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.

In this paper, we study the existence of periodic solutions with prescribed minimal period for superquadratic autonomous second order Hamiltonian systems defined on Rn with no convexity assumptions. We use the direct variational approach for this problem on a W1,2-space of even functions, and prove new iteration inequalities on Morse indices. Using these tools and the saddle point theorem, we obtain results under precisely Rabinowitz' superquadratic condition on potential functions. We show that for every T>0 the above mentioned system possesses a T-periodic even solution with minimal period not smaller than T/(n + 2).

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 that on every compact C2 hypersurface in R2n bounding a convex set with non-empty interior, either there exists a sequence of variationally visible hyperbolic closed characteristcs with their minimal periods tending to infinity, or there exists at least one variationally visible nonhyperbolic closed characteristic.

In this paper the integer valued ω-index theory parameterized by all ω on the unit circle for paths in the symplectic group Sp(2n) is established. Based on this index theory, the Bott formula of the Malsov-type index theory for iterated paths in Sp(2n) is estalished, the mean index for periodic solutions of Hamiltonian systems is defined, and the increasing estimate of the iterated Maslov-type index in terms of the mean index is proved.

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.

Let