Research Article On Generalized Strong Vector Variational-Like Inequalities in Banach Spaces

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 vec-tor 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 prob-lem proposed by Chen and Hou in 2000, and also extend and improve the corresponding results of Fang and Huang (2006). 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. 1. Introduction an