## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- 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

### 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.

#### Article information

**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