PERIODIC SOLUTION FOR NON-AUTONOMOUS SECOND ORDER HAMILTONIAN SYSTEMS ON TIME SCALES

We consider the following non-autonomous second order Hamiltonian system on time scales T of the form    u ∆∆(ρ(t)) = ▽H(t, u(t)) ∆-a.e. t ∈ [0, T ]T, u(0) − u(T ) = u ∆(ρ(0)) − u ∆(ρ(T )) = 0. As is well known, it is very difficult to use the Hilger’s integral to consider the existence of periodic solutions of some second order Hamiltonian systems on time scales since it is only concerned with antiderivatives. Therefore, in this paper, we use a new integral on time scales T defined by Rynne (J. Math. Anal. Appl. 328 (2007) 1217–1236), and establish a new existence result for periodic solutions in H1 T (T, R n) space of the above-mentioned second order Hamiltonian system on time scales T by applying variational methods and critical theory. As an application, an example is given to illustrate the result.