On existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite-dimensional space.
type: Item
instance of: scholarly article
title: On existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite-dimensional space.
zbMATH Open document ID: 1034.34016
author: Grzegorz Gabor
author: Marc Quincampoix
published in: Journal of Differential Equations
publication date: 2003-03-18T00:00:00Z
review text: The authors study a Cauchy problem for a first-order differential inclusion of the type \[ \dot{x}(t) \in F(x(t)) \quad\text{for a.e. }t \geq 0, \qquad x(0)=x_0 \in \mathbb{R}^n, \] where \(F:\mathbb{R}^n \to \mathbb{R}^n\) is a multifunction, and, given a closed set \(K \subset \mathbb{R}^n\), they give sufficient conditions for the existence of points \(x_0 \in K\) and solutions to the problem which remain in \(K\), i.e., \(x(t) \in K, \forall t \geq 0\). The authors give different conditions for the solvability of the problem, in particular, they present two different, and often independent methods: in the first part, they prove an existence result by using homology theory and, in the second part, they use retracts. All results obtained are illustrated by examples and a particular case of equations is also considered.
reviewed by: Francesca Papalini
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 1884210
zbMATH Keywords: differential equations and inclusions
zbMATH Keywords: Wazewski topological principle
zbMATH Keywords: homology groups
zbMATH Keywords: retracts
zbMATH Keywords: strong deformation retracts
zbMATH Keywords: Lipschitz selection
zbMATH Keywords: viability theory.
MaRDI profile type: Publication
full work available at URL: jdeq.2002.4165
OpenAlex ID: W2007910372
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DOI: 10.1006/JDEQ.2002.4165
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description: scientific article; zbMATH DE number 1884210