Asymptotic Integration of Delay Differential Systems

The purpose of this paper is to completely prove a conjecture in J. R. Haddock and R. Sacker [l] and further extend a result on asymptotic integration obtained previously by 0. Arino and I. Gyori [24]. Asymptotic integration deals with non-autonomous evolution equations which asymptotically are autonomous, and aims at relating the asymptotic behavior of the solutions of these equations to the asymptotic behavior of the solutions of the limit equation. Classical results on this problem exist for ordinary differential equations (i.e., cf. [5-91). For delay differential equations, the earliest results are due to K. L. Cooke [lo], and some later results can be found in [l-4, 11&16]. In [ 11, in search of an extension of results by Hartman [S], Hartman and Wintner [6], Atkinson [7], and Harris and Lutz [S], notably for ordinary differential equations of the form