On the oscillation properties of first-order impulsive differential equations with a deviating argument
type: Item
instance of: scholarly article
title: On the oscillation properties of first-order impulsive differential equations with a deviating argument
zbMATH Open document ID: 0880.34070
DOI: 10.1007/BF02937333
author: Drumi Bainov
author: Yury Domshlak
author: P. S. Simeonov
published in: Israel Journal of Mathematics
publication date: 1997-11-10T00:00:00Z
review text: For first-order linear impulsive differential equations with a deviating argument oscillation Sturm-like theorems are proved. Various criteria for oscillation or nonoscillation of the solutions of these equations are found. The oscillatory properties of some concrete equations of the considered type are investigated. The following Sturm-like oscillation theorem from the paper gives a better idea of the considered equations and the obtained results. A brief explanation follows the theorem. Theorem. Let the intervals \(J_n=(\xi_n,\eta_n)\) with \(\lim_{n\to\infty}\xi_n=+\infty\) be regular positive hemicycles of equation \((2)\) and on each of them let conditions \((a)\) and \((b)\) hold. Then all solutions of equation \((1)\) are oscillating. The considered impulsive equation with deviating argument is of the type \[ L[x]:=x'(t)+\sum_{i=1}^m a_i(t)x[r_i(t)]=0\quad(t\neq t_j),\quad x(t_j^+)=\alpha_j(t_j^-).\tag{1} \] The solutions of (1) are compared with the solutions of \[ \tilde L[y]:=-y'(t)+\sum_{i=1}^m q_i'(t)b_i[q_i(t)]y[q_i(t)]=0\quad(t\neq t_j),\quad y(t_j^+)=\beta_j(t_j^-).\tag{2} \] The finite interval \(J=(\xi,\eta)\) is said to be a regular positive hemicycle of equation (2) if \(q_i(\eta)>\xi,\) \(q_i(\xi)<\eta\) \((i=1,2,\dots,m)\) and there exists a solution \(y(t)\) in the interval \(J\) such that \(y(\xi^+)=y(\eta^-)=0\), \(y(t)>0\) \((t\in J)\) and \(y(t)\leq 0\) \((t\in E\text{ ext}(J):=\left(\cup_{j=1}^m q_i(J)\right)\setminus J ).\) Conditions \((a)\) and \((b)\) are the following. \((a)\) \(b_i(t)\geq0\) \((t\in q_i(J)\setminus J\), \(i=1,2,\dots m)\), \(a_i(t)\geq0\) \((t\in J\setminus q_i(J)\), \(i=1,2,\dots m)\), \(a_i(t)\geq b_i(t)\) \((t\in J\cap q_i(J)\), \(i=1,2,\dots m)\), \(1-\alpha_j\beta_j\geq0\) \((t_j\in J)\). \((b)\) At least one of the inequalities \(1-\alpha_j\beta_j\geq0\) is strict or at least one of the remaining inequalities in condition \((a)\) is strict in some subinterval of the respective sets where they are supposed to be…
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34A37
zbMATH DE Number: 1033784
zbMATH Keywords: differential equations with impulses
zbMATH Keywords: deviating argument
zbMATH Keywords: oscillations
reviewed by: Ivan Ginchev
MaRDI profile type: Publication
cites work: Sturmian comparison theory for impulsive differential inequalities and equations
cites work: Q3995625
cites work: Q4001772
cites work: Q4325523
cites work: Q3993076
cites work: Q3324291
cites work: Q4726625
cites work: Sturm-type comparison theorems for first- and second-order differential equations with delays of variable sign
cites work: On delay differential equations with impulses
cites work: Q4000352
cites work: The functional-differential equation $y'\left( x \right) = ay\left( {\lambda x} \right) + by\left( x \right)$
cites work: Q3759218
cites work: Q5676559
cites work: Oscillatory behaviour of an equation arising from an industrial problem
full work available at URL: https://doi.org/10.1007/bf02937333
OpenAlex ID: W1985543779
description: scientific article; zbMATH DE number 1033784