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

Myjak, J.; Sampalmieri, R.

doi:10.1090/S0002-9939-1992-1087466-7

On the porosity of the set of $\omega$-nonexpansive mappings without fixed points
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by J. Myjak and R. Sampalmieri PDF

Proc. Amer. Math. Soc. 114 (1992), 357-363 Request permission

Abstract:

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

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