Stability and almost periodicity of trajectories of periodic processes
type: Item
instance of: scholarly article
title: Stability and almost periodicity of trajectories of periodic processes
zbMATH Open document ID: 0815.34050
published in: Journal of Differential Equations
publication date: 1995-01-31T00:00:00Z
review text: Based on the De Leeuw-Glicksberg decomposition theorem, the author proves that if the monodromy operator \(V\) is power-bounded and its spectrum \(\sigma (V)\) has countable intersection with the unit circle \(\Gamma\), and the point spectrum \(P \sigma (V^*)\) of \(V^*\) satisfies a certain condition, then all solutions on \(\mathbb{R}_ +\) of (1) \(u'(t)=A(t)u(t)\), where \(A(t)\) are closed linear operators in a Banach space \(E\), which depend on \(t\) in a periodic manner, are asymptotically almost periodic. In particular, if \(P \sigma (V^*) \cap \Gamma = \emptyset\), then all solutions on \(\mathbb{R}_ +\) of (1) converge to 0 as \(t \to \infty\). It is also proved that if \(\sigma (V) \cap \Gamma\) is countable and \(f\) is almost periodic, then every bounded uniformly continuous mild solution on \(\mathbb{R}\) of (2) \(u'(t) = A(t)u(t) + f(t)\) is almost periodic, provided an additional condition. Some examples of applications and conditions under which nontrivial bounded solution on \(\mathbb{R}\) of (1) are given.
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 716262
zbMATH Keywords: De Leeuw-Glicksberg decomposition theorem
zbMATH Keywords: Banach space
zbMATH Keywords: asymptotically almost periodic
author: Vu Quoc Phong
reviewed by: A. Rhandi
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1020
OpenAlex ID: W2043327213
DOI: 10.1006/JDEQ.1995.1020
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description: scientific article; zbMATH DE number 716262