Fixed point and selection theorems in hyperconvex spaces
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- by M. A. Khamsi, W. A. Kirk and Carlos Martinez Yañez PDF
- Proc. Amer. Math. Soc. 128 (2000), 3275-3283 Request permission
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.References
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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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 3275-3283
- MSC (1991): Primary 47H04, 47H10, 54H25; Secondary 47H09, 54E40
- DOI: https://doi.org/10.1090/S0002-9939-00-05777-4
- MathSciNet review: 1777578