TAILIEUCHUNG - Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 748918,

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 748918, 17 pages doi: Research Article An Implicit Hierarchical Fixed-Point Approach to General Variational Inequalities in Hilbert Spaces L. C. Zeng,1, 2 Ching-Feng Wen,3 and J. C. Yao3 1 2 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China 3 Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan Correspondence should be addressed to Ching-Feng Wen, cfwen@ Received 13 December 2010; Accepted 5 March 2011 Academic Editor: Jong Kim Copyright q 2011 L. C. Zeng et al. This is an open. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 748918 17 pages doi 2011 748918 Research Article An Implicit Hierarchical Fixed-Point Approach to General Variational Inequalities in Hilbert Spaces L. C. Zeng 1 2 Ching-Feng Wen 3 and J. C. Yao3 1 Department of Mathematics Shanghai Normal University Shanghai 200234 China 2 Scientific Computing Key Laboratory of Shanghai Universities Shanghai 200234 China 3 Center for General Education Kaohsiung Medical University Kaohsiung 807 Taiwan Correspondence should be addressed to Ching-Feng Wen cfwen@ Received 13 December 2010 Accepted 5 March 2011 Academic Editor Jong Kim Copyright 2011 L. . 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. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F C H be a K-Lipschitzian and -strongly monotone operator with constants K n 0 V T C C be nonexpansive mappings with Fix T 0 where Fix T denotes the fixed-point set of T and f C H be a p-contraction with coefficient p e 0 1 . Let 0 p 2y K2 and 0 Y T where T 1 -yj 1 - p 2p - pK2 . For each s t e 0 1 let xs t be a unique solution of the fixed-point equation xst PCỊsỵf xS t I - spFfitV 1 - t T xS t . We derive the following conclusions on the behavior of the net xs t along the curve t t s i if t s O s as s 0 then xS f s zTO strongly which is the unique solution of the variational inequality of finding zTO e Fix T such that pF - Yf l I - V zTO x - zTO 0 for all x e Fix T and ii if t s s TO as s 0 then xS f s xTO strongly which is the unique solution of some hierarchical variational inequality problem. 1. Introduction Let C be a nonempty closed convex subset of a real Hilbert space H. Throughout this paper we write xn x to indicate that the sequence xn converges weakly to x. xn x implies that xn

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