Convergence theorems for nonexpansive semigroups in spaces (Q640131)

Let \(C\) be a closed convex subset of a complete CAT(0) space \(X\) and \(T_n:C\to C\), \(n\geq 1\), be a family of uniformly asymptotically regular and nonexpansive maps such that \(F:=\bigcap_nF(T_n)\neq \emptyset\). Define an iterative process \((x_n)\) by \[ \begin{aligned} x_1&\in C,\\ x_{n+1}&=\alpha_nT_nx_n\oplus (1-\alpha_n)x_n,\;n\geq 1.\end{aligned} \] In addition, assume that either \(\lim_nd(T_{n+1}x_n,T_nx_n)=0\) or \(\lim_nd(T_{n+1}x_{n+1},T_nx_{n+1})=0\). Then \((x_n)\) \(\Delta\)-converges to some point of \(F\). A~counterpart of this result for one-parameter continuous semigroups \(\{T_t: t\geq 0\}\) of nonexpansive maps over \(C\) is also provided.