In this paper, suppose L(t,x,p) =12A(x)p·p + V (t, x), A is positive definite and symmetric, and both A and V are C3 and 1-periodic in all of their variables. We prove that the Poincare map (i.e. the time-1-solution map) of the Lagrangian system d dt Lx˙(t, x, x˙)−Lx(t, x, x˙) =0 possesses infinitely many periodic points on Tn produced by contractible integer periodic solutions.

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous periodic and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems and unify known results under various convexity conditions.

In this paper we prove the existence of infinitely many distinct T-periodic solutions for the perturbed second order Hamiltonian system q + V 0(q) = f(t)under the conditions that V: RN→R is continuously differentiable and superquadratic, and that f is square integrable and T-periodic. In the proof we use the minimax method of the calculus of variation combining with a priori estimates on minimax values of the corresponding functionals.