Solvability of linear equations in the Besicovitch and Bohr classes of almost periodic functions

An example is given of a finite-dimensional equation $u'+A(t)u=f(t)$, where $A(t)$ and $f(t)$ are Bohr almost periodic elements, having bounded solutions but not almost periodic solutions (the question of a similar example was already posed and discussed in Favard's original papers). On the other hand, solvability in the Besicovitch class does not require subtle separability or stability conditions. General theorems of such a kind are provided in this note.