Multiple periodic solutions of some forced Hamiltonian systems and the generalized saddle point theorem (Q1318622)

The author proves the existence of geometrically distinct periodic solutions of the Hamiltonian system \[ J \dot u + \nabla H(t,u)=0 \] where \(H(t,x)\) is periodic with respect to \(t,x_ 1, \dots,x_ p\) and goes to zero uniformly with respect to \((t,x_ 1, \dots,x_ p)\) when \((x_{p+1}, \dots,x_{2N})\) goes to infinity. It is assumed that \(H(t,x):\mathbb{R} \times \mathbb{R}^{2N} \to \mathbb{R}\) is a continuously differentiable function, periodic in \(t\) with period \(T>0\). The author establishes that this system has at least \((p+1)\) \(T\)-periodic solutions.