An implicit iteration process for nonexpansive semigroups

Let C be a closed convex subset of a Banach space E. Let {T (t) : t >= 0} be a strongly continuous semigroup of nonexpansive mappings on C such that boolean AND(t >= 0) F(T(t)) not equal empty set. Let {alpha(n)} and {t(n)} be sequences of real numbers satisfying appropriate conditions, then for arbitrary x(0) is an element of C, the Mann type implicit iteration process {x(n)} given by x(n) = alpha(n)x(n-1) + (1 - alpha(n))T(t(n))x(n), n >= 0, weakly (strongly) converges to an element of boolean AND(t >= 0)F(T(t)). (C) 2011 Elsevier Ltd. All rights reserved.