Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by where 0 ≤ α _n ≤ 1, lim _n→∞ α _n = 0, Σ _n=0 ^∞ α _n = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {x _n} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results. Supported by the Natural Science Foundation of the Educational Dept. of Zhejiang Province (20020868).