Abstract and Applied AnalysisVolume 2011 (2011), Article ID 562689, 24 pagesdoi:10.1155/2011/562689

Research Article

A New Iterative Algorithm for the Set of Fixed-Point Problems of Nonexpansive Mappings and the Set of Equilibrium Problem and Variational Inequality Problem

Atid Kangtunyakarn

Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Received 7 October 2010; Revised 25 January 2011; Accepted 11 February 2011

Academic Editor: Norimichi Hirano

Copyright © 2011 Atid Kangtunyakarn. Atid Kangtunyakarn. 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.

We introduce a new iterative scheme and a new mapping generated by infinite family of nonexpansive mappings and infinite real number. By using both of these ideas, we obtain strong convergence theorem for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of fixed-point problems of infinite family of nonexpansive mappings. Moreover, we apply our main result to obtain strong convergence theorems for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of common fixed point of pseudocontractive mappings.

1. Introduction

Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be a nonlinear mapping and let be a bifunction. A mapping of into itself is called nonexpansive if . We denote by the set of fixed points of (i.e., ). Goebel and Kirk [1] showed that is always closed convex, and also nonempty provided has a bounded trajectory.

A bounded linear operator on is called strongly positive with coefficient if there is a constant with the property

The equilibrium problem for is to find , such that The set of solutions of (1.2) is denoted by . Many problems in physics, optimization, and economics are seeking some elements of , see [2, 3]. Several iterative methods have been proposed to solve the equilibrium problem, see, for instance, [2–4]. In 2005, Combettes and Hirstoaga [3] introduced an iterative scheme for finding the best approximation to the initial data when EP() is nonempty and proved a strong convergence theorem.

The variational inequality problem is to find a point , such that The set of solutions of the variational inequality is denoted by . Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games reduce to find element of (1.2) and (1.3).

A mapping of into is called inverse-strongly monotone, see [5], if there exists a positive real number , such that for all .

The problem of finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed-point sets of a family of nonexpansive mapping (see [6, 7]).

The problem of finding a common element of and the set of all common fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and importance. Many iterative methods are purposed for finding a common element of the solutions of the equilibrium problem and fixed-point problem of nonexpansive mappings, see [8–10].

In 2007, S. Takahashi and W. Takahashi [10] introduced a general iterative method for finding a common element of and . They defined in the following way: where , and proved strong convergence of the scheme (1.5) to , where in the framework of a Hilbert space, under some suitable conditions on , and bifunction .

In this paper, by motive of (1.5), we prove strong convergence theorem for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of fixed-point problems by using a new mapping generated by infinite family of nonexpansive mapping and infinite real number. Moreover, we apply our main result to obtain strong convergence theorems for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of common fixed point of pseudocontractive mappings.

2. Preliminaries

In this section, we collect and give some useful lemmas that will be used for our main result in the next section.

Let be closed convex subset of a real Hilbert space , and let be the metric projection of onto , that is, for , satisfies the property The following characterizes the projection .

Lemma 2.1 (see [11]). Given and , then if and only if there holds the inequality

Lemma 2.2 (see [12]). Let be a uniformly convex Banach space, let be a nonempty closed convex subset of , and let be a nonexpansive mapping, then is demiclosed at zero.

Lemma 2.3 (see [13]). Let be a sequence of nonnegative real numbers satisfying where is a sequence in and is a sequence, such that (1), (2) then .

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions: (A1),(A2), that is, ,(A3)for all ,(A4)for all is convex and lower semicontinuous.

The following lemma appears implicitly in [2].

Lemma 2.4 (see [2]). Let be a nonempty closed convex subset of , and let be a bifunction of into satisfying (A1)–(A4). Let and , then there exists , such that for all .

Lemma 2.5 (see [3]). Assume that satisfies (A1)–(A4). For and , define a mapping as follows: for all , then the following hold: (1) is single valued, (2) is firmly nonexpansive, that is, (3), (4) is closed and convex.

Lemma 2.6 (see [14]). Let be a Hibert space, let be a nonempty closed convex subset of , and let be a mapping of into . Let , then for , where is the metric projection of onto .

Definition 2.7. Let be a nonempty convex subset of a real Hilbert space. Let be mappings of into itself. For each , let where and . For every , we define the mapping as follows: This mapping is called S-mapping generated by and .

Lemma 2.8. Let be a nonempty closed convex subset of a real Hilbert space. Let be nonexpansive mappings of into itself with , and let , where , and . For every , let be -mapping generated by and , then for every and exists.

Proof. Let and . Fix , then for every with , we have It follows that where and .For any , we have Since , we have . From (2.12), we have that is a Cauchy sequence. Hence, exists.

For every and , we define mapping and as follows: Such a mapping is called -mapping generated by and .

Remark 2.9. For each is nonexpansive and for every bounded subset of . To show this, let and be a bounded subset of , then we have Then, we have that is also nonexpansive, Indeed, observe that for each , By (2.11), we have This implies that for and , By letting , for any , we have It follows that

Lemma 2.10. Let be a nonempty closed convex subset of a real Hilbert space. Let be nonexpansive mappings of into itself with , and let , where , and . For every , let and be -mappings generated by and and , and , respectively, then .

Proof. It is easy to see that . For every , let and , then we have For and (2.20), we have as . This implies that .Again by (2.20), we have as . Hence, From , and (2.23), we obtain that . This implies that .

3. Main Result

Theorem 3.1. Let be a nonempty closed convex subset of a Hilbert space . Let be bifunctions from into satisfying (A1)–(A4). Let be a -inverse-strongly monotone mapping. Let be infinite family of nonexpansive mappings with , and let , where , and . For every , let and be -mappings generated by and and , and , respectively. Let be sequences generated by and where , such that . Assume that (i), (ii), (iii),then the sequence converge strongly to .

Proof. First, we show that is nonexpansive. Let . Since is -inverse-strongly monotone and , we have Thus, is nonexpansive. We will divide our proof into 5 steps.Step 1. We shall show that the sequence is bounded. Since By Lemma 2.5, we have and .Let . By nonexpansiveness of and , we have By induction, we can prove that is bounded and so is .Step 2. We will show that . By definition of , we have Since , by definition of , we have Similarly, From (3.6) and (3.7), we obtain By (3.8), we have It follows that This implies that It follows that It follows that Putting , then is bounded. By definition of , we have Substituting (3.13) and (3.14) into (3.5), we have where . By (3.15), Lemma 2.3, and conditions (i)–(iii), we obtain Step 3. We shall show that . Let . Since and is firmly nonexpansive, we have Hence, By (3.18), we have it implies that By (3.16) and condition (i), we have Let and by nonexpansiveness of , we have It implies that By (3.16) and condition (i), we have Since by (3.24) and (3.21), we have Since by (3.24), (3.16), and condition (i), we have Since again by (3.24), (3.16), and condition (i), we have Since by (3.21) and (3.30), we have Step 4. Putting , we will show that To show this inequality, take a subsequence of , such that Without loss of generality, we may assume that where . By nonexpansiveness of , (3.28), and Lemma 2.2, we have . By Lemma 2.6, we obtain that . Since as , we have . Since By (A2), we have In particular, By condition (A4), is lower semicontinuous and convex, and thus weakly semicontinuous. By (3.21) imply that in norm. Therefore, letting in (3.37), we have Replacing with , , we have , and using (A1), (A4), and (3.38), we obtain Hence, and for all . Letting and using assumption (A3), we can conclude that Therefore, .We will show that . By Lemma 2.10, we have . Assume that . Using Opial property, (3.32), , and Remark 2.9, we have This is a contradiction, then . Hence, .Since and , we have Step 5. Finally, we show that and converse strongly to . Putting , by nonexpansiveness of , , and , we have

From Step 4 and Lemma 2.3, we obtain that converse strongly to . By using (3.21), we have converse strongly to .

4. Application

Using our main theorem (Theorem 3.1), we obtain the following strong convergence theorems involving infinite family of -strict pseudocontractions.

To prove strong convergence theorem in this section, we need definition and lemma as follows.

Definition 4.1. A mapping is said to be a -strongly pseudocontraction mapping, if there exists , such that

Lemma 4.2 (see [15]). Let be a nonempty closed convex subset of a real Hilbert space and a -strict pseudocontraction. Define by for each . Then, as is nonexpansive, such that .

Theorem 4.3. Let be a nonempty closed convex subset of a Hilbert space . Let be bifunctions from into satisfying (A1)–(A4). Let be a -inverse-strongly monotone mapping. Let be infinite family of -pseudocontractions mappings with . Define a mapping by , , and let , where ,, and . For every , let and be -mappings generated by and and , and , respectively. Let be sequences generated by and where , such that . Assume that (i), (ii), (iii), then the sequence converges strongly to .

Proof. For every , by Lemma 4.2, we have that is nonexpansive mappings. From Theorem 3.1, we could have the desired conclusion.

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