Porosity results for sets of strict contractions on geodesic metric spaces
Keywords
Banach space, hyperbolic space, metric space, nonexpansive mapping, porous set, set-valued mapping, star-shaped set, strict contractionAbstract
We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(\kappa)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $\sigma$-porous subset. For certain separable and complete metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases.Published
2017-05-21
How to Cite
1.
BARGETZ, Christian, DYMOND, Michael and REICH, Simeon. Porosity results for sets of strict contractions on geodesic metric spaces. Topological Methods in Nonlinear Analysis. Online. 21 May 2017. Vol. 50, no. 1, pp. 89 - 124. [Accessed 29 March 2024].
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