正曲率空间上典型非扩张映射不动点的存在性

C. Bargetz, Michael Dymond, Emir Medjic, S. Reich
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引用次数: 4

摘要

我们证明了CAT($\kappa$)空间的一个足够小的子集上的典型非扩展映射是Rakotch意义上的收缩。我们所说的典型是指不具有这种性质的非膨胀映射集是$\sigma$ -多孔集,因此也属于第一类Baire范畴。此外,我们还展示了在非扩张映射空间中严格收缩不密集的度量空间。在某些情况下,我们证明了所有连续自映射都有一个不动点。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature
We show that the typical nonexpansive mapping on a small enough subset of a CAT($\kappa$)-space is a 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.
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