Abstract and Applied Analysis

Volume 2013 (2013), Article ID 952021, 6 pages

http://dx.doi.org/10.1155/2013/952021

Research Article

Existence Solutions of Vector Equilibrium Problems and Fixed Point of Multivalued Mappings

Kanokwan Sitthithakerngkiet and Somyot Plubtieng

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 30 September 2012; Accepted 15 December 2012

Academic Editor: Qamrul Hasan Ansari

Copyright © 2013 Kanokwan Sitthithakerngkiet and Somyot Plubtieng. 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.

Let be a nonempty compact convex subset of a topological vector space. In this paper-sufficient conditions are given for the existence of such that , where is the set of all fixed points of the multivalued mapping and is the set of all solutions for vector equilibrium problem of the vector-valued mapping . This leads us to generalize and improve some existence results in the recent references.

1. Introduction

Let be a Hausdorff topological vector space, a fixed point of such a multi-valued mapping , where denotes, the family of subsets of means a point in such that and the set of all fixed points of is denoted by . The study of fixed point theorems of multi-valued mappings started from von Neumann [1] in case of continuous mappings to multi-valued mappings. Since then, various notions of the fixed point theorems for the multi-valued mappings have been studied in [2–4]. Recently, the fixed point theorems for multi-valued mapping were generalized and improved by many authors: see, for example, [5–10].

On the other hand, the equilibrium problems were introduced by Blum and Oettli [11] and by Noor and Oettli [12] in 1994 as generalizations of variational inequalities and optimization problems. The equilibrium problem theory provides a novel and united treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. This theory has had a great impact and influence in the development of several branches of pure and applied sciences. Classical examples of equilibrium problems are variational inequalities, optimization problems, and complementarity problems. Presently, many results on the existence of solutions for vector variational inequalities (in short, VVI) and vector equilibrium problems (in short, VEP) have been established (see, e.g., [13–20]). For the more generalized from of (VEP) and (VVI) as special case, we assume that and are Hausdorff topological vector spaces, is a nonempty convex subset of , and is a pointed closed convex cone in with int . Let be a set-valued mapping and for a given vector valued mapping such that for each , the vector quasi-equilibrium problem (VQEP) and the vector quasi-variational inequality (VQVI), respectively: find such that

where , and is denoted by the space of all continuous linear operators for to : see, for example, [21–26] and references therein.

From our mentions on the importance of the fixed point theorems and the equilibrium problems, we have an inspired question to find when the set of the solution of both problems will have a joint solution, not an empty set.

To answer the question, we assume that is a set-valued mapping and for a given vector-valued mapping such that for each , let us present the fixed point problem of multi-valued mapping together with the vector equilibrium problem; in particular, it is to find such that

This problem shows the relationship in sense of intersection between fixed points of the multi-valued mapping and the vector equilibrium problem so the set of all solutions of the problem (2) is denoted by . This problem includes vector quasi-equilibrium problems (in short, VQEP) and vector quasi-variational inequalities (in short, VQVI) as special cases.

The main purpose of this paper, we provide sufficient conditions and prove the existence solutions of intersection between the set of all fixed points of the multi-valued mapping and the set of all solutions for vector equilibrium problem by using the generalization of the Fan-Browder fixed point theorem. We also study the existence solutions of intersection between the set of all fixed points of the multi-valued mapping and the set of all solutions for vector variational inequality. Consequently, our results extend the existence theorems of vector quasi-equilibrium problems and vector quasi-variational inequalities.

2. Preliminaries

Throughout this paper, unless otherwise specified, we assume that , and are Hausdorff topological vector spaces, is a nonempty convex subset of and is a pointed closed convex cone in with int . From problem (2), some special cases are as follows.

(I) If for all , then problem (2) reduces to the vector equilibrium problem (in short, ): find such that

The set of all solutions for vector equilibrium problem is denoted by .

Moreover, if we set and , then reduces to the equilibrium problem that the set of all solutions is denoted by : find such that

(II) If and , then problem (2) reduces to the problem: find such that

Moreover, this problem includes the quasi-equilibrium problems (in short, QEPs) considered and studied by Lin and Park [27] which is to find such that

(III) If for all , problem (3) reduces to the vector variational inequality (in short, ): find such that

The solution sets of problem (7) is denoted by , where , and is denoted by the space of all continuous linear operators for to .

Furthermore, if and , then reduces to the variational inequality that the set of all solutions is denoted by : find such that

(IV) If for all , then problem (2) reduces to the following problems that is to find such that

This problem shows the relationship in sense of intersection between fixed points of the multi-valued mapping and the vector variational inequality so the set of all solutions of the problem (9) is denoted by .

Let us recall some concepts and properties that are needed in this sequel. Given the multi-valued mappping , the inverse of is the multi-valued map from , the range of , to defined by

The mapping is continuous at some point if and only if for any neighborhood of , there is a neighborhood of such that .

Definition 1 (see [28]). Let and be topological vector spaces. Let be a nonempty subset of and be a point closed convex cone in with , where denotes the topological interior of . A bifunction is said to be -strongly pseudomonotone if, for any ,

A mapping is said to be -convex if for all and for all ,

And the mapping is said to be hemicontinuous if, for all and for all ,

Remark 2. If and , then

(1)the -strongly pseudomonotonicity of reduces to the monotonicity of (i.e., for all ). In fact, , this implies that and it is equivalence to ;(2)the -convexity of reduces to the convexity of (i.e., ).

Definition 3 (see [29]). Let be a topological space and let be a set. A map is said to have the local intersection property if for each with there exists an open neighborhood of such that .

The following lemma is useful in what follows and can be found in [30].

Lemma 4. Let be a topological space and let be a set. Let be a map with nonempty values. Then, the following are equivalent.

(i) has the local intersection property. (ii)There exists a map s.t. for each , is open for each and .

Subsequently, Browder [2] obtained in 1986 the following fixed point theorem.

Theorem 5 (Fan-Browder fixed point theorem). Let be a nonempty compact convex subset of a Hausdorff topological vector space and be a map with nonempty convex values and open fibers (i.e., for , is called the fiber of on ). Then has a fixed point.

The generalization of the Fan-Browder fixed point theorem was obtained by Balaj and Muresan [31] in 2005 as follows.

Theorem 6. Let be a compact convex subset of a and be a map with nonempty convex values having the local intersection property. Then has a fixed point.

3. Main Theorem

In this section, the existence solutions of the fixed point for multi-valued mappings and the vector equilibrium problems will be presented. To do this, the following lemma is necessary.

Lemma 7. Let be a nonempty and convex subset of . Let be a set-valued mapping such that for any , is nonempty convex subset of . Assume that be a hemicontinuous in the first argument, -convex in the second argument, and -strong pseudomonotone. Then the following statements are equivalent.

(i)Find such that and . (ii)Find such that and .

Proof. (i)(ii) It is clear by the -strong pseudomonotone.

(ii)(i) Let such that

For any and , we set and so we have because is convex. By the assumption, we conclude that

Since is -convex in the second argument and by (15), we get

This implies that and since is a convex cone then we have . Since is a hemicontinuous in the first argument and as , we have for all . Therefore we obtain that

This completes the proof.

Theorem 8. Let be a nonempty compact convex subset of and let be a -strong pseudomonotone, hemicontinuous in the first argument and -convex, l.s.c. in the second argument such that for all . Let be a set-valued mapping such that for any is nonempty convex subset of and for any , is open in . Assume the set is open in and for any , . Then .

Proof. For any , we define the set-valued mapping by

Also, we define the set-valued mapping by

Then we have is convex. Indeed, let and . Since is -convex in the second argument, we have

Then and hence is convex. Since is convex, then is also convex.

By the defining of , we see that has no fixed point Indeed, suppose that there is such that . It is impossible for , then and so . Thus , a contradiction with . Using the contrapositive of Theorem 6, we obtain that has no local intersection property. Define the set-valued mapping by

From the -strong pseudmonotonicity of , we have for any . Next, we will show that for each is open in . For any , we denote the complement of by . Since is closed and is l.s.c. in the second argument, we have is closed in and so is open in . We note that

Since for any , , , and are open, we have is open in . Thus, by the contrapositive of Lemma 4, we have

Hence, there exists such that for all . That is . If then , which contradicts with the assumption. Therefore, and . This implies that and for all . This completes the proof by Lemma 7.

The following example guarantees the assumption that the set , where .

Example 9. Let , , and . For any , we define two mappings and by

Clearly, is nonempty convex subset of and is open in . If for all , then and it implies that for , . This shows that is -strong pseudomonotone. Let and and since , we obtain that

Then is -convex in the second argument and it is easy to see that is a hemicontinuous in the first argument and l.s.c. in the second argument.

Note that

If , then which includes . Also is contained for all . Otherwise, for any . This is to confirm the set for each (see Figure 1). Moreover, this example asserts that the set is open in because it is equal to the set which is open in .

Figure 1

Taking for all in Theorem 8, we have the following results.

Corollary 10. Let be a nonempty compact convex subset of and be a -strong pseudomonotone, hemicontinuous in the first argument and -convex, l.s.c. in the second argument such that for all . Then, has a solution.

If we set the vector-valued mapping , then Theorem 8 reduces to the following corollary introduced by Browder (see [2, Theorem 1]).

Corollary 11. Let be a nonempty compact convex subset of . Let be a set-valued mapping such that for any is a nonempty convex subset of and for any is open in . Then there exists in such that .

If we set and in Theorem 8 together with Remark 2, we have the following result.

Corollary 12. Let be a nonempty compact convex subset of and let be a monotone, hemicontinuous in the first argument and convex, l.s.c. in the second argument such that for all . Let be a set-valued mapping such that for any is a nonempty convex subset of and for any , is open in . Assume the set is open in and for any , . Then .

Let be a space of all linear continuous operators from to . A mapping is said to be -strong pseudomonotone if it satisfies

and it is called hemicontinuous if, for all and for all , the mapping is continuous at .

As a direct consequence of Theorem 8, we obtain the following result.

Theorem 13. Let be a nonempty compact convex subset of and let be a -strong pseudomonotone and hemicontinuous. Let be a set-valued mapping such that for any , is nonempty convex subset of and for any is open in . Assume the set is open in and for any , . Then there exists such that

Proof. We define the vector value mapping by

We will show that satisfies all conditions in Theorem 8. Clearly and by the assumptions of , we have is -strong pseudomonotone and hemicontinuous in the first argument. Let be fixed. For any and , we obtain that

Then is -convex in the second argument.

Next, we will show that is l.s.c. in the second argument. Let be fixed. Let and be a neighborhood of . Since the linear operator is continuous, there exists an open neighborhood of such that for all , because is a neighborhood of . Thus for all , . Hence, is continuous in the second argument and so it is l.s.c. in the second argument. Then all hypotheses of the Theorem 8 hold and hence, there exists such that

This completes the proof.

If we take for all in Theorem 13, we have the following corollary.

Corollary 14. Let be a nonempty compact convex subset of and let be a -strong pseudomonotone and hemicontinuous. Then, has a solution.

If we set and in Theorem 13, we have the following result.

Corollary 15. Let be a nonempty compact convex subset of and let be a monotone and hemicontinuous in the first argument. Let be a set-valued mapping such that for any , is a nonempty convex subset of and for any , is open in . Assume the set is open in and for any , . Then .

Remark 16. (1) Theorems 8 and 13 are the extensions of vector quasi-equilibrium problems and vector quasi-variational inequalities, respectively.

(2) If is a real Banach space, then Corollary 10 comes to be Theorem  2.3 in [28].

Acknowledgment

The first author would like to thank the Office of the Higher Education Commission, Thailand for supporting by a grant fund under the program Strategic Scholarships for the Join Ph.D. Program Thai Doctoral degree for this research under Grant CHE-Ph.D.-SW-RG/41/2550, Thailand.

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