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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 $\left(X,\rho \right)$ 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 $c:ℝ\to X$ with the property $\rho \left(c\left(s\right),c\left(t\right)\right)=|s-t|$ $\left(s,t\in ℝ\right)$, and

(b) $\rho \left(\frac{1}{2}x\oplus \frac{1}{2}y,\frac{1}{2}w\oplus \frac{1}{2}z\right)\le \frac{1}{2}\left(\rho \left(x,w\right)+\rho \left(y,z\right)\right)$ $\left(x,y,z,w\in X\right)$ ($\left(1-t\right)x\oplus ty$ is defined as a point $z$ for which $\rho \left(x,z\right)=t\rho \left(x,y\right)$ and $\rho \left(z,y\right)=\left(1-t\right)\rho \left(x,y\right)$; 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:
 47H09 Mappings defined by “shrinking” properties 47J25 Iterative procedures (nonlinear operator equations)