Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

Abstract

This paper studies the convergence of the sequence defined by \(x_0 \in C,x_{n + 1} = \alpha _n u + (1 - \alpha _n )Tx_n ,n = 0,1,2,...,\) where 0 ≤ αn ≤ 1, limn→∞αn = 0, Σn=0∞αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} 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.