Oscillations of Differential-Difference Systems of Neutral Type

Oscillations of Differential-Difference Systems of Neutral Type

Journal of Mathematical Analysis and Applications 253, 274᎐289 Ž2001. doi:10.1006rjmaa.2000.7113, available online at http:rrwww.idealibrary.com on O...

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Journal of Mathematical Analysis and Applications 253, 274᎐289 Ž2001. doi:10.1006rjmaa.2000.7113, available online at http:rrwww.idealibrary.com on

Oscillations of Differential-Difference Systems of Neutral Type Jose ´ M. Ferreira1 Departamento de Matematica, Instituto Superior Tecnico, A¨ . Ro¨ isco Pais, ´ ´ 1096 Lisboa-codex, Portugal

and Ana M. Pedro Departamento de Matematica, Faculdade de Ciencias e Tecnologia, Uni¨ ersidade No¨ a ´ ˆ de Lisboa, Quinta da Torre, 2825 Monte de Caparica, Portugal Submitted by George Leitmann Received July 6, 1999

This note is concerned with the oscillatory behavior of a linear differentialdifference system of neutral type. Criteria for oscillations are obtained for autonomous and nonautonomous systems. 䊚 2001 Academic Press

1. INTRODUCTION The aim of this note is to obtain new explicit oscillation criteria of the linear delay differential-difference system of neutral type d dt

m

xŽ t. y

m

Ý Bj x Ž t y ␶ j . js1

s

Ý A j x Ž t y rj . ,

Ž 1.

js1

where x Ž t . g ⺢ n, the A j and Bj Ž j s 1, . . . , m. are real n-by-n matrices, and the ␶ j and r j Ž j s 1, . . . , m. are positive real numbers such that r 1 - ⭈⭈⭈ - rm and ␶ 1 - ⭈⭈⭈ - ␶m . 1 Partially supported by FCT ŽPortugal. under Projects PRAXIS XXI Project 2r2.1r MATr199r94 and PRAXISrPCEXrPrMATr36r96.

274 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

DIFFERENTIAL-DIFFERENCE SYSTEMS

275

As usual in many papers of delay equations, the analysis of the oscillatory behavior of solutions is based upon the existence or absence of real zeros of the corresponding characteristic equation. For neutral scalar equations that analysis has been made by many authors Žsee w10x and references therein . always restricted to the case where the corresponding first member of Ž1. has a single delay. For functional systems and, in particular, to a system like Ž1., Kirchner and Stroinski w6x get explicit oscillation conditions with no restriction on the number of delays. A similar study relative to an N-order differential system with a single delay in the left-hand member of Ž1. is made by Erbe and Kong w2x Žsee also w3x.. In both papers is excluded the possibility of the Bj Ž j s 1, . . . , m. having eigenvalues with negative real part. This aspect will not have any relevance in the criteria given here. More recently Stroinski w9x obtained a result which for systems is not very easy to apply, but that for scalar equations seems to be rather helpful. For systems there isn’t yet an uniformity in the literature, regarding the definition of an oscillatory solution. For that purpose we will follow the book by Gyori ¨ and Ladas w5x: with R s max rm , ␶m 4, a solution x: wyR, q⬁wª ⺢ n of Ž1., x Ž t . s w x 1Ž t ., . . . , x nŽ t .xT , is said to be oscillatory componentwise if every component, x i Ž t ., i s 1, . . . , n, has arbitrary large zeros. Whenever all solutions of the system Ž1. are oscillatory componentwise we will say that Ž1. is totally oscillatory componentwise. In Section 3 we discuss the nonautonomous case of system Ž1. and, by the reasons pointed out in w5x, a weaker concept of an oscillatory solution is adopted. In Section 2 we investigate when the system Ž1. is totally oscillatory componentwise. We will provide some spectral bound conditions on the matrices A j , Bj Ž j s 1, . . . , m., involving the delays r j and ␶ j Ž j s 1, . . . , m., which imply that Ž1. be totally oscillatory componentwise. We will make use of the matrix measures, of a matrix, already introduced on this subject by several authors. For a matter of completeness we recall in the following its definition and main properties. Denoting by ⺢ n= n the Banach space of all n-by-n real matrices, for a given norm 5 ⭈ 5 in ⺢ n= n we define a corresponding matrix measure ␮ : ⺢ n= n ª ⺢ as

␮ Ž C . s limq ␪ª0

5 I q ␪C5 y 1



,

where C is a matrix in ⺢ n= n and I is the identity matrix. We will also take into account the value ␯ Ž C . defined as ␯ Ž C . s y␮ŽyC .. Independently of the considered norm, a matrix measure has always the following properties Žsee w1x.:

276

FERREIRA AND PEDRO

ŽI. y5 C 5 F ␮ Ž C . F 5 C 5; ŽII. ␮ Ž C1 q C2 . F ␮ Ž C1 . q ␮ Ž C2 . Ž C1 , C2 g ⺢ n= n .; ŽIII. ␮ Ž␥ C . s ␥␮ Ž C ., for every ␥ G 0; ŽIV. ␮ Ž␥ C . s ␥␯ Ž C ., for every ␥ F 0; ŽV. ␯ Ž C . F 5 Cy1 5y1 , if C is nonsingular. Denoting by ␴ Ž C . the spectrum of any matrix C, and introducing the upper and lower bounds of the set Re ␴ Ž C . s  Re ␭: ␭ g ␴ Ž C .4 , which are given by s Ž C . s max  Re z : z g ␴ Ž C . 4

l Ž C . s min  Re z : z g ␴ Ž C . 4 ,

and

we have:

␯ Ž C . F lŽ C . F sŽ C . F ␮Ž C . .

Ž VI .

2. TOTALLY OSCILLATORY BEHAVIOR In order to obtain a sufficient condition which assures that Ž1. is totally oscillatory we first state the following lemmas. LEMMA 1.

For pj , q j , gx0, q⬁w Ž j s 1, . . . , m., let m

F Ž ␭. s ␭ q ␭

Ý pj ey␭␶

m

j

q

js1

Ý q j ey ␭ r . j

js1

If rm ) ␶m , and m

Ý

m

pj q 1 F e

js1

Ý

q j Ž r j y ␶m . ,

Ž 2.

jsm 0

where m 0 G 1 is such that r j ) ␶m for j s m 0 , . . . , m, and ␶m G r j for j s 1, . . . , m 0 y 1, then F Ž ␭. has no negati¨ e real zeros. Proof. Taking the functions

␾ Ž ␭ . s ey␭␶ m q

m

Ý p j e ␭ Ž␶ y ␶ j

m.

and

js1

␺ Ž ␭. s

1



m

Ý q j e ␭Ž r y ␶ j

m.

,

js1

F Ž ␭. has no real negative zero if and only if the equation

␾ Ž ␭. s ␺ Ž ␭. has no real positive root.

Ž 3.

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277

For j s 1, . . . , m 0 y 1, we have inf

qj

½␭

e ␭Ž r jy ␶ m . : ␭ ) 0 s 0,

5

while for j s m 0 , . . . , m, min

qj

½␭

e ␭Ž r jy ␶ m . : ␭ ) 0 s q j Ž r j y ␶m . e,

5

being this minimum attained at ␭ s 1rŽ r j y ␶m .. Therefore assuming that Ž3. has a real positive root ␭0 , we have m

e

Ý

m

q j Ž r j y ␶m . F ␺ Ž ␭ 0 . s ␾ Ž ␭ 0 . - ␾ Ž 0 . s 1 q

jsm 0

Ý pj , js1

since ␾ Ž ␭. is strictly decreasing on x0, q⬁w. This fact is in contradiction with the condition Ž2.. Remark 1. Notice that if rm F ␶m , the function ␺ Ž ␭. above is such that ␺ Ž0q. s q⬁ and ␺ Žq⬁. s 0. On the other hand for the function ␾ Ž ␭. we have ␾ Žq⬁. s pm ) 0. So in such a case we can conclude that F Ž ␭. has at least a positive real root. LEMMA 2.

For pj , q j , gx0, q⬁w Ž j s 1, . . . , m., let m

F Ž ␭. s ␭ y ␭

pj ey␭␶ j q

Ý js1

m

Ý q j ey ␭ r . j

js1

F Ž ␭. has no real positi¨ e zeros pro¨ iding that one of the following assumptions is satisfied: m

Ý pj F 1,

Ž 4.

js1 m

Ý js1

pj

␶j

-

e m

m

Ý js1

1 rj

log Ž mq j r j e . .

Proof. Assume that F Ž ␭ 0 . s 0 for some real ␭0 ) 0.

Ž 5.

278

FERREIRA AND PEDRO

Relatively to Ž4., if Ý mjs1 pj F 1 then Ý mjs1 pj ey ␭ 0␶ j - 1 and so m

Ý q j ey␭ ␭0 s y

0 rj

js1 m

Ý pj ey ␭ a

1y

0

- 0, j

js1

which is contradictory. With respect to Ž5., let’s consider the functions m

f Ž ␭. s ␭ q

Ý q j ey␭ r

m

g Ž ␭. s ␭

and

j

js1

Ý pj ey ␭␶ . j

js1

The function f Ž ␭. can be rewritten as m

f Ž ␭. s

Ý js1

ž

1 m

␭ q q j ey␭ r j .

/

For each j s 1, . . . , m, we have min

½

1 m

␭ q q j ey␭ r j : ␭ g ⺢ s

5

1 mr j

log Ž mq j r j e . ,

being this minimum attained at ␭ s Ž1rr j .logŽ mq j r j .. On the other hand, for j s 1, . . . , m, we have max  ␭ pj ey ␭␶ j : ␭ ) 0 4 s

pj

␶j e

,

being this maximum attained at ␭ s 1r␶ j . Therefore we have necessarily m

Ý js1

1 mr j

log Ž mq j r j e . F f Ž ␭0 . s g Ž ␭0 . F

1 e

m

Ý js1

pj

␶j

,

which is in contradiction with Ž5.. For the matrix m

AŽ ␭. s ␭ I y

Ý ey␭ r A j j

js1

we state the following characteristics.

Ž ␭ g ⺢. ,

Ž 6.

DIFFERENTIAL-DIFFERENCE SYSTEMS

279

LEMMA 3. For any gi¨ en norm, 5 ⭈ 5, and the corresponding matrix measure, ␮ , assume that ␮ Ž A j . F 0, for j s 1, . . . , m, and m

Ý r j ␮ Ž A j . - y1re. js1

Then for e¨ ery real ␭, the matrix AŽ ␭. is nonsingular and 5 A Ž ␭ . y1 5 F y

1

.

␮ Ž yA Ž ␭ . .

Proof. The assumptions on the matrices A j , for j s 1, . . . , m, imply that Žsee w7x. det A Ž ␭ . / 0,

for every real ␭ .

On the other hand, by ŽIV. and ŽV. we have, for every real ␭, y␮ Ž yA Ž ␭ . . F

1

.

5 A Ž ␭ . y1 5

So we only have to show that y␮ŽyAŽ ␭.. is positive, for every real ␭. For that purpose, since by ŽII., ŽIII., and the fact that ␮ Ž"I . s "1, m

Ý ␮ Ž A j . ey␭ r ,

␮ Ž yA Ž ␭ . . F y␭ q

j

js1

we will prove further that m

Ý ␮ Ž A j . ey␭ r ) 0,

h Ž ␭. s ␭ y

for every real ␭ .

j

js1

In fact suppose that hŽ ␭. assumes nonpositive values. Then, as hŽ0. G 0, there will be a real value ␭0 such that hŽ ␭0 . s 0. Therefore m

␭0 m

žÝ

js1

␮Ž Aj .

Ý ␮ Ž A j . ey␭ s

js1

␮Ž Aj .

js1



G exp y␭0

m

Ý js1

␮Ž Aj .

m

s

m

/ žÝ

0 rj

Ý js1

/

žÝ

␮Ž Aj .

js1

␮Ž Aj .

js1

␮Ž Aj .

0

rj ,

m

žÝ

ey ␭ 0 r j

m

/

/

280

FERREIRA AND PEDRO

by the convexity of the function u ¬ ey␭ 0 u. Through the multiplication of both members of this inequality by Ý mjs1 r j ␮ Ž A j . - 0, we obtain that

rj ␮ Ž A j .

m

␭0

Ý js1

m

ž

Ý ␮Ž Aj . js1



/

␮ Ž A j . rj

m

exp ␭0

Ý js1

m

ž

Ý ␮Ž Aj . js1

/

0

m

F

Ý rj ␮ Ž A j . . js1

However, the function u ¬ ue u has an unique minimum attained at u s y1, and so the left hand member of this inequality is larger or equal to yey1 . This means that then

y

1 e

m

F

Ý rj ␮ Ž A j . , js1

and a contradiction is obtained. Hence hŽ ␭. ) 0, for every real ␭, which completes the proof of the lemma. Now jointly with the matrix AŽ ␭. defined in Ž6., we shall consider the matrix m

B Ž ␭. s ␭

Ý ey␭␶ Bj j

Ž ␭ g ⺢. .

js1

According to w5, Chap. 6x, all bounded solutions of system Ž1. are oscillatory componentwise if and only if the equation det A Ž ␭ . y B Ž ␭ . s 0,

Ž 7.

has no roots on x y ⬁, 0x. Moreover system Ž1. is totally oscillatory componentwise if and only if Eq. Ž7. has no real roots. As a consequence of these characterizations on the oscillatory behavior of Ž1., we state the following theorems. THEOREM 1.

Let rm ) ␶m and m 0 F m be the lowest integer such that r j ) ␶m ,

for j s m 0 , . . . , m.

Ž 8.

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281

If m

␮ Ž A j . F 0 Ž j s 1, . . . , m . , and 1 q

Ý

m

5 Bj 5 F e

js1

Ý

< ␮ Ž A j . < Ž r j y ␶m . ,

jsm 0

Ž 9. then all bounded solutions of Ž1. are oscillatory componentwise. Proof. Since ␮ Ž A j . F 0, for j s 1, . . . , m, then m m

Ý ␮ Ž A j . Ž r j y ␶m .

m

Ý rj ␮ Ž A j . F Ý js1

jsm 0

rj ␮ Ž A j . -

m

1q

jsm 0

Ý

5 Bj 5

Fy

1 e

.

js1

Therefore by Lemma 3, AŽ ␭. is nonsingular, for every real ␭, and by consequence all bounded solutions of Ž1. are oscillatory if and only if det I y A Ž ␭ .

y1

B Ž ␭. / 0

for every ␭ gx y ⬁, 0x, that is, if and only if 1 f ␴ Ž AŽ ␭.

y1

for every ␭ g x y ⬁, 0 x .

B Ž ␭. .

By properties ŽI. and ŽVI., this occurs, in particular, whenever 5 A Ž ␭ . y1 B Ž ␭ . 5 - 1,

Ž 10 .

for every ␭ gx y ⬁, 0x. On the other hand we have, for every real ␭, 5 A Ž ␭ . y1 5 F y

1

␮ Ž yA Ž ␭ . .

,

still be Lemma 3, where through the analysis made in its proof, in view of the condition Ý mjs1 r j ␮ Ž A j . - y1re, one has m

y␮ Ž yA Ž ␭ . . G ␭ y

Ý ␮ Ž A j . ey␭ r ) 0, j

js1

282

FERREIRA AND PEDRO

for every real ␭. Thus, for every real ␭, we have 5 A Ž ␭ . y1 B Ž ␭ . 5 F 5 A Ž ␭ . y1 5 5 B Ž ␭ . 5 F < ␭< F

y␮ Ž yA Ž ␭ . .

m

Ý 5 Bj 5 ey␭␶

j

js1 m

␭y

5 B Ž ␭. 5

,

Ý ␮ Ž A j . ey␭ r

j

js1

and condition Ž10. will be fulfilled if, for every ␭ gx y ⬁, 0x, one has

␭ y < ␭<

m

Ý 5 Bj 5 ey␭␶

m

j

y

js1

Ý ␮ Ž A j . ey ␭ r ) 0.

Ž 11 .

j

js1

Considering the function m

G Ž ␭. s ␭ q ␭

Ý 5 Bj 5 ey␭␶

m

j

y

js1

Ý ␮ Ž A j . ey ␭ r , j

js1

on x y ⬁, 0x, as G Ž0. ) 0, all bounded solutions of Ž1. will be oscillatory provided that G Ž ␭. has no negative real zeros. But regarding the function F Ž ␭. of the Lemma 1, corresponding to pj s 5 Bj 5

and

qj s < ␮ Ž A j . <,

for j s 1, . . . , m, we have G Ž ␭. s F Ž ␭. . Hence the conclusion of the theorem follows directly from Lemma 1. Remark 2. As was noticed in the preceding proof, condition Ž9. implies that Ý mjs1 r j ␮ Ž A j . - y1re. This means that, implicitly, we are requiring the totally oscillatory behavior of the differential-difference system dx Ž t . dt

m

s

Ý A j x Ž t y rj . . js1

DIFFERENTIAL-DIFFERENCE SYSTEMS

283

THEOREM 2. Let the matrices A j , Bj Ž j s 1, . . . , m. and the delays r j , ␶ j , satisfy Ž8., Ž9. and at least one of the conditions m

Ý 5 Bj 5 F 1,

Ž 12 .

js1 m

Ý

5 Bj 5

js1

␶j

-

e m

m

Ý js1

1 rj

log Ž m < ␮ Ž A j . < r j e . .

Ž 13 .

Then Ž1. is totally oscillatory componentwise. Proof. Proceeding as in the proof of the theorem before, replacing ␭ gx y ⬁, 0x by ␭ g ⺢, we obtain, analogously, that system Ž1. is totally oscillatory if the inequality Ž11. is satisfied for every real ␭. To complete the proof it remains to analyze the case where ␭ ) 0. Taking now m

G Ž ␭. s ␭ y ␭

5 Bj 5 ey␭␶ j y

Ý js1

m

Ý ␮ Ž A j . ey ␭ r ) 0, j

js1

Ž11. holds for every ␭ ) 0, if GŽ ␭. has no positive real zeros. This corresponds to having in Lemma 2 G Ž ␭. s F Ž ␭. , for pj s 5 Bj 5

and

q j s < ␮ Ž A j . < Ž j s 1, . . . , m . .

Then by that lemma the theorem follows. The application of these results requires a previous formulation of the norm and of the corresponding matrix measure in ⺢ n= n. As is well known, a given norm, < ⭈ <, in ⺢ n, induces a norm in ⺢ n=n through the relation 5 C 5 s max  < Cu < : < u < s 1 4

Ž C g ⺢ n= n . .

The norms in ⺢ n, < u<1 s

n

Ý < ui < js1

and

< u < ⬁ s max  < u i < : i s 1, . . . , n4

284

FERREIRA AND PEDRO

Ž u ' w u1 , . . . , u n xT . induce, respectively, the following norms in ⺢ n= n, 5 C 5 1 s max

n

Ý < ci k < :

½ ½Ý

5 C 5 ⬁ s max

n

5 5

k s 1, . . . , n ,

is1

< c i k < : i s 1, . . . , n

ks1

Ž C ' w c i k x g ⺢ n= n .. For these norms in ⺢ n= n, the corresponding matrix measures are, respectively, given by

½ Ý < c <: k s 1, . . . , n5 , ␮ Ž C . s max ½ c q Ý < c < : i s 1, . . . , n 5 . ␮ 1 Ž C . s max c k k q

ik

i/k



ii

ik

k/i

For the euclidean norm in ⺢ n, < u < 2 s 'uT u Ž u ' w u1 , . . . , u n xT ., the corresponding induced norm in ⺢ n= n, is given by 5 C 5 2 s s Ž C T C . and the matrix measure associated to this norm is explicitly given by ␮ 2 Ž C . s sŽ 12 Ž C q C T ... For the norms introduced above, Theorem 2 is illustrated in the following example.

'

EXAMPLE 1. Consider the system d dt

x Ž t . y B1 x Ž t y 1 . y B2 x Ž t y 3 . s A1 x Ž t y 2 . q A 2 x Ž t y 4 . ,

Ž 14 . where B1 s A1 s

y 4e 0 y2 e 3

e 4

1

,

y1 y2 ,

B2 s

y

A2 s

1 2 e 4

y

1 3 e 4

,

ye

e 3

y 21

ye

.

With respect to the three norms above, 5 ⭈ 5 1 and 5 ⭈ 5 ⬁ are in this case of easiest application. For the latter one we have 5 B1 5 ⬁ s 5 B2 5 ⬁ s 2e , ␮⬁Ž A1 . s y1, and ␮⬁Ž A 2 . s y 23e . Conditions Ž8., Ž9., and Ž13. of Theorem 2 are easily satisfied and so Ž14. is totally oscillatory componentwise. Notice that since the matrix B1 has a negative eigenvalue, it is not possible to apply w6x to this system.

DIFFERENTIAL-DIFFERENCE SYSTEMS

285

Remark 3. The preceding theorems can obviously be applied to the scalar case of Ž1. corresponding to having A j , Bj g ⺢ Ž j s 1, . . . , m.. For that purpose just observe that < ⭈ < 1 s < ⭈ < 2 s < ⭈ < ⬁ s 5 ⭈ 5 1 s 5 ⭈ 5 2 s 5 ⭈ 5 ⬁ s < ⭈ <, the usual absolute value of a real number, and that then ␮ 1Ž A j . s ␮ 2 Ž A j . s ␮⬁Ž A j . s A j Ž j s 1, . . . , m.. The interest of Theorem 2 in this case is expressed in the following example. EXAMPLE 2. By Theorem 2 one easily concludes that the equation d dt

xŽ t. y

1 2

x ty

ž

1 2

/

q

1 2

x Ž t y 1 . s ax Ž t y 1 . q bx Ž t y 2 . ,

is totally oscillatory for every a F 0 and b F y2 ey1 . As before w6x cannot be applied to this equation. The same happens to w9, Theorem 4.5x whenever yey1 - a F 0. However, when a F yey1, through w9, Theorem 4.5x the parameter b can run in a larger range: the interval x y ⬁, yey1 x.

3. NONAUTONOMOUS SYSTEMS Now we shall consider the nonautonomous system d dt

m

xŽ t. y

Ý Bj Ž t . x Ž t y ␶ j . js1

m

s

Ý A j Ž t . x Ž t y rj . ,

Ž 15 .

js1

where the A j Ž t . and the Bj Ž t . Ž j s 1, . . . , m. are continuous functions in w0, q⬁w with values in ⺢ n= n and the delays ␶ j and r j Ž j s 1, . . . , m. are as before: 0 - r 1 - ⭈⭈⭈ - rm , 0 - ␶ 1 - ⭈⭈⭈ - ␶m . Following w5x, define the value sgn x, for any x g ⺢, by sgn x s 0, if x s 0, sgn x s 1, if x ) 0, and sgn x s y1, if x - 0; a solution x: wyR, q⬁wª ⺢ n Ž R s max rm , ␶m 4. of Ž15., x Ž t . s w x 1Ž t ., . . . , x nŽ t .xT , will be said to be oscillatory if it is eventually trivial or if at least one of its components, x i Ž t ., i g  1, . . . , n4 , is such that sgn x i Ž t . is not eventually constant. Whenever all solutions of the system Ž1. are oscillatory, we will say that Ž15. is totally oscillatory. This concept coincides for autonomous systems like Ž1. and scalar equations, with the one given in Section 1 of oscillatory componentwise. But for nonautonomous systems the latter seems to be stronger. Through a comparison method with a family of totally oscillatory autonomous systems introduced in w8x, we will obtain a sufficient condition for Ž15. to be totally oscillatory, which extends and corrects w4, Theorem 8x. For that purpose, given two n-by-n real matrices, D s w d i k x and C s w c i k x, we introduce the notation C F D with the meaning that d i k F c i k , i, k s 1, . . . , n.

286

FERREIRA AND PEDRO

THEOREM 3. Let, for each j g  1, . . . , m4 , A j Ž t . and Bj Ž t . be bounded matrix ¨ alued functions on w0, q⬁w and denote by A j , Bj , A j and Bj , n-by-n real matrices such that Aj F Aj Ž t . F Aj ,

Bj F Bj Ž t . F Bj ,

for e¨ ery t G 0. If, for A j F A j F A j and Bj F Bj F Bj , j s 1, . . . , m, each system d dt

m

xŽ t. y

m

Ý Bj x Ž t y ␶ j .

s

js1

Ý A j x Ž t y rj . , js1

is totally oscillatory Ž componentwise., then Ž15. is totally oscillatory. Proof. Assume that Ž15. is not totally oscillatory. Then it has a solution noneventually trivial, x Ž t . s w x 1Ž t ., . . . , x nŽ t .xT , such that, for some real value t 0 G R and all i g  1, . . . , n4 , sgn x i Ž t . is constant for every t G t 0 . With no loss of generality we may assume that for some i 0 g  1, . . . , n4 one has x i 0Ž t . ) 0 for every t G t 0 . The boundedness of the matrix valued functions A j Ž t . and Bj Ž t . Ž j s 1, . . . , m. implies the existence of a real ␻ such that, for any norm < ⭈ < in ⺢ n, t G 0 and j s 1, . . . , m, we have < x Ž t .< s O Ž e ␻ t ., < Bj Ž t . x Ž t y r j .< s O Ž e ␻ t ., and < A j Ž t . x Ž t y r j .< s O Ž e ␻ t .. Then we can conclude that both m

Ý Bj Ž t . x Ž t y ␶ j .

uŽ t . s x Ž t . y

js1

and its derivative u⬘Ž t . are such that < uŽ t .< s O Ž e ␻ t ., < u⬘Ž t .< s O Ž e ␻ t . for t G 0. Therefore the Laplace transforms of x Ž t ., uŽ t ., and u⬘Ž t . are all well defined for every real ␭ ) ␻ . Moreover for the Laplace transforms of x Ž t . and uŽ t ., X Ž ␭. s

q⬁

Ht

ey␭ t x Ž t . dt ,

U Ž ␭. s

0

q⬁

Ht

ey ␭ t u Ž t . dt ,

0

respectively, we have for every real ␭ ) ␻ , m

␭U Ž ␭ . y u Ž t 0 . s ␭ X Ž ␭ . y ␭ m

y x Ž t0 . q

q⬁

ÝH

js1 t 0

ey␭ t Bj Ž t . x Ž t y ␶ j . dt y m

q⬁

Ý Bj Ž t 0 . x Ž t 0 y ␶ j . s Ý H

js1

js1 t 0

ey␭ tA j Ž t . x Ž t y r j . dt.

DIFFERENTIAL-DIFFERENCE SYSTEMS

287

For j s 1, . . . , m, and every real ␭ ) ␻ , each one of the n components of the vectors q⬁

Ht

q⬁

ey␭ tA j Ž t . x Ž t y r j . dt ,

Ht

0

ey ␭ t Bj Ž t . x Ž t y ␶ j . dt ,

0

is sum of n terms like q⬁

Ht

c Ž t . ey␭ t x i Ž t y ␥ j . dt ,

0

where i s 1, . . . , n, ␥ j s r j or ␶ j , and cŽ t . denotes, respectively, an entry of the matrix function A j Ž t . or Bj Ž t .. Since sgnŽ ey␭ t x i Ž t y r j .. is constant for every t G t 0 , by an extension of the first mean value theorem, there exists a real number c such that q⬁

Ht

q⬁

c Ž t . ey␭ t x i Ž t y ␥ j . dt s c

Ht

0

ey ␭ t x i Ž t y ␥ j . dt.

0

Thus for j s 1, . . . , m, there exist n-by-n real matrices, A j , Bj , such that A j F A j F A j , Bj F Bj F Bj , and m

q⬁

ÝH

js1 t 0 m

ey␭ t Bj Ž t . x Ž t y ␶ j . dt s

m js1

q⬁

ÝH

js1 t 0

ey␭ tA j Ž t . x Ž t y r j . dt s

q⬁

Ý B jH

t0

m

ey ␭ t x Ž t y ␶ j . dt

q⬁

Ý A jH

js1

t0

ey ␭ t x Ž t y r j . dt.

Then through a change of variables, we obtain, for every real ␭ ) ␻ , that ⌬Ž ␭. X Ž ␭. s ␸ Ž ␭., where m

⌬ Ž ␭. s ␭ I y

ž

Ý Bj ey␭␶ js1

m

␸ Ž ␭. s ␭

t0

Ý Bj ey␭␶ H j

js1

m

y

t 0y ␶ j

m

j

/

y

Ý A j ey ␭ r , j

js1

ey ␭ t x Ž t . dt q x Ž t 0 . m

t0

Ý Bj Ž t 0 . x Ž t 0 y ␶ j . q Ý A j ey␭ r H j

js1

js1

t 0yr j

ey ␭ t x Ž t . dt.

Since det ⌬Ž ␭. / 0, for every real ␭, by making ⌿ Ž ␭. s ⌬Ž ␭.y1␸ Ž ␭., we have for every real ␭ ) ␻ , X Ž ␭. s ⌿ Ž ␭..

288

FERREIRA AND PEDRO

Taking ⌬Ž ␭.y1 s Adj ⌬Ž ␭.rdet ⌬Ž ␭., notice that det ⌬Ž ␭. can be written as an exponential polynomial of the type N

P Ž ␭. y

Ý Q j Ž ␭. ey␭ m ⭈␳ ,

Ž 16 .

j

js1

where P Ž ␭. is a polynomial having ␭ n as a higher order term, the Q j Ž ␭. are polynomials with degree less than or equal to n, the m j are nonzero 2 m-tuples of nonnegative integers, and m j ⭈ ␳ means the inner product of m j with ␳ ' Ž␶ 1 , . . . , ␶m , r 1 , . . . , rm .. Since det ⌬Ž ␭. ª q⬁ as ␭ ª q⬁, one must have then det ⌬Ž ␭. ª q⬁ as ␭ ª y⬁. This enables us to conclude the existence of some ␣ F 0 such that 1rdet ⌬Ž ␭. F 1, for every ␭ F ␣ . On the other hand each term of the adjugate matrix, Adj ⌬Ž ␭., of ⌬Ž ␭., is also an exponential polynomial like Ž16. where now the polynomials P Ž ␭. and Q j Ž ␭. have independent degrees. Thus we can conclude the existence of ␻ 1 , ␻ 2 gx0, q⬁w such that 5Adj ⌬Ž ␭.5 s O Ž ey␭ ␻ 1 . Ž ␭ g x y ⬁, 0x., for any norm 5 ⭈ 5 of ⺢ n= n, and < ␸ Ž ␭.< s O Ž ey␭ ␻ 2 . Ž ␭ gx y ⬁, 0x., for every norm < ⭈ < in ⺢ n. Thus ⌿ Ž ␭. is an analytic function all over the real line, which equals X Ž ␭. at least for ␭ gx ␻ , q⬁w and such that for ␻ 0 s ␻ 1 q ␻ 2 , e ␭␻ 0 ⌿ Ž ␭. is a bounded function on x y ⬁, ␣ x. Considering the Laplace transform, X i 0Ž ␭., of the component function Ž x i 0 t . of x Ž t ., its abscissa of convergence, y⬁ F ␥ i 0 F ␻ , and the i 0-component ⌿i 0Ž ␭. of ⌿ Ž ␭. one has X i 0Ž ␭. s ⌿i 0Ž ␭., for every real ␭ gx␥ i 0 , q⬁w, and proceeding as in w8x one obtains a contradiction. EXAMPLE 3. Let d dt

x Ž t . y B Ž t . x Ž t y 1. s AŽ t . x Ž t y 2. ,

where AŽ t . s

y2 1 2

y t q1 1 y3

and

BŽ t. s

y1 y1

With respect to the theorem before, we have As Bs

y2 1 2

y1 , y3

As

y1 y1

1 , y 32

Bs

y2 1 2

y1 y1

0 , y3 1 , 0

1 . y 32 eyt

DIFFERENTIAL-DIFFERENCE SYSTEMS

289

and for y1 F a F 0, y 32 F b F 0, As

y2

a , y3

1 2

Bs

and

y1 y1

1 , b

it is 3 3 ␮ 1 Ž A . s max y , y3 y a s y 2 2

½

5

5 B 5 1 s max  2, 1 y b 4 F

5 2

.

Therefore 1 q 5 B 51 F 5 B 51 F

7 2 5 2

-

3 2 e 2

e s e < ␮1Ž A . < Ž r y ␶ . , log Ž 3e . s

␶e r

log Ž re < ␮ 1 Ž A . < . ,

and Theorem 2 enables us to conclude that the system above is totally oscillatory.

REFERENCES 1. C. A. Desoer and M. Vidyasagar, ‘‘Feedback Systems: Input-Output Properties,’’ Academic Press, San Diego, 1975. 2. L. Erbe and Q. Kong, Explicit conditions for oscillation of neutral differential systems, Hiroshima Math. J. 24 Ž1994., 317᎐329. 3. L. Erbe, Q. Kong, and B. G. Zhang, ‘‘Oscillation Theory for Functional Differential Equations,’’ Dekker, New York, 1995. 4. J. M. Ferreira and A. Pedro, Oscillations of delay difference systems, J. Math. Anal. Appl. 221 Ž1998., 364᎐383. 5. I. Gyori ¨ and G. Ladas, ‘‘Oscillation Theory of Delay Differential Equations,’’ Oxford Univ. Press, London, 1991. 6. J. Kirchner and U. Stroinski, Explicit oscillation criteria for systems of neutral differential equations with distributed delay, Differential Equations Dynam. Systems 3 Ž1995., 101᎐120. 7. Q. Kong, Oscillation for systems of functional differential equations, J. Math. Anal. Appl. 198 Ž1996., 608᎐619. 8. Q. Kong and H. I. Freedman, Oscillations in delay differential systems, Differential Integral Equations 6 Ž1993., 1325᎐1336. 9. U. Stroinski, Order and oscillation in delay differential systems, J. Math. Anal. Appl. 207 Ž1997., 158᎐171. 10. J. Ziwen, Oscillation of first order neutral delay differential equations, J. Math. Anal. Appl. 196 Ž1995., 800᎐813.