An implicit iteration process for nonexpansive semigroups

Let CC be a closed convex subset of a Banach space EE. Let {T(t):t⩾0}{T(t):t⩾0} be a strongly continuous semigroup of nonexpansive mappings on CC such that ∩t⩾0F(T(t))≠0̸. Let {αn}{αn} and {tn}{tn} be sequences of real numbers satisfying appropriate conditions, then for arbitrary x0∈Cx0∈C, the Mann type implicit iteration process {xn}{xn} given by xn=αnxn−1+(1−αn)T(tn)xn,n⩾0xn=αnxn−1+(1−αn)T(tn)xn,n⩾0, weakly (strongly) converges to an element of ∩t⩾0F(T(t)).