This article deals with nonexpansive operators in hyperbolic metric spaces. A metric space is called hyperbolic if
(a) contains a family of metric lines such that for each pair of , there is a unique metric line in which passes through and ; metric line, by definition, is the image of a metric embedding with the property , and
(b) ( is defined as a point for which and ; 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 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 .