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The Scientific World Journal

Volume 2013 (2013), Article ID 150130, 6 pages

http://dx.doi.org/10.1155/2013/150130
Research Article

Existence Theorems for Vector Equilibrium Problems via Quasi-Relative Interior

Technical University of Cluj-Napoca, Department of Mathematics, Cluj-Napoca, G. Bariţiu 25-28, 400027 Jud. Cluj, Romania

Received 7 April 2013; Accepted 25 April 2013

Academic Editors: L. B. Ciric, M. Hajarian, and Y. Xia

Copyright © 2013 Capătă Adela Elisabeta. 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

The aim of this paper is to present new existence theorems for solutions of vector equilibrium problems, by using weak interior type conditions and weak convexity assumptions.

1. Introduction

Starting with the paper of Blum and Oettli [1], the field of equilibrium problems is intensively studied by researchers, and many papers dealing with vector equilibrium problems were written. The interest of the researchers on this topic is due to the fact that equilibrium problems represent a natural and unified framework for other problems, such as optimization problems, variational inequality problems, and saddle-point problems, problems which until now were separately studied and have applications in physics, economics, and so forth, (see, for instance, [2]). Since then, a large variety of vector equilibrium problems were considered and the authors studied the existence of solutions (see, for instance, [310]), well posedness (see, for instance, [11, 12]), and sensitivity analysis (see, for instance, [13, 14]).

Most of the existence results are based on the hypothesis of nonemptyness of the ordering cone. In this paper, based on weak convexity assumptions defined by the means of quasi-relative interior of a convex set introduced in [15] and by using a quite recent separation theorem, whose statement was given in [16], we establish existence theorems for solutions of vector equilibrium problems. Then, the results are applied to vector optimization problems and to vector variational inequalities.

The paper is organized as follows. In Section 2 we recall some notions and auxiliary results that we need throughout this paper. Then, in Section 3 we present the main results of this paper, whose statements are given in the terms of quasi-relative interior and weak convexity assumptions. These results are applied to a vector optimization problem and to vector variational inequalities in Sections 4 and 5, respectively.

2. Preliminaries

Let be a separated locally convex space, and let be a nontrivial convex cone. Having , the dual space of , the dual cone of is

We recall the definitions of quasi-interior and quasi-relative interior of a convex set and some useful properties of the quasi-relative interior notion.

Definition 1. Let be a nonempty convex subset of . The quasi interior of is the set

Definition 2 (see [17]). Let be a nonempty convex subset of . The quasi-relative interior of is the set

In a separable Banach space the quasi-relative interior of any nonempty closed convex set is nonempty (cf. [17]). For the proof of the following properties and other useful properties of the quasi-relative interior of a convex set, we refer the reader to [17, 18], respectively.

Proposition 3. Let and be nonempty convex subsets of , , and . Then the following statements are true:(i)qri qri qri ;(ii) qri qri ;(iii) qri qri qri ;(iv) qri qri ;If qri , then (v) qri ;(vi) qri .

The normal cone of a convex subset of at is defined as By means of the normal cone, the next characterizations of the abovementioned interior notions hold, and they can be found in [16, 17].

Theorem 4. Let be a nonempty convex subset of   and . Then if and only if .

Theorem 5. Let be a nonempty convex subset of   and . Then if and only if is a linear subspace of .

For other generalizations of the classical interior we refer the reader to [1922]. Whenever the interior of the set is nonempty, then int (see [17, Corollary 2.14]). If is finite dimensional, then ri (see [17]), where by ri we understand the relative interior of , that is, the interior with respect to the affine hull.

The following characterization for the quasi-interior of a convex cone holds, and it can be found in [23].

Proposition 6. Let be a convex cone in a locally convex space , and let . Then if and only if

Lemma 7 (see [17]). Let be a convex cone in a separable locally convex space . Then if and only if and .

Lemma 8 (see [17]). Let be a convex cone in a locally convex space . Then for all .

The statement of the next theorem is due to [16], where it was proved for normed spaces, and, later on, it was proved for separated locally convex spaces by [24].

Theorem 9. Let be a nonempty convex subset of and . Then, there exists such that for all .

For other separation theorems which involve the quasi-relative interior we refer the reader to [25].

Definition 10. A function is said to be generalized -subconvexlike on if cone is convex.

Proposition 11. If is convex, then cone is also convex.

Proof. Let and let . So, there exist , , and such that

Case 1. If and , then , and

Case 2. If or , denote So, we have In what follows we will prove that . For this, we evaluate Dividing this equality by the positive number it yields By the above equalities and the convexity of , we have Thus, whence multiplying this relation by we obtain the conclusion.

3. Existence Results

Throughout this paper we study the following strong vector equilibrium problem: where and are non-empty sets, , and has a non-empty quasi-relative interior.

Definition 12. We say that a point is a quasi-relative solution of if

Theorem 13. Suppose that the following conditions are satisfied: (i)for every , the function is generalized -subconvexlike on ;(ii) ;(iii)for every , is not a linear subspace of . Then, problem admits a quasi-relative solution.

Proof. Suppose by contradiction that has no solutions; that is, for any there exists such that This leads us to which implies that Assumption (i) assures the convexity of the set , and, together with the relation (18) and assumption (iii), it gives The separation Theorem 9 of convex sets assures the existence of a nonzero functional such that So Taking , by and (21) we have which gives Further, for any , dividing by relation (21) it implies Now letting in the above relation we get Since , by assumption (ii), Lemma 7, and Proposition 6, we get which is a contradiction to (25), and this completes the proof.

This theorem allows us to obtain existence results for important practical spaces, whose ordering cones have empty interiors, but nonempty quasi-relative interiors. This is the case of the Banach space with the positive cone where is a real constant, is an interval, is a -finite measure space, and .

The next result deals with stronger assumptions than the ones presented in Theorem 13.

Corollary 14. Suppose that the following conditions are satisfied: (i)for every , the function is generalized -subconvexlike on ;(ii) ;(iii)for every , . Then, problem admits a quasi-relative solution.

Proof. Since and for every , by the definition of the quasi-relative interior, we get is not a linear subspace of . So all the assumption of Theorem 13 are satisfied, and the conclusion follows now by this theorem.

Remark 15. In Theorem 13 and Corollary 14, according to Proposition 11, the generalized -subconvexlike assumption of the function on can be replaced by the convexity of the set .

4. Existence Results for Vector Optimization Problems

Let , and let the function . In this section we study the vector optimization problem, According to [26], the point is called quasi-relative minimal point of the set , that is, while is a quasi-relative minimizer of , that is,

By Theorem 13 and Corollary 14 we have the following results.

Theorem 16. Suppose that the following conditions are satisfied: (i)for every , the function is generalized -subconvexlike on ;(ii) ;(iii)for every , is not a linear subspace of . Then, problem admits a quasi-relative solution.

Proof. Define the function by It is easy to see that all the assumptions of the Theorem 13 are satisfied by this function . So, problem admits a solution, which implies that problem has a solution, and the proof is completed.

Corollary 17. Suppose that the following conditions are satisfied: (i)for every , the function is generalized subconvexlike on ;(ii) ;(iii)for every , . Then, problem admits a quasi-relative solution.

To show that the set of functions which satisfies the assumptions of Corollary 17 is nonempty, we give the following example.

Example 18. Let , , , and . We have that Since and is a convex set, we deduce that is also convex and that, by Proposition 11, the first assumption of Corollary 17 is verified. The second assumption is obviously satisfied, and we still have to verify its third assumption. Because , then it is equal to . The set for all , while for Thus, and assumption (iii) is checked.

5. Existence Results for Vector Variational Inequalities

Let be a non-empty convex subset of a vector space, let , and let the operator , where denotes the set of all linear and continuous functions defined on with values on . Throughout this section we study the Minty vector variational inequality:

Definition 19. We say that a point is a quasi-relative solution of if

By Theorem 13 and Corollary 14 we have the following results.

Theorem 20. Suppose that the following conditions are satisfied: (i) ;(ii)for every , is not a linear subspace of . Then, problem admits a quasi-relative solution.

Proof. Let the function be defined by Obviously fulfills assumptions (ii) and (iii) of Theorem 13. It remains to show that the first assumption of this theorem is also verified by this function .

Since is a convex set, by the definition of the function we deduce that is a convex set for every . By this we get convex, which together with Proposition 11 gives the conclusion, and proof is completed.

Corollary 21. Suppose that the following conditions are satisfied:(i) ;(ii)for every , . Then, problem admits a quasi-relative solution.

In what follows we turn our attention to existence results for the Stampacchia vector variational inequality

Definition 22. We say that a point is a quasi-relative solution of if

Next we recall a definition concerning vector variational inequalities (see [27]).

Definition 23. Let . The operator is said to be -upper hemicontinuous if is a convex set, and, for all the function is upper semicontinuous at .

Theorem 24. Let , let , and suppose that the following conditions are satisfied: (i) for all ;(ii) is -upper hemicontinuous. Then, problem admits a quasi-relative solution.

Proof . By assumption (i) whence, by using Lemma 8, we deduce that that is, is a quasi-relative solution of . In what follows we will prove that is also a quasi-relative solution of .

Because is convex, then for all and all , and by (43), we get For each , there exists, according to (ii), a number such that This, together with (46), gives Since the inequality holds for each , we deduce that whence, by using Lemma 8, we get which completes the proof.

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