Existence Results for Strong Mixed Vector Equilibrium Problem for Multivalued Mappings

Abstract

We consider a strong mixed vector equilibrium problem in topological vector spaces. Using generalized Fan-Browder fixed point theorem (Takahashi 1976) and generalized pseudomonotonicity for multivalued mappings, we provide some existence results for strong mixed vector equilibrium problem without using KKM-Fan theorem. The results in this paper generalize, improve, extend, and unify some existence results in the literature. Some special cases are discussed and an example is constructed.

1. Introduction

The minimax inequalities of Fan [1] are fundamental tools in proving many existence theorems in nonlinear analysis. Their equivalence to the equilibrium problems was introduced by Takahashi [2], Blum and Oettli [3], and Noor and Oettli [4]. The equilibrium theory provides a novel and unified treatment of a wide class of problems which arises in economics, finance, transportation, elasticity, optimization, and so forth. The generalization of equilibrium problem for vector valued mappings is known as vector equilibrium problem and has been studied vastly by many authors; see, for example, [5–8].

Recently, Kum and Wong [9] considered a multivalued version of generalized equilibrium problem which extends the strong vector variational inequality studied by Fang and Huang [10] in real Banach spaces. From Brouwer’s fixed point theorem and Fan-Browder fixed point theorem, they derived existence results for generalized equilibrium problem with and without monotonicity in general Hausdorff topological vector spaces.

The main motivation of this paper is to establish some existence results for strong mixed vector equilibrium problem which is combination of vector equilibrium problem and a vector variational inequality. Proposition 3.3 of Ahmad and Akram [11] related to the core of a set is extended for multivalued mappings and used to prove an existence result for strong mixed vector equilibrium problem. We also prove our results with and without monotonicity assumptions.

2. Preliminaries

Throughout this paper, let and be the topological vector spaces. Let be a nonempty convex subset of and a pointed closed convex cone with . Let and be the multivalued mappings, where is space of all continuous and bounded mappings. We consider the following problem.

Find , such that, for all , We call problem (1) strong mixed vector equilibrium problem for multivalued mappings.

2.1. Special Cases

(1)If , then problem (1) reduces to the problem of finding such that Problem (2) is called multivalued generalized system and this problem was considered and studied by Kum and Wong [9].(2)If , , , and is single-valued, then problem (1) reduces to the classical equilibrium problem introduced and studied by Blum and Oettli [3] which is to find such that (3)If and are single-valued and ,  , then problem (1) reduces to the generalized equilibrium problem of finding such that which was studied by S. Takahashi and W. Takahashi [12].(4)If and is single-valued, then (1) reduces to strong vector variational inequality problem which is to find such that Problem (5) was considered and studied by Fang and Huang [10].

It is clear that the problem under consideration is much more general than the other problems that exist in the literature.

Let us recall some definitions and results that are needed to prove the main results of this paper.

Definition 1 (see [13]). Let and be the topological vector spaces, and let be a multivalued mapping. Then one has the following: (i) is said to be upper semicontinuous at , if, for each and each open set in with , there exists an open neighborhood of in such that , for each ;(ii) is said to be lower semicontinuous at , if, for each and each open set in with , there exists an open neighborhood of in such that , for each ;(iii) is said to be continuous on , if it is at the same time upper semicontinuous and lower semicontinuous on . It is also known that is lower semicontinuous if and only if, for each closed set in , the set is closed in .

Definition 2. A multivalued mapping is said to be generalized -strongly pseudomonotone, if, for any ,

Definition 3. A multivalued mapping is said to be (i)generalized -strongly pseudomonotone, if, for any , there exists such that implies that there exists such that (ii)generalized hemicontinuous, if, for any given and for , the mapping is upper semicontinuous at .

Definition 4 (see [9]). Let be a multivalued mapping. Then is said to be (i)-convex, if, for all and for , (ii)generalized hemicontinuous, if, for all and for , the mapping is upper semicontinuous at .

Definition 5 (see [14]). Let be a multivalued mapping. Then is said to have local intersection property, if, for each with , there exists an open neighborhood of such that .

Lemma 6 (see [14]). Let and be the topological spaces and a multivalued mapping. Then the following conditions are equivalent: (i) has the local intersection property;(ii)there exists a multivalued mapping such that , for each ; is open in for each and .

Theorem 7 (Fan-Browder fixed point theorem [15]). Let be a nonempty, compact, and convex subset of a Hausdorff topological vector space and let be a mapping 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 [15] was derived by Balaj and Muresan [16] as follows.

Theorem 8. Let be a nonempty, compact, and convex subset of a topological vector space and let be a mapping with nonempty convex values having the local intersection property. Then has a fixed point.

Definition 9 (see [3]). Let and be convex subsets of with . The core of relative to , denoted by , is the set defined by if and only if and , for all .

3. Existence Results

In this section, we prove some existence results for strong mixed vector equilibrium problem for multivalued mappings.

Theorem 10. Let be a nonempty compact convex subset of and a closed convex pointed cone in . Let and be the multivalued mappings. Suppose that the following conditions hold: (i)for all , ;(ii) is generalized -strongly pseudomonotone and -convex in the second argument;(iii) is generalized hemicontinuous in the first argument and lower semicontinuous;(iv) is generalized -strongly pseudomonotone and generalized hemicontinuous.
Then, there exists , such that

First, we prove the following lemma which is required to prove Theorem 10 for which the assumptions remain the same as in Theorem 10.

Lemma 11. The following two problems are equivalent. (I)Find , such that ,  for all  .(II) Find , such that ,  for all  .

Proof. Suppose holds. Then by using generalized -strongly pseudomonotone of and , follows.
Conversely, assume that holds; that is, For any , set , for . Obviously , and there exists such that Since is -convex in the second argument and , using (12) we have which implies that As is a convex cone, we get from (14) Since is generalized hemicontinuous in the first argument and is generalized hemicontinuous, therefore we have for Therefore, we get , such that and hence follows.

Proof of Theorem 10. Consider the multivalued mappings for any as follows: Clearly, and are nonempty sets as . By the generalized -strong pseudomonotonicity of and , we have .
We claim that is convex. Indeed, letting , then we have Since is -convex in the second argument, therefore, for any , we have This implies that , and hence is convex.
By definition of , we see that has no fixed point. Indeed, suppose that there exists an such that . Thus, we have which is a contradiction to hypothesis .
Next, we show that is open in . For any , we denote the complement of by Let be a net in such that . Then Since is lower semicontinuous and is continuous, we get Since is open, then there exists such that, for all , which contradicts (23). Hence , for and therefore . Thus is closed and accordingly is open.
From the contrapositive of generalized Fan-Browder fixed point theorem and Lemma 6, we have Hence, there exists such that which contradicts the fact that is nonempty and hence By applying Lemma 11, we get that there exists , such that This completes the proof.

Example 12. Let , , and . Define by Also is given by
We see that is generalized -strongly pseudomonotone. Indeed, suppose that . Therefore, , which implies that . It follows that . Hence, is generalized -strongly pseudomonotone.
Similarly, to show that is generalized -strongly pseudomonotone, assume that , which implies that . It follows that . Therefore, is generalized -strongly pseudomonotone.
Let and . Then, we see that So, is -convex in the second argument.
It is clear that is a solution of strong mixed vector equilibrium problem (1) as .
The following lemma is an extension of Proposition 3.3 [11] related to the core of a set for multivalued mappings.

Lemma 13. Let and be the convex subsets of with . Let be -convex, ; , and , for all . Then, , for all .

Proof. On the contrary, suppose that , for some . Then, there is such that .
Since , then there exists such that .
Suppose , for . Then . By using -convexity of , we have Then, there exists such that, for some , we have Therefore, Since , so we have a point . By (34), we have , a contradiction to the hypothesis. Thus, , for all .

Theorem 14. Let be a nonempty convex subset of and a closed convex pointed cone in . Let and be the mappings satisfying the conditions which are the same as in Theorem 10. In addition, suppose that the following condition holds: there exists a nonempty convex compact subset of such that for and , Then, there exists a point such that, for all ,

Proof. By Theorem 10, it follows that there exists , such that Set . Then is -convex and , for all .
If , then choose . If , then choose , where is the same as in the hypothesis. In both cases, and . Hence by Lemma 13, it follows that , for all , which implies that Thus, there exists at least one and such that This completes the proof.

Now, we prove the existence results for strong mixed vector equilibrium problem for multivalued mappings without monotonicity.

Theorem 15. Let be a nonempty compact convex subset of . Let and be the mappings such that is -convex in the second argument and . Assume that, for each , the set is open. Then the strong mixed vector equilibrium problem (1) has a solution.

Proof. Using the same assertions in Kum and Wong [9], one can easily prove this theorem.

Theorem 16. Let be a nonempty convex subset of . Let and be the mappings such that is -convex in the second argument and . Also, assume that (i) is locally compact and there is an and , such that, for all , , and , for ;(ii)for each , the set is open.
Then, there exists , such that

Proof. Let . Since is locally compact, therefore is compact. By applying Theorem 10, we can see that there exists and such that To show that is the solution of problem (41), consider the following two cases. (I)If , then, by assumption (i), we have, for , Now, for any , set , for . Obviously, and it follows that for Since is -convex in the second argument, we have Therefore, using (42) we conclude that which implies that (II)If , then, for any , set , for . Clearly, and it follows that for Using -convexity of in the second argument and , we have which implies that, for , we have This completes the proof.

Conflict of Interests

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

Acknowledgments

This joint work was carried out when Professor Rais Ahmad visited University Putra Malaysia as a visiting scientist from May 2 to May 31, 2014. Thus, he is very grateful to the staff of the Institute for Mathematical Research (INSPEM) for providing local hospitalities. Further the authors also gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme project no. 5527068.