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 thefollowing 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 continuousmap from the interval $[0,T]$ to the set of $N$-order symmetric matrices.We propose sufficient conditions only on the potential $F$. Moreprecisely, we assume that $G$ satisfies only a usual growth conditionwhich allows us to use a variational approach.\end{abstract}