An extragradient-like approximation method for variational inequality problems and fixed point problems Lu-Chuan Ceng a,1 , Jen-Chih Yao b, * ,2 a Department of Mathematics, Shanghai Normal University, Shanghai 200234, China b Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan Abstract The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of a non- expansive mapping and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous map- ping. We introduce an extragradient-like approximation method which is based on so-called extragradient method and viscosity approximation method. We establish a strong convergence theorem for two iterative sequences generated by this method. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Extragradient-like approximation method; Variational inequality; Fixed point; Monotone mapping; Nonexpansive mapping; Strong convergence 1. Introduction Let H be a real Hilbert space whose inner product and norm are denoted by h; i and kk, respectively. Let C be a nonempty closed convex subset of H. A mapping A of C into H is called monotone if hAx  Ay ; x  y i P 0 8x; y 2 C: A is called b-inverse-strongly monotone (see [1,3]) if there exists a constant b > 0 such that hAx  Ay ; x  y i P bkAx  Ay k 2 8x; y 2 C: It is clear that a b-inverse-strongly monotone mapping A is monotone and Lipschitz continuous. 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.021 * Corresponding author. E-mail addresses: zenglc@hotmail.com (L.-C. Ceng), yaojc@math.nsysu.edu.tw (J.-C. Yao). 1 This research was partially supported by the Dawn Program Foundation in Shanghai and the Shanghai Leading Academic Discipline Project (No. T0401). 2 This research was partially supported by a grant from the National Science Council. Applied Mathematics and Computation 190 (2007) 205–215 www.elsevier.com/locate/amc