Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by χ0 ∈ C, χn+1 = αnu +(1 - αn)Txn, n = 0,1,2, ⋯, where 0 ≤ αn ≤ 1, limn→∞ αn = 0, ∑n=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 {χn} 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. © Editorial Committee of Applied Mathematics 2007.