Exponential dichotomies for linear differential equations with delays: Periodic and autonomous equations

Abstract

We consider a homogeneous linear differential equation with delays u.+Mu=0,where u takes values in a Banach space and M is a memory with bounded recall under natural Caratheodory conditions, i.e., a linear mapping from the continuous to the locally Bochner-integrable functions such that Mu on [a, b] depends only on u on [a−l, b] for a suitable recall bound l, and M satisfies a reasonable boundedness condition. The presence of the conditional exponential stability behaviour called an exponential dichotomy—which is known to be related to the existence of bounded solutions of the inhomogeneous equation for right-hand sides in a suitable function space — is examined when the equation is periodic; it is shown to be equivalent to a splitting of the spectrum of the transition operators corresponding to a period. More detailed results are obtained when the equation is autonomous.