The pendulum equation: from periodic to almost periodic forcings (Q637029)

The author considers the differential equation \[ \ddot x +a \sin x = p(t), \] where \(a>0\) is a parameter and \(p\:\mathbb {R}\to \mathbb {R}\) is an almost periodic function. Almost periodic is understood in the classical sense defined by H. Bohr. The author works with the Banach space \(\text{LPer}_{T}\) with the \(L^{\infty }\)-norm, which is defined as the space of functions obtained as uniform limits of periodic functions with periods of the type \(T, 2T, \dots , NT,\dots \). The space \(\text{LPer}_{T}\) consists of limit periodic functions and is a subclass of the space of almost periodic functions. The almost periodic function \(p\) has the mean value defined as \[ M\{p\}=\lim _{\tau \to \infty }\frac {1}{\tau }\int _a^{a+\tau } p(t)\, dt \] uniformly in \(a\in \mathbb {R}\). Using it, the Banach space \(X=\{p\in \text{LPer}_{T} : M\{p\}=0\}\) is introduced. The following main result is proved. Given \(a>0\) there exists an open and dense set \(G\subset X\) such that the above equation has a solution in \(\text{LPer}_{T}\) for each \(p\in G\). A variational approach is used in the proof.