Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 94092, 13 pages
On Generalized Strong Vector Variational-Like Inequalities in Banach Spaces
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
Academic Editor: Yeol Je Cho
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.
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).