Proceedings of the Centre for Mathematics and its Applications

Almost periodic behaviour of unbounded solutions of differential equations

Bolis Basit

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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.

Article information

Source
International Conference on Harmonic Analysis and Related Topics. Xuan Thinh Duong and Alan Pryde, eds. Proceedings of the Centre for Mathematics and its Applications, v. 41. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 2003), 17-34

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1416322424

Mathematical Reviews number (MathSciNet)
MR1994512

Zentralblatt MATH identifier
1151.42302

Citation

Basit, Bolis. Almost periodic behaviour of unbounded solutions of differential equations. International Conference on Harmonic Analysis and Related Topics, 17--34, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 2003. https://projecteuclid.org/euclid.pcma/1416322424


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