Journal of Inequalities and Applications 
Volume 2007 (2007), Article ID 94092, 13 pages
doi:10.1155/2007/94092
Research Article

On Generalized Strong Vector Variational-Like Inequalities in Banach Spaces

Lu-Chuan Ceng,1 Yen-Cherng Lin,2 and Jen-Chih Yao3

 1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2General Education Center, China Medical University, Taichung 404, Taiwan
3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
Received 8 December 2006; Accepted 9 February 2007

Recommended by Yeol Je Cho

Abstract

The purpose of this paper is to study the solvability for a class of generalized strong vector variational-like inequalities in reflexive Banach spaces. Firstly, utilizing Brouwer's fixed point theorem, we prove the solvability for this class of generalized strong vector variational-like inequalities without monotonicity assumption under some quite mild conditions. Secondly, we introduce the new concept of pseudomonotonicity for vector multifunctions, and prove the solvability for this class of generalized strong vector variational-like inequalities for pseudomonotone vector multifunctions by using Fan's lemma and Nadler's theorem. Our results give an affirmative answer to an open problem proposed by Chen and Hou in 2000, and also extend and improve the corresponding results of Fang and Huang (2006).