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ALMOST PERIODIC BEHAVIOUR OF UNBOUNDED SOLUTIONS OF DIFFERENTIAL EQUATIONS

A key result in describing the asymptotic behaviour of bounded solutions of differential equations is the classical result of Bohl-Bohr: If φ : R → C is almost periodic and Pφ(t) = ∫ t 0 φ(s) ds is bounded then Pφ is almost periodic too. In this paper we reveal a new property of almost periodic functions: If ψ(t) = tφ(t) where φ is almost periodic and Pψ(t)/(1 + |t|) is bounded then Pφ 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 solutions of differential equations of the form ∑m j=0 bju (t) = tφ(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.