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

Recommended by Donal O'Regan

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.