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∈Rλ∈R, T   is a real positive number, A:[0,T]→RN×NA:[0,T]→RN×N is a continuous map from the interval [0,T][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.