Asymptotic behaviour for an almost-orbit of nonexpansive semigroups in Banach spaces
Jong Kyu Kim, Gang Li · Bulletin of the Australian Mathematical Society · 2000
In this paper, by using the technique of product nets, we are able to prove a weak convergence theorem for an almost-orbit of right reversible semigroups of nonexpansine mappings in a general Banach spaceXwith Opial's condition. This includes many well known results as special cases. LetCbe a weakly compact subset of a Banach spaceXwith Opial's condition. LetGbe a right reversible semitopological semigroup, = {T(t):t∈G} a nonexpansive semigroup onC, andu(·) an almost-orbit of . Then {u(t):t∈G} is weakly convergent (to a common fixed point of ) if and only if it is weakly asymptotically regular (that is, {u(ht) −u(t)} converges to 0 weakly for everyh∈G).