Approximating fixed points of non-self asymptotically nonexpansive mappings

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptoti-cally nonexpansive mapping with {kn} ⊂ [1,∞) such that n=1(kn − 1) < ∞ and F(T) is nonempty, where F(T) denotes the fixed points set of T. Let {αn}, {α′n}, and {α′′n} be real sequences in (0,1) and ≤ αn,α′n,α′′n ≤ 1 − for all n ∈N and some > 0. Starting from arbitrary x1 ∈ K, define the sequence {xn} by x1 ∈ K, zn = P(α′′n T(PT)n−1xn + (1− α′′n)xn), yn = P(α′nT(PT)n−1zn + (1−α′n)xn), xn+1 = P(αnT(PT)n−1yn + (1−αn)xn). (i) If the dual E ∗ of E has the Kadec-Klee property, then {xn} converges weakly to a fixed point p ∈ F(T); (ii) if T satisfies condition (A), then {xn} converges strongly to a fixed point p ∈ F(T). Copyright © 2006 Y. Su and X. Qin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction an