Open Access

An Implicit Hierarchical Fixed-Point Approach to General Variational Inequalities in Hilbert Spaces

Fixed Point Theory and Applications20112011:748918

https://doi.org/10.1155/2011/748918

Received: 13 December 2010

Accepted: 5 March 2011

Published: 20 March 2011

Abstract

Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian and -strongly monotone operator with constants , be nonexpansive mappings with where denotes the fixed-point set of , and be a -contraction with coefficient . Let and , where . For each , let be a unique solution of the fixed-point equation . We derive the following conclusions on the behavior of the net along the curve : (i) if , as , then strongly, which is the unique solution of the variational inequality of finding such that and (ii) if , as , then strongly, which is the unique solution of some hierarchical variational inequality problem.

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Authors’ Affiliations

(1)
Department of Mathematics, Shanghai Normal University
(2)
Scientific Computing Key Laboratory of Shanghai Universities
(3)
Center for General Education, Kaohsiung Medical University

Copyright

© L. C. Zeng et al. 2011

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.