Abstract
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.
Citation
Christian Bargetz. Michael Dymond. Simeon Reich. "Porosity results for sets of strict contractions on geodesic metric spaces." Topol. Methods Nonlinear Anal. 50 (1) 89 - 124, 2017. https://doi.org/10.12775/TMNA.2017.013