Unbounded Solutions of Almost Periodically Forced Pendulum-Type Equations

Abstract

In this paper, we prove that the forced pendulum-type equation ${\ddot x} + G_x(x, t) = p(t)$, where G(x,t) ∈ C ¹ (R ²) with 1-periodicity in x satisfies the conditions: $\sup _{{{\left( {x,t} \right)} \in R^{2} }} {\left| {G_{x} {\left( {x,t} \right)}} \right|} < + \infty$ and $\lim \;\sup _{{t \to \infty }} {\left\{ {\sup _{{x \in R}} {\left| {\frac{{G_{t} {\left( {x,t} \right)}}} {t}} \right|}} \right\}} = 0$, possesses infinitely many unbounded solutions on a cylinder S ¹×R for any almost periodic function p(t) with nonvanishing mean value.