Existence of periodic solutions to second-order Hamiltonian systems with potential indefinite in sign (Q5919914)

The authors study the existence of periodic solutions of the second-order Hamiltonian system \[ x''(t) + A(t)x(t)+ b(t)V'(x(t)) = 0, \] where \(A: \mathbb{R}\rightarrow \mathbb{R}^{n\times n}\) is continuous, symmetric and \(T-\)periodic; \(b : \mathbb{R}\rightarrow \mathbb{R}\) is continuous and \(T-\)periodic and \(V:\mathbb{R}^n \rightarrow \mathbb{R}\) is \(C^2\) and with a superquadratic behavior. If \(\int_0^T b(s) \;ds > 0,\) they improve some related previous results. In the proof, variational methods are used, specially linking theorems.