Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application
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Research Article Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application Ren-You Zhong,1 Nan-Jing Huang,1 and Yeol Je Cho2 1 2
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea
Correspondence should be addressed to Yeol Je Cho, [email protected] Received 25 October 2010; Accepted 19 January 2011 Academic Editor: K. Teo Copyright q 2011 Ren-You Zhong 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. This paper is devoted to the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed by different parameters. By using the properties of recession cones, several equivalent characterizations are given for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. As an application, the stability of solution set for the set-valued vector equilibrium problem in a reflexive Banach space is also given. The results presented in this paper generalize and extend some known results in Fan and Zhong 2008, He 2007, and Zhong and Huang 2010.
1. Introduction Let X and Y be reflexive Banach spaces. Let K be a nonempty closed convex subset of X. Let F : K × K → 2Y be a set-valued mapping with nonempty values. Let P be a closed convex pointed cone in Y with int P / ∅. The cone P induces a partial ordering in Y , which was defined by y1 ≤P y2 if and only if y2 − y1 ∈ P . We consider the following set-valued vector equilibrium problem, denoted by SVEPF, K, which consists in finding x ∈ K such that F x, y ∩ − int P ∅,
∀y ∈ K.
1.1
2
Journal of Inequalities and Applications
It is well known that 1.1 is closely related to the following dual set-valued vector equilibrium problem, denoted by DSVEPF, K, which consists in finding x ∈ K such that F y, x ⊂ −P ,
∀y ∈ K.
1.2
We denote the solution sets of SVEPF, K and DSVEPF, K by S and SD , respectively. Let Z1 , d1 and Z2 , d2 be two metric spaces. Suppose that a nonempty closed convex set L ⊂ X is perturbed by a parameter u, which varies over Z1 , d1 , that is, L : Z1 → 2X is a set-valued mapping with nonempty closed convex values. Assume that a set-valued mapping F : X × X → 2Y is perturbed by a parameter v, which varies over Z2 , d2 , that is, F : X × X × Z2 → 2Y . We consider a parametric set-valued vector equilibrium problem, denoted by SVEPF·, ·, v, Lu, which consists in finding x ∈ Lu such that F x, y, v ∩ − int P ∅,
∀y ∈ Lu.
1.3
Similarly, we consider the parameterized dual set-valued vector equilibrium problem, denoted by DSVEPF·, ·, v, Lu, which consists in finding x ∈ L
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