Weak Convergence and Non-linear Ergodic Theorems for Reversible Semigroups of Non-Lipschitzian Mappings

Abstract

LetGbe a semitopological semigroup. LetCbe a closed convex subset of a uniformly convex Banach spaceEwith aFréchetdifferentiable norm and T={Tt:t∈G} be a continuous representation ofGas asymptotically nonexpansive type mappings ofCinto itself such that the common fixed point setF(T) of (T) inCis nonempty. We prove in this paper that ifGis right reversible, then for every almost-orbitu(·) of T,[formula]consists of at most one point. Further,[formula]is nonempty for eachx∈Cif and only if there exists a nonexpansive retractionPofContoF(T) such thatPTs=TsP=Pfor alls∈GandP(x) is in the closed convex hull of {Tsx:s∈G},x∈C. This result is applied to study the problem of weak convergence of the net {u(t):t∈G} to a common fixed point of T.