Higher Order Analysis on the Existence of Periodic Solutions in Continuous Differential Equations via Degree Theory

Abstract

Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the coincidence degree theory for nonlinear operator equations, we perform a higher order analysis of continuous (non-Lipschitz) perturbed differential equations and derive sufficient conditions for the existence of periodic solutions for such systems. We apply our results to study continuous (non-Lipschitz) higher order perturbations of a harmonic oscillator.