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.
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