Limit cycles for fourth-order autonomous differential equations

We provide sufficient conditions for the existence of periodic solutions of the fourth-order differential equation $$ ddddot x -(lambda+mu) dddot x +(1+lambda mu)ddot x -(lambda+mu)dot x+lambda mu x = varepsilon F(x, dot x, ddot x, dddot x), $$ where $lambda$, $mu$ and $varepsilon$ are real parameters, varepsilon is small and F is a nonlinear function.