Periodic Solutions in the Golden Sixties: The Birth of a Continuation Theorem

Abstract

This chapter describes the genesis of a continuation theorem introduced in the late sixties, for proving the existence of periodic solutions for ordinary differential equations. In the early sixties, the study of T-periodic solutions of T-periodic ordinary differential equations or systems was divided into two almost separated worlds: the case of weakly nonlinear differential equations, and the case of strongly nonlinear differential equations. In the case of systems with a small parameter, it was necessary to distinguish between the nonresonant case, in which the associated linear system. Topological degree techniques were less than popular among experts in differential equations, with the exception of M.A. Krasnosel'skii and his school in Voronezh, whose important ideas had not yet really penetrated the Western world. Leray-Schauder method had been successfully applied to problems of T-periodic solutions of some ordinary differential equations. The method introduced in this chapter allows finding, under general mathematical conditions, the existence theorems proved by the author with the help of Cesari's method.”