Almost periodic upper and lower solutions. (Q1412350)

Consider the second-order equation \((*) \;\ddot{u} = f(t,u,\dot{u}),\) where \(f\) is continuous and \(T\)-periodic in \(t\). If \(f\) satisfies a Nagumo condition, then the method of upper and lower solutions is a powerful tool to establish the existence of \(T\)-periodic solutions of \((*)\). The authors prove that there is no simple extension of this approach to the case of almost-periodic solutions. Especially, they prove that to given \(\omega \in \mathbb R \backslash Q\) there exists a function \(g \in C(\;T \times \mathbb R,\mathbb R)\) and numbers \(c>0, \;\delta > 0, \;\alpha < \beta\) such that for each \(t \in \mathbb R\) the inequalities \(g(t,\omega t,\alpha)\leq - \delta < 0 < \delta \leq g (t,\omega t, \beta)\) hold (that means \(\alpha\) and \(\beta\) are lower and upper solutions), but the equation \(\ddot{u} + c\dot{u} = g(t,\omega t,u)\) has no almost-periodic solution.