On the solvability of a periodic problem for nonlinear ordinary differential equation of the second order

The article deals with a periodic problem for a nonlinear ordinary differential equation of the second order with the main positively homogeneous part selected. The paper uses the research scheme previously implemented by the authors in the study of the third two-point boundary value problem for a nonlinear ordinary differential equation of the second order. According to the research scheme, first in terms of the properties of the main positively homogeneous part, the conditions for a priori estimation of periodic solutions are found. And then in terms of a priori estimation theorems on the solvability of the periodic problem are formulated and proved using methods for calculating the rotation of vector fields. The results obtained can subsequently be generalized for systems of nonlinear ordinary differential equations of the second order.