Almost periodic solutions for delay-differential equations with infinite delays

Abstract

One of the most widely methods for establishing the existence of almost periodic (a.p. for short) solutions of systems or ordinary differential equations is based on a general result due to Amerio [ 1 J, a generalization of an earlier result for linear a.p. systems due to Favard. In essence, this basic method consists of showing that if a solution is in a compact set in the statespace for all t and has a certain separation property with respect to other such solutions, then if this same property holds for all systems in the socalled hull of the given system, this solution will be a.p. This result in fact relates to a necessary as well as sufficient stability condition for a fairly general dynamical system to have an a.p. solution; cf. [2, Chap. 51. In fact Miller [3] used the idea of embedding the a.p. system in a more general dynamical system and was thus able to obtain a local stability condition for the existence of a.p. solutions without using the separation conditions due to Favard and Amerio. Most stability conditions for the existence of a.p. solutions known prior to Miller’s result were in terms of global stability conditions. For a fairly complete and comprehensive discussion of a.p. solutions for a.p. systems, the books by Fink [4] and Yoshizawa [5] are recommended. For a more general discussion of the idea of embedding a.p. systems in dynamical systems, cf. Sell [6]. For ordinary differential equations with fixed finite time delays, much of the method based on Amerio’s separation result can and has been adapted to obtain similar stability conditions for the existence of a.p. solutions; cf. [7], for example. Miller in [3] shows that local stability conditions can also be used for such delay-differential equations. Results for delay equations with infinite time delays based on Amerio’s separation condition have also been obtained; cf., for example, Hino [B]. However, for such systems which do not involve substantially fading memory, not much seems to have been done; one of the main difficulties in such infinite delay systems arises from the fact that bounded solutions may no longer remain in compact subsets of the state