Periodic solutions of a second order differential equation with a large parameter

We consider a second-order differential equation containing a large parameter. Such an equation can be interpreted as the equation of forced oscillations of a mechanical system with one degree of freedom in the case when the natural frequency of the system is much greater than the external frequency. We present a new way of proving the existence of a periodic solution of this equation close to the periodic solution of the corresponding degenerate equation. The original proof, obtained earlier by one of the authors of the paper, was reduced to solving a system of integral equations constructed using the Green's function of a periodic boundary-value problem for the linearized and transformed initial equation. This method of proof was proposed by Lichtenstein and is an alternative to the Poincaré method, based on the implicit function theorem. In the case of singularly perturbed differential equations, the Liechtenstein method seems to be more economical. Nevertheless, it is interesting to see how the Poincare method can be applied in a singularly perturbed problem. The proof given below is obtained by the Poincaré method.