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Uniformly Lipschitzian group actions on hyperconvex spaces


Authors: Andrzej Wiśnicki and Jacek Wośko
Journal: Proc. Amer. Math. Soc. 144 (2016), 3813-3824
MSC (2010): Primary 47H10, 54H25; Secondary 37C25, 47H09
Published electronically: March 30, 2016
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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.


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Additional Information

Andrzej Wiśnicki
Affiliation: Department of Mathematics, Rzeszów University of Technology, Al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland
Email: awisnicki@prz.edu.pl

Jacek Wośko
Affiliation: Institute of Mathematics, Maria Curie-Skłodowska University, 20-031 Lublin, Poland
Email: jwosko@hektor.umcs.lublin.pl

DOI: https://doi.org/10.1090/proc/13016
Keywords: Uniformly Lipschitzian mapping, group action, hyperconvex space, H\"older continuous retraction, fixed point, involution
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
Article copyright: © Copyright 2016 American Mathematical Society