Selecting families and their applications
type: Item
instance of: scholarly article
title: Selecting families and their applications
zbMATH Open document ID: 1154.54007
author: Lai-Jiu Lin
author: Mircea Balaj
publication date: 2008-06-26T00:00:00Z
review text: Extending the result of \textit{P.\,Deguire} and \textit{M.\,Lassonde} [Topol.\ Methods Nonlinear Anal.\ 5, No.\,2, 261--269 (1995; Zbl 0877.54017)], the authors present the following theorem. Let \(X\) be a paracompact space and \(\{Y_i\}_{i \in I}\) a family of convex subsets of Hausdorff topological vector spaces. If \(\mathcal T=\{T_i:X \multimap Y_i\}_{i \in I}\) is a family of multimaps with convex values satisfying the condition \[ X = \bigcup_{i \in I} \bigcup_{y_i \in Y_i} \operatorname{int} T^-_i(y_i), \] then \(\mathcal T\) has a selecting family, i.e., there exists a family of continuous maps \(\{f_i: X \to Y_i\}_{i \in I}\) such that for each \(x \in X\), there exists \(i \in I\) such that \(f_i(x) \in T_i(x)\). As application, some fixed component theorems of Browder-Fan type, an intersection result, a maximal element theorem, and a minimax inequality are considered.
reviewed by: Valerii V. Obukhovskij
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 5293421
zbMATH Keywords: multivalued map
zbMATH Keywords: continuous selection
zbMATH Keywords: selecting family
zbMATH Keywords: fixed point
zbMATH Keywords: minimax inequality
zbMATH Keywords: maximal element
MaRDI profile type: Publication
full work available at URL: j.camwa.2007.06.011
OpenAlex ID: W1976303244
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DOI: 10.1016/J.CAMWA.2007.06.011
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 5293421