Abstract

A constrained weak Nash-type equilibrium problem with multivalued payoff functions is introduced. By virtue of a nonlinear scalarization function, some existence results are established. The results extend the corresponding one of Fu (2003). In particular, if the payoff functions are singlevalued, our existence theorem extends the main results of Fu (2003) by relaxing the assumption of convexity.

1. Introduction

For a long time, real valued functions have played a central role in game theory. More recently, motivated by applications to real-world situations, many authors have studied the existence of solutions of Pareto equilibria of multiobjective game with vector payoff functions; for example, see [1–4] and the references therein. Notice that most payoffs may be one collection of things from many collections of things in the real world; reference [5] studied the constrained Nash-type equilibrium problem with multivalued payoff functions and proved the existence results.

In the paper, let be an index set, a real topological vector space, and a Hausdorff topological space. Let and . For each , let and denote the th coordinate of and the projection of on , respectively. In the sequel, we may write . For all , let be a convex, closed, and pointed cone of , with apex at the origin and with nonempty interior; let and . We consider a class of constrained weak Nash-type equilibrium problems with multivalued payoff functions.

Finding an such that, for each , , and , there exists satisfying Then, is a solution of .

The following problems are special cases of .(i)If, for each , is a singlevalued function, , and , reduces to the Nash equilibrium problem [6].(ii)Let , , and be real Hausdorff topological vector spaces, and let and be two nonempty subsets of and , respectively. Let be a closed convex and pointed cone with , let and be two set-valued mappings, and let be two vector-valued mappings. The problem reduces to a class of symmetric vector quasiequilibrium problems (for short, SVQEP) that consists in finding such that , , and  which was considered by Fu [7].

In this paper, we obtain the existence result for . Our existence theorem extends the main result of [6] from singlevalued case to multivalued case. In particular, if the payoff functions are singlevalued, our existence theorem extends the corresponding result in [7] by relaxing the assumption of convexity.

The rest of the paper is organized as follows. In Section 2, we state some notations and preliminary results for multivalued mappings. We recall the nonlinear scalarization function and its properties. In Section 3, we show existence result for .

2. Preliminaries

Let us first recall some definitions of continuity for set-valued mappings. Let and be two topological spaces. is a set-valued mapping. is said to be upper semicontinuous at if, for each open set containing , there is an open set containing such that, for each , . It is said to be upper semicontinuous if it is upper semicontinuous at every point . is said to be lower semicontinuous at if, for each open set with , there is an open set containing such that, for each , . It is said to be lower semicontinuous on if it is lower semicontinuous at every point . is said to be continuous at if it is both upper semicontinuous and lower semicontinuous at . It is said to be continuous on if it is continuous at every point .

From [7, Lemma 2], is l.s.c. at if and only if, for any and any net , , there is a net such that and . is closed if and only if, for any net , , and any net , , , one has .

Definition 1. Assume that is a Hausdorff topological space and is a real topological vector space. Let be a nonempty convex subset of , let be a set-valued mapping, and let be a closed convex and pointed cone with . is said to be generalized Luc's quasi--convex on if, for every , , , and , there exist and such that where is the set of all upper bounds of and ; that is,

Remark 2. Definition 1 is a generalization of the concept of Luc's quasi--convexity in [8].

Now we recall the definition of the nonlinear scalarization function [9, 10] as follows.

Definition 3. Let be a real topological vector space, and let be a closed convex and pointed cone with . The nonlinear scalarization function is defined by

Lemma 4 (see [9]). The nonlinear scalarization function has the following main properties: (i) is continuous and convex;(ii) is subadditive; that is, ;(iii) is strictly monotone; that is, if , then .

3. Existence for the Solution of (CWNEP)

Throughout this section, let be a locally convex Hausdorff topological vector space, and let be a real Hausdorff topological vector space. Let be a nonempty, compact convex subset of , respectively. Let be a closed convex and pointed cone with . Suppose that is a continuous set-valued mapping with compact convex values and is a continuous set-valued mapping with compact values. For every , set .

Lemma 5 (see [11]). Let be a nonempty compact convex subset of a locally convex Hausdorff topological space . If is upper semicontinuous and, for each , is a nonempty, closed, and convex subset, then there exists an such that .

Theorem 6. Suppose that the following conditions hold: (i) is continuous with compact convex values;(ii) are continuous with compact values;(iii)for each fixed , is generalized Luc's quasi--convex.Then, there exists an such that, for each , , and , there exists satisfying

Proof. We define a set-valued mapping by
It follows from [12, pages 110–119, Propositions 6 and 21] that is upper semicontinuous for each fixed . By [12, page 112, Proposition 11], the set is compact. Therefore, is nonempty for every .
Let We must show that . First, note that and then . As is upper semicontinuous and the set is compact, it follows that . Suppose that . Then, there exists a vector satisfying As is lower semicontinuous, there exists , such that . It follows from compactness of that there exists such that
It follows from the upper semicontinuity of and the compactness of that is compact. Hence, for the net , there exists a subnet of converging to . Without loss of generality, assume . Now we prove that Since the mapping is upper semicontinuous and the set is compact, we have .
Now, suppose that . Namely, there exists such that . As is lower semicontinuous, there exists such that . Since is continuous, for large enough, which is a contradiction to (11).
From the compactness of , we take such that By the compactness of , we can choose a converging subnet of , which is denoted without loss of generality by the original net . Assume . Similar to the preceding proof, we have Then, by (10), .
It follows from the continuity of that and . Therefore, , when is large enough. It is said that By the definition of and , we have This, however, contradicts the fact . Therefore, the mapping is closed.
Let , , and From the definition of , we have and As is convex-valued, .
According to the generalized Luc's quasi--convexity of , we get that, for all , there exist and such that Without loss of generality, suppose and , ; we have and . From (20), . By the monotonicity of , As therefore, .
Since is arbitrary, we have By the fact that is compact and is continuous, there exists such that Thus, . It follows from the definition of that Thus, ; namely, is a convex set.
Define by . Therefore, is a nonempty, convex, and closed subset of for each . Since is closed, so is , and since , is compact, by [12, page 111, Corollary 9], is upper semicontinuous. By Lemma 5, there exists a point such that .
By the definition of , we have
From (28), , By the compactness of and the continuity of , there exists , such that . Thus, for all , there exists such that . Then, it follows from the subadditivity of that By Lemma 4, we get So is a solution of (CWNEP) and this completes the proof.

Let , , and be real Hausdorff topological vector spaces, and let and be two compact subsets of and , respectively.

Corollary 7. Let , , and be real Hausdorff topological vector spaces, and let and be two nonempty subsets of and , respectively. Let be a closed convex and pointed cone with . Assume that (1) and are continuous and compact, and for each , and are nonempty, closed convex subsets;(2) are continuous;(3)for any fixed , is Luc's quasi--convex; for any fixed , is Luc's quasi--convex. Then there exists such that , , and

Remark 8. Since both the class of properly quasi--convex functions and the class of -convex functions (see [7]) are larger than the class of Luc's quasi--convex functions, Corollary 7 improves [7, Theorem].

Example 9. Suppose that , , and and let and be defined as and , respectively. For all , let It is clear that the mappings and are not properly quasi--convex (see [7]), but all the conditions of Corollary 7 hold. It is easy to see from [7] that both the class of properly quasi--convex functions and the class of -convex functions (see [7]) are larger than the class of Luc's quasi--convex functions, and then Corollary 7 improves [7, Theorem].

Conflict of Interests

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

Acknowledgment

This paper is supported by the Fundamental Research Funds for the Central Universities (JBK130401 and JBK140924).