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Proceedings of the American Mathematical Society
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
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Abstract:

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.


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Additional Information

M. A. Khamsi
Affiliation: Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas 79968-0514
Email: mohamed@math.utep.edu

W. A. Kirk
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
Email: kirk@math.uiowa.edu

Carlos Martinez Yañez
Affiliation: Institute of Mathematics, Universidad Catolica de Valparaiso, Valparaiso, Chile
Email: cmartine@ucv.cl

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05777-4
PII: S 0002-9939(00)05777-4
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



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