On first-order ordinary differential equations with nonnegative right-hand sides
type: Item
instance of: scholarly article
title: On first-order ordinary differential equations with nonnegative right-hand sides
zbMATH Open document ID: 1025.34003
DOI: 10.1016/S0362-546X(02)00291-2
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-04-02T00:00:00Z
review text: Some new results are given in this paper, concerning the existence of extremal solutions to the scalar initial value problem \[ x'(t)=f \bigl(t,x(t) \bigr),\;t\geq t_0,\;x(t_0)= x_0,\tag{1} \] with \(f:[t_0,T]\times [x_0,R] \to\mathbb{R}\), \(T>t_0\), and \(R>x_0\). Firstly, some preliminary notions are defined. We denote by \(AC([a,b])\) the set of all real functions which are absolutely continuous on the interval \([a,b]\). For a subset \(Y\subset AC([t_0, T])\), \(t_0<T_0\leq T\), we say that \(x_*\in Y\) is a minimal solution to (1) in \(Y\) if \(x_*\) is a solution to (1) and \(x_*(t)\leq x(t)\) for all \(t\in[t_0,T_0]\) and for any other solution \(x\in Y\). We define the maximal solution to (1) in \(Y\) by reversing the inequalities. When both the minimal and the maximal solutions to (1) exist, we call them extremal solutions in \(Y\). We say that \(x_-\) is a subfunction on \([t_0,T_0] \subset[t_0,T]\) for problem (1) if \(x_-\in AC([t_0, T_0])\), \(x_-(t_0)=x_0\) and \((x_-)'(t)\leq f(t,x_-(t))\) for a.e. \(t\in [t_0,T_0]\). If \(T_0=T\) we say that \(x_-\) is a subfunction. Analogously, we say that \(x_+\) is a superfunction on \([t_0,T_0] \subset [t_0,T]\) for (1) if \(x_+\in AC([t_0, T_0])\), \(x_+(t_0)=x_0\), and \((x_+)'(t)\geq f(t,x_+(t))\) for a.e. \(t\in[t_0,T_0]\). If \(T_0=T\) we say that \(x_+\) is a superfunction. There are assumed that for the given right-hand side \(f:[t_0,T] \times [x_0,R]\to\mathbb{R}\) there exists a null measure set \(NC[x_0,R]\) such that the following conditions hold: (f1) for all \(t\in [t_0,T]\), \(f(t,\bullet)\) is measurable on \([x_0,R]\); (f2) for all \(x\in[x_0,R] \setminus N\), \[ \liminf_{s\to t^-} f(s,x)\geq f(t,x)\text{ for all } t\in(t_0,T], \text{ and} \] \[ f(t,x)\geq\limsup_{s\to t^+}f(s,x)\text{ for all }t \in[t_0,T); \] (f3) there exists \(M\in L^1(x_0,R)\) such that for all \(x\in [x_0, R] \setminus N\) and all \(t\in[t_0,T]\), \(0<{1\over M(x)}\leq f(t,x)\). The main result on the local existence state…
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 1891308
zbMATH Keywords: subfunctions
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: Carathéodory conditions
zbMATH Keywords: discontinuous differential equations
zbMATH Keywords: singular differential equations
author: J. Ángel Cid
author: Rodrigo L. Pouso
reviewed by: Marian Ivanovici
MaRDI profile type: Publication
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full work available at URL: s0362-546x(02)00291-2
OpenAlex ID: W2080896974
description: scientific article; zbMATH DE number 1891308