Asymptotic behavior for commutative semigroups of asymptotically nonexpansive-type mappings (Q1576557)

The main result of the paper is the following weak convergence theorem for almost-orbits of commutative semigroups of asymptotically nonexpansive type mappings on a Banach space: Theorem 1. Suppose that \(X\) has the Opial's property and the norm of \(X\) is \(UKK\). Let \(C\) be a weakly compact convex subset of \(X\), \(S=\{T(t): t\in G\}\) be a commutative semigroup of asymptotically nonexpansive-type on \(C\), and let \(u(\cdot)\) be an almost-orbit of \(S\). Then \(\{u(t): t\in G\}\) is weakly convergent (to a fixed point) if and only if it is weakly asymptotically regular (i.e., \((u(t+ h)- u(t))\) converges to \(0\) weakly for every \(h\in G\)).