Table of Contents
International Journal of Computational Mathematics
Volume 2014, Article ID 643230, 6 pages
http://dx.doi.org/10.1155/2014/643230
Research Article

Extended -Vector Equilibrium Problem

Department of Mathematics, Integral University, Lucknow, Uttar Pradesh 226026, India

Received 25 June 2014; Revised 23 October 2014; Accepted 14 November 2014; Published 3 December 2014

Academic Editor: Sheung-Hung Poon

Copyright © 2014 Khushbu and Zubair Khan. 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 and study extended -vector equilibrium problem. By using KKM-Fan Theorem as basic tool, we prove existence theorem in the setting of Hausdorff topological vector space and reflexive Banach space. Some examples are also given.

1. Introduction

Equilibrium problems have been extensively studied in recent years; the origin of this can be traced back to Blum and Oettli [1] and Noor and Oettli [2]. The equilibrium problem is a generalization of classical variational inequalities and provides us with a systematic framework to study a wide class of problems arising in finance, economics, operations research, and so forth. General equilibrium problems have been extended to the case of vector-valued bifunctions, known as vector equilibrium problems. Vector equilibrium problems have attracted increasing interest of many researchers and provide a unified model for several classes of problems, for example, vector variational inequality problems, vector complementarity problems, vector optimization problems, and vector saddle point problems; see [14] and references therein. Many existence results for vector equilibrium problems have been established by several eminent researchers; see, for example, [513].

The generalized monotonicity plays an important role in the literature of equilibrium problems and variational inequalities. There are a substantial number of papers on existence results for solving equilibrium problems and variational inequalities based on different monotonicity notions such as monotonicity, pseudomonotonicity, and quasimonotonicity.

Let and be two Hausdorff topological vector spaces, let be a nonempty, closed, and convex subset of , and let be a pointed, closed, convex cone in with . Given a vector-valued mapping , the vector equilibrium problem consists of finding such that

Inspired by the concept of monotonicity, KKM-Fan Theorem, and the other work done in the direction of generalization of vector equilibrium problems (see [1416]) we introduce and study extended -vector equilibrium problem and prove some existence results in the setting of Hausdorff topological vector spaces and reflexive Banach spaces.

2. Preliminaries

The following definitions and concepts are needed to prove the results of this paper.

Definition 1. The Hausdorff topological vector space is said to be an ordered space denoted by if ordering relations are defined in by a pointed, closed, convex cone of as follows: If the interior of is , then the weak ordering relations in are also defined as follows: Throughout this paper, unless otherwise specified, we assume that is an ordered Hausdorff topological vector space with .

Definition 2. Let be a nonempty convex subset of a topological vector space . A set-valued mapping is said to be KKM-mapping, if, for each finite subset of , , where denotes the convex hull of .

The following KKM-Fan Theorem is important for us to prove the existence results of this paper.

Theorem 3 (KKM-Fan Theorem). Let be a nonempty convex subset of a Hausdorff topological vector space and let be a KKM-mapping such that is closed for all and is compact for at least one ; then

Lemma 4. Let be an ordered topological vector space with a pointed, closed, and convex cone . Then for all , one has the following:(i) and imply ;(ii) and imply ;(iii) and imply ;(iv) and imply .
Let and be the Hausdorff topological vector spaces and let be a nonempty, closed, convex subset of . Let be a pointed, closed, convex cone in with . Let be a vector-valued mapping and let be another mapping. We introduce the following extended -vector equilibrium problem.
Find such that for all and
If and , the problem (5) reduces to the vector equilibrium problem of finding such that Problem (6) was introduced and studied by Xuan and Nhat [13].
In addition, if and , then problem (5) reduces to the equilibrium problem of finding such that Problem (7) was introduced and studied by Blum and Oettli [1].
The fact that problem (5) is much more general than many existing equilibria, vector equilibrium problems, and so forth motivated us to study extended -vector equilibrium problem given by (5).

Definition 5. Let and be the Hausdorff topological vector spaces and let be a nonempty, closed, convex subset of and let be a pointed, closed, convex cone in with . Let and be mappings. Then, is said to be(i)-monotone with respect to , if and only if, for all , , (ii)-hemicontinuous, if and only if, for all , , the mapping is continuous at ;(iii)-pseudomonotone, if and only if, for all , , (iv)-generally convex, if and only if, for all , ,
In support of Definition 5, we have the following examples.

Example 6. Let , , , and .
Let and be mappings such that Then, that is, .
Hence, is -monotone with respect to .

Example 7. Let , , , and . Let be a mapping such that Let and be mappings such that Then, which implies that is continuous at . Hence, is -hemicontinuous.

Example 8. Let , , , and .
Let and be mappings such that Then, implies , so it follows that Hence, is -pseudomonotone with respect to .

Example 9. Let , , , and .
Let and be mappings such that Then, implies , and again implies , so it follows that Hence, is -generally convex.

Definition 10. A mapping is said to be affine in the first argument if and only if, for all and , Similarly, one can define the affine property of with respect to the second argument.

Definition 11. Let be an ordered topological vector space. A mapping is called -convex if and only if, for each pair and ,

3. Existence Results

We prove the following equivalence lemma which we need for the proof of our main results.

Lemma 12. Let be a Hausdorff topological vector space, let be a closed, convex subset of , and let be an ordered Hausdorff topological vector space with . Let be a vector-valued mapping which is -convex in the second argument, -monotone with respect to , positive homogeneous in the second argument, and-hemicontinuous and let be a continuous, affine mapping such that, for , , for all . Then for all and , the following statements are equivalent. Find such that (i), ;(ii), .

Proof. (i)(ii). For all and , let be a solution of (i); then we have Since is -monotone with respect to , we have Suppose to the contrary that (ii) is false. Then, there exists such that By (26), we obtain which contradicts (i). Thus, (i) (ii).
(ii)(i). Conversely, suppose that (ii) holds. Let be arbitrary and taking , so as is convex.
Therefore, As , we have Since is affine, we have Since is -convex in the second argument, therefore we have By (29), as and using Lemma 4, we have from (32) Since is -hemicontinuous, therefore, for , we have Thus, (ii)(i). This completes the proof.

Theorem 13. Let be a Hausdorff topological vector space and let be a compact and convex subset of and let be an ordered Hausdorff topological vector space with . Let be a vector-valued, affine mapping which is -monotone with respect to , positive homogeneous in the second argument, and -hemicontinuous and let be a continuous, affine mapping such that, for , , for all and let the mapping be continuous. Then problem (5) admits a solution; that is, for all and , there exists such that

Proof. For , we define Clearly , as . We divide the proof into three steps.
Step 1. We claim that is a KKM-mapping. If is not a KKM-mapping, then there exists such that, for all ,   with , we have Thus, we have Since is affine in the second argument and , we have It follows that which contradicts to the pointedness of and hence is a KKM-mapping.
Step 2. One has and is also a KKM-mapping.
If , then . By the -monotonicity of with respect to , we have Suppose that . Then, we have It follows from (40) that which contradicts the fact that . Therefore, ; that is, . Then, On the other hand, suppose that . We have By Lemma 12, we have That is, . Hence, So,
Also , since . From above, we know that and, by Step 1, we know that is a KKM-mapping. Thus, is also a KKM-mapping.
Step 3. For all , is closed.
Let be a sequence in such that converges to . Then, Since the mapping is continuous, we have We conclude that ; that is, is a closed subset of a compact set and hence compact.
By KKM-Theorem 3, and also . Hence, there exists ; that is, there exists such that Thus, is a solution of problem (5).

In support of Theorem 13, we give the following example.

Example 14. Let , , and .
Let and be mappings such that Then,  (i)for any , that is, is -monotone with respect to ; (ii) for any , that is, is positive homogeneous in the second argument; (iii) let be a mapping such that , ; then, which is a continuous mapping; that is, is continuous at . Hence, is -hemicontinuous. (iv)Let be a mapping such that then, which implies that is continuous mapping.
Hence, all the conditions of Theorem 13 are satisfied.
In addition, Thus, it follows that is a solution of problem (5) for all and .

Corollary 15. Let be a compact and convex subset of and let be an ordered topological vector space with . Let be a vector-valued mapping which is -pseudomonotone with respect to and let be a continuous, affine mapping such that, for , , for all . Let the mapping be continuous. Then, problem (5) is solvable.

Proof. By Step 1 of Theorem 13, it follows that is a KKM-mapping. Also it follows from -pseudomonotonicity of that ; thus is also a KKM-mapping. By Step 3 of Theorem 13, the conclusion follows.

Theorem 16. Let be a reflexive Banach space; let be an ordered topological vector space with . Let be a nonempty, bounded, and convex subset of . Let be a vector-valued mapping which is -monotone with respect to , positive homogeneous in the second argument, -hemicontinuous, and -generally convex on . Let be a continuous, affine mapping such that, for , , for all . Then, problem (5) admits a solution, that is, for all and , there exists such that

Proof. For each , let for all and .
From the proof of Theorem 13, we know that is closed and is a KKM-mapping. We also know that Since is a bounded, closed, and convex subset of a reflexive Banach space , therefore, is weakly compact.
Now, we show that is convex. Suppose that and with .
Then, Since is -generally convex, we have that is, , which implies that is convex. Since is closed and convex, is weakly closed.
As is a KKM-mapping, is weakly closed subset of ; therefore is weakly compact. By KKM-Theorem 3, there exists such that . That is, there exists such that Hence, problem (5) is solvable.

4. Conclusion

In this paper, some existence results for extended -vector equilibrium problem are proved in the setting of Hausdorff topological vector spaces and reflexive Banach spaces. The concept of monotonicity plays an important role in obtaining existence results. The results of this paper can be viewed as generalizations of some known equilibrium problems as explained by (6) and (7).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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