Acta Mathematica Scientia 2011,31B(2):716–726 http://actams.wipm.ac.cn VISCOSITY ITERATIVE METHODS FOR COMMON FIXED POINTS OF TWO NONEXPANSIVE MAPPINGS WITHOUT COMMUTATIVITY ASSUMPTION IN HILBERT SPACES ∗ Eknarin Jankaew Somyot Plubtieng Anutep Tepphun Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand. E-mail: Somyotp@nu.ac.th Abstract In this article, we introduce a new viscosity iterative method for two nonex- pansive mappings in Hilbert spaces. We also prove, without commutativity assumption, that the iterates converge to a common fixed point of the mappings which solves some variational inequality. The results presented extend the corresponding results of Shimizu and Takahashi [T. Shimizu, W. Takahashi, Strong convergence to common fixed point of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997), 71-83], and Yao and Chen [Y. Yao, R. Chen, Convergence to common fixed points of average mappings without commutativity assumption in Hilbert spaces, Nonlinear Analysis 67(2007), 1758-1763]. Key words Viscosity iterative method; common fixed points; nonexpansive mappings; variational inequality 2000 MR Subject Classification 41A50; 47H10; 54H25 1 Introduction Let H be a real Hilbert space, and C a closed convex subset of H . Recall that a mapping T : C → C is nonexpansive if ‖Tx - Ty‖≤‖x - y‖, for all x, y ∈ C. For a mapping T of C into itself, we denote F (T ) the set of fixed points of T . We also denote the sets of positive integers and nonnegative real numbers by N and R + , respectively. Baillon [1] proved the first nonlinear egodic theorem: Let C be a nonempty bounded closed convex subset of H and T be a nonexpansive mapping of C into itself. Then, for an arbitary x ∈ C,  1 n+1 n ∑ i=0 T i x  converges weakly to a fixed point of T . In contrast, Wittmann [2] studied * Received Febuary 18, 2008; revised August 24, 2009. The second author would like to thank the Thailand Research Fund for financial support under Grant BRG5280016.