Some generic properties of α-nonexpansive mappings

Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, $\text{E}$ a complete metric space formed by all α-nonexpansive mappings fC → A and $\text{M}$ a complete metric space formed by α-nonexpansive differentiable mappings fC → X. The following assertions are proved in this paper: (1) Properness of I ? f is a generic property in $\text{E}$ (2)the subset of $\text{E}$ formed by all α-contractive mappings is of Baire first category in $\text{E}$; and (3) for every y?X, the functional equation x ? f(x) = y has generically a finite number of solutions for f in $\text{M}$. Some applications to the fixed point theory and calculation of the topological degree are given.