Asymptotically diagonal delay differential systems
type: Item
instance of: scholarly article
title: Asymptotically diagonal delay differential systems
zbMATH Open document ID: 0876.34078
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-10-30T00:00:00Z
review text: The paper deals with the equation \[ x'(t)=(\tilde A+A(t))x(t)+(\tilde B+B(t))x(t-\tau), \] where \(\tau\) is a positive constant. Here \smallskip \((H_1)\) \(\tilde A,\tilde B\) are constant diagonal matrices \(\tilde A=\text{ diag }\big\{ \tilde a_1,\tilde a_2,\ldots,\tilde a_n\big\},\) \(\tilde B=\text{ diag }\big\{ \tilde b_1,\tilde b_2,\ldots,\tilde b_n\big\},\) such that \((|\tilde a_i|+|\tilde b_i|)\tau<1/e\) and the equation \(\lambda\tilde a_i+\tilde b_i e^{-\lambda\tau}\) has a unique real root \(\tilde\lambda_i\) in the interval \((\tilde a_i-1/\tau,\infty),\) \(i=1,2,\ldots,n.\) These roots do not coincide, i. e. \(\tilde\lambda_i\neq\tilde\lambda_j\) for \(i\neq j,\) \(i,j=1,2,\ldots,n.\) \smallskip \((H_2)\) \(A(\cdot)=(a_{ij}(\cdot))_{1\leq i,j\leq n},\) \(B(\cdot)=(b_{ij}(\cdot))_{1\leq i,j\leq n}\) are continuous matrix functions on \([0,\infty)\) such that \(\alpha(t):=\sup_{s\geq t}|A(s)|\in L^2,\) \(\beta(t):=\sup_{s\geq t}|B(s)|\in L^2,\) and \(a_{ii}(t)-{1\over\tau}\int_{t-\tau}^t a_{ii}(s) ds\in L^1,\) \(b_{ii}(t)-{1\over\tau}\int_{t-\tau}^t b_{ii}(s) ds\in L^1,\) \(i=1,2,\ldots,n.\) \medskip The main result is: Let hypotheses \((H_1)\) and \((H_2)\) be fulfilled. Then there exists a matrix function \(F\), \(F(t)\to0\) as \(t\to\infty,\) such that the following statements are valid. \smallskip \((A)\) For every constant vector \(c\), the function \(x\) defined by \[ x(t)=(I+F(t))\exp\left(\int_0^t\Lambda(s) ds\right)c, \] where \[ \Lambda(t)=\tilde\Lambda+\big(I+(\tilde\Lambda-\tilde A)\tau\big)^{-1} \big(\text{diag}\{A(t)\}+\text{diag}\{B(t)\}e^{-\tilde\Lambda\tau}\big) \] and \(\tilde\Lambda=\text{diag} \{\tilde\lambda_1,\tilde\lambda_2,\ldots,\tilde\lambda_n\},\) is a solution of the considered equation. \smallskip \((B)\) For every solution \(x\) there exist a constant vector \(c\) and a vector function \(f,\) \(f(t)\to0\) as \(t\to\infty\), such that \[ x(t)=(I+F(t))\exp\left(\int_0^t\Lambda(s) ds\right)(c+f(t)) \] and \[ \left|x(t)-(I+F(…
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34K99
zbMATH DE Number: 1005595
zbMATH Keywords: theorem of Hartman-Winter
zbMATH Keywords: asymptotic form of the fundamental matrix
zbMATH Keywords: asymptotic diagonal matrix
zbMATH Keywords: scalar delay differential equation
author: Ovide Arino
author: István Győri
author: Mihály Pituk
reviewed by: Ivan Ginchev
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0463
OpenAlex ID: W2008724907
DOI: 10.1006/JMAA.1996.0463
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description: scientific article; zbMATH DE number 1005595