  We propose the class of uniformly convex $W$-hyperbolic spaces with monotone
modulus of uniform convexity ($UCW$-hyperbolic spaces for short) as an
appropriate setting for the study of nonexpansive iterations. $UCW$-hyperbolic
spaces are a natural generalization both of uniformly convex normed spaces and
CAT(0)-spaces. Furthermore, we apply proof mining techniques to get effective
rates of asymptotic regularity for Ishikawa iterations of nonexpansive
self-mappings of closed convex subsets in $UCW$-hyperbolic spaces. These
effective results are new even for uniformly convex Banach spaces.
