Kakutani property of the polytopes implies Kakutani property of the whole space
type: Item
instance of: scholarly article
title: Kakutani property of the polytopes implies Kakutani property of the whole space
zbMATH Open document ID: 0646.54044
DOI: 10.1016/0022-247X(88)90388-5
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: Let X be a compact Hausdorff space and let \(\Phi:X\to X\) be an upper semicontinuous set-valued map such that, for any \(x\in X\), \(\Phi\) (x)\(\in {\mathcal K}\) where \({\mathcal K}\) is a family of closed subsets of X containing X as an element and which is closed under arbitrary intersections and satisfies another so-called regularity condition. The main result states that the map \(\Phi\) has a fixed point provided, for any finite subset F of X and any upper semicontinuous map \(\Psi\) :P\(\to P\) where P is the smallest element of \({\mathcal K}\) which contains F, \(\Psi\) has a fixed point. The mentioned result generalizes the fixed point theorems of \textit{Ky Fan} [Proc. Nat. Acad. Sci. U.S.A. 38, 121-126 (1952; Zbl 0047.351)] and \textit{A. D. Wallace} [Bull. Am. Math. Soc. 47, 757-760 (1941)]. Other results are proved for compactifications of a (non-compact) space and, moreover, reformulated in the order-theoretic language. It would be interesting to see whether the proven theorems can be used to solve any new nonartificial problems.
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 54H12
zbMATH DE Number: 4054593
zbMATH Keywords: closed convex subsets
zbMATH Keywords: polytopes
zbMATH Keywords: continuous lattices
zbMATH Keywords: Kakutani fixed point theorem
zbMATH Keywords: upper semicontinuous set-valued map
zbMATH Keywords: compactifications
author: Klaus Keimel
author: Andrzej Wieczorek
reviewed by: Wojciech Kryszewski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4054593