Intersection theorems and minimax theorems based on connectedness
type: Item
instance of: scholarly article
title: Intersection theorems and minimax theorems based on connectedness
zbMATH Open document ID: 0786.49007
author: Jürgen Kindler
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-01-13T00:00:00Z
review text: The problem is treated whether a given family \({\mathcal C}= \{C_ x: x\in X\}\) of nonvoid subsets of some set \(Y\) has a nonvoid intersection. To this end, a concept of abstract connectedness is used, which has previously been introduced into the theory of minimax theorems by the author [Arch. Math. 55, No. 6, 573-583 (1990; Zbl 0701.49017)], \textit{S. Simons} [Acta Math. Hungar., to appear], \textit{H. König} [Arch. Math. 59, No. 1, 55-64 (1992)] and \textit{H. Komiya} [`Fixed point theory and applications', Kok-Keong Tan (ed.) World Scientific Publ.]. A family \(\mathcal A\) of sets is called connected iff \(\{A,B,C\}\subset{\mathcal A}\), \(A\subset B\cup C\), and \(A\cap B\cap C=\emptyset\) implies \(A\subset B\) or \(A\subset C\). Set \(C(A)=\bigcap\{C_ x: x\in A\}\), \(A\subset X\), and \(C^*(B)=\bigcup\{C^*_ y: y\in B\}\), \(B\subset Y\). Theorem. The properties (i) \(\{C(A): A\subset X\}\) is connected, (ii) \(\{C^*(B): B\subset Y\}\) is connected, and (iii) \(\mathcal C\) is compact (i.e., every subfamily with the finite intersection property has nonvoid intersection) are sufficient for \(C(X)\) to be nonvoid. Moreover it is shown that condition (ii) can be slightly weakened such that the tree properties become sufficient and necessary. Specialized to the situation of topological connectedness one obtains generalized versions of previous topological intersection theorems of the author [Proc. Amer. Math. Soc. 117, 1003-1011 (1993; Zbl 0774.54016)]. Finally, versions of minimax theorems due to Ha, Horvath, König, König- Zartmann, Komiya, Komornik, Simons, Sion, Stacho, and Hoang Tuy are obtained as easy consequences of such intersection theorems.
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 54D05
zbMATH DE Number: 484033
zbMATH Keywords: intersecting sets
zbMATH Keywords: abstract connectedness
zbMATH Keywords: minimax theorems
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1323
OpenAlex ID: W2082245712
DOI: 10.1006/JMAA.1993.1323
description: scientific article; zbMATH DE number 484033