A Perron-type theorem for nonautonomous delay equations
type: Item
instance of: scholarly article
title: A Perron-type theorem for nonautonomous delay equations
zbMATH Open document ID: 1271.34072
author: Luís Barreira
author: Claudia Valls
published in: Central European Journal of Mathematics
publication date: 2013-06-14T00:00:00Z
review text: The authors show that the asymptotic exponential behavior of the solutions to a linear delay equation \[ x'=L(t)x_{t}, \] where \(x_{t}(\theta)=x(t+\theta)\), \(\theta\in [-r,0]\) for some fixed \(r>0\), persists under small perturbations, i.e., for an appropriate choice of \(f(t,x)\) in the equation \[ x'=L(t)x_{t}+f(t,x_{t}). \] More precisely, let \(X\) be a Banach space and \(r>0\), and define \(\hat{\mathbb{D}}\) as the space of functions \(\varphi:[-r,0]\to X\) such that the one-sided limits of \(\varphi\) at \(s\) exist and \(\varphi\) is right continuous at \(s\) for every \(s\in[-r,0]\). We write \(T(t,s)\) for the evolution operator with values in \(\hat{\mathbb{D}}\). One also assumes that \(\hat{\mathbb{D}}=F_{1}(t)\oplus F_2(t) \oplus \dotsb\oplus F_{p}(t)\) for every \(t\geq0\) and \(\lambda_1<\lambda_2<\dotsb<\lambda_p\) are real numbers which satisfy {\parindent=8mm \begin{itemize}\itemH1 For each \(t,s\geq0\) and \(i=1,\dotsc,p\), one has \(T(t,s)F_{i}(s)=F_{i}(t)\); \itemH2 Given \(\varepsilon>0\), there exists \(\rho>0\) such that \[ \rho^{-1}e^{(\lambda_i-\varepsilon)(t-s)-\varepsilon s}\left\|x\right\|\leq \left\|T(t,s)x\right\|\leq \rho e^{(\lambda_i+\varepsilon)(t-s)+\varepsilon s}\left\|x\right\|,\quad t\geq s\geq0, \] for every \(i=1,\dotsc,p\) and \(x\in F_{i}(s)\); \itemH3 For each \(i=1,\dotsc,p\), the projections \(P_i(t)\) associated to the decomposition of \(\hat{\mathbb{D}}\) satisfy \[ \lim_{t\to\infty}\frac{1}{t}\log \left\|P_{i}(t)\right\|=0. \] \end{itemize}} The previous hypotheses are widely satisfied and are also motivated by ergodic theory. The main result of the work states that, under the previous assumptions, if \(x\) is a solution of \[ x_t=T(t,s)x_s+\int_{s}^{t}T(t,\tau)X_0 f(\tau,x_{\tau})\,d\tau, \] where \((X_0 u)(0)=u\) and \((X_0 u)(\theta)=0\) for every \(\theta<0\), and also \(\left\|f(t,x_t)\right\|\leq \gamma(t)\left\|x_t\right\|\) for every \(t\geq0\), and for some continuous function \(\gamma:\mathb…
reviewed by: Alberto Lastra
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34K06
zbMATH DE Number: 6176172
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: nonautonomous delay equations
MaRDI profile type: Publication
full work available at URL: s11533-013-0244-6
OpenAlex ID: W2013650583
cites work: Q5431024
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DOI: 10.2478/S11533-013-0244-6
description: scientific article; zbMATH DE number 6176172