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

Let KK be a nonempty closed convex subset of a real Banach space EE. Let T≔{T(t):t∈R+}T≔{T(t):t∈R+} be a strongly continuous semi-group of asymptotically nonexpansive mappings from KK into KK with a sequence {Lt}⊂[1,∞){Lt}⊂[1,∞). Suppose F(T)≠0̸F(T)≠0̸. Then, for a given u0∈Ku0∈K and tn>0tn>0 there exists a sequence {un}⊂K{un}⊂K such that un=(1−αn)T(tn)un+αnu0un=(1−αn)T(tn)un+αnu0, for n∈Nn∈N such that {αn}⊂(0,1){αn}⊂(0,1) and Ltn−1<αnLtn−1<αn, where tn∈R+tn∈R+. Suppose, in addition, that EE is reflexive strictly convex with a uniformly Gâteaux differentiable norm and that limn→∞tn=∞limn→∞tn=∞, View the MathML sourcelimn→∞αn=limn→∞Ltn−1αn=0. Then the sequence {un}{un} converges strongly to a point of F(T)F(T). Moreover, it is proved that an explicit sequence {xn}{xn} generated from x1∈Kx1∈K by xn+1≔αnu+(1−αn)T(tn)xnxn+1≔αnu+(1−αn)T(tn)xn, n≥1n≥1, converges to a fixed point of TT.