich was generalized and extended by many authors, see for example, [1–5, 8, 9] for finding a common element of the set of fixed points and the set of solutions of the variational inequality.

We recall the following well-known result which is called the best approximation result or the projection lemma.

Lemma 1.1. For a given , satisfies the inequality where is the projection of onto a closed convex set .

Lemma 1.2. is a solution of the variational inequality if and only if satisfies the relation where is the projection of onto a closed convex set and is a constant.

Let be a nonempty closed convex subset of a real Hilbert space . Let for all be three mappings. In this paper, we focus on the problem of finding such that which is called a new general system of variational inequalities, where for all . In particular, if and , then problem (1.8) reduces to find such that which is called a general system of variational inequalities and defined by Ceng et al. [10]. If we add up the requirement that , then problem (1.9) reduces to find such that which is defined by Verma [11], and is called the new system of variational inequalities. Further, if we add up the requirement that and , then problem (1.10) reduces to the classical variational inequality . Ceng et al. [10] introduced and studied a relaxed extragradient method for finding a common element of the set of solutions of problem (1.9) for the and -inverse-strongly monotone mappings and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Some related works, we refer to see [9, 12–16].

Recently, in 2012, Ceng et al. [12] considered an iterative method for the system of problem (1.9) and obtained a strong convergence theorem for the two different systems of problem (1.9) and the set of fixed points of a strict pseudocontraction mapping in a real Hilbert space.

Let be a proper extended real-valued function and be a bifunction from to , where is the set of real numbers. Ceng and Yao [17] considered the following mixed equilibrium problem: The set of solutions of problem (1.11) is denoted by . It is easy to see that is a solution of problem (1.11) implies that .

If , then the problem (1.11) becomes the following equilibrium problem: The set of solution of (1.12) is denoted by .

If , then the problem (1.11) reduces to the convex minimization problem:

If and for all , where is a mapping from into , then problem (1.11) reduces to the classical variational inequality and . For solving problem (1.11), Ceng and Yao [17] introduced a hybrid iterative scheme for finding a common element of the set and the set of common fixed points of finite many nonexpansive mappings in a Hilbert space. Some related works, we refer to see [3, 5, 9, 15].

Recently, in 2012, Kumam and Katchang [14] introduced 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.

In this paper, motivated and inspired by the idea of Kumam and Katchang [14], we introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Starting with an arbitrary and let , and be the sequences generated by where for all , and . Using the demiclosedness principle for nonexpansive mappings, we show that the sequence converges strongly to a common element of those three sets under some control conditions. Our results extend and improve recent results announced by many others.

2. Preliminaries

In this section, we recall the well-known results and give some useful lemmas that are used in the next section.

Let be a nonempty closed convex subset of a real Hilbert space . For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies Obviously, this immediately implies that Recall that, is characterized by the following properties: and for all and ; see Goebel and Kirk [18] for more details.

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction and the set :(A1) for all ;(A2) is monotone, that is, for all ;(A3) For each , is weakly upper semicontinuous;(A4) For each , is convex;(A5) For each , is lower semicontinuous;(B1) For each and , there exist a bounded subset and such that for any , (B2) is a bounded set.

In the sequel we will need to use the following lemma.

Lemma 2.1 (see [19]). Let be a nonempty closed convex subset of . Let be a bifunction from to satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows: for all . Then the following conclusions hold: (1)for each , ;(2) is single-valued;(3) is firmly nonexpansive, that is, for any , (4);(5) is closed and convex.

We also need the following lemmas.

Lemma 2.2 (see [20]). Let be an inner product space. Then, for all and with , one has

Lemma 2.3. In a real Hilbert space , there holds the inequality

Lemma 2.4 (see [21]). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that (i); (ii) or . Then .

Lemma 2.5 (see [22]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .

Lemma 2.6 (see [18]). Demiclosedness principle. Assume that is a nonexpansive self-mapping of a nonempty closed convex subset of a real Hilbert space . If has a fixed point, then is demiclosed: that is, whenever is a sequence in converging weakly to some (for short, ), and the sequence converges strongly to some (for short, , it follows that . Here is the identity operator of .

In 2009, Kangtunyakarn and Suantai [23] introduced a new mapping called the -mapping. Let be a finite family of nonexpansive mappings of into itself. For each , let , where and . They defined the new mapping as follows: This mapping is called -mapping generated by and ,. Nonexpansivity of each ensures the nonexpansivity of .

Lemma 2.7 (see [23]). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a finite family of nonexpansive mappings of into itself with and let , , where , , for all , and for all . Let be the -mapping generated by and . Then .

3. Main Results

In this section, we prove strong convergence theorems of the iterative scheme (1.14) to a common element of the set of solutions of a new general system of variational inequalities for relaxed ()-cocoercive mappings, the set of fixed points of a nonexpansive mapping, and the set of solutions of a mixed equilibrium problem in a real Hilbert space.

The next lemmas are crucial for proving the main theorems.

Lemma 3.1. Let be a nonempty closed and convex subset of a real Hilbert space and let be -Lipschitzian and relaxed -cocoercive mappings for . Let defined by If , for , then is a nonexpansive mapping.

Proof. For all , we have It is well known that if is -Lipschitzian and relaxed -cocoercive, then is nonexpansive for all . By our assumption, we obtain is nonexpansive for all . It follows that is nonexpansive. Therefore, from (3.2), we obtain immediately that the mapping is nonexpansive.

Lemma 3.2. Let be a nonempty closed and convex subset of a real Hilbert space . Let be three possibly nonlinear mappings. For given , is a solution of problem (1.8) if and only if , and , where is the mapping defined as in Lemma 3.1.

Proof. Note that we can rewrite (1.8) as From Lemma 1.1, we can deduce that (3.3) is equivalent to It is easy to see that (3.4) is equivalent to , , and .

Throughout this paper, the set of fixed points of the mapping is denoted by .

Now we prove the strong convergence theorems of the algorithm (1.14) for solving problem (1.8), fixed point problem of nonexpansive mapping and mixed equilibrium problem.

Theorem 3.3. Let be a nonempty closed and convex subset of a real Hilbert space . Let be a function from to satisfying (A1)–(A5) and be a proper lower semicontinuous and convex function. Let the mappings be -Lipschitzian and relaxed -cocoercive for and be a nonexpansive self-mapping of such that . Assume that either (B1) or (B2) holds and that is an arbitrary point in . Let and be the sequences generated by where and , for and , are two sequences in such that (C1) and ;(C2);(C3) and . Then converges strongly to and is a solution of problem (1.8), where and .

Proof. Let and be a sequence of mappings defined as in Lemma 2.1. It follows from Lemma 3.2 that Put , and . Then , and By nonexpansiveness of , we have which implies that Thus, is bounded. Consequently, the sequences , , , , , , and are also bounded. Also, observe that On the other hand, from and , we have Putting in (3.11) and in (3.12), we have From the monotonicity of , we obtain that and hence Then, we have and hence It follows from (3.10) and (3.17) that Let for all . Then, we obtain By (3.18) and (3.19), we have This together with (C1)–(C3), we obtain that Hence, by Lemma 2.5, we get as . Consequently, Since we have that Next, we prove that . From Lemma 2.1 (3), we have Hence From Lemma 2.2, (3.8) and (3.26), we have It follows that From the conditions (C1), (C2) and (3.22), we obtain By (3.24) and (3.29), we have Next, we show that , and , as . From (3.8) and the fact that is -Lipschitzian and relaxed -cocoercive, we have Similarly, since is -Lipschitzian and relaxed -cocoercive mappings for , and , we can show that From (3.31) and (3.32), we have This together with (C1), (C2) and (3.2), we obtain that Next, we prove that as . From (2.2), (3.8) and nonexpansiveness of and , we get Therefore From (3.36), we have Hence This together with (C1), (C2), (21), and (3.34), we obtain Therefore From Lemma 2.3 and (2.3), it follows that This together with (3.30), (3.34), and (3.41), we obtain as . This together with (3.30) and (3.40), we obtain that Next, we show that where . Indeed, since and are two bounded sequences in , we can choose a subsequence of such that and Since , we obtain that as . Next, we show that .Since and , we obtain by Lemma 2.6 that .From (3.43) and (3.24), we obtain Furthermore, by Lemma 3.1, we have is nonexpansive. Then, we have hence . Again by Lemma 2.6, we have . Since and , we obtain that . From , we also obtain that . By using the same argument as that in the proof of [19, Theorem 3.1, pp. 1825], we can show that . Therefore .On the other hand, it follows from (2.4), (3.24), and as that Hence, we have which implies that This together with (C1) and (3.48), we have by Lemma 2.4 that converges strongly to . This completes the proof.

Theorem 3.4. Let be a nonempty closed and convex subset of a real Hilbert space . Let be a function from to satisfying (A1)–(A5) and be a proper lower semicontinuous and convex function. Let the mappings be -Lipschitzian and relaxed -cocoercive for and be a finite family of nonexpansive self-mappings of such that . Let , , where , , for all , and for all . Let be the -mapping generated by and . Assume that either (B1) or (B2) holds and that is an arbitrary point in . Let and be the sequences generated by If , for and , , are as in Theorem 3.3. Then converges strongly to and is a solution of problem (1.8), where and .

Proof. By Lemma 2.7, we obtain that is nonexpansive and . Hence, the result is obtained directly from Theorem 3.3.

If in Theorem 3.3, then, we obtain the following result.

Corollary 3.5. Let be a nonempty closed and convex subset of a real Hilbert space . Let be a function from to satisfying (A1)–(A5) and be a proper lower semicontinuous and convex function. Let the mappings be -Lipschitzian and relaxed -cocoercive for and be a nonexpansive self-mapping of such that . Let and be the sequences generated by If , for and the sequences , , and are as in Theorem 3.3, then converges strongly to and is a solution of problem (1.8), where and .

Acknowledgments

The author would like to thank Professor Dr. Suthep Suantai and the reviewer for careful reading, valuable comment, and suggestions on this paper. The author also would like to thank the Commission on Higher Education, the Thailand Research Fund, the Centre of Excellence in Mathematics, and Thaksin University for their financial support.

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