Abstract
Let K be a nonempty closed convex subset of a real Banach space E. Let T := {T(t) : t is an element of R+} be a strongly continuous semi-group of asymptotically nonexpansive mappings from K into K with a sequence {L-t} subset of [1, infinity). Suppose F(T) not equal empty set. Then, for a given u(0) is an element of K and t(n) > 0 there exists a sequence {u(n)} subset of K such that u(n) = (1 - alpha(n))T(t(n))u(n) + alpha(n)u(0), for n is an element of N such that {alpha(n)} subset of (0, 1) and L-tn - 1 < alpha(n), where t(n) is an element of R+. Suppose, in addition, that E is reflexive strictly convex with a uniformly Gateaux differentiable norm and that lim(n ->infinity) t(n) = infinity. lim(n ->infinity) alpha(n) = lim(n ->infinity) L-tn-1/alpha(n) = 0. Then the sequence {u(n)} converges strongly to a point of F(T). Moreover, it is proved that an explicit sequence {x(n)} generated from x(1) is an element of K by x(n+1) := alpha(n)u + (1 - alpha(n))T(t(n))x(n), n >= 1, converges to a fixed point of T. (C) 2009 Elsevier Ltd. All rights reserved.