Existence of periodic solutions for higher-order differential systems that are not of class D

Abstract

The concept of a system of class D or of a dissipative system for large +splacements has been introduced by N. Levinson [l] for differential equations of the second order, and this hypothesis has been widely used since then to obtain periodic solutions for such equations. This notion has been extended to differential systems of any order, and the corresponding definitions can be found in an interesting book by V. A. Pliss which is quoted in [Z]. In recent years some existence theorems for periodic solutions have been obtained by various authors in the field of second order difFerentia1 equations that,are not of class D, and we refer to [3] for a detailed bibliography. For higher-order systems, similar results are less numerous [2-4], but very interesting ones have been given by C. Corduneanu [5-71 whose proofs are based upon Schauder’s fixed point theorem. Other proofs are due to V. A. Pliss [2] and make use of the method of M. A. Krasnosel’skii and A. I. Perov P, 91. We have recently introduced [IO-121 a method to prove the existence of periodic solutions in nonlinear Lipschitzian differential systems. The basis of this process was the well-known algorithm of L. Cesari [13, 141 but our approach differed from that of Cesari in the analysis of the so-called determining equations: by use of a priori bounds for the periodic solutions, we had only to look for the determining equations of an associated weakly nonlinear system, generally a very easy task. The basic existence theorems of [14] have already been applied to a number of particular systems and have been extended in [Is], independently of Cesari’s method, to continuous differenti+ systems. Using one such theorem, we generalize in this paper some results of C. Corduneanu and relate them to a 2n-dimensional extension of an existence theorem due to L. Nirenberg [16].