On the Solvability of a Periodic Problem for a System of Second-Order Nonlinear Ordinary Differential Equations

The solvability of a periodic problem for a system of nonlinear second-order ordinarydifferential equations with a positively homogeneous main part is investigated. New conditions arefound that ensure an a priori estimate for the solutions of the periodic problem underconsideration. The conditions are stated in terms of the properties of the positively homogeneousmain part of the system. Under the a priori estimate, using and developing methods for calculatingthe mapping degree for vector fields, we prove a theorem on the solvability of the periodic problemthat generalizes the results previously obtained by the present authors on the study of the periodicproblem for systems of second-order nonlinear ordinary differential equations.