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 {k(n}ngreater than or equal to1) subset of [1, infinity), lim k(n) = 1, F(T) := {x is an element of K: Tx = x} not equal emptyset. Suppose {x(n)}(ngreater than or equal to1) is generated iteratively by x(1) is an element of K, x(n+1) = P((1 - alpha(n))x(n) + alpha(n)T(PT)(n-1) x(n)), n greater than or equal to 1, where {alpha(n)}(ngreater than or equal to1) subset of (0, 1) is such that epsilon < 1 - alpha(n) < 1 - epsilon for some epsilon > 0. It is proved that (1 - T) is demiclosed at 0. Moreover, if Sigma(ngreater than or equal to1)(k(n)(2) - 1) < infinity and T is completely continuous, strong convergence Of {x(n)} I to some x* is an element of F(T) is proved. If T is not assumed to be completely continuous but E also has a Frechet differentiable norm, then weak convergence Of {x(n)} I to some x* is an element of F(T) is obtained. (C) 2003 Elsevier Science (USA). All rights reserved.