Nonlinear oscillations and boundary value problems for Hamiltonian systems

We prove the existence of solutions of various boundary-value problems for nonautonomous Hamiltonian systems with forcing terms $$begin{gathered} dot x(t) = H'_p (t, x(t), p(t)) + g(t), hfill dot p(t) = - H'_x (t, x(t), p(t)) - f(t). hfill end{gathered} $$ Among these problems is the existence of T-periodic solutions, namely those satisfying x(t+T)=x(t) and p(t+T)+p(t). A special study is made of the classical case, where H(x, p)=1/2 |p|²+V(x). In the case of parametric oscillations, where (f, g)=(0, 0) and t ? H(t, x, p) is T-periodic, we give a lower bound for the true (minimal) period of the T-periodic solution (x, p) produced by our method, and we prove the existence of an infinite number of subharmonics.