Periodic solutions of differential equations with almost constant time lag

Abstract

Delay-differential equations, or functional differential equations as they are frequently called, have become important in many fields. These equations have been studied by many authors and a moderate sized literature has been developed by Hale and his associates for the case where the equations are linear or slightly perturbed from the linear case. Cooke [l] has pointed out the need for further study of delay-differential equations with perturbed lag, with particular need for investigating the existence of oscillations. Although Perello [2] has established procedures for studying the oscillations of perturbed linear systems, the procedure fails to apply to Cooke’s problem. It is the purpose of this paper to establish that by a slight modification in Perello’s banach spaces, his procedure may be used identically in this case. We will use the notation which Hale and his associates have made standard and will assume that the reader is thoroughly familiar with their theory for linear autonomous functional differential equations. An extensive treatment of this theory may be found in Hale [3] and a brief sketch in Perello [2]. Perello considered equations of the form