Journal of Applied MathematicsVolume 2012 (2012), Article ID 812783, 10 pagesdoi:10.1155/2012/812783

Research Article

Approximation of Solutions of an Equilibrium Problem in a Banach Space

Hecai Yuan¹ and Guohong Shi²

¹School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China²College of Science, Hebei University of Engineering, Handan 056038, China

Received 15 December 2011; Accepted 4 February 2012

Academic Editor: Rudong Chen

Copyright © 2012 Hecai Yuan and Guohong Shi. 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.

An equilibrium problem is investigated based on a hybrid projection iterative algorithm. Strong convergence theorems for solutions of the equilibrium problem are established in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property.

1. Introduction

Equilibrium problems which were introduced by Fan [1] and Blum and Oettli [2] have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. It has been shown [3–8] that equilibrium, problems include variational inequalities, fixed point, the Nash equilibrium, and game theory as special cases. A number of iterative algorithms have recently been studying for fixed point and equilibrium problems, see [9–26] and the references therein. However, there were few results established in the framework of the Banach spaces. In this paper, we suggest and analyze a projection iterative algorithm for finding solutions of equilibrium in a Banach space.

2. Preliminaries

In what follows, we always assume that is a Banach space with the dual space . Let be a nonempty, closed, and convex subset of . We use the symbol to stand for the normalized duality mapping from to defined by

where denotes the generalized duality pairing of elements between and .

Let be the unit sphere of . is said to be strictly convex if for all with . It is said to be uniformly convex if for any there exists such that for any ,

It is known that a uniformly convex Banach space is reflexive and strictly convex; for details see [27] and the references therein.

Recall that a Banach space is said to have the Kadec-Klee property if a sequence of satisfies that , where denotes the weak convergence, and , where denotes the strong convergence, and then . It is known that if is uniformly convex, then enjoys the Kadec-Klee property; for details see [26] and the references therein.

is said to be smooth provided exists for all . It is also said to be uniformly smooth if the limit is attained uniformly for all .

It is well known that if is strictly convex, then is single valued; if is reflexive, and smooth, then is single valued and demicontinuous; for more details see [27, 28] and the references therein.

It is also well known that if is a nonempty, closed, and convex subset of a Hilbert space , and is the metric projection from onto , then is nonexpansive. This fact actually characterizes the Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [29] introduced a generalized projection operator in the Banach spaces which is an analogue of the metric projection in the Hilbert spaces.

Let be a smooth Banach space. Consider the functional defined by Notice that, in a Hilbert space , (2.3) is reduced to for all . The generalized projection is a mapping that is assigned to an arbitrary point , the minimum point of the functional ; that is, , where is the solution to the following minimization problem:

The existence and uniqueness of the operator follow from the properties of the functional and the strict monotonicity of the mapping ; see, for example, [27, 28]. In the Hilbert spaces, . It is obvious from the definition of the function that

Let be a mapping. Recall that a point in is said to be an asymptotic fixed point of if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . is said to be relatively nonexpansive if

The asymptotic behavior of a relatively nonexpansive mapping was studied in [27, 29, 30].

Let be a bifunction from to , where denotes the set of real numbers. In this paper, we consider the following equilibrium problem. Find such that We use to denote the solution set of the equilibrium problem (2.3). That is,

Given a mapping , let

Then if and only if is a solution of the following variational inequality. Find such that

To study the equilibrium problem (2.8), we may assume that satisfies the following conditions:

(A1) ;

(A2) is monotone, that is, ;

(A3)

(A4) for each , is convex and weakly lower semicontinuous.

In this paper, we study the problem of approximating solutions of equilibrium problem (2.8) based on a hybrid projection iterative algorithm in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property. To prove our main results, we need the following lemmas.

Lemma 2.1. Let be a strictly convex and uniformly smooth Banach space and a nonempty, closed, and convex subset of . Let be a bifunction from to satisfying (A1)–(A4). Let and . Then (a)(see [2]). There exists such that (b)(see [31]). Define a mapping by Then the following conclusions hold: (1) is single valued;(2) is a firmly nonexpansive-type mapping; that is, for all , (3); (4) is closed and convex;(5) is relatively nonexpansive.

Lemma 2.2 (see [29]). Let be a reflexive, strictly convex, and smooth Banach space and a nonempty, closed, and convex subset of . Let , and . Then if and only if

Lemma 2.3 (see [29]). Let be a reflexive, strictly convex, and smooth Banach space and a nonempty, closed, and convex subset of , and . Then

Lemma 2.4 (see [27]). Let be a reflexive, strictly convex, and smooth Banach space. Then one has the following

3. Main Results

Theorem 3.1. Let be a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property and a nonempty, closed, and convex subset of . Let be a bifunction from to satisfying (A1)–(A4) such that . Let be a sequence generated by the following manner: where is a real number sequence in , where is some positive real number. Then the sequence converges strongly to .

Proof. In view of Lemma 2.1, we see that is closed and convex. Next, we show that is closed and convex. It is not hard to see that is closed. Therefore, we only show that is convex. It is obvious that is convex. Suppose that is convex for some . Next, we show that is also convex for the same . Let and , where . It follows that where . From the above two inequalities, we can get that where . It follows that is closed and convex. This completes the proof that is closed, and convex.Next, we show that . It is obvious that . Suppose that for some . For any , we see from Lemma 2.1 that On the other hand, we obtain from (2.6) that Combining (3.4) with (3.5), we arrive at which implies that . This shows that . This completes the proof that .Next, we show that is a convergent sequence and strongly converges to , where . Since , we see from Lemma 2.2 that It follows from that By virtue of Lemma 2.3, we obtain that This implies that the sequence is bounded. It follows from (2.5) that the sequence is also bounded. Since the space is reflexive, we may assume that . Since is closed and convex, we see that . On the other hand, we see from the weakly lower semicontinuity of the norm that which implies that as . Hence, as . In view of the Kadec-Klee property of , we see that as . Notice that . It follows that Since and , we arrive at . This shows that is nondecreasing. It follows from the boundedness that exists. It follows that By virtue of , we find that It follows that In view of (2.5), we see that Since , we find that It follows that This implies that is bounded. Note that both and are reflexive. We may assume that . In view of the reflexivity of , we see that there exists an element such that . It follows that Taking on the both sides of the equality above yields that That is, , which in turn implies that . It follows that . Since enjoys the Kadec-Klee property, we obtain from (3.17) that . Since is demicontinuous, we find that . This implies from (3.16) and the Kadec-Klee property of that . This in turn implies that . Since is uniformly norm-to-norm continuous on any bounded sets, we find that Next, we show that . In view of Lemma 2.1, we find from that It follows from condition (A2) and (3.20) that In view of condition (A4), we obtain from (3.17) that For and , define . It follows that , which yields that . It follows from conditions (A1) and (A4) that That is, Letting , we find from condition (A3) that , . This implies that . This shows that .Finally, we prove that . Letting in (3.8), we see that In view of Lemma 2.2, we can obtain that . This completes the proof.

In the framework of the Hilbert spaces, we have the following.

Corollary 3.2. Let be a Hilbert space and a nonempty, closed, and convex subset of . Let be a bifunction from to satisfying (A1)–(A4) such that . Let be a sequence generated by the following manner: where is a real number sequence in , where is some positive real number. Then the sequence converges strongly to .

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