Open Access

Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

Journal of Inequalities and Applications20062006:90295

DOI: 10.1155/JIA/2006/90295

Received: 30 April 2004

Accepted: 29 August 2004

Published: 18 January 2006

Abstract

Let be a nonlinear mapping from a nonempty closed invex subset of an infinite-dimensional Hilbert space into . Let be proper, invex, and lower semicontinuous on and let be continuously Fréchet-differentiable on with , the gradient of , -strongly monotone, and -Lipschitz continuous on . Suppose that there exist an , and numbers , , such that for all and for all , the set defined by is nonempty, where and is -Lipschitz continuous with the following assumptions. (i) , and . (ii) For each fixed , map is sequentially continuous from the weak topology to the weak topology. If, in addition, is continuous from equipped with weak topology to equipped with strong topology, then the sequence generated by the general auxiliary problem principle converges to a solution of the variational inequality problem (VIP): for all .

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Authors’ Affiliations

(1)
Department of Mathematics, University of Toledo

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© Verma 2006

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.