Periodic solutions of second-order differential equations in Hilbert spaces (Q821547)

Classical existence results for the scalar periodic problem \[\begin{cases} \ddot x=f(t,x), \\ x(0)=x(T),\;\; \dot x(0)=\dot x(T), \end{cases} \] where \(f:[0,T]\times \mathbb{R}\to \mathbb{R}\) is continuous, are extended to systems, both in a finite-dimensional setting (\(f:[0,T]\times \mathbb{R}^N\to \mathbb{R}^N\)) and in an infinite- dimensional setting (\(f:[0,T]\times H\to H\), \(H\) is a separable Hilber space). The method of lower/upper solutions is used in both the well-ordered and non-well-ordered cases.