Existence of solutions for a class of discontinuous differential equations in \(\mathbb{R}^n\)
type: Item
instance of: scholarly article
title: Existence of solutions for a class of discontinuous differential equations in \(\mathbb{R}^n\)
zbMATH Open document ID: 0930.34005
author: Witold Rzymowski
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-02-16T00:00:00Z
review text: The author investigates the Cauchy problem \[ x_i'= g_i(t, q_{i1}(x_1(t)), q_{i2}(x_2(t)),\dots, q_{in}(x_n(t))),\quad x_i(0)= 0,\quad i= 1,2,\dots, n.\tag{1} \] It is assumed that \(g_i: [0,\infty)\times \mathbb{R}^n\to [a_i,b_i]\) \((0\not\in[a_i, b_i])\) satisfies the Carathéodory conditions and \(q_{ij}:\Delta_{ij}\to \overline{\mathbb{R}}\) are measurable \((i,j= 1,2,\dots, n)\), where \(\Delta_{ij}= [0,\infty)\) for \(a_i>0\) and \(\Delta_{ij}= (-\infty,0]\) if \(a_i< 0\). Under the above assumptions it is proved that problem (1) has an absolutely continuous solution on the interval \([0,\infty)\). A scalar case is also considered. Interesting examples illustrating the obtained results are given. The results extend several ones obtained earlier by D. C. Biles, A. Bressan and P. Binding.
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 1295269
zbMATH Keywords: Cauchy problem
zbMATH Keywords: absolutely continuous solution
reviewed by: Józef Banaś
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6324
OpenAlex ID: W2074367168
cites work: Existence of solutions for discontinuous differential equations
cites work: The differential equation x=fox
cites work: Unique Solutions for a Class of Discontinuous Differential Equations
cites work: Uniqueness for discontinuous ODE and conservation laws
cites work: Extremal solutions of a discontinuous scalar differential equation
DOI: 10.1006/JMAA.1999.6324
description: scientific article; zbMATH DE number 1295269