Asymptotic behavior of solutions to abstract functional differential equations
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions to abstract functional differential equations
zbMATH Open document ID: 1209.34094
author: Qing Liu
author: Rong Yuan
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-18T00:00:00Z
review text: The authors consider ergodicity and spectra of functions taking values in a Banach space \(X\) and defined on \(\mathbb{R}\) or \(\mathbb{R}^+\). Let \(G\) be a closed translation-invariant subspace of \(BUC(\mathbb{R}; X)\) and \(E\subset BUC(\mathbb{R}^+; X)\) a translation bi-invariant subspace. For \(\nu\in BUC(\mathbb{R}, X)\), define the spectrum \(sp^{\mathbb{R}}_G\) of \(\nu\) with respect to \(G\) as the set of \(\eta\in\mathbb{R}\) such that, for any \(\varepsilon> 0\), there exists \(f\in L^1(\mathbb{R})\) such that the support of the Fourier transform of \(f\) is contained in \((\eta- \varepsilon, \eta+ \varepsilon)\) and \(f\) convolved with \(\nu\) is not in \(G\). For \(u\in BUC(\mathbb{R}^+, X)\), define the the spectrum \(sp^{\mathbb{R}^+}_E\) of \(u\) with respect to \(E\) as the set of \(\eta\in\mathbb{R}\) such that, for any \(\varepsilon> 0\), there exists \(f\in L^1(\mathbb{R})\) such that the support of the Fourier transform of \(f\) is contained in \((\eta-\varepsilon,\eta+\varepsilon)\) and the restriction to \(\mathbb{R}^+\) of \(f\) convolved with \(u\) is not in \(E\). The authors prove that, if \(C^+_0(\mathbb{R};X)\subset G\) and \(E= G\), where \(G\) restricted to \(\mathbb{R}^+\), and if \(u=\nu\) restricted to \(\mathbb{R}^+\) with \(\nu\in BUC(\mathbb{R}; X)\), then \(\text{sp}^{\mathbb{R}^+}_E(u)= sp^{\mathbb{R}}_G(\nu)\). This result is applied to prove that, if \(u\in BUC(\mathbb{R}; X)\) is such that \(sp^{\mathbb{R}}_G(u)\) is countable and if for some real \(s\) the translation \(u_s\) (restricted to \(\mathbb{R}^+\)) is totally ergodic on \(isp^{\mathbb{R}}_G(u)\) and for any \(\eta\in sp^{\mathbb{R}}_G(u)\) one has \(M_\eta(u_s)\in E\), then \(u\in G\). Here \(M_\eta\) denotes the Cesàro limit and \(u_s\) is restricted to \(\mathbb{R}^+\). These results are used to describe the asymptotic behavior of mild solutions of the equation \[ u'(t)= Au(t)+ Bu(t)+ f(t),\qquad t\in \mathbb{R}. \] The work is related to and extends resu…
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34K14
Mathematics Subject Classification ID: 44A99
zbMATH DE Number: 5566243
zbMATH Keywords: Tauberian theorems
zbMATH Keywords: almost periodic functions
zbMATH Keywords: abstract differential equations
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.03.029
OpenAlex ID: W2057630921
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DOI: 10.1016/J.JMAA.2009.03.029
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description: scientific article; zbMATH DE number 5566243