Uniformly Lipschitzian group actions on hyperconvex spaces

Wiśnicki, Andrzej; Wośko, Jacek

doi:10.1090/proc/13016

Uniformly Lipschitzian group actions on hyperconvex spaces
HTML articles powered by AMS MathViewer

by Andrzej Wiśnicki and Jacek Wośko

Proc. Amer. Math. Soc. 144 (2016), 3813-3824

DOI: https://doi.org/10.1090/proc/13016

Published electronically: March 30, 2016

PDF | Request permission

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

E. Alvoni and P. L. Papini, Perturbation of sets and centers, J. Global Optim. 33 (2005), no. 3, 423–434. MR 2225378, DOI 10.1007/s10898-005-0539-7
N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439. MR 84762, DOI 10.2140/pjm.1956.6.405
Jean-Bernard Baillon, Nonexpansive mapping and hyperconvex spaces, Fixed point theory and its applications (Berkeley, CA, 1986) Contemp. Math., vol. 72, Amer. Math. Soc., Providence, RI, 1988, pp. 11–19. MR 956475, DOI 10.1090/conm/072/956475
Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
M. S. Brodskiĭ and D. P. Mil′man, On the center of a convex set, Doklady Akad. Nauk SSSR (N.S.) 59 (1948), 837–840 (Russian). MR 0024073
S. Dhompongsa, W. A. Kirk, and Brailey Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal. 65 (2006), no. 4, 762–772. MR 2232680, DOI 10.1016/j.na.2005.09.044
Andreas W. M. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. in Math. 53 (1984), no. 3, 321–402. MR 753872, DOI 10.1016/0001-8708(84)90029-X
Andreas W. M. Dress, A note on compact groups, acting on $\textbf {R}$-trees, Beiträge Algebra Geom. 29 (1989), 81–90. MR 1036092
Rafael Espínola and Aurora Fernández-León, Fixed point theory in hyperconvex metric spaces, Topics in fixed point theory, Springer, Cham, 2014, pp. 101–158. MR 3203910, DOI 10.1007/978-3-319-01586-6_{4}
R. Espínola and M. A. Khamsi, Introduction to hyperconvex spaces, in [18].
Rafa Espínola and Pepa Lorenzo, Metric fixed point theory on hyperconvex spaces: recent progress, Arab. J. Math. (Springer) 1 (2012), no. 4, 439–463 (English, with English and Arabic summaries). MR 3041058, DOI 10.1007/s40065-012-0044-z
K. Goebel and W. A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973), 135–140. MR 336468, DOI 10.4064/sm-47-2-134-140
K. Goebel and E. Złotkiewicz, Some fixed point theorems in Banach spaces, Colloq. Math. 23 (1971), 103–106, 176. MR 303367, DOI 10.4064/cm-23-1-103-106
Jarosław Górnicki and Krzysztof Pupka, Fixed point theorems for $n$-periodic mappings in Banach spaces, Comment. Math. Univ. Carolin. 46 (2005), no. 1, 33–42. MR 2175857
M. A. Khamsi, W. A. Kirk, and Carlos Martinez Yañez, Fixed point and selection theorems in hyperconvex spaces, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3275–3283. MR 1777578, DOI 10.1090/S0002-9939-00-05777-4
J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65–76. MR 182949, DOI 10.1007/BF02566944
W. A. Kirk, A fixed point theorem for mappings with a nonexpansive iterate, Proc. Amer. Math. Soc. 29 (1971), 294–298. MR 284887, DOI 10.1090/S0002-9939-1971-0284887-3
William A. Kirk and Brailey Sims (eds.), Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht, 2001. MR 1904271, DOI 10.1007/978-94-017-1748-9
Urs Lang, Injective hulls of certain discrete metric spaces and groups, J. Topol. Anal. 5 (2013), no. 3, 297–331. MR 3096307, DOI 10.1142/S1793525313500118
E. A. Lifšic, A fixed point theorem for operators in strongly convex spaces, Voronež. Gos. Univ. Trudy Mat. Fak. Vyp. 16 Sb. Stateĭ po Nelineĭnym Operator. Uravn. i Priložen. (1975), 23–28 (Russian). MR 0477901
T.-C. Lim and Hong Kun Xu, Uniformly Lipschitzian mappings in metric spaces with uniform normal structure, Nonlinear Anal. 25 (1995), no. 11, 1231–1235. MR 1350742, DOI 10.1016/0362-546X(94)00243-B
William O. Ray and Robert C. Sine, Nonexpansive mappings with precompact orbits, Fixed point theory (Sherbrooke, Que., 1980) Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 409–416. MR 643018
R. C. Sine, On nonlinear contraction semigroups in sup norm spaces, Nonlinear Anal. 3 (1979), no. 6, 885–890. MR 548959, DOI 10.1016/0362-546X(79)90055-5
Paolo M. Soardi, Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), no. 1, 25–29. MR 512051, DOI 10.1090/S0002-9939-1979-0512051-6
P. Szeptycki and F. S. Van Vleck, Centers and nearest points of sets, Proc. Amer. Math. Soc. 85 (1982), no. 1, 27–31. MR 647891, DOI 10.1090/S0002-9939-1982-0647891-6
Andrzej Wiśnicki, On the structure of fixed-point sets of asymptotically regular semigroups, J. Math. Anal. Appl. 393 (2012), no. 1, 177–184. MR 2921659, DOI 10.1016/j.jmaa.2012.03.036

AMS Website Down

Proceedings of the American Mathematical Society

Uniformly Lipschitzian group actions on hyperconvex spaces
HTML articles powered by AMS MathViewer

Abstract: