Periodic solutions of time-dependent perturbed Hamiltonian systems

We consider time-periodic Hamiltonians of the form H(t, Q, P, & varepsilon;) where & varepsilon; is a small parameter: The unperturbed function H-0(Q, P) = H(t, Q, P, 0) is autonomous, integrable and has periodic solutions. It is assumed that these Hamiltonian functions can be written in convenient symplectic coordinates in the form H(t, theta, phi, q, I, J, p, & varepsilon;) = H-0(I, J) + & varepsilon;H-1(t, theta, phi, q, I, J, p) + O(& varepsilon;(2)), where theta, phi is an element of T, I, J is an element of R, q, p is an element of R-n. The aim of this paper is to show the existence of periodic solutions of the previous family of time-dependent 2 pi-periodically perturbed Hamiltonian systems under different approaches.