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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 373462, 9 pages
http://dx.doi.org/10.1155/2012/373462
Research Article

Noncompact Equilibrium Points and Applications

1Research Chair on Applied and Actuarial Mathematics, Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received 16 February 2012; Accepted 28 March 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Zahra Al-Rumaih et al. 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 prove an equilibrium existence result for vector functions defined on noncompact domain and we give some applications in optimization and Nash equilibrium in noncooperative game.

1. Introduction

Let be a subset of a vector space and , with . It is well known in the literature that a point satisfying the property: is called an equilibrium point. This notion of equilibrium plays an important role in various areas such as optimization, variational inequalities, and Nash equilibrium problems.

We recall that an equilibrium point in this formulation was first introduced and studied by Blum and Oettli [1], who, inspired by the very known work of Allen [2] and of Fan [3], proved the existence of an equilibrium point by using some hypothesis concerning continuity, convexity, compactness, and monotonicity.

Recently, many authors have investigated the existence of such equilibrium points in different context. In some references, different generalizations of monotonicity condition are used to prove the existence of equilibrium (see, e.g., [46]); while some other references studied this equilibrium problem under generalized convexity condition (see, e.g., [5]). The main objective of our work is to study this equilibrium problem by using a generalized coercivity-type condition.

In Section 3 of this paper, we prove the existence of equilibrium points when is a noncompact subset of a Hausdorff real-topological vector space and is a vector function that takes its values in another Hausdorff real topological vector space. The order on will be defined by a cone . Formally, we obtain the existence of a point , which will be called a weak equilibrium point, that satisfies the following condition: where denotes the interior of the cone in . The existence of what we refer to as equilibrium point, satisfying the following condition: is then deduced. The results that we obtain in this section generalize the corresponding results obtained in the classical formulation by Fan in [3, 7], Blum and Oettli in [1] as well as the corresponding results obtained in noncompact case by Tan and Tinh in [8].

In Section 4 and as applications, we prove the existence of saddle points for vector functions defined on noncompact domain. We also prove the existence of Nash equilibrium for an infinite set of players game in which every player has a noncompact strategy set and vector loss function.

2. Preliminaries

In this section, we will recall some notions, definitions, and some properties from the literature that will be used in the paper. Let and be real Hausdorff topological vector spaces. Let be a nonempty closed convex subset and a pointed convex closed cone. The cone can define a partial order on , denoted by , as follows: if and only if . We will write if and only if , in the case .

We say that the cone satisfies condition (*) if there is a pointed convex closed cone such that .

Let be a mapping. is said to be convex (resp., concave) with respect to if for all and , the following condition is satisfied:(resp.: . It is clear that if and are two convex closed cones in with and is convex (resp., concave) with respect to , then is also convex (resp., concave) with respect to .

The mapping is said to be lower semicontinuous, in brief l.s.c. (resp., upper semi-continuous, in brief u.s.c.), at with respect to if for any neighborhood of in , there exists a neighborhood of in such that

(resp. ).

Note that following Lemma  2.11 in [8], if is l.s.c. with respect to , then the set is closed. The mapping is said to be continuous with respect to at a point in if it is l.s.c. and u.s.c. with respect to at this point.

The mapping is said to be monotone with respect to if for all , the following condition is satisfied:

In this paper, we will use the definition of coercing family borrowed from [9].

Definition 2.1. Consider a subset of a topological vector space and a topological space . A family of pair of sets is said to be coercing for a set-valued map if and only if the following hold.(i)For each , is contained in a compact convex subset of and is a compact subset of .(ii)For each , there exists such that .(iii)For each , there exists with .

Remark 2.2. Definition 2.1 can be reformulated by using the “dual’’ set-valued map defined for all by . Indeed, a family is coercing for if and only if it satisfies conditions (i), (ii) of Definition 2.1 and the following one:

Note that in case where the family is reduced to one element, condition (iii) of Definition 2.1 and in the sense of Remark 2.2 appeared first in this generality (with two sets and ) in [10] and generalizes condition of Karamardian [11] and Allen [2]. Condition (iii) is also an extension of the coercivity condition given by Fan [7]. For other examples of set-valued maps admitting a coercing family that is not necessarily reduced to one element, see [9].

The following generalization of KKM principle obtained in [9] will be used in the proof of the main result of this paper.

Proposition 2.3. Let be a Hausdorff topological vector space, a convex subset of , a nonempty subset of , and a KKM map with compactly closed values in (i.e., for all , is closed for every compact set of ). If admits a coercing family, then .

3. The Main Result

The main result of this paper is the following equilibrium theorem for vector valued maps.

Theorem 3.1. Let be a nonempty closed convex subset of a Hausdorff topological vector space , a Hausdorff topological vector space, and be two functions satisfying the following conditions.(1) is monotone function.(2)For any fixed , the function is convex, l.s.c. with respect to on .(3) for all .(4)For any fixed , the function is u.s.c. with respect to on .(5)For any fixed , the function is convex.(6)There exists a family satisfying conditions (i) and (ii) of Definition 2.1 and the following one: For each , there exists such that
Then, there exists a point such that:

Proof. For any , we consider the set-valued map: We have the following.
(i) For all , is closed in , then has compactly closed values.
(ii) Let be a finite subset of and . We want to show that By absurdity, suppose that with , and , it means that for all , , By assumption (1) and (2), we obtain Further, it implies from assumptions (3) and (5) that It follows that Or We deduce that so we have a contradiction.
(iii) Hypothesis (6) implies that the family satisfies the following condition: for all , there exists with and hence it is a coercing family for .
satisfies all hypothesis of Proposition 2.3, so Take in this intersection, then , for all .

Corollary 3.2. Let , , , and satisfy all assumptions of Theorem 3.1 and the additional following conditions.(a) for all .(b)For any fixed , the function is u.s.c. with respect to at .
Then, there exists a point such that In addition, if satisfies condition (*), then

Proof. By Theorem 3.1, there exists with Since , by applying Lemma  3.3 in [8], we obtain This proves the first assertion of Corollary 3.2. Now, let satisfy condition (*) and let be a pointed convex closed cone in such that . It is easy to see that , , , , and satisfy all assumptions of Theorem 3.1, by the first assertion, we have Since , it follows that

Let be a convex subset of . The core of relative to , denoted by , is the set defined by if and only if and for all , where .

The following result can be deduced from Theorem 3.1.

Corollary 3.3. Let , , , and satisfy hypothesis (1–5) of Theorem 3.1 and the following condition(6′)There exists a nonempty convex compact subset of such that, for any , one can find a point such that
Then, there exists a point such that In addition, if satisfies condition (*), then

Proof. We just prove the first assertion. By taking for all , , which is a convex compact set we can see that by using hypothesis (6′) that admits a coercing family in the sense of Remark 2.2.

Remark 3.4. Note that if is a compact convex subset of , then condition (6) of Theorem 3.1 is automatically satisfied. Hence, Theorem 3.1 extends Theorem  1 in [1]. Corollary 3.2 extends also Lemma  3.2 in [8] obtained in the noncompact case and Corollary 3.3 corresponds to Theorem  3.1 in [1]. In case of real-valued function , those results coincide with the corresponding results obtained in [3, 7].

4. Applications

Let be a (possibly infinite) set of players. If each player has a nonempty strategy subset of a Hausdorff topological vector space and a loss function, depending on the strategies of players. For , we denote and .

A point is said to be a weak Nash equilibrium if and only if for all , holds for all .

A point is said to be a Nash equilibrium if and only if for all ,

holds for all .

Proposition 4.1. Let , , be as above. Assume that, for all , the following conditions are satisfied.(1) is continuous with respect to .(2)For any fixed , the function is convex.(3)There exists a family satisfying condition (a) and (b) of Definition 2.1 and the following one:
Then, there exists a weak Nash equilibrium. In addition, if satisfies condition (*), then there exists a Nash equilibrium.

Proof. Define , for all , , by We can easily verify that , , , and as above satisfy all assumptions of Corollary 3.2. Applying the first part of this corollary, we conclude that there exists a point with or
If for , we choose in such a way that , then
This completes the proof of the first assertion of the proposition. The second assertion of Corollary 3.2 implies that and this gives us the second assertion.

Let and be two Hausdorff topological vector spaces, and be two nonempty convex closed subsets of and , respectively. Let and be as above and .

A point is called a weak saddle point of with respect to if

holds for all .

A point is called a saddle point of with respect to if

holds for all .

Proposition 4.2. Let , , and be as above and . Assume that the following hold.(1)For any fixed , is a convex l.s.c. function with respect to .(2)For any fixed , is a concave u.s.c. function with respect to .(3)There exists a family satisfying condition (a) and (b) of Definition 2.1 and the following one. For each , there exists such that
Then, there exists a weak saddle point of . In addition, if satisfies condition (*), then there exists a saddle point of .

Proof. Consider the function , where , defined for all by Apply the first part of Corollary 3.2 for and the null function 0, we conclude that there exists a point with , for all . This follows: The second assertion of Corollary 3.2 implies that and this completes the proof.

Acknowledgments

The third author extends his appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the visiting professor program (VPP).

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