We investigate an Ishikawa iteration process in the set up of generalized $\alpha$- nonexpansive mappings. Approximation of these two mappings to a common fixed point by $\mathrm{\Delta }-$convergence and strong convergence of the scheme in hyperbolic space are also illustrated. The presented results amplify and polish many recent ideas put forward in uniformly convex Banach spaces, including CAT(0) spaces.