Three periodic solutions for perturbed second order Hamiltonian systems

In this paper we study the existence of three distinct solutions for the following problem − u ¨ + A ( t ) u = ∇ F ( t , u ) + λ ∇ G ( t , u ) a.e. in [ 0 , T ] , u ( T ) − u ( 0 ) = u ˙ ( T ) − u ˙ ( 0 ) = 0 , where λ ∈ R , T is a real positive number, A : [ 0 , T ] → R N × N is a continuous map from the interval [ 0 , T ] to the set of N-order symmetric matrices. We propose sufficient conditions only on the potential F. More precisely, we assume that G satisfies only a usual growth condition which allows us to use a variational approach.