Some generic properties of \(\alpha\)-nonexpansive mappings
A property P is said to be generic in a Baire space E if P holds in a residual subset of E. Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X and \(\alpha\) (\(\cdot)\) the Kuratowski measure of noncompactness. Denote by \({\mathfrak E}\) the complete metric space formed by all \(\alpha\)-nonexpansive mappings f:C\(\to A\), endowed with the topology of uniform convergence, and by \({\mathfrak M}^ a \)complete metric space formed by \(\alpha\)-nonexpansive differentiable mappings f:C\(\to X\). The following assertions are proved in this paper: (1) Properness of I-f is a generic property in \({\mathfrak E}\); (2) the subset of \({\mathfrak E}\) formed by all \(\alpha\)-contractive mappings is of Baire first category in \({\mathfrak E}\); and (3) for every \(y\in X\) the functional equation \(x-f(x)=y\) has generically a finite number of solutions for f in \({\mathfrak M}\). Some applications to the fixed point theory are given. Further information about (1) and (2) can be found in a later author's paper [Proc. Am. Math. Soc. 93, 252-254 (1985)].
scientific article; zbMATH DE number 3883071
scientific article