Existence criterion for periodic solutions to one class of systems of second order nonlinear ordinary differential equations

Abstract

In this paper we study the existence of periodic solutions for a class of systems of second order ordinary differential equations with a separated main nonlinear part. Taking into consideration the structure of zero set for the main nonlinear part, we find new conditions ensuring an apriori estimate for periodic solutions. Under the apriori estimate, we formulate and prove a criterion for the existence of periodic solutions under any perturbation from a given class. The proof is made by using methods for calculating mapping degree of the vector field and employing the invariance of the existence of periodic solutions under a continuous varying of the main nonlinear part.