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Convergence of Krasnoselskii-Mann iterations of nonexpansive operators. (English) Zbl 0977.47046

This article deals with nonexpansive operators in hyperbolic metric spaces. A metric space (X,ρ) is called hyperbolic if

(a) X contains a family M of metric lines such that for each pair of x,yX, xy there is a unique metric line in M which passes through x and y; metric line, by definition, is the image of a metric embedding c:X with the property ρ(c(s),c(t))=|s-t| (s,t), and

(b) ρ(1 2x1 2y,1 2w1 2z)1 2(ρ(x,w)+ρ(y,z)) (x,y,z,wX) ((1-t)xty is defined as a point z for which ρ(x,z)=tρ(x,y) and ρ(z,y)=(1-t)ρ(x,y); such a point exists due to (a)).

The main results of the article are three theorems in which it is proved that a generic nonexpansive operator A on a closed and convex (but not necessarily bounded) subset of a hyperbolic space has a unique fixed point which attracts the Krasnosel’skii-Mann iterations of A.

47H09Mappings defined by “shrinking” properties
47J25Iterative procedures (nonlinear operator equations)