Some applications of the Kakutani fixed point theorem
type: Item
instance of: scholarly article
title: Some applications of the Kakutani fixed point theorem
zbMATH Open document ID: 0612.54055
DOI: 10.1016/0022-247X(87)90242-3
author: Won Kyu Kim
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: The famous Ky Fan theorem, leading to many non-trivial applications in nonlinear functional analysis, is based on the theorem on closed coverings of a simplex. The authors prove a likewise theorem using open coverings. The main theorem is as follows. Let C be a nonempty closed convex subset in a Hausdorff topological vector space E and \(F: C\to 2^ C\) be a map such that F(x) is closed for each \(x\in C\) and \(F^{-1}(y)\) is convex for each \(y\in C\), and \(C\subset \cup F(x_ i)\) for some finite set \(\{x_ 1,...,x_ n\}\). Then there exists w such that \(w\in F(w)\).
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 3990375
zbMATH Keywords: theorem on closed coverings of a simplex
zbMATH Keywords: Brouwer fixed point theorem
zbMATH Keywords: Sperner lemma
zbMATH Keywords: Knaster-Kuratowski-Mazurkiewicz theorem
zbMATH Keywords: intersection properties of open covers
zbMATH Keywords: Kakutani fixed point theorem
reviewed by: Olga Kosheleva
MaRDI profile type: Publication
cites work: Q4179433
cites work: A generalization of Tychonoff's fixed point theorem
cites work: Q4057217
cites work: Some properties of convex sets related to fixed point theorems
cites work: Q4064159
cites work: Ein Fixpunktsatz
full work available at URL: 0022-247x(87)90242-3
OpenAlex ID: W2028030405
description: scientific article; zbMATH DE number 3990375