Connectedness and Compactness of Weak Efficient Solutions for Set-Valued Vector Equilibrium Problems
- PDF / 138,477 Bytes
- 16 Pages / 600.05 x 792 pts Page_size
- 38 Downloads / 228 Views
Research Article Connectedness and Compactness of Weak Efficient Solutions for Set-Valued Vector Equilibrium Problems Bin Chen, Xun-Hua Gong, and Shu-Min Yuan Department of Mathematics, Nanchang University, Nanchang 330047, China Correspondence should be addressed to Xun-Hua Gong, [email protected] Received 1 November 2007; Revised 17 July 2008; Accepted 5 September 2008 Recommended by C. E. Chidume We study the set-valued vector equilibrium problems and the set-valued vector HartmanStampacchia variational inequalities. We prove the existence of solutions of the two problems. In addition, we prove the connectedness and the compactness of solutions of the two problems in normed linear space. Copyright q 2008 Bin Chen 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.
1. Introduction We know that one of the important problems of vector variational inequalities and vector equilibrium problems is to study the topological properties of the set of solutions. Among its topological properties, the connectedness and the compactness are of interest. Recently, Lee et al. 1 and Cheng 2 have studied the connectedness of weak efficient solutions set for single-valued vector variational inequalities in finite dimensional Euclidean space. Gong 3–5 has studied the connectedness of the various solutions set for single-valued vector equilibrium problem in infinite dimension space. The set-valued vector equilibrium problem was introduced by Ansari et al. 6. Since then, Ansari and Yao 7, Konnov and Yao 8, Fu 9, Hou et al. 10, Tan 11, Peng et al. 12, Ansari and Flores-Baz´an 13, Lin et al. 14 and Long et al. 15 have studied the existence of solutions for set-valued vector equilibrium and set-valued vector variational inequalities problems. However, the connectedness and the compactness of the set of solutions to the set-valued vector equilibrium problem remained unstudied. In this paper, we study the existence, connectedness, and the compactness of the weak efficient solutions set for set-valued vector equilibrium problems and the set-valued vector Hartman-Stampacchia variational inequalities in normed linear space.
2
Journal of Inequalities and Applications
2. Preliminaries Throughout this paper, let X, Y be two normed linear spaces, let A be a nonempty subset of X, let F : A × A → 2Y be a set-valued map, and let C be a closed convex pointed cone in Y . We consider the following set-valued vector equilibrium problem SVEP: find x ∈ A, such that Fx, y ∩ −int C ∅ ∀y ∈ A.
2.1
Definition 2.1. Let int C / ∅. A vector x ∈ A satisfying Fx, y ∩ −int C ∅ ∀y ∈ A
2.2
is called a weak efficient solution to the SVEP. Denote by Vw A, F the set of all weak efficient solutions to the SVEP. Let Y ∗ be the topological dual space of Y . Let C∗ {f ∈ Y ∗ : fy ≥ 0 ∀y ∈ C}
2.3
be the dual cone of C. Definition 2.2. Let f ∈ C∗ \ {0}.
Data Loading...
