Strong and weak convergence theorems for 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 asymptotically nonexpansive nonself-map with sequence {kn}n 1 ⊂ [1,∞), lim kn = 1, F(T ) := {x ∈ K: T x = x} =∅. Suppose {xn}n 1 is generated iteratively by x1 ∈ K, xn+1 = P (1 − αn)xn + αnT (PT)n−1xn , n 1, where {αn}n 1 ⊂ (0, 1) is such that < 1 − αn < 1 − for some > 0. It is proved that (I − T ) is demiclosed at 0. Moreover, if n 1(k2 n − 1) < ∞ and T is completely continuous, strong convergence of {xn} to some x∗ ∈ F(T ) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x∗ ∈ F(T ) is obtained.  2003 Elsevier Science (USA). All rights reserved