Porosity results for sets of strict contractions on geodesic metric spaces

Abstract

Abstract. We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic CAT(κ)-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 σ-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. Mathematics Subject Classification (2010). 47H09, 47H04, 54E52

Fixed points of terminating mappings in partial metric spaces

“…More on applications of fixed point theory can be found in [3][4][5][6][7][8][9]. Furthermore, spaces and mappings are very important when studying fixed points; metric spaces, partial metric spaces, fuzzy spaces, smooth spaces, contractive mappings, monotone mappings and so on, see [1,2,4,[10][11][12][13][14][15][16].…”

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

“…More precisely, they showed that the set of strict contractions, that is, the set of nonexpansive mappings f : C → C for which there is an L < 1 satisfying ρ(f (x), f (y)) ≤ Lρ(x, y) for all x, y ∈ C is not only of the first Baire category, but even σ-porous. In [1] this result was generalised to general Banach spaces and in [2] to a class of metric spaces.…”

“…For more details on CAT(κ)-spaces, we refer the reader to [4,6]. In [2] the concept of weakly hyperbolic spaces was introduced. The class of weakly hyperbolic spaces contains the class of hyperbolic spaces and the class of all CAT(κ)-spaces for all κ ∈ R. In [2] it is shown that the typical nonexpansive mapping on a closed and star-shaped subset of a weakly hyperbolic space has the maximal possible Lipschitz constant one.…”

Porosity phenomena of non-expansive, Banach space mappings

“…The problems discussed above have also been studied in more general metric spaces and for more general classes of mappings; see, for example, [11], [3] and [2].…”