Continuation theorems and periodic solutions of ordinary differential equations

Abstract

Continuation methods of the Leray-Schauder type play an important role in the theory of differential equations. In these lectures, we present some classical and recent continuation theorems and their application to the existence and multiplicity of periodic solutions of ordinary differential equations. Using the Poincare operator, and following M.A. Krasnosel’skii and H. Amann, we apply continuation techniques to the computation of the Brouwer degree of some gradient mappings and then describe some recent developments in the method of guiding functions jointly obtained with A.M. Krasnosel’skii, M.A. Krasnosel’skii and A.I. Pokrovskii. We then prove some continuation theorems in spaces of continuous functions and use them in the study of periodic solutions of some complex-valued differential equations in the line of some joint work with R. Manasevitch and F. Zanolin. We finally present some continuation theorems in the absence of a priori bounds, based upon the behavior of some associated functionals, and recently developed with A. Capietto and F. Zanolin. We use them, together with the time-map method, to prove sharp non-resonance conditions for the existence of periodic solutions of perturbed conservative equations of the second order.