JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
204, 701]728 Ž1996.
0463
Asymptotically Diagonal Delay Differential Systems O. Arino Laboratoire de Mathematiques Appliquees, ´ ´ Uni¨ ersite´ de Pau et des Pays de l’Adour, I.P.R.A., A¨ enue de l’Uni¨ ersiste, ´ 64000 Pau, France
and I. Gyori ˝ and M. Pituk Department of Mathematics and Computing, Uni¨ ersity of Veszprem, ´ 8201 Veszprem, ´ P.O. Box 158, Hungary Submitted by Gerry Ladas Received October 2, 1995
1. INTRODUCTION A recent monograph by Eastham w8x contains a survey of the results on the asymptotic integration of ordinary differential equations. One of the classical results is the following theorem. THEOREM A ŽThe Hartman]Wintner Theorem.. Consider the system x9 s Ž D Ž t . . q R Ž t . . x,
Ž 1.1.
where D, R are continuous n = n matrix functions on w0, `.. Suppose that DŽ t . is a diagonal matrix, D Ž t . s diag d1 Ž t . , d 2 Ž t . , . . . , d n Ž t . 4 , and suppose there exists a d ) 0 such that di Ž t . y d j Ž t . G d
Ž 1.2.
on some inter¨ al w t 0 , `. for each i / j, i, j g 1, 2, . . . , n4 . Let Rg Lp
for some p g Ž 1, 2 .
Ž 1.3.
701 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
702
ARINO, GYORI ˝ , AND PITUK
Then Eq. Ž1.1. has a fundamental system of the form X Ž t . s Ž I q F Ž t . . exp
t
žH Ž t0
D Ž s . q diag R Ž s . 4 . ds
as t ª `,
/
Ž 1.4. where I denotes the identity matrix and F is a matrix function, F Ž t . ª 0 as t ª `. In an effort to extend the Hartman]Wintner theorem to delay differential systems Haddock and Sacker w11x conjectured an asymptotic formula for the solutions of the equation x9 Ž t . s Ž A˜ q A Ž t . . x Ž t . q B Ž t . x Ž t y t . ,
Ž 1.5.
where t ) 0, A˜ s diag a ˜1 , a˜2 , . . . , a˜n4 with a˜i / a˜j for i / j and A, B are L2-matrix functions. They stated that there exists a matrix function F, F Ž t . ª 0 as t ª `, such that for each solution x of Ž1.5. there exist a constant vector c and a vector function f, f Ž t . ª 0 as t ª `, such that x Ž t . s Ž I q F Ž t . . exp
t
žH
t0
L Ž s . ds
/Ž
c q f Ž t.. ,
Ž 1.6.
where ˜
L Ž t . s A˜ q diag A Ž t . 4 q diag B Ž t . 4 eyA t . Haddock and Sacker w11x proved formula Ž1.6. in the scalar case Žwith F ' 0.. In the case of ‘‘quasi-triangular’’ systems formula Ž1.6. was proved by the first two authors w2x. A complete proof of the conjecture was given by Ai w1x. For further extension allowing A, B g L p with p ) 2, we refer to the work of Cassel and Hou w4x. In the above-mentioned works the delay differential equation is considered as a perturbation of an Žautonomous. ordinary differential equation. The situation becomes more complicated when the limiting equation is a delay differential equation Žsee w3, 5, 12x.. A first result of this type concerning the scalar equation x9 Ž t . s Ž a ˜ q aŽ t . . x Ž t . q Ž ˜b q b Ž t . . x Ž t y t .
Ž 1.7.
Žt ) 0, a, ˜ ˜b s const., aŽ t ., bŽ t . ‘‘small’’. can be found in the classical book by Bellman and Cooke w3x. Assuming that the characteristic equation
l s a˜ q ˜ beylt
Ž 1.8.
703
ASYMPTOTICALLY DIAGONAL DELAY SYSTEMS
of the limiting equation x9 Ž t . s ax ˜ Ž t . q ˜bx Ž t y t . has a dominant real root and under additional technical assumptions, they proved an asymptotic formula for the solutions of Ž1.7. Žformula Ž1.13. below.. Recently, the last two authors established the following improvement of w3, Theorem 9.2x. THEOREM B w10, Theorem 1x.
Assume that in Eq. Ž1.7.,
eya˜t ˜ bt ) y a Ž t . ª 0,
1
,
Ž 1.9.
b Ž t . ª 0 as t ª `,
Ž 1.10.
a g L2 ,
e
b g L2 ,
Ž 1.11.
and aŽ t . y
1
t
H aŽ s . ds g L , t ty t 1
bŽ t . y
1
t
H b Ž s . ds g L . t ty t 1
Ž 1.12.
Then the following statements are ¨ alid.
˜ in the inter¨ al Ž a˜ y Ži. Equation Ž1.8. has a unique real solution l 1rt , `.. Žii. For e¨ ery solution x of Ž1.7. the limit j w x x s lim x Ž t . exp y tª`
ž
t
Ht lŽ s . ds 0
/
Ž 1.13.
exists and is finite, where
˜ q 1 q Ž l˜ y a˜. t lŽ t . s l
ž
y1
˜ ylt
/ Ž aŽ t . q e
bŽ t . . .
Žiii. There exists a solution x of Ž1.7. for which j w x x / 0. We remark that condition Ž1.12. is actually independent of t . A necessary and sufficient condition for a locally integrable function a to satisfy relation Ž1.12. is that it can be written in the form a s a q a , where a is a continuously differentiable function with a 9 g L1 and a g L1. Thus, Ž1.12. is satisfied if a9 g L1 Žmod L1 .. The proof of the above result is given in the Appendix.
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ARINO, GYORI ˝ , AND PITUK
The aim of the present paper is to give an extension of Theorem B to a system of the form x9 Ž t . s Ž A˜ q A Ž t . . x Ž t . q Ž B˜ q B Ž t . . x Ž t y t . ,
Ž 1.14.
˜ B˜ are constant diagonal matrices and A, B are ‘‘small’’ matrix where A, functions. The main theorem concerning Eq. Ž1.14. is formulated in Section 3. The method developed in this paper is new compared to those used in the above-mentioned works on asymptotic integration of systems w1, 2, 4, 11x. Our method is based on previous qualitative results showing that if the delay term is ‘‘small’’ then the solutions of the delay differential system are asymptotic to some member of an n-parameter family of special solutions. ŽResults of this type were initiated by Ryabov w17x. For further extension see w7, 9, 15, 16x.. Roughly speaking, this means that there exist n linearly independent solutions which asymptotically characterize the other solutions. These linearly independent solutions form a fundamental system of a related ordinary differential equation. In general, we do not know the special solutions, therefore it is important to find a way to determine the above-mentioned ordinary differential equation in terms of the coefficients and the delays. This result which is of independent interest is formulated in Theorem 2.4 in Section 2. In Section 3, we use the above result to reduce the problem of asymptotic integration for Eq. Ž1.14. to a similar problem for an ordinary differential equation to which the Hartman]Wintner theorem ŽTheorem A. can be applied. 2. ASYMPTOTIC PROPERTIES OF LINEAR SYSTEMS WITH SMALL DELAYS Let R n denote the n-dimensional space of real vectors with any convenient norm < ?
