Asymptotic behavior of almost-orbits of semigroups of Lipschitzian mappings in Banach spaces

Abstract

Let C be a nonempty closed convex subset of a uniformly convex Banach space E, G a right reversible semitopological semigroup and S={S(t) : t∈G} a continuous representation of G as Lipschitzain self-mappings on C. We consider the asymptoic behavior of an almost-orbit {u(t) : t∈G} of S={S(t) : (t)∈G}. We show that if E has a Fréchet differentiable norm and if lim_t sup k_t{≤}1, then the closed convex set
\underset{s∈G}∩\overline{co}{u(t) : t{≥}s}∩F(S)
consists of at most one point, where k_t is the Lipschitzian constant of S(t). This result is applied to study the problem of weak convergence of the net {u(t) : t∈G}.