Linearized oscillation theory for a nonlinear delay impulsive equation.
type: Item
instance of: scholarly article
title: Linearized oscillation theory for a nonlinear delay impulsive equation.
zbMATH Open document ID: 1045.34039
author: E. Braverman
author: Leonid Berezansky
published in: Journal of Computational and Applied Mathematics
publication date: 2004-01-26T00:00:00Z
review text: The authors provide sufficient conditions for the oscillatory and nonoscillatory of the solutions of the following impulsive delay nonlinear differential equation \[ \dot x(t)+\sum_{k=1}^{m}r_{k}(t)f_{k}[x(h_{k}(t))]=0, \quad t\not=\tau_{j},\quad x(\tau_{j})=I_{j}x(\tau_{j}^{-})), j=1,2,\dots, \] where \(r_{k}(t)\geq 0\), \(h_{k}(t)\leq t\), \(\lim_{j\to\infty}\tau_{j}=\infty\). It is shown that oscillatory and nonoscillatory properties of the solutions of the above nonlinear equation are deduced from the following auxiliary linear impulsive differential equation \[ \dot x(t)+\sum_{k=1}^{m}r_{k}(t)a_{k}(t)x(h_{k}(t))=0, \quad t\not=\tau_{j},\quad x(\tau_{j})=b_{j}x(\tau_{j}^{-}),\quad j=1,2,\dots\;. \] Some applications for impulsive models of mathematical biology and numerical simulations for impulsive logistic equations are presented. The results of the present paper continue recent ones of the authors on the linearized theory for nonimpulsive delay equations in [J. Comput. Appl. Math. 151, No. 1, 119--127 (2003; Zbl 1025.34069)].
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34K45
zbMATH DE Number: 2032608
zbMATH Keywords: oscillation
zbMATH Keywords: delay impulsive equations
zbMATH Keywords: linearization
reviewed by: Mouffak Benchohra
MaRDI profile type: Publication
full work available at URL: j.cam.2003.06.004
OpenAlex ID: W1979790453
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DOI: 10.1016/J.CAM.2003.06.004
description: scientific article; zbMATH DE number 2032608