STRONG CONVERGENCE THEOREMS FOR CONTINUOUS SEMIGROUPS OF ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

Let K be a nonempty closed convex and bounded subset of a real Banach space E. Let T := {T(t) : t is an element of R+} be a strongly continuous semigroup of asymptotically nonexpansive self-mappings on K with a sequence {L-t} subset of [1, infinity). Then, for a given u(0) is an element of K and s(n) is an element of (0, 1), 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 each n is an element of N, satisfying parallel to u(n) - T(t)u(n)parallel to -> 0 as n -> infinity, for any t is an element of R+, where alpha(n) := (1 - s(n)/L-tn). If, in addition, E is uniformly convex with uniformly Gateaux differentiable norm, then it is proved that F (T) not equal empty set and the sequence {u(n)} converges strongly to a point of F(T) under certain mild conditions on {L-t}, {t(n)} and {s(n)}. 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(0) + (1 - alpha(n)) T(t(n))x(n), n >= 1, converges to a fixed point of T under appropriate assumption imposed upon the sequence {x(n)}.