Uniformly Lipschitzian group actions on hyperconvex spaces
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- by Andrzej Wiśnicki and Jacek Wośko PDF
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Abstract:
Suppose that $\{T_{a}:a\in G\}$ is a group of uniformly $L$-Lipschitzian mappings with bounded orbits $\left \{ T_{a}x:a\in G\right \}$ acting on a hyperconvex metric space $M$. We show that if $L<\sqrt {2}$, then the set of common fixed points $\mathrm {Fix} G$ is a nonempty Hölder continuous retract of $M.$ As a consequence, it follows that all surjective isometries acting on a bounded hyperconvex space have a common fixed point. A fixed point theorem for $L$-Lipschitzian involutions and some generalizations to the case of $\lambda$-hyperconvex spaces are also given.References
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Additional Information
- Andrzej Wiśnicki
- Affiliation: Department of Mathematics, Rzeszów University of Technology, Al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland
- MR Author ID: 360658
- Email: awisnicki@prz.edu.pl
- Jacek Wośko
- Affiliation: Institute of Mathematics, Maria Curie-Skłodowska University, 20-031 Lublin, Poland
- MR Author ID: 292234
- Email: jwosko@hektor.umcs.lublin.pl
- Received by editor(s): April 2, 2015
- Received by editor(s) in revised form: October 26, 2015
- Published electronically: March 30, 2016
- Communicated by: Kevin Whyte
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3813-3824
- MSC (2010): Primary 47H10, 54H25; Secondary 37C25, 47H09
- DOI: https://doi.org/10.1090/proc/13016
- MathSciNet review: 3513540