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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On multivalued fixed-point free maps on $ \mathbb{R}^n$


Author: Raushan Z. Buzyakova
Journal: Proc. Amer. Math. Soc. 140 (2012), 2929-2936
MSC (2010): Primary 54H25, 58C30, 54B20
Published electronically: December 13, 2011
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Abstract: To formulate our results let $ f$ be a continuous multivalued map from $ \mathbb{R}^n$ to $ 2^{\mathbb{R}^n}$ and $ k$ a natural number such that $ \vert f(x)\vert\leq k$ for all $ x$. We prove that $ f$ is fixed-point free if and only if its continuous extension $ \tilde f:\beta \mathbb{R}^n\to 2^{\beta \mathbb{R}^n}$ is fixed-point free. If one wishes to stay within metric terms, the result can be formulated as follows: $ f$ is fixed-point free if and only if there exists a continuous fixed-point free extension $ \bar f: b\mathbb{R}^n\to 2^{b\mathbb{R}^n}$ for some metric compactificaton $ b\mathbb{R}^n$ of $ \mathbb{R}^n$. Using the classical notion of colorablity, we prove that such an $ f$ is always colorable. Moreover, a number of colors sufficient to paint the graph can be expressed as a function of $ n$ and $ k$ only. The mentioned results also hold if the domain is replaced by any closed subspace of $ \mathbb{R}^n$ without any changes in the range.


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Additional Information

Raushan Z. Buzyakova
Affiliation: Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, North Carolina 27402
Email: rzbouzia@uncg.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11166-3
PII: S 0002-9939(2011)11166-3
Keywords: Fixed point, hyperspace, multivalued function
Received by editor(s): November 22, 2010
Received by editor(s) in revised form: March 7, 2011
Published electronically: December 13, 2011
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.