Unbounded solutions to almost-periodically forced pendulum-type equations (Q5946538)

The author studies the Lagrange instability of the pendulum-type equation \(\ddot x(t)+G_x(x(t),t)=p(t)\), where the external force \(p\) is almost-periodic and the \(t\)-dependent potential \(G\) is \(1\)-periodic in \(x\). The equation has infinitely many unbounded solutions on a cylinder \(S^1\times \mathbb{R}\) provided that \(p\) has a nonvanishing mean value, whereas \[ \sup_{t\in \mathbb{R}}\sup_{x\in \mathbb{R}}|G_x(x,t)|<+\infty\quad\text{and}\quad \lim_{t\to\infty}\sup_{x\in \mathbb{R}}|G_t(x,t)/t|=0. \]