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

Abstract

Let C be a nonempty closed convex subset of a uniformly convex Banach space E with a Fréchet differentiable norm, G a right reversible semitopological semigroup, and 𝒮={S(t):t∈G} a continuous representation of G as mappings of asymptotically nonexpansive type of C into itself. The weak convergence of an almost-orbit {u(t):t∈G} of 𝒮={S(t):t∈G} on C is established. Furthermore, it is shown that if P is the metric projection of E onto set F(S) of all common fixed points of 𝒮={S(t):t∈G}, then the strong limit of the net {Pu(t):t∈G} exists.

Keywords

• almost-orbit
• fixed point
• reversible semitopological semigroup
• semigroup of asymptotically nonexpansive type
• uniformly convex Banach space.