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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 296759,

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 296759, 16 pages doi:10.1155/2010/296759 Research Article Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces Somyot Plubtieng and Sukanya Chornphrom Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence should be addressed to Somyot Plubtieng, somyotp@nu.ac.th Received 30 June 2010; Revised 10 October 2010; Accepted 13 December 2010 Academic Editor: Brailey Sims Copyright q 2010 S. Plubtieng and S. Chornphrom. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 296759 16 pages doi 10.1155 2010 296759 Research Article Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces Somyot Plubtieng and Sukanya Chornphrom Department of Mathematics Faculty of Science Naresuan University Phitsanulok 65000 Thailand Correspondence should be addressed to Somyot Plubtieng somyotp@nu.ac.th Received 30 June 2010 Revised 10 October 2010 Accepted 13 December 2010 Academic Editor Brailey Sims Copyright 2010 S. Plubtieng and S. Chornphrom. 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 an iterative method for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonspreading mapping in a Hilbert space. Then we prove a strong convergence theorem which is connected with the work of S. Takahashi and W. Takahashi 2007 and Iemoto and Takahashi 2009 . 1. Introduction Let H be a real Hilbert space with inner product and norm II II respectively and let C be a closed convex subset of H. Let F C X C X be bifunction where R is the set of real numbers. The equilibrium problem for F C X C R is to find x C such that F x y 0 Vy e C. 1.1 The set of solution of 1.1 is denoted by EP F . Given a mapping A C H let F x y Ax y - x for all x y e C. Then z e EP F if and only if Az y - z 0 for all y e C that is z is a solution of the variational inequality. Numerous problems in physics optimization and economics reduce to find a solution of 1.1 see for example 1-9 and the references therein. A mapping T of C into itself is said to be nonexpansive if Tx - Ty x - y for all x y e C and a mapping F is said to be firmly nonexpansive if Fx - Fytf2 x - y Fx - Fy for all x y e C. Let E be a smooth strictly .