On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature

Christian Bargetz, Michael Dymond, Emir Medjic, Simeon Reich

DOI: http://dx.doi.org/10.12775/TMNA.2020.040

Abstract


We show that the typical nonexpansive mapping on a small enough subset of a $\CAT(\kappa)$-space is a \textit{contraction in the sense of Rakotch}. By typical we mean that the set of nonexpansive mapppings without this property is a $\sigma$-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contractions are not dense in the space of nonexpansive mappings. In some of these cases we show that all continuous self-mappings have a fixed point nevertheless.

Keywords


Nonexpansive mappings; Rakotch contractions; spaces of positive curvature; $\sigma$-porous sets

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References


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