On the solvability of a periodic problem for a system of ordinary differential equations with quasi-homogeneous nonlinearity

In this paper, we examine the solvability of a periodic problem for a system of ordinary differential equations whose principal nonlinear part is a quasi-homogeneous mapping. We prove that if an unperturbed system with quasi-homogeneous nonlinearity has no nonzero bounded solutions, then the periodic problem admits an a priori estimate. The results obtained are of interest from the point of view of the application and development of methods of nonlinear analysis in the theory of differential and integral equations.