On the porosity of the set of 𝜔-nonexpansive mappings without fixed points

Abstract

Let C C be a nonempty closed convex bounded subset of a Banach space E E . Let M \mathcal {M} denote the family of all multivalued mappings from C C into E E which are nonempty weakly compact convex valued, ω \omega -nonexpansive and weakly-weakly u.s.c., endowed with the metric of uniform convergence. Let M 0 {\mathcal {M}_0} be the set of all F ∈ M F \in \mathcal {M} for which the fixed point problem is well posed. It is proved that the set M ∖ M 0 \mathcal {M}\backslash {\mathcal {M}_0} is σ \sigma -porous (in particular meager). A similar result is given for weak properness.