Levitan Almost Periodic Solutions of Linear Differential Equations

The known Levitan's Theorem states that the linear differential equation $$x'=A(t)x+f(t) \ \ \ (*)$$ with Bohr almost periodic coefficients $A(t)$ and $f(t)$ admits at least one Levitan almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations $$x'=A(t)x\ .\ \ \ (**)$$ In this paper we prove that linear differential equation (*) with Levitan almost periodic coefficients has a Levitan almost periodic solution, if it has at least one bounded solution. In this case, the separation from zero of bounded solutions of equation (**) is not assumed. The analogue of this result for difference equations also is given. We study the problem of existence of Bohr/Levitan almost periodic solutions for equation (*) in the framework of general nonautonomous dynamical systems (cocycles).