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Convergence theorems for nonexpansive semigroups in spaces. (English) Zbl 1237.47070
Let $C$ be a closed convex subset of a complete CAT(0) space $X$ and $T_n:C\to C$, $n\ge 1$, be a family of uniformly asymptotically regular and nonexpansive maps such that $F:=\bigcap_nF(T_n)\ne \emptyset$. Define an iterative process $(x_n)$ by $$\align x_1&\in C,\\ x_{n+1}&=\alpha_nT_nx_n\oplus (1-\alpha_n)x_n,\ n\ge 1.\endalign$$ 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 $\{T_t: t\ge 0\}$ of nonexpansive maps over $C$ is also provided.

47J25Iterative procedures (nonlinear operator equations)
47H20Semigroups of nonlinear operators
54H25Fixed-point and coincidence theorems in topological spaces
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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