Journal of Differential Equations 170, 381407 (2001) doi:10.1006jdeq.2000.3824, available online at http:www.idealibrary.com on

More on Linear Differential Systems with Small Delays Ovide Arino Departement Mathematiques Recherche, Universite de Pau, Avenue de l 'Universite, 64000 Pau, France

and Mihaly Pituk 1 Department of Mathematics and Computing, University of Veszprem, P.O. Box 158, 8201 Veszprem, Hungary Received June 7, 1999; revised November 29, 1999

1. INTRODUCTION Let R n and R n * denote the n-dimensional space of real column vectors and the n-dimensional space of real row vectors, respectively, with any convenient norm | } |. For r>0, let C=C([&r, 0], R n ) denote the Banach space of continuous functions from [&r, 0] into R n with the supremum norm, |,| =sup &r%0 |,(%)| for , # C. The space C*=C([0, r], R n *) is defined similarly. Adopting the usual conventions, if x: [t&r, t] Ä R n and y: [t, t+r] Ä n R * are continuous functions, then the new functions x t # C and y t # C* are defined by x t(%)=x(t+%)

for &r%0

and y t(!)= y(t+!)

for 0!r,

respectively. Consider a linear homogeneous system of nonautonomous delay differential equations x$(t)=

|

0

d %['(t, %)] x(t+%),

(1.1)

&r

1 Partially supported by Hungarian National Foundation for Scientific Research Grants F 023772 and I 31935, and by the Janos Bolyai Research Grant of the Hungarian Academy of Sciences.

381 0022-039601 35.00 Copyright  2001 by Academic Press All rights of reproduction in any form reserved.

382

ARINO AND PITUK

where '(t, %) is an n_n matrix-valued function on R_R, measurable in (t, %) # R_R, normalized so that '(t, %)=0

for %0,

'(t, %)='(t, &r)

for %&r,

(1.2)

and '(t, %) is continuous from the left in % on (&r, 0) and has bounded variation in % on [&r, 0] for any t # R. Also, assume that there exists a positive constant K such that Var [&r, 0] '(t, } )K

for a.e. (almost every) t # R.

(1.3)

The above hypotheses on ' guarantee that for any , # C, System (1.1) has a unique solution on [&r, ) with initial value , at zero. That is, a unique continuous function x: [&r, ) Ä R n exists such that x 0 =,, x is locally absolutely continuous on [0, ) and satisfies System (1.1) for a.e. t0 (see [8, Chap. 6, Theorem 1.1]). A key tool in the analysis of System (1.1) with small delay is a class of ``special solutions'' introduced by Ryabov [15] and investigated by several authors (see, e.g., [1, 4, 6, 10, 14]). In the following theorems we summarize some known results in the form close to that given by Driver [4]. Theorem 1.1 [4, Theorem 3]. If Kre&r.

Thus, Var [&r, 0] '(t, } )= |A| and the smallness condition (1.4) has the form |A| re