New existence and multiplicity theorems of periodic solutions for non-autonomous second order Hamiltonian systems

In the present paper, the non-autonomous second order Hamiltonian systems (1) { u ̈ ( t ) = ∇ F ( t , u ( t ) ) , a.e.  t ∈ [ 0 , T ] u ( 0 ) − u ( T ) = u ̇ ( 0 ) − u ̇ ( T ) = 0 , are studied and a new existence theorem and a new multiplicity theorem of periodic solutions are obtained.