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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 971479,

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 971479, 28 pages doi:10.1155/2011/971479 Research Article Hybrid Algorithms of Common Solutions of Generalized Mixed Equilibrium Problems and the Common Variational Inequality Problems with Applications Thanyarat Jitpeera, Uamporn Witthayarat, and Poom Kumam Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 5 January 2011; Accepted 20 February 2011 Academic Editor: Tomonari Suzuki Copyright q 2011 Thanyarat Jitpeera et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 971479 28 pages doi 10.1155 2011 971479 Research Article Hybrid Algorithms of Common Solutions of Generalized Mixed Equilibrium Problems and the Common Variational Inequality Problems with Applications Thanyarat Jitpeera Uamporn Witthayarat and Poom Kumam Department of Mathematics Faculty of Science King Mongkut s University of Technology Thonburi KMUTT Bangmod Bangkok 10140 Thailand Correspondence should be addressed to Poom Kumam poom.kum@kmutt.ac.th Received 5 January 2011 Accepted 20 February 2011 Academic Editor Tomonari Suzuki Copyright 2011 Thanyarat Jitpeera et al. 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 new iterative algorithms by hybrid method for finding a common element of the set of solutions of fixed points of infinite family of nonexpansive mappings the set of common solutions of generalized mixed equilibrium problems and the set of common solutions of the variational inequality with inverse-strongly monotone mappings in a real Hilbert space. We prove the strong convergence of the proposed iterative method under some suitable conditions. Finally we apply our results to complementarity problems and optimization problems. Our results improve and extend the results announced by many others. 1. Introduction Throughout this paper let H be a real Hilbert space with inner product and norm II II and let C be a nonempty closed convex subset of H. A mapping T C C is called nonexpansive if Tx - Ty x - yU for all x y e C. The set of fixed points of T denoted by F T that is F T x e C Tx x . If C c H is bounded closed and convex and T is a nonexpansive mapping of C into itself then F T f 0 see for instance 1 . Let F be a bifunction of C X C into R where R is the set of real numbers A C H a .