Asymptotic behavior of almost-orbits of semigroups of Lipschitzian mappings in Banach spaces
Wataru Takahashi, Peijun Zhang · Kodai Mathematical Journal · 1988
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 Lipschitzian self-mappings on C. We consider the asymptotic behavior of an almost-orbit {u(t) : t<=G} of S={S(t) : ίeG}.We show that if E has a Frechet differentiable norm and if limsup&ί^l, then the closed convex set Γ\cδ{u(t):t^s}Γ\F(S) SGG 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}.