On Ky Fan's section theorem
type: Item
instance of: scholarly article
title: On Ky Fan's section theorem
zbMATH Open document ID: 0928.47042
author: K. Appert
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-01-04T00:00:00Z
review text: The authors generalize Ky Fan's classical section theorem to the context of \(H\)-spaces. A pair \((X,\Gamma)\) is called an \(H\)-space if \(X\) is a topological space and \(\Gamma\) is a mapping which assigns to each non-empty finite subset \(A\subset X\) a non-empty contractible subset \(\Gamma_A\subset X\) such that \(\Gamma\) is monotone with respect to inclusions. A subset \(D\subset X\) is said to be \(H\)-convex if \(\Gamma_A\subset D\) whenever \(A\subset D\) is a finite set. The \(H\)-convex hull \(H\text{-conv}K\) of a set \(K\) is the smallest \(H\)-convex set containing \(K\). If finally, \(X\) and \(Y\) are sets and \(M\subset X\times Y\) then \(M[y]:=\{x\in X| \;(x,y)\in M\}\). The authors' main result can then be stated as follows: Let \(X\) be a Hausdorff space and \((Y,\Gamma)\) an \(H\)-space. Assume that \(M\) and \(N\) are subsets of \( X\times Y\) such that the following conditions hold: there is a compact set \(K\subset X\) and a subset \(P\) of \(M\) which is closed in \(X\times Y\) such that \(H\text{-conv}\{y\in Y| \;(x,y)\notin N\}\subset \{y\in Y| \;(x,y)\notin M\}\) whenever \(x\in K\), \(\bigcap_{y\in Y}\text{cl}N[y]=\bigcap_{y\in Y}N[y]\) and such that \(\{x\in K| \;(x,y)\in P\}\) is a nonempty acyclic set for each \( y\in Y\). The conclusion is that there is an \(x_0\in K\) such that \(\{x_0\}\times Y\subset N\).
reviewed by: Christian Fenske
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 49J35
zbMATH DE Number: 1236165
zbMATH Keywords: section theorem
zbMATH Keywords: multivalued mapping
zbMATH Keywords: H-space
zbMATH Keywords: minimax theorem
zbMATH Keywords: Ky Fan's section theorem
MaRDI profile type: Publication
OpenAlex ID: W2071913304
cites work: Coincidence theorems, minimax theorems, and variational inequalities
cites work: Some coincidence theorems and applications
cites work: A generalization of Tychonoff's fixed point theorem
cites work: Applications of a theorem concerning sets with convex sections
cites work: Minimax and fixed point theorems
cites work: Q3041976
cites work: Q5186383
cites work: Q3756152
cites work: On a form of KKM principle and SupInfSup inequalities of von Neumann and of Ky Fan type
cites work: Minimax and Variational Inequalities for Compact Spaces
cites work: Q4020526
cites work: Geometric Properties, Minimax Inequalities, and Fixed Point Theorems on Convex Spaces
cites work: On general minimax theorems
cites work: Q3828582
cites work: Q4022832
cites work: Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity
cites work: On Nonlinear Variational Inequalities
cites work: Minimax theorems and saddling transformations
DOI: 10.1006/JMAA.1998.6081
description: scientific article; zbMATH DE number 1236165