In this paper, a class of second-order Hamiltonian systems is studied. Under the assumption of superquadratic and subquadratic for the nonlinear term, the existence of six periodic solutions and nine periodic solutions are obtained by using the variational method and space decomposition. Finally, two examples are given to verify the feasibility of the new criteria.

The goal of this paper is to study fractional differential equations involving instantaneous and non-instantaneous impulses with Sturm-Liouville boundary conditions. By using critical point theory and variational approach, infinitely many solutions are obtained. The interesting point is that the potential has an oscillating asymptotic behavior. Also one example is presented to illustrate the main result.