The Existence of Periodic Solutions in a Certain Class of Nonlinear Fourth Order Autonomous ODE’s

In this paper we study the existence of periodic solutions in a certain class of autonomous nonlinear ordinary differential equations of fourth order. Our method is based on the topological degree theory and the homotopy invariance property of the topological degree. For this purpose, first we prove a lemma which provides some sufficient conditions for the existence of periodic solutions in a system of two second order ordinary differential equations (ODEs). Then, we prove a theorem which provides the sufficient conditions needed for existence of periodic solutions in the ODEs under consideration. Finally, we present two numerical examples to show the applicability of the proposed theorem to the ODEs under consideration.