Nonlinear ergodic theorems of almost-orbits of non-Lipschitzian semigroups
Let \(G\) be a semitopological semigroup and let \(\{S(t):t\in G\}\) be a continuous representation of \(G\) in the family of continuous mappings of a nonempty closed convex subset \(C\) of a Banach space into itself. The representation is said to be a semigroup of weakly asymptotically nonexpansive type if for each \(x\in C\) and each bounded \(D\subset C\) \[ \limsup_ t\sup_{y\in D}(| S(t)x-S(t)y|-| x-y|)\leq 0. \] If \(G\) is right reversible then a function \(u:G\to C\) is an almost orbit of \(\{S(t):t\in G\}\) if \[ \lim_ t\bigl(\sup_{s\in G}| u(st)-S(s)u(t)|\bigr)=0. \] The author of the paper under review proves theorems on fixed points of almost-orbits in the case of uniformly convex Banach spaces. Also some results on the weak convergence of almost-orbits and ergodic theorems are proved. The theorems generalize previously known commutative versions.
semigroup of weakly asymptotically nonexpansive type
scientific article; zbMATH DE number 167946
scientific article