Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces

In this paper, by using some new analysis techniques, we study the approximation problems of common fixed points of Halpern’s iterative sequence for a class of finite nonexpansive mappings in strictly convex and reflexive Banach spaces by using Banach’s limit. The main results presented in this paper generalize, extend and improve the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 202 (1996) 150–159], Halpern [Fixed points of nonexpansive maps, Bull. Am. Math. Soc. 73 (1967) 957–961], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Am. Math. Soc. 125 (1997) 3641–3645], Takahashi et al. [Approximation of common fixed points of a family of finite nonexpansive mappings in Banach spaces, Sci. Math. Jpn. 56 (2002) 475–480], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491], Xu [Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65 (2002) 109–113, Remarks on an iterative method for nonexpansive mappings, Commun. Appl. Nonlinear Anal. 10 (2003) 67–75] and others.