Applications of a Theorem of Dubrovskii to the Periodic Responses of Nonlinear Systems

In 1939 V. M. Dubrovskii1 proved the following result: Theorem 1: If A is a completely continuous operator which maps a Banach space X into itself, with the property that lim l'll^'T|ll = 0, x £ X, IIx I I II x II then for each scalar X and y c X, the equation x = y + Ax has at least one solution x c X. Dubrovskii's theorem was stated in the long review article of M. A. Krasnoselskii" on problems of nonlinear analysis, but except for a recent application,3 it seems to have gone largely unnoticed. It is the purpose of this paper to indicate some applications of the basic idea in the theorem to integral equations (and systems thereof) that arise in the study of nonlinear electrical networks and automatic control systems. The applications to be made all center around the existence and uniqueness of periodic responses of nonlinear systems to periodic driving signals. These properties of the equations governing nonlinear systems are frequently taken for granted. The fact is, though, that these are by no means universal properties of such equations, as simple examples 2855