The continuity of inf sup, with applications
type: Item
instance of: scholarly article
title: The continuity of inf sup, with applications
zbMATH Open document ID: 0608.49007
DOI: 10.1007/BF01189636
author: Stephen Simons
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Let X and Y be nonempty sets and a,b: \(X\times Y\to {\mathbb{R}}\). We give conditions under which the map on \({\mathbb{R}}^ 2_ +\) defined by \((\lambda,\mu)\to \inf _{\chi}\sup _{\gamma}(\lambda a+\mu b)\) is continuous. We deduce an infinite dimensional generalization of a result of Fan on the equilibrium value of a system of convex and concave functions. We also deduce an infinite dimensional generalization of a result of Aubin on eigenvalues of a multifunction. Our results have applications to matrix theory (the von Neumann-Kemeney theorem and the Perron-Frobenius theorem). We use the Hahn-Banach theorem and do not use any fixed-point related concepts.
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 46A22
zbMATH DE Number: 3983974
zbMATH Keywords: Perron-Frobenius theorem
zbMATH Keywords: Hahn-Banach theorem
zbMATH Keywords: minimax theorem
zbMATH Keywords: sublinear functional
MaRDI profile type: Publication
cites work: Q3900561
cites work: Q3782303
cites work: On the equilibrium vakue of a system of convex and concave functions
cites work: Über das von-Neumannsche Minimax-Theorem
cites work: On certain applications of the Hahn-Banach and minimax theorems
description: scientific article; zbMATH DE number 3983974