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VOL. 41 | 2003 Almost periodic behaviour of unbounded solutions of differential equations

Abstract

A key result in describing the asymptotic behaviour of bounded solutions of differential equations is the classical result of Bohl-Bohr: If $phi : \mathbb{R \rightarrow C$ is almost periodic and $P\phi(t) = \int_t^0 \phi(s)ds$ is bounded then $P\phi$ is almost periodic too. In this paper we reveal a new property of almost periodic functions: If $\psi(t) = t^N \phi(t)$ where $phi$ is almost periodic and $P\psi(t)/(1 + |t|)^N is bounded then $P\phi$ is bounded and hence almost periodic. As a consequence of this result and a theorem of Kadets, we obtain results on the almost periodicity of the primitive of Banach space valued almost periodic functions. This allows us to resolve the asymptotic behaviour of unbounded P solutions of differential equations of the form \sum_{j=0}^m b_ju^{(j)}(t) = t^N \phi(t). The results are new even for scalar valued functions. The techniques include the use of reduced Beurling spectra and ergodicity for functions of polynomial growth.

Information

Published: 1 January 2003
First available in Project Euclid: 18 November 2014

zbMATH: 1151.42302
MathSciNet: MR1994512

Rights: Copyright © 2003, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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