  In this paper we obtain a quadratic bound on the rate of asymptotic
regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in
CAT(0)-spaces, whereas previous results guarantee only exponential bounds. The
method we use is to extend to the more general setting of uniformly convex
hyperbolic spaces a quantitative version of a strengthening of Groetsch's
theorem obtained by Kohlenbach using methods from mathematical logic (so-called
``proof mining'').
