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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 $(X,\rho)$ is called hyperbolic if (a) $X$ contains a family $M$ of metric lines such that for each pair of $x,y\in X$, $x\ne y$ 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 $c:\bbfR\to X$ with the property $\rho(c(s),c(t))= |s- t|$ $(s,t\in\bbfR)$, and (b) $\rho({1\over 2} x\oplus{1\over 2} y,{1\over 2} w\oplus{1\over 2} z)\le{1\over 2}(\rho(x, w)+ \rho(y,z))$ $(x,y,z,w\in X)$ ($(1- t)x\oplus ty$ is defined as a point $z$ for which $\rho(x,z)= t\rho(x,y)$ and $\rho(z,y)= (1- t)\rho(x,y)$; 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 $A$.

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