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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 :CC, n1, be a family of uniformly asymptotically regular and nonexpansive maps such that F:= n F(T n ). Define an iterative process (x n ) by

x 1 C,x n+1 =α n T n x n (1-α n )x n ,n1·

In addition, assume that either lim n d(T n+1 x n ,T n x n )=0 or lim n d(T n+1 x n+1 ,T n x n+1 )=0. Then (x n ) Δ-converges to some point of F. A counterpart of this result for one-parameter continuous semigroups {T t :t0} of nonexpansive maps over C is also provided.

MSC:
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
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