Ordinary differential equations with a non-Lipschitz right-hand side
type: Item
instance of: scholarly article
title: Ordinary differential equations with a non-Lipschitz right-hand side
zbMATH Open document ID: 1264.34017
DOI: 10.1134/S1064562412040382
published in: Doklady Mathematics
publication date: 2012-11-15T00:00:00Z
review text: Consider the initial value problem \[ u_t(t, x)=f(t, u(t, x)), u(0, x)=x\in F,\tag{1} \] where \(f(t, x)=(f^1, f^2, \dotsc, f^m)(t, x)\in {\mathcal C}(I_{\tau}\times \overline{Q}, \mathbb R^m)\), \(\sup_{(t, x)\in I_{\tau}\times \overline{Q}}||f(t, x)||<\infty\), \(Q\) is an open domain in \(\mathbb R^m\), \(F\subset Q\) is a compact set, \(I_{\tau}=[0, \tau)\). The author proves that problem (1) has a general solution \(u(t, x)\) such that the functions \(x\mapsto u(t, x)\) and \(u_t(t, x)\) lie in \({\mathcal B}(F, {\mathcal C}(I_T, \mathbb R^m))\), and, if \(u\in {\mathcal B}(F, {\mathcal C}(I_T, \mathbb R^m))\) is a general solution of (1), then the map \(t\mapsto u(t, x)\) is in \({\mathcal C}^1(I_T, B(F))\). Here, \({\mathcal B}(F, {\mathcal C}(I_T, \mathbb R^m))\) is the set of Borel measurable functions of \(F\) to \({\mathcal C}(I_T, \mathbb R^m)\) and \(B(F)\) is the space of \(F\)-bounded and Borel measurable mappings \(v:F\to \mathbb R^m\), \(T\in (0, \tau]\).
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 6106511
zbMATH Keywords: Peano existence theorem
zbMATH Keywords: non-Lipschitz nonlinearity
zbMATH Keywords: non-uniqueness
zbMATH Keywords: IVP
zbMATH Keywords: Cauchy problem
author: Oleg Zubelevich
reviewed by: Svetlin Georgiev
MaRDI profile type: Publication
cites work: Ordinary differential equations, transport theory and Sobolev spaces
cites work: Q4255465
cites work: Peano's theorem in Banach spaces
cites work: On dichotomies for solutions of \(n\)-th order linear differential equations
cites work: Kamke's Uniqueness Theorem
cites work: Systems of differential equations with nonunique solutions to the Cauchy problem
cites work: Q3874718
full work available at URL: s1064562412040382
OpenAlex ID: W2083302856
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description: scientific article; zbMATH DE number 6106511