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On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature. (English) Zbl 1491.47041

Summary: We show that the typical nonexpansive mapping on a small enough subset of a \(\mathrm{CAT} (\kappa)\)-space is a contraction in the sense of E. Rakotch [Proc. Am. Math. Soc. 13, 459–465 (1962; Zbl 0105.35202)]. 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.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54E52 Baire category, Baire spaces
54E40 Special maps on metric spaces

Citations:

Zbl 0105.35202
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References:

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