Selections of multifunctions of two variables
type: Item
instance of: scholarly article
title: Selections of multifunctions of two variables
zbMATH Open document ID: 0552.54010
DOI: 10.1216/RMJ-1984-14-3-503
author: Biagio Ricceri
published in: Rocky Mountain Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: We establish a very general selection theorem for multifunctions of two variables. From it, of which we cannot give here the extremely complex statement, a long series of consequences is then derived. We obtain, for instance, the following continuous selection theorem, here stated in a simplified version. Theorem. Let T, X be two paracompact topological spaces, Y a Banach space and F a multifunction from \(T\times X\) into Y, with non-empty, closed and convex values. Assume that, for each fixed \(x\in X\), the set F(T,x) is bounded, the multifunction F(.,x) is lower semicontinuous and, for each of its continuous selections g, one has \[ \lim_{z\to x}\sup_{t\in T}d(g(t),F(t,z))=0, \] where \(d(g(t),F(t,z))=\inf \{\| g(t)-v\|:\) \(v\in F(t,z)\}\). Under such hypotheses, F admits a continuous selection on \(T\times X\). This theorem is not a direct consequence of the classical Michael selection theorem, since, in general, the product of two paracompact spaces fails to be paracompact.
Mathematics Subject Classification ID: 54C65
zbMATH DE Number: 3879694
zbMATH Keywords: existence theorem of differential inclusions in Banach spaces
zbMATH Keywords: continuous selection theorem
zbMATH Keywords: paracompact topological spaces
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3879694