Asymptotic behaviour for an almost-orbit of nonexpansive semigroups in Banach spaces

Jong Kyu Kim, Gang Li
2000 Bulletin of the Australian Mathematical Society  
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 space X with Opial's condition. This includes many well known results as special cases. Let C be a weakly compact subset of a Banach space X with Opial's condition. Let G be a right reversible semitopological semigroup, S -{T(t) : t £ G} a nonexpansive semigroup on C, and «(•) an almost-orbit of <S.
more » ... Then {u(t) : t £ G} is weakly convergent (to a common fixed point of S) if and only if it is weakly asymptotically regular (that is, {u(ht) -u(t)} converges to 0 weakly for every h £ G).
doi:10.1017/s0004972700022358 fatcat:v2pcsxgo3jbbriwwnatxbweo4a