Topological Methods in Nonlinear Analysis

Fixed points and non-convex sets in ${\rm CAT}(0)$ spaces

Bożena Piątek and Rafa Espínola

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Abstract

Dropping the condition of convexity on the domain of a nonexpansive mapping is a difficult and unusual task in metric fixed point theory. Hilbert geometry has been one of the most fruitful at which authors have succeeded to drop such condition. In this work we revisit some of the results in that direction to study their validity in ${\rm CAT}(0)$ spaces (geodesic spaces of global nonpositive curvature in the sense of Gromov). We show that, although the geometry of ${\rm CAT}(0)$ spaces resembles at certain points that one of Hilbert spaces, much more than the ${\rm CAT}(0)$ condition is required in order to obtain counterparts of fixed point results for non-convex sets in Hilbert spaces. We provide significant examples showing this fact and give positive results for spaces of constant negative curvature as well as $R$-trees.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 41, Number 1 (2013), 135-162.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461253859

Mathematical Reviews number (MathSciNet)
MR3086537

Zentralblatt MATH identifier
1290.58007

Citation

Piątek, Bożena; Espínola, Rafa. Fixed points and non-convex sets in ${\rm CAT}(0)$ spaces. Topol. Methods Nonlinear Anal. 41 (2013), no. 1, 135--162. https://projecteuclid.org/euclid.tmna/1461253859


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References

  • A.G. Aksoy and M.A. Khamsi, A selection theorem in metric trees , Proc. Amer. Math. Soc., 134 (2006), 2957–2966 \ref
  • M.R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Springer–Verlag, Berlin, Heidelberg (1999) \ref
  • S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly lipschitzian mappings , Nonlinear Anal., 65 (2006), 762–772 \ref
  • R. Espínola and A. Fernández-León, $\CAT(\kappa)$-spaces, weak convergence and fixed points , J. Math. Anal. Appl., 353 (2009), 410–427 \ref
  • R. Espínola and W.A. Kirk, Fixed point theorems in $\Bbb{R}$-trees with applications to graph theory , Topology Appl., 153 (2006), 1046–1055 \ref
  • K. Goebel and R. Schöneberg, Moons, bridges, birds$\ldots$ and nonexpansive mappings in Hilbert space , Bull. Austral. Math. Soc., 17 (1977), 463–466 \ref
  • K. Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge studies in advanced mathematics 28, Cambridge University Press, Cambridge (1990) \ref
  • W.A. Kirk, Hyperconvexity of $R$-trees , Fund. Math., 156 (1998), 67–72 \ref ––––, Geodesic geometry and fixed point theory , Proceedings, Universities of Malaga and Seville, September 2002–February 2003 (D. Girela, G. López and R. Villa, eds.), Universidad de Sevilla, Sevilla (2003), 195–225 \ref ––––, Geodesic geometry and fixed point theory \romII, Fixed Point Theory and Applications (J. García-Falset, E. Llorens-Fuster and B. Sims, eds.), Yokohama Publ. (2004), 113–142 \ref
  • W.A. Kirk and B. Panyanak, Best approximation in R-trees , Numer. Funct. Anal. Optim., 28 (2007), 681–690 \ref ––––, A concept of convergence in geodesic spaces , Nonlinear Anal., 68 (2008), 3689–3696 \ref
  • W.A. Kirk and B. Sims, Handbook of Metric Fixed Point Theory, Kluwer Academic Publishers (2001) \ref
  • M.D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen , Fund. Math., 22 (1934), 77–108 \ref
  • Y. Komura, Differentiability of nonlinear semigroups , J. Math. Soc. Japan, 21 (1969), 375–402 \ref
  • T. Kuczumow and A. Stachura, Extensions of nonexpansive mappings in the Hilbert ball with the hyperbolic metric \romI, \romII, Comment. Math. Univ. Carolin., 29 (1988), 399–402, 403–410 \ref
  • U. Lang and V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature , Geom. Funct. Anal., 7 (1997), 535–560 \ref
  • U. Lang and V. Schroeder, Jung's theorem for Alexandrov spaces of curvature bounded above , Ann. Global Anal. Geom., 15 (1997), 263–275 \ref
  • B.D. Rouhani, On the fixed point property for nonexpansive mappings and semigroups , Nonlinear Anal., 30 (1997), 389–396 \ref
  • B.D. Rouhani, Remarks on asymptotically non-expansive mappings in Hilbert space , Nonlinear Anal., 49 (2002), 1099–1104 \ref
  • K.T. Sturm, Probability measures on metric spaces of nonpositive curvature , Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Lecture Notes from a quater program on heat kernels, random walks, and analysis on manifolds and graphs, April 16–July 13, 2002 Paris, France. (P. Auscher at al., eds.), Providence, RI: Amer. Math. Soc. (AMS) Contemp. Math. 338 (2003) \ref
  • F.A. Valentine, On the extension of a function so as to preserve a Lipschitz condition , Bull. Amer. Math. Soc., 49 (1943), 100–108