Asymptotic behavior of almost-orbits of semigroups of Lipschitzian mappings in Banach spaces
Let G be a right-reversible semitopological semigroup, and let C be a nonempty closed convex subset of a uniformly convex Banach space E; let \(S=\{S(t):\) \(t\in G\}\) be a family of Lipschitzian mappings \(C\to C\) with Lipschitzian constants \(k_ t\) such that \(\lim_{t} \sup k_ t\leq 1\) and with semigroup properties (i) \(S(st)x=S(s)S(t)x\) and (ii) \(u(t)=S(t)\) x is continuous function on G. The authors show that the set F(S) of common fixed points of S(t) is a closed and convex subset of C. Moreover, having \(\{\) u(t): \(t\in G\}\) an almost-orbit of S they prove that, if E has a Fréchet-differentiable norm than the set \(M=\cap_{s\in G}\overline{co}\{u(t):\) \(t\geq s\}\) contains at most one point of F(S). (An almost-orbit is defined as a function u: \(G\to C\) such that \(\lim_{t}\sup_{s}| u(st)- S(s)u(t)| =0.)\) Using this fact and other properties of M and F(S), they establishes some theorems concerning weak convergence of an almost-orbit and prove strong convergence of its metric projection on F(S).
right-reversible semitopological semigroup
strong convergence of its metric projection
scientific article; zbMATH DE number 4101950