Lyapunov regularity of impulsive differential equations
type: Item
instance of: scholarly article
title: Lyapunov regularity of impulsive differential equations
zbMATH Open document ID: 1206.34073
author: Luís Barreira
author: Claudia Valls
published in: Journal of Differential Equations
publication date: 2010-09-29T00:00:00Z
review text: The authors consider a linear impulsive differential equation in \(\mathbb{R}^n\) of the form \[ x'= A(t)x,\quad t\neq\tau_i,\quad\Delta x|_{t=\tau_i}= B_i x.\tag{1} \] \(A(t)\) are \(n\times n\) matrices varying continuously for \(t\geq 0\), \(B_1,B_2,\dots\) are given matrices and \(0<\tau_1< \tau_2<\cdots\)\,. A solution \(x\) of (1) is a function of class \(C^1\) except the times \(\tau_i\) and having discontinuities of the first kind at these times with \[ x'(t)= A(t) x\qquad (t\neq \tau_i), \] \[ \Delta x|_{t=\tau_i}= x(\tau^+_i)- x(\tau^-_i)\quad\text{for }i\in\mathbb{N}. \] The notion of Lyapunov regularity for an impulsive differential equation is introduced. Sufficiently conditions for the existence of stable and unstable invariant manifolds for the solution of the nonlinear differential equation \[ x'= A(t) x+ f(t,x) \] are derived. By means of so-called exponential contraction sufficient conditions for the existence of a solution \(x(t)\) of the equation \[ x'= A(t)x+ f(t,x(t)),\quad t\neq \tau_i,\quad\Delta x|_{t=\tau_i}= B_i x+ q_i(x) \] are found. The following estimate is valid: \[ \| x(t)\|\leq Ke^{\widetilde a(t- s)+ as}\| x(s)\|,\quad t\geq s, \] \(K\), \(\widetilde a\), \(a\)-const.
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 34A37
zbMATH DE Number: 5793315
zbMATH Keywords: linear impulse differential equations in \(\mathbb{R}^n\)
zbMATH Keywords: dichotomy
zbMATH Keywords: Lyapunov regularity
zbMATH Keywords: nonuniform exponential contraction
reviewed by: S. Kostadinov
MaRDI profile type: Publication
full work available at URL: j.jde.2010.07.016
OpenAlex ID: W2025437244
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DOI: 10.1016/J.JDE.2010.07.016
description: scientific article; zbMATH DE number 5793315