A generalization of the Levinson-Massera’s equalities

Abstract

In his study of non-linear differential equations of the second order, N. Levinson [3] defined the dissipative systems (D-systems) which arise in many important cases in practice. To a dissipative system a transformation T: R2 → R2 called the Poincaré transformation is associated. Levinson used the Poincaré transformation in the qualitative study of dissipative systems, and he [3] and Massera [5] obtained certain equalities between the number of subharmonic solutions of a dissipative systems under suitable conditions. We call these the Levinson-Massera’s equalities.

The number of periodic points of smooth maps

“…We give a condition on n and on the homotopy class of the map under which v{n) is even. This generalizes results in [14,15,18].…”

“…We give a condition on n and on the homotopy class of the map under which v{n) is even. This generalizes results in [14,15,18].In the second application, we generalize some results of Franks [8,9] on the existence of infinitely many periodic points.In the last section, we generalize the Dold's equalities to isolated sets of periodic points. …”

“…Theorem 2 follows from Shiraiwa [18,Theorem 3] in the case where / is a diffeomorphism with L(f) = L(f') for any i.…”

“…He gave a classification of periodic points and proved certain equalities between the cardinal numbers of the classes of periodic points. His equalities were improved by Massera [15] and extended to compact manifolds by Shiraiwa [18]. Recently, Dold obtained an extension of these equalities to smooth maps on manifolds in [5] (Theorem 1 below).…”

The number and linking of periodic solutions of periodic systems

“…It is known that this condition is satisfied by many dissipative systems as Duffing's equation. See for example [6].…”