Convergence theorems for strongly continuous semi-groups of asymptotically nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E. Let T≔{T(t):t∈R+} be a strongly continuous semi-group of asymptotically nonexpansive mappings from K into K with a sequence {Lt}⊂[1,∞). Suppose F(T)≠0̸. Then, for a given u0∈K and tn>0 there exists a sequence {un}⊂K such that un=(1−αn)T(tn)un+αnu0, for n∈N such that {αn}⊂(0,1) and Ltn−1<αn, where tn∈R+. Suppose, in addition, that E is reflexive strictly convex with a uniformly Gâteaux differentiable norm and that limn→∞tn=∞, limn→∞αn=limn→∞Ltn−1αn=0. Then the sequence {un} converges strongly to a point of F(T). Moreover, it is proved that an explicit sequence {xn} generated from x1∈K by xn+1≔αnu+(1−αn)T(tn)xn, n≥1, converges to a fixed point of T.