Iterative approximation of fixed points of nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm and T :K → K be a nonexpansive mapping with F(T ) := {x ∈ K: T x = x} = ∅. For a fixed δ ∈ (0, 1), define S :K →K by Sx := (1 − δ)x + δT x, ∀x ∈ K. Assume that {zt } converges strongly to a fixed point z of T as t →0, where zt is the unique element of K which satisfies zt = tu+(1−t)T zt for arbitrary u ∈ K. Let {αn} be a real sequence in (0, 1) which satisfies the following conditions: C1: limαn = 0; C2: αn =∞. For arbitrary x0 ∈ K, let the sequence {xn} be defined iteratively by xn+1 = αnu+(1−αn)Sxn. Then, {xn} converges strongly to a fixed point of T . © 2005 Elsevier Inc. All rights reserved.