Sturmian comparison theory for impulsive differential inequalities and equations
type: Item
instance of: scholarly article
title: Sturmian comparison theory for impulsive differential inequalities and equations
zbMATH Open document ID: 0856.34033
DOI: 10.1007/BF01196165
author: P. S. Simeonov
author: Drumi Bainov
author: Yury Domshlak
published in: Archiv der Mathematik
publication date: 1997-02-19T00:00:00Z
review text: The authors generalize the Sturmian theory to second-order impulsive differential equations \((*)\) \(x''(t)+ p(t) x(t)= 0\), \(t\neq \tau_k\), \(\Delta x(\tau_k)=0\), \(\Delta x'(\tau_k)+p_k x(\tau_k)= 0\). Particularly, a comparison theorem, oscillation and non-oscillation theorems as well as a zero-separation theorem are proved. (Note that all solutions of an impulsive system, given in the special form \((*)\), are continuous.) Using comparison results and considering various simple (e.g., periodic, with constant coefficients) test systems, the authors present various sufficient conditions for oscillation and non-oscillation in \((*)\). On the other hand, this theory is also used here in the inverse order, to construct impulsive systems of the form \((*)\) with previously known oscillatory properties. The last section of the paper contains some applications of the main results to nonlinear impulsive differential equations.
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 34A40
zbMATH DE Number: 938573
zbMATH Keywords: Sturmian theory
zbMATH Keywords: second-order impulsive differential equations
zbMATH Keywords: comparison theorem
zbMATH Keywords: non-oscillation theorems
reviewed by: Sergei Trofimchuk
MaRDI profile type: Publication
cites work: Q3995625
cites work: Exponential stability of the solutions of the initial value problem for systems with an impulse effect
cites work: Q4325523
cites work: On delay differential equations with impulses
full work available at URL: https://doi.org/10.1007/bf01196165
OpenAlex ID: W1973411745
description: scientific article; zbMATH DE number 938573