Selecting families
type: Item
instance of: scholarly article
title: Selecting families
zbMATH Open document ID: 0877.54017
DOI: 10.12775/TMNA.1995.017
published in: Topological Methods in Nonlinear Analysis
publication date: 1996-05-09T00:00:00Z
review text: Let \(X\) be a topological space and \(\{Y_i\mid i\in I\}\) a family of spaces. For a given family of multimaps \(\{A_i\mid i\in I\}\), a family of continuous maps \(\{f_i\mid i\in I\}\) is said to be a selecting family provided for every \(x\in X\) there exists \(i\in I\) such that \(f_i(x)\in A_i(x)\). The main result is an analogue of the Michael-Browder theorem stating the existence of a selecting family for a family \(\{A_i\mid i\in I\}\) with convex values \(A_i(x)\), open fibres \(A_i^{-1}(y)\) and such that for every \(x\in X\) there exists \(i\in I\) with \(A_i(x)\neq \emptyset\). As applications the authors obtain versions of the Ky Fan-Browder fixed point theorem for systems of multimaps and minimax inequalities of Ky Fan and von Neumann-Sion for systems of functions.
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 49K35
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 856951
zbMATH Keywords: selecting family
zbMATH Keywords: Michael-Browder theorem
zbMATH Keywords: Ky Fan-Browder fixed point theorem
zbMATH Keywords: minimax inequalities
author: Paul Deguire
author: Marc Lassonde
reviewed by: Valerii V. Obukhovskij
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 856951