Open Access

On Generalized Strong Vector Variational-Like Inequalities in Banach Spaces

Journal of Inequalities and Applications20072007:094092

https://doi.org/10.1155/2007/94092

Received: 8 December 2006

Accepted: 9 February 2007

Published: 10 April 2007

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).

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Authors’ Affiliations

(1)
Department of Mathematics, Shanghai Normal University
(2)
General Education Center, China Medical University
(3)
Department of Applied Mathematics, National Sun Yat-sen University

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Copyright

© Ceng et al. 2007

This article is published under license to BioMed Central Ltd. 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.