Periodic orbits of the fourth-order non-autonomous differential equation u⁗+qu″+pu=εF(t,u,u′,u″,u‴)

We provide sufficient conditions for the existence of periodic solutions of the fourth-order differential equationu⁗+qu″+pu=εF(t,u,u′,u″,u‴),where q,p and ε are real parameters, ε is small and F is a nonlinear non-autonomous periodic function with respect to t. Moreover we provide some applications.