Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 61794, 13 pages
doi:10.1155/2007/61794
Research Article

Generalized Vector Equilibrium-Like Problems without Pseudomonotonicity in Banach Spaces

Lu-Chuan Ceng,1 Sy-Ming Guu,2 and Jen-Chih Yao3

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Department of Business Administration, College of Management, Yuan-Ze University, Chung-Li City 330, Taoyuan Hsien, Taiwan
3Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan

Received 10 January 2007; Accepted 21 March 2007

Academic Editor: Donal O'Regan

Copyright © 2007 Lu-Chuan Ceng 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

Let X and Y be real Banach spaces, D a nonempty closed convex subset of X, and C:D→2Y a multifunction such that for each u∈D, C(u) is a proper, closed and convex cone with intC(u)≠∅, where intC(u) denotes the interior of C(u). Given the mappings T:D→2L(X,Y), A:L(X,Y)→L(X,Y), f:L(X,Y)×D×D→Y, and h:D→Y, we study the generalized vector equilibrium-like problem: find u0∈D such that f(As0,u0,v)+h(v)−h(u0)∉−intC(u0) for all v∈D for some s0∈Tu0. By using the KKM technique and the well-known Nadler result, we prove some existence theorems of solutions for this class of generalized vector equilibrium-like problems. Furthermore, these existence theorems can be applied to derive some existence results of solutions for the generalized vector variational-like inequalities. It is worth pointing out that there are no assumptions of pseudomonotonicity in our existence results.