Three periodic solutions for pertubed second order Hamiltonian system

Three periodic solutions for perturbed second order Hamiltonian systems \begin{abstract} In this paper we study the existence of three distinct solutions for the following problem \begin{displaymath} \begin{array}{ll} -\ddot{u}+A(t)u=\nabla F(t,u)+\lambda \nabla G(t,u) & \mbox{a.e\ in\ } [0,T] \\ u(T)-u(0)=\dot{u}(T)-\dot{u}(0)=0, \end{array} \end{displaymath} where $\lambda\in \mathbb{R}$, $T$ is a real positive number, $A:[0,T]\rightarrow \mathbb{R}^{N}\times \mathbb{R}^{N}$ 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. \end{abstract}