Generic properties of nonexpansive mappings on unbounded domains

We investigate typical properties of nonexpansive mappings on unbounded complete hyperbolic metric spaces. For two families metrics uniform convergence bounded sets, we show that the mapping is a Rakotch contraction every subset and there set which mapped into itself by this mapping. In particular, obtain in setting has unique fixed point can be reached iterating Nevertheless, it turns out not whole space maximal possible Lipschitz constant one residual its domain. By mean complement with property $\sigma$-$\phi$-porous, is, small sense. pointwise convergence, contractions meagre.