Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space

Some sufficient and necessary conditions for the iterative sequence converging to a common fixed points for a family of nonexpansive mappings in Banach spaces are obtained. The results presented in this paper not only give an affirmative answer to Halpern's open question and a partial answer to the Reich's open question but also extend and improve some recent results of Bauschke [J. Math. Anal. Appl. 202 (1996) 150-159], Halpern [Bull. Amer. Math. Soc. 73 (1967) 957-961], Lions [C. R. Acad. Sci. Paris, Sér. A 284 (1977) 1357-1359], Reich [J. Math. Anal. Appl. 75 (1980) 128-292; PanAmer. Math. J. 4 (1994) 23-28], Shioji and Takahashi [Proc. Amer. Math. Soc. 125(12) (1997) 3641-3645], Takahashi et al. [J. Approximation Theory 91 (1997) 386-397; Sci. Math. Jap. 56(3) (2002) 417-422], Wittmann [Arch. Math. 58 (1992) 486-491] and Xu [Bull. Austral. Math. Soc. 65 (2002) 109-113; Commun. Appl. Nonlinear Anal. 10 (2003) 67-75]. As applications, at the end of the paper, we utilize our results to study the feasibility problem.