  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.
