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ArticleOriginal scientific text

Title

Hyperconvexity of ℝ-trees

Authors 1

Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419, U.S.A.

Abstract

It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete ℝ-tree; (ii) M is hyperconvex and has unique metric segments.

Keywords

ENG
hyperconvex metric space, ℝ-tree, fixed point, nonexpansive mapping

Bibliography

  1. N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439.
  2. J. B. Baillon, Nonexpansive mappings and hyperconvex spaces, in: Fixed Point Theory and its Applications, R. F. Brown (ed.), Contemp. Math. 72, Amer. Math. Soc., Providence, R.I., 1988, 11-19.
  3. L. M. Blumenthal, Distance Geometry, Oxford Univ. Press, London, 1953.
  4. J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 439-447.
  5. E. Jawhari, D. Misane and M. Pouzet, Retracts: graphs and ordered sets from the metric point of view, in: Combinatorics and Ordered Graphs, I. Rival (ed.), Contemp. Math. 57, Amer. Math. Soc., Providence, R.I., 1986, 175-226.
  6. M. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106 (1989), 723-726.
  7. M. A. Khamsi, KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204 (1996), 298-306.
  8. W. A. Kirk and S. S. Shin, Fixed point theorems in hyperconvex spaces, Houston J. Math. 23 (1997), 175-188.
  9. J. Kulesza and T. C. Lim, On weak compactness and countable weak compactness in fixed point theory, Proc. Amer. Math. Soc. 124 (1996), 3345-3349.
  10. H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, Berlin, 1974.
  11. R. Mańka, Association and fixed points, Fund. Math. 91 (1976), 105-121.
  12. J. W. Morgan, Λ-trees and their applications, Bull. Amer. Math. Soc. 26 (1992), 87-112.
  13. F. Rimlinger, Free actions on ℝ-trees, Trans. Amer. Math. Soc. 332 (1992), 313-329.
  14. R. Sine, On nonlinear contractions in sup norm spaces, Nonlinear Anal. 3 (1979), 885-890.
  15. P. Soardi, Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25-29.
  16. F. Sullivan, Ordering and completeness of metric spaces, Nieuw Arch. Wisk. (3) 29 (1981), 178-193.
  17. J. Tits, A "Theorem of Lie-Kolchin" for trees, in: Contributions to Algebra: a Collection of Papers Dedicated to Ellis Kolchin, Academic Press, New York, 1977, 377-388.
Fundamenta Mathematicae
1998/156/1
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Pages:
67-72
Main language of publication
English
Received
1997-01-16
Accepted
1997-10-14
Published
1998
Exact and natural sciences