The Perron integral in ordinary differential equations. (English) Zbl 0784.34006

Schwabik, Štefan

Differ. Integral Equ. 6, No. 4, 863-882 (1993).

A finite sequence of numbers \(D=\{\alpha_ 0,\tau_ 1,\alpha_ 1,\cdots, \alpha_{k-1},\tau_ k,\alpha_ k\}\) is called a partition of an interval \([a,b]\) if \(a=\alpha_ 0<\alpha_ 1< \cdots<\alpha_ k=b\) and \(\alpha_{j-1}\leq \tau_ j\leq\alpha_ j\), \(j=1,2,\dots,k\). Given a positive function \(\delta: [a,b]\to (0,\infty)\), called a gauge, the partition \(D\) is called \(\delta\)-fine if \([\alpha_{j-1},\alpha_ j]\subset [\tau_ j- \delta(\tau_ j), \tau_ j+\delta(\tau_ j)]\), \(j=1,2,\dots,k\). Let a function \(f:[a,b]\times \mathbb{R}^ n\to \mathbb{R}^ n\) satisfy the following conditions: (1) \(f(t,\cdot)\) is continuous for almost all \(t\in[a,b]\), (2) the integral \(\int_ a^ b f(s,z)ds\) in the sense of Perron exists for every \(z\in\mathbb{R}^ n\), (3) there is a compact set \(S\subset\mathbb{R}^ n\) and a gauge \(\delta\) on \([a,b]\) such that for all \(\delta\)-fine partitions \(\{\alpha_ 0,\tau,\dots,\alpha_ k\}\) and all functions \(w:[a,b]\to\mathbb{R}^ n\) we have: \(\sum_{i=1}^ k f(\tau_ i, w(\tau_ i)) (\alpha_ i- \alpha_{i-1})\in S\). The author shows that under these conditions for every \(v\in\mathbb{R}^ n\) and \(\alpha\in[a,b]\) there is a function \(y:[a,b]\to \mathbb{R}^ n\) such that \(y(t)=v+ \int_ \alpha^ t f(s,y(s))ds\) for \(t\in[a,b]\), where the integral is understood in the sense of Perron.

Reviewer: L.Janos (Los Angeles)

MSC:

34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems
26A39	Denjoy and Perron integrals, other special integrals

Keywords:

partition; gauge; integral; in the sense of Perron