On generalized impulsive piecewise constant delay differential equations
type: Item
instance of: scholarly article
title: On generalized impulsive piecewise constant delay differential equations
zbMATH Open document ID: 1333.34118
author: K. S. Chiu
published in: Science China. Mathematics
publication date: 2015-10-26T00:00:00Z
review text: This paper deals with the following impulsive differential equations with piecewise constant delay of generalized type: \[ \begin{aligned} & x^{\prime }(t)=a(t)x(t)+b(t)x(\beta (t)), x(\tau )=c_{0},~t\neq t_{k},\\ & \left.\Delta x\right | _{t=t_{k}}=d_{k}x(t_{k}^{-}), k\in \mathbb{Z}, \end{aligned}\tag{1} \] \[ \begin{aligned} & y^{\prime }(t)=a(t)y(t)+b(t)y(\beta (t))+g(t),~y(\tau )=c_{0}, t\neq t_{k},\\ & \left. \Delta y\right | _{t=t_{k}}=d_{k}y(t_{k}^{-}), k\in\mathbb{Z},\end{aligned}\tag{2} \] \[ \begin{aligned} & z^{\prime }(t)=a(t)z(t)+b(t)z(\beta (t))+f(t,z(t),z(\beta (t))),~t\neq t_{k},\\ & \left. \Delta z\right | _{t=t_{k}}=J_{k}z(t_{k}^{-}), k\in\mathbb{Z},z(\tau )=c_{0}.\end{aligned}\tag{3} \] First, a representation formula for solutions of equation (1) is obtained. Later, Green's function is established and a representation formula which corresponds to variation of parameters formula for solutions of equation (2) is stated. Similarly, variation of parameters formula for equation (3) is also established. Moreover, Gronwall type integral inequalities are obtained. Finally, oscillatory properties as well as stability of solutions are investigated and some examples are given to illustrate the results.
reviewed by: Fatma Karakoç
Mathematics Subject Classification ID: 34K45
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 6498015
zbMATH Keywords: impulsive differential equation
zbMATH Keywords: piecewise constant delay of generalized type
zbMATH Keywords: variation of parameters formula
zbMATH Keywords: Gronwall integral inequality
zbMATH Keywords: Green function
zbMATH Keywords: oscillation
zbMATH Keywords: nonoscillation
zbMATH Keywords: stability of solutions
MaRDI profile type: Publication
full work available at URL: s11425-015-5010-8
OpenAlex ID: W2272092609
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DOI: 10.1007/S11425-015-5010-8
description: scientific article; zbMATH DE number 6498015