Research Article Hybrid Viscosity Iterative Method for Fixed Point, Variational Inequality and Equilibrium Problem | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 264628 13 pages doi 2010 264628 Research Article Hybrid Viscosity Iterative Method for Fixed Point Variational Inequality and Equilibrium Problems Yi-An Chen and Yi-Ping Zhang College of Mathematics and Statistics Chongqing Technology and Business University Chongqing 400067 China Correspondence should be addressed to Yi-An Chen chenyian1969@ Received 27 December 2009 Revised 3 May 2010 Accepted 1 June 2010 Academic Editor Simeon Reich Copyright 2010 . Chen and . Zhang. 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 scheme by the viscosity iterative method for finding a common element of the solution set of an equilibrium problem the solution set of the variational inequality and the fixed points set of infinitely many nonexpansive mappings in a Hilbert space. Then we prove our main result under some suitable conditions. 1. Introduction Let H be a real Hilbert space with the inner product and the norm being denoted by and II II respectively. Let C be a nonempty closed and convex subset of H and let F be a bifunction of C X C into R where R denotes the real numbers. The equilibrium problem for F C X C R is to find x e C such that F x y 0 Vy e C. The solution set of is denoted by EP F . Let A C H be a mapping. The classical variational inequality denoted by VI A C is to find x e C such that Ax v - x 0 Vv e C. 2 Fixed Point Theory and Applications The variational inequality has been extensively studied in the literature see . 1-3 . The mapping A is called a-inverse-strongly monotone if Au - Av u - v aỊỊAu - AvỊỊ2 Vu v e C where a is a positive real number. A mapping T C C is called strictly pseudocontractive if there exists k with 0 k 1 such that II .
