On the Existence Result for System of Generalized Strong Vector Quasiequilibrium Problems

Abstract

We introduce a new type of the system of generalized strong vector quasiequilibrium problems with set-valued mappings in real locally convex Hausdorff topological vector spaces. We establish an existence theorem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem. The results presented in the paper improve and extend the main results of Long et al. (2008).

1. Introduction

The equilibrium problem is a generalization of classical variational inequalities. This problem contains many important problems as special cases, for instance, optimization, Nash equilibrium, complementarity, and fixed-point problems (see [1–3] and the references therein). Recently, there has been an increasing interest in the study of vector equilibrium problems. Many results on existence of solutions for vector variational inequalities and vector equilibrium problems have been established (see, e.g., [4–16]).

Let and be real locally convex Hausdorff space, a nonempty subset and be a closed convex pointed cone. Let be a given set-valued mapping. Ansari et al. [17] introduced the following set-valued vector equilibrium problems (VEPs) to find such that

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or to find such that

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If is nonempty, and satisfies (1.1), then we call a weak efficient solution for (VEP), and if satisfies (1.2), then we call a strong solution for VEP. Moreover, they also proved an existence theorem for a strong vector equilibrium problem (1.2) (see [17]).

In 2000, Ansari et al. [5] introduced the system of vector equilibrium problems (SVEPs), that is, a family of equilibrium problems for vector-valued bifunctions defined on a product set, with applications in vector optimization problems and Nash equilibrium problem [11] for vector-valued functions. The (SVEP) contains system of equilibrium problems, systems of vector variational inequalities, system of vector variational-like inequalities, system of optimization problems and the Nash equilibrium problem for vector-valued functions as special cases. But, by using (SVEP), we cannot establish the existence of a solution to the Debreu type equilibrium problem [7] for vector-valued functions which extends the classical concept of Nash equilibrium problem for a noncooperative game. Moreover, Ansari et al. [18] introduced the following concept of system of vector quasiequilibrium problems.

Let be any index set and for each , let be a topological vector space. Consider a family of nonempty convex subsets with . We denote by and . For each , let be a topological vector space and let and be multivalued mappings and be a bifunction. The system of vector quasiequilibrium problems (SVQEPs), that is, to find such that for each ,

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If for all , then (SVQEP) reduces to (SVEP) (see [5]) and if the index set is singleton, then (SVQEP) becomes the vector quasiequilibrium problem. Many authors studied the existence of solutions for systems of (vector) quasiequilibrium problems, see, for example, [19–23] and references therein.

On the other hand, it is well known that a strong solution of vector equilibrium problem is an ideal solution, It is better than other solutions such as efficient solution, weak efficient solution, proper efficient solution and supper efficient solution (see [13]). Thus, it is important to study the existence of strong solution and properties of the strong solution set. In general, the ideal solutions do not exist.

Very recently, the generalized strong vector quasiequilibrium problem (GSVQEPs) is introduced by Long et al. [16]. Let , , and are real locally convex Hausdorff topological vector spaces, and are nonempty compact convex subsets, and is a nonempty closed convex cone. Let , and are three set-valued mappings. They considered the GSVQEP: finding , such that and

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Moreover, they gave an existence theorem for a generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak* compact base.

Motivated and inspired by research works mentioned above, in this paper, we introduce a different kind of systems of generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak* compact base. Let , , and are real locally convex Hausdorff topological vector spaces, and are nonempty compact convex subsets, and is a nonempty closed convex cone. We also suppose that , and are set-valued mappings. We consider the following system of generalized strong vector quasiequilibrium problem (SGSVQEPs): finding and , such that , satisfying

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We call this a strong solution for the (SGSVQEP).

At a quick glance, our required solution seems to be similar to such a thing of Ansari et al. [5, 18], in the case of and . In fact, however, the main different point comes from the independent choice of coordinate. In this paper, we establish an existence theorem of strong solution set for the system of generalized strong vector quasiequilibrium problem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of the solution set. Moreover, we apply our result to obtain the result of Long et al. [16].

2. Preliminaries

Throughout this paper,we suppose that , , and are real locally convex Hausdorff topological vector spaces, and are nonempty compact convex subsets, and is a nonempty closed convex cone. We also suppose that , , and are set-valued mappings.

Definition 2.1.

Let and be two topological vector spaces and a nonempty subset of and let be a set-valued mapping.

(i) is called upper -continuous at if, for any neighbourhood of the origin in , there is a neighbourhood of such that, for all ,

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(ii) is called lower -continuous at if, for any neighbourhood of the origin in , there is a neighbourhood of such that, for all ,

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Definition 2.2.

Let and be two topological vector spaces and a nonempty convex subset of . A set-valued mapping is said to be properly -quasiconvex if, for any and , we have

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Definition 2.3.

Let and be two topological vector spaces, and be a set-valued mapping.

(i) is said to be upper semicontinuous at if, for any open set containing , there exists an open set containing such that, for all , ; is said to be upper semicontinuous on if it is upper semicontinuous at all .

(ii) is said to be lower semicontinuous at if, for any open set with , there exists an open set containing such that, for all , ; is said to be lower semicontinuous on if it is lower semicontinuous at all .

(iii) is said to be continuous on if, it is at the same time upper semicontinuous and lower semicontinuous on .

(iv) is said to be closed if the graph, , of , that is, , is a closed set in .

Lemma 2.4 (see [12]).

Let be a nonempty compact subset of locally convex Hausdorff vector topology space . If is upper semicontinuous and for any , is nonempty, convex and closed, then there exists an such that .

Lemma 2.5 (see [24]).

Let and be two Hausdorff topological vector spaces and be a set-valued mapping. Then, the following properties hold:

(i)if is closed and is compact, then is upper semicontinuous, where and denotes the closure of the set ,

(ii)if is upper semicontinuous and for any , is closed, then is closed,

(iii) is lower semicontinuous at if and only if for any and any net , , there exists a net such that and .

3. Main Results

In this section, we apply Kakutani-Fan-Glicksberg fixed-point theorem to prove an existence theorem of strong solutions for the system of generalized strong vector quasiequilibrium problem. Moreover, we also prove the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem.

Theorem 3.1.

For each , let be continuous set-valued mappings such that for any , are nonempty closed convex subsets of . Let be upper semi continuous set-valued mappings such that for any are nonempty closed convex subsets of and be set-valued mappings satisfy the following conditions:

(i)for all , ,

(ii)for all , are properly -quasiconvex,

(iii) are upper -continuous,

(iv)for all , are lower -continuous.

Then, SGSVQEP has a solution. Moreover, the set of all strong solutions is closed.

Proof.

For any , define set-valued mappings by

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Step 1.

Show that and are nonempty.

For any , we note that and are nonempty. Thus, for any , we have and are nonempty.

Step 2.

Show that and are convex subsets of .

Let and . Put . Since and is convex set, we have . By (ii), is properly -quasiconvex. Without loss of generality, we can assume that

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We claim that . In fact, if , then there exists such that

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It follows that

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which contradicts to . Therefore and hence is a convex subset of . Similarly, we have is convex subset of .

Step 3.

Show that and are closed subsets of .

Let be a sequence in such that . Thus, we have . Since is a closed subset of , it follows that . By the lower semicontinuity of and Lemma 2.5(iii), for any and any net , there exists a net such that and . This implies that

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Since is lower -continuous, for any neighbourhood of the origin in , there is a subnet of such that

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From (3.5) and (3.6), we have

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We claim that . Assume that there exists and . Thus, we note that and is closed. Hence is open and . Since is a locally convex space, there exists a neighbourhood of the origin such that is convex and . This implies that , that is, , which is a contradiction. Therefore . This mean that and so is a closed subset of . Similarly, we have is a closed subset of .

Step 4.

Show that and are upper semicontinuous.

Let be given such that , and let such that . Since and is upper semicontinuous, it follows by Lemma 2.5(ii) that . We now claim that . Assume that . Then, there exists such that

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which implies that there is a neighbourhood of the origin in such that

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Since is upper -continuous, for any neighbourhood of the origin in , there exists a neighbourhood of such that

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Without loss of generality, we can assume that . This implies that

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Thus there is such that

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which contradicts to . Hence and, therefore, is a closed mapping. Since is a compact set and is a closed subset of , we note that is compact. Then, is also compact. Hence, by Lemma 2.5(i), is an upper semicontinuous mapping. Similarly, we note that is an upper semicontinuous mapping.

Step 5.

Show that SGSVQEP has a solution.

Define the set-valued mapping and by

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Then, and are upper semicontinuous and, for all , , and are nonempty closed convex subsets of .

Define the set-valued mapping by

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Then, is also upper semicontinuous and, for all , is a nonempty closed convex subset of . By Lemma 2.4, there exists a point such that , that is

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This implies that , , , and . Then, there exists and , such that , ,

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Hence SGSVQEP has a solution.

Step 6.

Show that the set of solutions of SGSVQEP is closed.

Let be a net in the set of solutions of SGSVQEP such that . By definition of the set of solutions of SGSVQEP, we note that there exist , , , and satisfying

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Since and are continuous closed valued mappings, we obtain and . Let and . Since and are upper semicontinuous closed valued mappings, it follows by Lemma 2.5(ii) that and are closed. Thus, we note that and . Since and are lower -continuous, we have

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This means that belongs to the set of solutions of SGSVQEP. Hence the set of solutions of SGSVQEP is closed set. This completes the proof.

If we take , , and . Then, from Theorem 3.1, we derive the following result.

Corollary 3.2.

Let be a continuous set-valued mapping such that for any , is nonempty closed convex subset of . Let be an upper semicontinuous set-valued mapping such that for any , is a nonempty closed convex subset of and be set-valued mapping satisfy the following conditions:

(i)for all ,,

(ii)for all , is properly -quasiconvex,

(iii) is an upper -continuous,

(iv)for all , is a lower -continuous,

(v)if and then .

Then, GSVQEP has a solution. Moreover, the set of all solution of GSVQEP is closed.

Now we give an example to explain that Theorem 3.1 is applicable.

Example 3.3.

Let , , and . For each , let , and , . We consider the set-valued mappings defined by

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Then, it is easy to check that all of condition (i)–(iv) in Theorem 3.1 are satisfied. Hence, by Theorem 3.1, SGSVQEP has a solution. Let be the set of all strong solutions for SGSVQEP. Then, we note that

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