tailieunhanh - Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 274820,

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 274820, 19 pages doi: Research Article Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping Atid Kangtunyakarn Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Correspondence should be addressed to Atid Kangtunyakarn, beawrock@ Received 8 November 2010; Accepted 14 December 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 Atid Kangtunyakarn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution,. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 274820 19 pages doi 2011 274820 Research Article Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping Atid Kangtunyakarn Department of Mathematics Faculty of Science King Mongkut s Institute of Technology Ladkrabang Bangkok 10520 Thailand Correspondence should be addressed to Atid Kangtunyakarn beawrock@ Received 8 November 2010 Accepted 14 December 2010 Academic Editor Qamrul Hasan Ansari Copyright 2011 Atid Kangtunyakarn. 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. The purpose of this paper is to prove the strong convergence theorem for finding a common element of the set of fixed point problems of strictly pseudocontractive mapping in Hilbert spaces and two sets of generalized equilibrium problems by using the hybrid method. 1. Introduction Let C be a closed convex subset of a real Hilbert space H and let F C X C R be a bifunction. Recall that the equilibrium problem for a bifunction F is to find x e C such that F x y 0 Vy e C. The set of solutions of is denoted by EP F . Given a mapping T C H let F x y Tx y - x for all x y e C. Then z e EP F if and only if Tz y - z 0 for all y e C that is z is a solution of the variational inequality. Let A C H be a nonlinear mapping. The variational inequality problem is to find a u e C such that v - u Au 0 2 Fixed Point Theory and Applications for all v e C. The set of solutions of the variational inequality is denoted by VI C A . Now we consider the following generalized equilibrium problem Find z e C such that F z y Az y - z 0 Vy e C. The set of z e C is denoted by EP F A that is EP F A z e C F z y Az y - z 0 Vy e C . In .