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On the porosity of the set of $ \omega$-nonexpansive mappings without fixed points


Authors: J. Myjak and R. Sampalmieri
Journal: Proc. Amer. Math. Soc. 114 (1992), 357-363
MSC: Primary 47H09; Secondary 47H04, 47H10
DOI: https://doi.org/10.1090/S0002-9939-1992-1087466-7
MathSciNet review: 1087466
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1087466-7
Keywords: Multifunctions, fixed point, well-posedness, weak properness, porosity
Article copyright: © Copyright 1992 American Mathematical Society

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