Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 276859, 10 pages doi: Research Article An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces Nguyen Buong1 and Nguyen Thi Quynh Anh2 1 Vietnamese Academy of Science and Technology, Institute of Information Technology, 18, Hoang Quoc Viet, Cau Giay, Ha Noi 122100, Vietnam 2 Department of Information Technology, Thai Nguyen National University, Thainguye 842803, Vietnam Correspondence should be addressed to Nguyen Buong, nbuong@ Received 17 December 2010; Accepted 7 March 2011 Academic Editor: Jong Kim Copyright q. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 276859 10 pages doi 2011 276859 Research Article An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces Nguyen Buong1 and Nguyen Thi Quynh Anh2 1 Vietnamese Academy of Science and Technology Institute of Information Technology 18 Hoang Quoc Viet Cau Giay Ha Noi 122100 Vietnam 2 Department of Information Technology Thai Nguyen National University Thainguye 842803 Vietnam Correspondence should be addressed to Nguyen Buong nbuong@ Received 17 December 2010 Accepted 7 March 2011 Academic Editor Jong Kim Copyright 2011 N. Buong and N. T. Quynh Anh. 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. We introduce a new implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces. 1. Introduction Let C be a nonempty closed and convex subset of a real Hilbert space H with inner product and norm II II and let F H H be a nonlinear mapping. The variational inequality problem is formulated as finding a point p e C such that F ip p - p 0 Vp e C. Variational inequalities were initially studied by Kinderlehrer and Stampacchia in 1 and ever since have been widely investigated since they cover as diverse disciplines as partial differential equations optimal control optimization mathematical programming mechanics and finance see 1-3 . 2 Fixed Point Theory and Applications It is well known that if F is an L-Lipschitz continuous and n-strongly monotone that is F satisfies the following conditions llF x - F y II LHx - y F x - F y x - y n x - y 2
