An infinite-dimensional extension of a Poincar\'e's result concerning the continuation of periodic orbits

We study the existence of periodic solutions for the infinite-dimensional second order system $\ddot x=V_{x},\ x\in{\Bbb T}^{{\Bbb Z}_+}.$ Using the Implicit-Function-Theorem, we prove the existence of time-periodic solutions at \lq\lq high frequencies"; no \lq\lq smallness condition" on $V(x)$ is required