Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Fixed point and selection theorems in hyperconvex spaces

Authors: M. A. Khamsi, W. A. Kirk and Carlos Martinez Yañez
Journal: Proc. Amer. Math. Soc. 128 (2000), 3275-3283
MSC (1991): Primary 47H04, 47H10, 54H25; Secondary 47H09, 54E40
Published electronically: April 28, 2000
MathSciNet review: 1777578
Full-text PDF

Abstract | References | Similar Articles | Additional Information


It is shown that a set-valued mapping $T^{\ast}$ of a hyperconvex metric space $M$ which takes values in the space of nonempty externally hyperconvex subsets of $M$ always has a lipschitzian single valued selection $T$ which satisfies $d(T(x),T(y))\leq d_{H}(T^{\ast}(x),T^{\ast}(y))$ for all $x,y\in M $. (Here $d_{H}$ denotes the usual Hausdorff distance.) This fact is used to show that the space of all bounded $\lambda$-lipschitzian self-mappings of $M $ is itself hyperconvex. Several related results are also obtained.

References [Enhancements On Off] (What's this?)

  • 1. Aronszajn, N., and Panitchpakdi, P., Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6(1956), 405-439. MR 18:917c; correction MR 19:1069h
  • 2. Baillon, J. B., Nonexpansive mappings and hyperconvex spaces, Contemp. Math. 72(1988), 11-19. MR 89k:54068
  • 3. Herrlich, H., Hyperconvex hulls of metric spaces, Topology Appl. 44(1992), 181-187. MR 93f:54043
  • 4. Isbell, J. R., Six theorems about injective metric spaces, Comment. Math. Helvetici 39(1964), 65-76. MR 32:431
  • 5. Isbell, J. R., Injective envelopes of of Banach spaces are rigidly attached, Bull. Amer. Math. Soc. 70(1984), 727-729. MR 32:1537
  • 6. Khamsi, M. A., Lin, M., and Sine, R., On the fixed points of commuting nonexpansive maps in hyperconvex spaces, J. Math. Anal. Appl. 168(1992), 372-380. MR 93j:47080
  • 7. Kirk, W. A., Hyperconvexity of $\mathbb{R} $-trees, Fund. Math. 156(1998), 67-72. MR 98k:54060
  • 8. Lacey, H. E., The Isometric Theory of Classical Banach Spaces, Springer-Verlag, New York, Heidelberg, Berlin, 1974. MR 58:12308
  • 9. Lin, M., and Sine, R., On the fixed point set of order preserving maps, Math. Zeit. 203(1990), 227-234. MR 91a:47074
  • 10. Sine, R., On nonlinear contraction semigroups in sup norm spaces, Nonlinear Analysis - Theory, Methods & Applications 3(1979), 885-890. MR 80i:47082
  • 11. Sine, R., Hyperconvexity and approximate fixed points, Nonlinear Analysis - Theory, Methods & Applications 13(1989), 863-869. MR 90g:54041
  • 12. Sine, R., Hyperconvexity and nonexpansive multifunctions, Trans. Amer. Math. Soc. 315(1989), 755-767. MR 90a:54054
  • 13. Soardi, P., Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73(1979), 25-29. MR 80c:47051
  • 14. Sullivan, F., Ordering and completeness of metric spaces, Nieuw Arch. Wisk. 29(1981), 178-193. MR 84d:54079

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47H04, 47H10, 54H25, 47H09, 54E40

Retrieve articles in all journals with MSC (1991): 47H04, 47H10, 54H25, 47H09, 54E40

Additional Information

M. A. Khamsi
Affiliation: Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas 79968-0514

W. A. Kirk
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419

Carlos Martinez Yañez
Affiliation: Institute of Mathematics, Universidad Catolica de Valparaiso, Valparaiso, Chile

Keywords: Hyperconvex metric spaces, fixed points, selection theorems, fixed points
Received by editor(s): December 17, 1998
Published electronically: April 28, 2000
Additional Notes: This research was carried out while the first two authors were visiting the Universidad Catolica de Valparaiso on the occasion of the XXV Semana de la Matematica, October, 1998. They express their thanks to the sponsors for generous support and hospitality. The research of the third author was partially supported by FONDECYT grant no. 1980431.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society