A note on amost periodic variational equations

The variational equation of a nonautonomous differential equation (x) over dot = F(t, x) along a solution mu is given by (x) over dot = DxF(t, mu(t))x. We consider the question whether the variational equation is almost periodic provided that the original equation is almost periodic by a discussion of the following problem: Is the derivative DxF almost periodic whenever F is almost periodic? We give a negative answer in this paper, and the counterexample relies on an explicit construction of a scalar almost periodic function whose derivative is not almost periodic. Moreover, we provide a necessary and sufficient condition for the derivative DxF to be almost periodic.