A Peano type theorem for a class of nonconvex-valued differential inclusions
type: Item
instance of: scholarly article
title: A Peano type theorem for a class of nonconvex-valued differential inclusions
zbMATH Open document ID: 1170.34308
author: Jordan Goblet
published in: Set-Valued Analysis
publication date: 2009-03-16T00:00:00Z
review text: This note is concerned with the existence of continuously differentiable solutions for the differential inclusion \[ x'(t) \in \{f_{1}(t,x(t)),\dots, f_{k}(t,x(t))\}, \quad x(0) = x_{0}, \tag{1} \] where \(f_{1},\dots,f_{k}:\Omega\subset \mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}\) and \(\Omega\) is an open set containing \((0,x_{0}).\) The existence of a continuously differentiable solution is proved under a continuity condition on the associated multiple-valued function in the sense of Almgren \[ f=\sum_{1}^{k}\mu(f_{i})\to Q_{k}(\mathbb{R}^n). \] Here, \[ Q_{k}(\mathbb{R}^n)=\{\sum_{1}^{k}\mu(x_{i}): x_{i}\in\mathbb{R}^{n}, i= 1,\dots,k\}, \] where \(\mu(x_{i})\) denotes the Dirac measure at \(x_{i}.\) \(Q_{k}(\mathbb{R}^n)\) is equipped with the metric \[ S(\sum_{1}^{k}\mu(x_{i}),\sum_{1}^{k}\mu(y_{i}))=\min \{\max_{i=1,\dots,k}|x_{i}-y_{\sigma(i)}|: \sigma\, \text{ix a permutation of}\, \{1,\dots,k\}\}. \] Theorem. Let the the associated multiple-valued function in the sense of Almgren \[ f=\sum_{1}^{k}\mu(f_{i})\to Q_{k}(\mathbb{R}^n) \] be continuous. Then there exist \(T >0\) and a continuously differentiable solution \(x:[0,T]\to \mathbb{R}^n\) to the differential inclusion (1).
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 5529274
zbMATH Keywords: Almgren's multiple-valued functions
zbMATH Keywords: Peano's theorem
MaRDI profile type: Publication
full work available at URL: s11228-008-0084-x
OpenAlex ID: W2014169187
cites work: Q4515158
cites work: Regular selections for multiple-valued functions
cites work: Q5483188
cites work: Q5682591
DOI: 10.1007/S11228-008-0084-X
description: scientific article; zbMATH DE number 5529274