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Periodic first order linear neutral delay differential equations
Applied Mathematics and Computation 117 (2001) 203±222 www.elsevier.com/locate/amc
Periodic ®rst order linear neutral delay dierential equations Ch.G. Philos *, I.K. Purnaras Department of Mathematics, University of Ioannina, P.O. Box 1186, 451 10 Ioannina, Greece
Abstract Some new asymptotic and stability results are given for a ®rst order linear neutral delay dierential equation with periodic coecients and constant delays. The asymptotic behavior of the solutions and the stability of the trivial solution are described by the use of an appropriate real root of an equation, which is in a sense the corresponding characteristic equation. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Dierential equation; Neutral delay dierential equation; Periodic coecients; Solution; Asymptotic behavior; Non-oscillation; Stability
1. Introduction A neutral delay dierential equation is a dierential equation in which the highest-order derivative of the unknown function appears both with and without delays. This paper deals with the asymptotic behavior of the solutions and the stability of the trivial solution for a ®rst order linear neutral delay dierential equation with periodic coecients and constant delays, where the coecients have a common period and the delays are multiples of this period. An asymptotic criterion for the solutions is obtained. Moreover, an estimate of the solutions is established and sucient conditions for the stability, the asymp-
*
Tel.: +30-651-98288; fax: +30-651-46361. E-mail address: [email protected] (C.G. Philos).
0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 9 ) 0 0 1 7 4 - 5
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totic stability and the instability of the trivial solution are given. Our results are derived by the use of a real root (with an appropriate property) of the corresponding (in a sense) characteristic equation. The very recent results given by Philos [11] for periodic ®rst order linear (non-neutral) delay dierential equations can be obtained (as a special case) from the results of the present paper. Also, the results given here contain essentially ones obtained very recently by Kordonis et al. [7] for the particular case of autonomous ®rst order linear neutral delay dierential equations. The techniques applied in obtaining our results are originated in a combination of the methods used in [7,11]. The results given in [7,11] extend and improve some signi®cant results on the asymptotic behavior and the stability for a ®rst order linear (non-neutral) delay dierential equation with constant coecients and one constant delay, which have been obtained by Driver et al. [2]. See Driver [1] for some similar asymptotic and stability results for ®rst order linear (non-neutral) delay dierential equations with in®nitely many distributed delays. The results in [7,11] as well as the results given in this paper are motivated by those in [2]. For some related results we refer to the paper by Gy ori [3]. Note that linear delay dierential equations with periodic coecients have been studied (with respect to oscillation) by Philos [9,10] and Zhang [12]; some results (concerning the oscillation) on linear neutral delay dierential equations with periodic coecients have been obtained by Huang and Chen [6] and by Ladas et al. [8]. The theory of neutral delay dierential equations is of both theoretical and practical interest. For a large class of electrical networks containing lossless transmission lines, the describing equations can be reduced to neutral delay dierential equations; such networks arise in high speed computers where nearly lossless transmission lines are used to interconnect switching circuits. Also, neutral delay dierential equations appear in the study of vibrating masses attached to an elastic bar and also as the Euler equation in some variational problems. It must be noted that the theory of neutral delay differential equations presents some additional complications, which are not presented in the theory of the corresponding delay dierential equations. For the basic theory of neutral delay dierential equations, the reader is referred to the books by Hale [4] and by Hale and Verduyn Lunel [5]. Consider the neutral delay dierential equation " E
x t
X i2I
#0 ci x t ÿ ri
a tx t
X
bj tx t ÿ sj ;
j2J
where I and J are initial segments of natural numbers, ci for i 2 I are real numbers, a and bj for j 2 J are continuous real-valued functions on the interval 0; 1, ri for i 2 I are positive real numbers such that ri1 6 ri2 for i1 ; i2 2 I with
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205
i1 6 i2 , and sj for j 2 J are positive real numbers such that sj1 6 sj2 for j1 ; j2 2 J with j1 6 j2 . The functions bj for j 2 J are assumed to be not identically zero on 0; 1. Moreover, it will be supposed that the coecients a and bj for j 2 J are periodic functions with a common period T > 0 and that there exist positive integers li for i 2 I and mj for j 2 J such that ri li T
for i 2 I;
and
sj m j T
for j 2 J :
De®ne r max ri ; i2I
s max sj ; j2J
and r maxfr; sg:
(r, s and r are positive real numbers.) As usual, a continuous real-valued function x de®ned on the interval ÿr; 1 Pis said to be a solution of the dierential equation (E) if the function x t i2I ci x t ÿ ri is continuously dierentiable for t P 0 and x satis®es (E) for all t P 0. In the sequel, by C ÿr; 0; R we will denote the set of all continuous realvalued functions on the interval ÿr; 0. This set is a Banach space endowed with the usual sup-norm k k de®ned by k/k sup j/ tj t2ÿr;0
for / 2 C ÿr; 0; R:
It is well-known (see, for example, Hale [4] or Hale and Verduyn Lunel [5]) that, for any given initial function / 2 C ÿr; 0; R, there exists a unique solution x of the dierential equation (E) which satis®es the initial condition C
x t / t for t 2 ÿr; 0:
We will call this function x the solution of the initial problem (E)±(C) or, more brie¯y, the solution of (E)±(C). Throughout the paper, we will use the notation Z Z 1 T 1 T a t dt; and Bj bj t dt for j 2 J A T 0 T 0 as well as the notation Z 1 T b jbj tj dt Bj T 0
for j 2 J :
b j > 0 for j 2 J are real constants.) We obviously (Clearly, A, Bj for j 2 J , and B have bj jBj j 6 B
for j 2 J :
b j for j 2 J in the case where each one of the coMoreover, we have jBj j B ecients bj for j 2 J is assumed to be of one sign on the interval 0; 1.
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With the dierential equation (E) we associate the equation ! X X ÿkri ci e Bj eÿksj ; I k 1 A i2I
j2J
which will be called the characteristic equation of (E). This terminology comes from the autonomous case where each one of the coecients a and bj for j 2 J is identically equal to a real constant. In what follows, by a~ and b~j for j 2 J we will denote the T-periodic extensions of the coecients a and bj for j 2 J respectively on the interval ÿr; 1. The notions of the stability, the asymptotic stability and the instability of the trivial solution of the neutral delay dierential equation (E) will be considered in the usual sense. For the sake of completeness, we will give here the de®nitions of these notions. The trivial solution of the dierential equation (E) is said to be stable (at 0) if for each > 0 there exists d d > 0 such that, for any / 2 C ÿr; 0; R with k/k < d, the solution x of (E)±(C) satis®es jx tj < for all t P ÿ r: Otherwise, the trivial solution of (E) is said to be unstable (at 0). Moreover, the trivial solution of (E) is called asymptotically stable (at 0) if it is stable (at 0) in the above sense and, in addition, there exists d0 > 0 such that, for any / 2 C ÿr; 0; R with k/k < d0 , the solution x of (E)±(C) satis®es lim x t 0:
t!1
In the particular case where ci 0 for all i 2 I (and the delays ri for i 2 I are arbitrary), the dierential equation (E) reduces to the (non-neutral) delay dierential equation X x0 t a tx t bj tx t ÿ sj : E0 j2J
As it concerns the (non-neutral) equation (E0 ), since the delays ri , i 2 I can be considered to be arbitrarily small so that r 6 s, we have the number s instead of r. The characteristic equation of (E0 ) is I0
kA
X
Bj eÿksj :
j2J
Before ending this section, let us especially consider the autonomous case, i.e. the special case of the neutral delay dierential equation
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
" E0
x t
X
#0 ci x t ÿ ri
ax t
i2I
X
207
bj x t ÿ sj ;
j2J
where I and J are initial segments of natural numbers and ci 2 R
for i 2 I;
ri 2 0; 1
a 2 R;
for i 2 I;
bj 2 R ÿ f0g
sj 2 0; 1
ri1 6 ri2
for i1 ; i2 2 I with i1 6 i2 ;
sj1 6 sj2
for j1 ; j2 2 J with j1 6 j2 :
for j 2 J ;
for j 2 J ;
The constant coecients a and bj for j 2 J of the diferential equation E0 can be considered to be T-periodic functions, for each real number T > 0. The characteristic equation of E0 is ! X X 0 ÿkri k 1 ci e bj eÿksj : a I i2I
j2J
Note that, as it is well-known, the trivial solution of the autonomous neutral delay dierential equation E0 is uniformly stable (respectively, uniformly asymptotically stable) if and only if it is stable (at 0) (resp., asymptotically stable (at 0)). The (non-neutral) delay dierential equation E0 has been studied in [11], while the autonomous neutral delay dierential equation E0 has been treated in [7]. A detailed analysis of the connection between our results and ones in [7,11] will be given at the end of the next section. 2. Statement of the main results and comments The main results of this paper are Theorems 1 and 2 below. Theorem 1 constitutes an asymptotic criterion for the solutions of the dierential equation (E). An estimate of the solutions of (E) and a stability criterion for the zero solution of (E) are established by Theorem 2. Theorem 1. Let k0 be a real root of the characteristic equation I and set X ci eÿk0 ri : qk0 1 i2I
Define fk0 t a~ t
X j2J
b~j teÿk0 sj
for t P ÿ r
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Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
and set 1 F^k0 T
Z
T 0
fk t dt: 0
Assume that the root k0 has the following property: X X b j sj eÿk0 sj < qk : B jci j jqk0 j F^k0 ri eÿk0 ri qk0 P k0 0 i2I
j2J
Then, for any / 2 C ÿr; 0; R, the solution x of (E)±(C) satisfies Z t 1 Lk / fk u du 0 ; lim x t exp ÿ t!1 qk0 0 0 1 ck0 where
X ci / ÿ ri Lk0 / / 0 i2I
Z Z s 1 ÿk0 ri 0 1 e fk0 s/ sexp ÿ fk u du ds ÿ qk0 qk0 0 0 ÿri Z 0 Z s X 1 ÿk0 sj ~ e fk u du ds bj s/ s exp ÿ qk0 0 0 ÿsj j2J and ck0
X
ci 1 ÿ k0 ri eÿk0 ri
i2I
X
Bj sj eÿk0 sj :
j2J
Note. Property P k0 guarantees that qk0 > 0 and 1 ck0 > 0. Let us assume that X bj 0 on 0; 1 Q a j2J
and
X i2I
jci j
X
b j sj < 1: B
j2J
P From the ®rst assumption of (Q) it follows immediately that A j2J Bj 0 and hence k0 0 is a (real) root of the characteristic equation (I). By using again the ®rst assumption of (Q), we see that, for k0 0, we have fk0 0 on the interval ÿr; 1, and Fbk0 0. Furthermore, because of the second assumption of (Q), it is not dicult to verify that the root k0 0 of (I) has the property (P k0 ). Thus, an application of Theorem 1 with k0 0 leads to the following result: Let condition (Q) be satisfied. Then, for any / 2 C ÿr; 0; R, the solution x of (E)±(C) satisfies
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
lim x t
/ 0
P
209
P R0 ci / ÿri j2J ÿsj b~j s/ s ds P P : 1 i2I ci j2J Bj sj
i2I
t!1
Another immediate consequence of Theorem 1 is the following nonoscillation result: Let k0 be a real root of the characteristic equation (I) and let qk0 , fk0 and Fbk0 be as in Theorem 1. Assume that k0 has the property (P k0 ). Then, for any / 2 C ÿr; 0; R, the solution x of (E)±(C) will be non-oscillatory (i.e., either eventually positive or eventually negative), except possibly if / is such that Lk0 / 0, where Lk0 / is defined as in Theorem 1. As a complement of the last result, we can prove, by using the same method with that in [1] (see also [11]), that the set of all / 2 C ÿr; 0; R which satisfy Lk0 / 0 is a nowhere dense subset of the Banach space C ÿr; 0; R with the sup-norm. Theorem 2. Let k0 be a real root of the characteristic equation (I) and let qk0 , fk0 and Fbk0 be as in Theorem 1. Assume that the root k0 has the property (P k0 ). Consider ck0 as in Theorem 1 and set lk0
X i2I
! X Fbk0 b j sj eÿk0 sj : B jci j 1 ri eÿk0 ri qk0 j2J
Then, for any / 2 C ÿr; 0; R, the solution x of (E)±(C) satisfies jx tj 6 Nk0 k/k exp
1 qk0
Z 0
t
fk0 u du
for all t P 0;
where Nk0
Z s 1 lk0 1 lk0 1 1 fk0 u du : lk0 max exp ÿ s2ÿr;0 qk0 0 1 ck0 1 ck0
The constant Nk0 is greater than 1. Moreover, the trivial solution of the differential equation (E) is: (i) Stable (at 0) if Z t lim sup fk0 u du < 1 G1 k0 t!1
0
(and, in particular, if condition (Q) holds).
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Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
(ii) Asymptotically stable (at 0) if Z t lim fk0 u du ÿ1: G2 k0 t!1
0
(iii) Unstable (at 0) if Z t lim fk0 u du 1: G3 k0 t!1
0
Let us consider the particular case of the (non-neutral) delay dierential equation (E0 ). In this case, property (P(k0 )) of a real root k0 of the characteristic equation I0 becomes X b j sj eÿk0 sj < 1: B P0 k0 j2J
So, by applying our results to the dierential equation (E0 ), we are led to an improved version of the results of Philos [11]. The hypothesis that each one of the coefficients bj for j 2 J is of one sign on 0; 1 is posed in [11]; in this case, b j jBj j for j 2 J . However, this hypothesis is not needed for our we have B results (and those in [11]) to hold. Next, let us concentrate our interest to the special case of the autonomous neutral delay dierential equation (E0 ). Let k0 be a real root of the character0 istic equation I . We immediately have X bj eÿk0 sj k0 qk0 for t P ÿ r fk0 t a j2J
and Fbk0 jk0 jjqk0 j: Moreover, we see that the assumption that k0 has the property (P(k0 )) is equivalent to the assumption that k0 has the property: X X P0 k0 jci j 1 jk0 jri eÿk0 ri jbj jsj eÿk0 sj < 1: i2I
j2J
Thus, an application of the results given in this paper to the special case of the autonomous equation E0 leads to the results obtained by Kordonis et al. [7], under the following additional assumption: (Y) There exist a number T > 0 and positive integers li for i 2 I and mj for j 2 J so that r i li T
for i 2 I;
and
sj m j T
for j 2 J :
It must be noticed that the results given in [7] remain valid without the assumption (Y).
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211
It is an interesting problem to ®nd sucient conditions (on the coecients and the delays of the neutral delay dierential equation (E)) for the characteristic equation (I) to have a real root k0 with the property (P(k0 )). This problem has been investigated in detail by Philos [11] in the particular case of the characteristic equation I0 of the (non-neutral) delay dierential equation (E0 ) and by Kordonis et al. [7] in the special case of the characteristic equation 0 I of the autonomous neutral delay dierential equation E0 . 3. Proof of Theorem 1 From property (P(k0 )) it follows immediately that jqk0 j
X i2I
jci jeÿk0 ri < jqk0 j;
i:e: jqk0 j 1 ÿ
X
jci jeÿk0 ri
! > 0:
i2I
So, we always have X jci jeÿk0 ri > 0: 1ÿ i2I
But, by the de®nition of qk0 , we get X X ci eÿk0 ri P 1 ÿ jci jeÿk0 ri qk0 1 i2I
i2I
and consequently qk0 is necessarily positive. Hence, (P(k0 )) becomes ! X X Fbk0 b j sj eÿk0 sj < 1: B jci j 1 ri eÿk0 ri q k0 i2I j2J
3:1
Next, we will establish some equalities needed below. The T-periodicity of the functions a~ and b~j for j 2 J implies that the function fk0 is also T-periodic. So, by taking into account the fact that ri li T for i 2 I, we obtain for i 2 I and t P 0 Z ri Z ri Z t 1 fk0 u du fk0 u du fk0 u du ri ri 0 tÿri 0 Z T 1 fk0 u du ri T ( 0Z ) X Z T 1 T 1 ÿk0 sj a u du bj u du e ri T 0 T 0 j2J ! X ÿk0 sj A Bj e ri : j2J
212
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
Thus, since k0 is a root of (I), we have Z t 1 fk u du k0 ri for every t P 0 and all i 2 I: qk0 tÿri 0
3:2
In a similar way, by taking into account the fact that sj mj T for j 2 J and using again the hypothesis that k0 is a root of (I), we can obtain Z t 1 fk u du k0 sj for every t P 0 and all j 2 J : 3:3 qk0 tÿsj 0 Moreover, by taking again into account the fact that ri li T for i 2 I, we get for i 2 I and t P 0 Z ri Z ri Z t 1 jfk0 uj du jfk0 uj du jfk0 uj du ri ri 0 tÿri 0 Z T 1 jfk0 uj du ri : T 0 So, it holds Z t jfk0 uj du Fbk0 ri tÿri
for every t P 0 and all i 2 I:
3:4
Furthermore, for each index j 2 J , we can use the assumption that the function b~j is T-periodic and that sj mj T to obtain for t P 0 Z sj Z T Z sj Z t 1 1 ~ bj u du bj u du bj u du sj bj u du sj : sj 0 T 0 tÿsj 0 Hence, Z
t
tÿsj
b~j u du Bj sj
for every t P 0 and all j 2 J :
In a similar manner, one can verify that Z t b j sj for every t P 0 and all j 2 J : jb~j uj du B tÿsj
3:5
3:6
By using (3.2) and (3.4) for a point t t0 P 0 and an index i0 2 I, we obtain Z Z t0 Fbk 1 t0 1 fk0 u du 6 jfk0 uj du 0 ; jk0 j qk0 ri0 t0 ÿri qk0 qk0 ri0 t0 ÿri0 0
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
213
i.e. jk0 j 6
Fbk0 : qk0
Thus, by the de®nition of ck0 , we have jck0 j 6 6
X
jci j 1 jk0 jri eÿk0 ri
i2I
X i2I
!
jci j 1
X
jBj jsj eÿk0 sj
j2J
X Fbk0 b j sj eÿk0 sj lk ; B ri eÿk0 ri 0 qk0 j2J
where lk0 is de®ned as in Theorem 2. We have thus proved that jck0 j 6 lk0 . But, in view of (3.1), lk0 < 1 and so we always have 1 ck0 > 0. Now, consider an arbitrary function / 2 C ÿr; 0; R. Let x be the solution of (E)±(C) and de®ne Z t 1 fk0 u du y t x t exp ÿ qk0 0
for t P ÿ r:
Then, by using (3.2), we obtain for every t P 0 ( ) Z t X X 1 ci x t ÿ ri y t ci y t ÿ ri exp ÿ fk u du x t qk0 tÿri 0 i2I i2I Z t 1 fk u du exp qk0 0 0 # " Z t X 1 ÿk0 ri ci e y t ÿ ri exp fk0 u du : y t qk0 0 i2I Also, by virtue of (3.3), we get for any t P 0 ( " #) Z t X X 1 bj tx t ÿ sj bj ty t ÿ sj exp ÿ fk u du qk0 tÿsj 0 j2J j2J Z t 1 fk0 u du exp qk0 0 " # Z t X 1 ÿk0 sj bj te y t ÿ sj exp fk0 u du : qk0 0 j2J
214
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
So, for every t P 0, we have 8" 9 #0 < = X X ci x t ÿ ri ÿ a tx t ÿ bj tx t ÿ sj x t : ; i2I j2J #0 " Z t X 1 ÿk0 ri fk u du y t ci e y t ÿ ri exp ÿ qk0 0 0 i2I " # X 1 ÿk0 ri fk t y t ci e y t ÿ ri ÿ a ty t qk0 0 i2I " #0 X X ÿk0 sj ÿk0 ri bj te y t ÿ sj y t ci e y t ÿ ri ÿ j2J
"
1 fk t y t qk0 0 ÿ
X
X i2I
#i2I " ci e
ÿk0 ri
y t ÿ ri ÿ fk0 t ÿ "
bj teÿk0 sj y t ÿ sj y t
j2J
X
X
# bj te
j2J
ÿk0 sj
y t
#0
ci eÿk0 ri y t ÿ ri
i2I
X 1 1 fk0 t ci eÿk0 ri y t ÿ ri ÿ 1 fk0 ty t qk0 qk0 i2I " # X X bj teÿk0 sj y t ÿ bj teÿk0 sj y t ÿ sj j2J
j2J
#0
! 1 X ÿk0 ri ci e y t ÿ ri ÿ ci e fk0 ty t y t qk0 i2I i2I " # X X 1 ÿk0 ri ÿk0 sj fk t ci e y t ÿ ri bj te y t qk0 0 i2I j2J X bj teÿk0 sj y t ÿ sj ÿ "
"
X
j2J
ÿk0 ri
#0
X 1 ci eÿk0 ri y t ÿ ri ÿ fk0 t ci eÿk0 ri y t ÿ y t ÿ ri y t qk0 i2I i2I X bj teÿk0 sj y t ÿ y t ÿ sj : X
j2J
Thus, the fact that x satis®es (E) for all t P 0 is equivalent to #0 " X X 1 ci eÿk0 ri y t ÿ ri fk0 t ci eÿk0 ri y t ÿ y t ÿ ri y t qk0 i2I i2I X bj teÿk0 sj y t ÿ y t ÿ sj for t P 0: 3:7 ÿ j2J
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215
Moreover, the initial condition (C) takes the following equivalent form
1 y t / t exp ÿ qk0
Z 0
t
fk0 u du
for t 2 ÿr; 0:
3:8
Furthermore, by taking into account the fact that the functions fk0 and b~j for j 2 J are T-periodic and that the delays ri , i 2 I and sj , j 2 J are multiples of T, we can verify that (3.7) is equivalent to Z 1 X ÿk0 ri t ci e y t ÿ ri ci e fk0 sy s ds y t qk0 i2I tÿri i2I Z t X eÿk0 sj b~j sy s ds K for t P 0; ÿ X
ÿk0 ri
tÿsj
j2J
3:9
where K y 0
X
ci e
ÿk0 ri
i2I
X
e
ÿk0 sj
X
0
ÿsj
j2J
y 0
Z
Z 1 X ÿk0 ri 0 y ÿri ÿ ci e fk0 sy s ds qk0 i2I ÿri
b~j sy s ds
Z X 1 ÿk0 ri 0 ci eÿk0 ri y ÿ ri ÿ e fk0 sy s ds qk0 ÿri i2I e
ÿk0 sj
Z
0
ÿsj
j2J
b~j sy s ds:
By (3.2) and (3.8) and the de®nition of Lk0 /, we can see that K Lk0 /. So, (3.9) can be written as follows: Z 1 X ÿk0 ri t ci e y t ÿ ri ci e fk0 sy s ds y t qk0 i2I tÿri i2I Z t X eÿk0 sj b~j sy s ds Lk0 / for t P 0: ÿ X
ÿk0 ri
j2J
tÿsj
Next, we de®ne z t y t ÿ
Lk0 / 1 ck 0
for t P 0:
3:10
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Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
Then, by using (3.2) and (3.5) and the way of de®nition of ck0 , it is not dicult to verify that (3.10) is equivalent to the following equation: z t
X
ci e
ÿk0 ri
i2I
ÿ
X
eÿk0 sj
Z 1 X ÿk0 ri t z t ÿ ri ci e fk0 sz s ds qk0 i2I tÿri
Z
t
tÿsj
j2J
b~j sz s ds for t P 0:
3:11
On the other hand, the initial condition (3.8) can be equivalently written
1 z t / t exp ÿ qk0
Z 0
t
fk0 u du ÿ
Lk0 / 1 ck0
for t 2 ÿr; 0:
3:12
By the de®nitions of y and z, what we have to prove is that lim z t 0:
3:13
t!1
In the rest of the proof we will establish (3.13). Set Z t 1 Lk / : fk0 u du ÿ 0 Mk0 / max / t exp ÿ t2ÿr;0 q 1c k0
0
k0
3:14
In view of (3.12), from (3.14) it follows that jz tj 6 Mk0 /
for t 2 ÿr; 0:
3:15
We will show that Mk0 / is a bound of z on the whole interval ÿr; 1, namely jz tj 6 Mk0 /
for all t P ÿ r:
3:16
To this end, let us consider an arbitrary number > 0. We claim that jz tj < Mk0 /
for every t P ÿ r:
3:17
Otherwise, because of (3.15), there exists a point t0 > 0 such that jz tj < Mk0 /
for t 2 ÿr; t0 ;
and
jz t0 j Mk0 / :
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217
Then, by using (3.1), (3.4) and (3.6), from (3.11) we obtain
# Z t0 1 Mk0 / jz t0 j 6 jci j jz t0 ÿ ri j jfk sjjz sj ds eÿk0 ri qk0 t0 ÿri 0 i2I Z t0 X eÿk0 sj jb~j sjjz sj ds "
X
( 6
j2J
X i2I
"
"
1 jci j 1 qk0
X
ÿk0 sj
e
j2J
X i2I
t0 ÿsj
Z
Z
t0 t0 ÿsj
t0
t0 ÿri
# jfk0 sj ds eÿk0 ri
) ~ jbj sj ds Mk0 /
! # X F^k0 ÿk0 ri ÿk0 sj b jci j 1 ri e B j sj e Mk0 / q k0 j2J
< Mk0 / :
This is a contradiction and so (3.17) holds true. Since (3.17) is satis®ed for all numbers > 0, (3.16) is always ful®lled. Next, in view of (3.4), (3.6) and (3.16), from (3.11) we get for every t P 0 Z t X 1 jci j jz t ÿ ri j jfk0 sjjz sj ds eÿk0 ri jz tj 6 q tÿr k i 0 i2I Z t X eÿk0 sj jb~j sjjz sj ds ( 6
"
tÿsj
j2J
Z t 1 jci j 1 jfk0 sj ds eÿk0 ri qk0 tÿri i2I ) Z t X ÿk0 sj e jb~j sj ds Mk /
X
j2J
X i2I
tÿsj
0
! # X Fbk0 b j sj eÿk0 sj Mk /: B jci j 1 ri eÿk0 ri 0 qk0 j2J
Hence, we have jz tj 6 lk0 Mk0 /
for all t P 0;
3:18
where lk0 is de®ned as in Theorem 2. By using (3.4), (3.6) and (3.11) and taking into account (3.16) and (3.18), one can show, by an easy induction, that z satis®es
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Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
ÿ n jz tj 6 lk0 Mk0 /
for all t P nr ÿ r
n 0; 1; . . .:
3:19
But, (3.1) guarantees that 0 < lk0 < 1. Thus, from (3.19) it follows immediately that z tends to zero as t ! 1, i.e. (3.13) holds. The proof of the theorem is complete.
4. Proof of Theorem 2 Let / be an arbitrary function in C ÿr; 0; R and x be the solution of (E)± (C). Moreover, let Lk0 / be de®ned as in Theorem 1, and y and z be as in the proof of Theorem 1. Also, let Mk0 / be de®ned by (3.14) (in the proof of Theorem 1). As in the proof of Theorem 1 , we show that z satis®es (3.18). By the de®nition of z, from (3.18) it follows that jy tj 6
jLk0 /j lk0 Mk0 / for t P 0: 1 ck0
4:1
On the other hand, (3.14) gives Z s 1 jLk /j fk0 u du 0 : Mk0 / 6 k/k max exp ÿ s2ÿr;0 qk0 0 1 ck 0 For convenience, we put Rk0 max
s2ÿr;0
Z s 1 exp ÿ fk0 u du : qk0 0
Then the last equation can be written as follows: Mk0 / 6 k/kRk0
jLk0 /j : 1 ck0
4:2
Furthermore, a combination of (4.1) and (4.2) yields jy tj 6
1 lk0 jLk /j k/klk0 Rk0 1 ck 0 0
for t P 0:
4:3
But, by using (3.4) and (3.6) (which have been established in the proof of Theorem 1), we obtain
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
219
Z 1 ÿk0 ri 0 e jfk0 sjj/ sj qk0 ÿri i2I Z s 1 fk u du ds exp ÿ qk0 0 0 Z 0 Z s X 1 eÿk0 sj jb~j sjj/ sj exp ÿ fk0 u du ds qk0 0 ÿsj j2J
jLk0 /j 6 j/ 0j
X
jci j j/ ÿ ri j
Z 0 1 ÿk0 ri jci j 1 e jfk0 sj ds Rk0 qk0 ÿri i2I ( "Z #) 0 X eÿk0 sj jb~j sj ds Rk k/k
6 k/k k/k
X
ÿsj
j2J
k/k k/k
X i2I
k/k
X
Fbk jci j 1 0 ri eÿk0 ri Rk0 qk0 !
0
!
b j sj eÿk0 sj Rk : B 0
j2J
But, in view of (3.2), we have for any i 2 I Z s 1 fk0 u du eÿk0 ri Rk0 eÿk0 ri max exp ÿ s2ÿr;0 qk0 0 Z ÿri 1 ÿk0 ri exp ÿ fk0 u du Pe qk0 0 Z 0 1 ÿk0 ri exp fk u du e qk0 ÿri 0 eÿk0 ri exp k0 ri 1: Thus, we get
"
jLk0 /j 6 k/k k/k k/k
X
X j2J
i.e.
i2I
! # F^k0 ÿk0 ri jci j 1 ri e Rk 0 qk0 !
b j sj eÿk0 sj Rk ; B 0
ÿ jLk0 /j 6 k/k 1 lk0 Rk0 :
4:4
220
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By combining (4.3) and (4.4), we take 1 lk0 ÿ 1 lk0 Rk0 lk0 Rk0 k/k jy tj 6 1 ck0
for every t P 0;
which, by the de®nitions of the function y and the constants Rk0 and Nk0 , gives Z t 1 fk u du for all t P 0: 4:5 jx tj 6 Nk0 k/k exp qk0 0 0 As in the proof of Theorem 1, we can see that jck0 j 6 lk0 and hence we immediately obtain Nk0 > 1. It remains to prove the stability criteria (i), (ii) and (iii). Assume ®rst that (G1 k0 )) holds and set Z t 1 fk0 u du : hk0 sup exp qk0 0 tP0 hk0 is obviously a real constant with hk0 P 1. Moreover, we put Hk0 hk0 Nk0 . Since Nk0 > 1, we also have Hk0 > 1. Let / be an arbitrary function in C ÿr; 0; R and x be the solution of (E)±(C). Then (4.5) gives jx tj 6 Hk0 k/k
for every t P 0:
Thus, since Hk0 > 1, we also have jx tj 6 Hk0 k/k
for all t P ÿ r:
This establishes that the trivial solution of (E) is stable (at 0). In particular, let us consider the case where condition (Q) holds. Then, as in Section 2, we see that k0 0, and fk0 0 on the interval ÿr; 1. In this case, (G1 k0 )) is always satis®ed. Next, let us suppose that (G2 k0 )) is ful®lled. Then, (G1 k0 )) is also satis®ed and hence the trivial solution of (E) is stable (at 0). Furthermore, let / 2 C ÿr; 0; R and x be the solution of (E)±(C). By (G2 k0 )), from (4.5) it follows that lim x t 0:
t!1
Hence, the trivial solution of (E) is asymptotically stable (at 0). Finally, let (G3 k0 )) be satis®ed. Then the trivial solution of (E) is unstable (at 0). Otherwise, there exists a number d d 1 > 0 such that, for any / 2 C ÿr; 0; R with k/k < d, the solution x of (E)±(C) satis®es jx tj < 1
for all t P ÿ r:
De®ne /0 t exp
1 qk0
Z
t 0
4:6
fk0 u du
for t 2 ÿr; 0:
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
221
Clearly, /0 6 0. Furthermore, by the de®nition of Lk0 /, we have Z 0 Z X 1 1 ÿk0 ri 0 ci exp ÿ fk0 u du ÿ e fk0 s ds Lk0 /0 1 qk0 ÿri qk0 ÿri i2I Z 0 X eÿk0 sj b~j s ds ÿsj
j2J
1
X
ci 1 ÿ k0 ri eÿk0 ri
i2I
X
Bj sj eÿk0 sj
j2J
1 ck0 > 0: Let / 2 C ÿr; 0; R be de®ned by /
d1 /; k/0 k 0
where d1 is a number with 0 < d1 < d. Moreover, let x be the solution of (E)± (C). From Theorem 1 it follows that x satis®es Z t 1 Lk / d1 =k/0 kLk0 /0 fk0 u du 0 lim x t exp ÿ t!1 qk0 0 1 ck 0 1 ck0 d1 > 0: k/0 k But, we have k/k d1 < d and hence from (4.6) and condition (G3 k0 )) it follows that Z t 1 fk u du 0: lim x t exp ÿ t!1 qk0 0 0 This is a contradiction. The proof of our theorem is complete.
References [1] R.D. Driver, Some harmless delays, in: Delay and Functional Dierential Equations and their Applications, Academic Press, New York, 1972, pp. 103±119. [2] R.D. Driver, D.W. Sasser, M.L. Slater, The equation x0 t ax t bx t ÿ s with ``small'' delay, Amer. Math. Monthly 80 (1973) 990±995. [3] I. Gy ori, Invariant cones of positive initial functions for delay dierential equations, Appl. Anal. 35 (1990) 21±41. [4] J.K. Hale, Theory of Functional Dierential Equations, Springer, New York, 1977. [5] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Dierential Equations, Springer, New York, 1993. [6] Q. Huang, S. Chen, Oscillation of neutral dierential equations with periodic coecients, Proc. Amer. Math. Soc. 110 (1990) 997±1001.
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[7] I.-G.E. Kordonis, N.Th. Niyianni, Ch.G. Philos, On the behaviour of the solutions of scalar ®rst order linear autonomous neutral delay dierential equations, Arch. Math. (Basel) 71 (1998) 454±464. [8] G. Ladas, Ch.G. Philos, Y.G. S®cas, Oscillations in neutral equations with periodic coecients, Proc. Amer. Math. Soc. 113 (1991) 123±134. [9] Ch.G. Philos, On the oscillation of dierential equations with periodic coecients, Proc. Amer. Math. Soc. 111 (1991) 433±440. [10] Ch.G. Philos, Oscillations of some delay dierential equations with periodic coecients, J. Math. Anal. Appl. 162 (1991) 452±475. [11] Ch.G. Philos, Asymptotic behaviour, nonoscillation and stability in periodic ®rst-order linear delay dierential equations, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 1371±1387. [12] S. Zhang, Asymptotic behaviour of solutions of periodic delay dierential equations, Proc. Roy. Soc. Edinburgh Sect. A 98 (1984) 167±181.
Periodic ®rst order linear neutral delay dierential equations Ch.G. Philos *, I.K. Purnaras Department of Mathematics, University of Ioannina, P.O. Box 1186, 451 10 Ioannina, Greece
Abstract Some new asymptotic and stability results are given for a ®rst order linear neutral delay dierential equation with periodic coecients and constant delays. The asymptotic behavior of the solutions and the stability of the trivial solution are described by the use of an appropriate real root of an equation, which is in a sense the corresponding characteristic equation. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Dierential equation; Neutral delay dierential equation; Periodic coecients; Solution; Asymptotic behavior; Non-oscillation; Stability
1. Introduction A neutral delay dierential equation is a dierential equation in which the highest-order derivative of the unknown function appears both with and without delays. This paper deals with the asymptotic behavior of the solutions and the stability of the trivial solution for a ®rst order linear neutral delay dierential equation with periodic coecients and constant delays, where the coecients have a common period and the delays are multiples of this period. An asymptotic criterion for the solutions is obtained. Moreover, an estimate of the solutions is established and sucient conditions for the stability, the asymp-
*
Tel.: +30-651-98288; fax: +30-651-46361. E-mail address: [email protected] (C.G. Philos).
0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 9 ) 0 0 1 7 4 - 5
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totic stability and the instability of the trivial solution are given. Our results are derived by the use of a real root (with an appropriate property) of the corresponding (in a sense) characteristic equation. The very recent results given by Philos [11] for periodic ®rst order linear (non-neutral) delay dierential equations can be obtained (as a special case) from the results of the present paper. Also, the results given here contain essentially ones obtained very recently by Kordonis et al. [7] for the particular case of autonomous ®rst order linear neutral delay dierential equations. The techniques applied in obtaining our results are originated in a combination of the methods used in [7,11]. The results given in [7,11] extend and improve some signi®cant results on the asymptotic behavior and the stability for a ®rst order linear (non-neutral) delay dierential equation with constant coecients and one constant delay, which have been obtained by Driver et al. [2]. See Driver [1] for some similar asymptotic and stability results for ®rst order linear (non-neutral) delay dierential equations with in®nitely many distributed delays. The results in [7,11] as well as the results given in this paper are motivated by those in [2]. For some related results we refer to the paper by Gy ori [3]. Note that linear delay dierential equations with periodic coecients have been studied (with respect to oscillation) by Philos [9,10] and Zhang [12]; some results (concerning the oscillation) on linear neutral delay dierential equations with periodic coecients have been obtained by Huang and Chen [6] and by Ladas et al. [8]. The theory of neutral delay dierential equations is of both theoretical and practical interest. For a large class of electrical networks containing lossless transmission lines, the describing equations can be reduced to neutral delay dierential equations; such networks arise in high speed computers where nearly lossless transmission lines are used to interconnect switching circuits. Also, neutral delay dierential equations appear in the study of vibrating masses attached to an elastic bar and also as the Euler equation in some variational problems. It must be noted that the theory of neutral delay differential equations presents some additional complications, which are not presented in the theory of the corresponding delay dierential equations. For the basic theory of neutral delay dierential equations, the reader is referred to the books by Hale [4] and by Hale and Verduyn Lunel [5]. Consider the neutral delay dierential equation " E
x t
X i2I
#0 ci x t ÿ ri
a tx t
X
bj tx t ÿ sj ;
j2J
where I and J are initial segments of natural numbers, ci for i 2 I are real numbers, a and bj for j 2 J are continuous real-valued functions on the interval 0; 1, ri for i 2 I are positive real numbers such that ri1 6 ri2 for i1 ; i2 2 I with
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
205
i1 6 i2 , and sj for j 2 J are positive real numbers such that sj1 6 sj2 for j1 ; j2 2 J with j1 6 j2 . The functions bj for j 2 J are assumed to be not identically zero on 0; 1. Moreover, it will be supposed that the coecients a and bj for j 2 J are periodic functions with a common period T > 0 and that there exist positive integers li for i 2 I and mj for j 2 J such that ri li T
for i 2 I;
and
sj m j T
for j 2 J :
De®ne r max ri ; i2I
s max sj ; j2J
and r maxfr; sg:
(r, s and r are positive real numbers.) As usual, a continuous real-valued function x de®ned on the interval ÿr; 1 Pis said to be a solution of the dierential equation (E) if the function x t i2I ci x t ÿ ri is continuously dierentiable for t P 0 and x satis®es (E) for all t P 0. In the sequel, by C ÿr; 0; R we will denote the set of all continuous realvalued functions on the interval ÿr; 0. This set is a Banach space endowed with the usual sup-norm k k de®ned by k/k sup j/ tj t2ÿr;0
for / 2 C ÿr; 0; R:
It is well-known (see, for example, Hale [4] or Hale and Verduyn Lunel [5]) that, for any given initial function / 2 C ÿr; 0; R, there exists a unique solution x of the dierential equation (E) which satis®es the initial condition C
x t / t for t 2 ÿr; 0:
We will call this function x the solution of the initial problem (E)±(C) or, more brie¯y, the solution of (E)±(C). Throughout the paper, we will use the notation Z Z 1 T 1 T a t dt; and Bj bj t dt for j 2 J A T 0 T 0 as well as the notation Z 1 T b jbj tj dt Bj T 0
for j 2 J :
b j > 0 for j 2 J are real constants.) We obviously (Clearly, A, Bj for j 2 J , and B have bj jBj j 6 B
for j 2 J :
b j for j 2 J in the case where each one of the coMoreover, we have jBj j B ecients bj for j 2 J is assumed to be of one sign on the interval 0; 1.
206
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
With the dierential equation (E) we associate the equation ! X X ÿkri ci e Bj eÿksj ; I k 1 A i2I
j2J
which will be called the characteristic equation of (E). This terminology comes from the autonomous case where each one of the coecients a and bj for j 2 J is identically equal to a real constant. In what follows, by a~ and b~j for j 2 J we will denote the T-periodic extensions of the coecients a and bj for j 2 J respectively on the interval ÿr; 1. The notions of the stability, the asymptotic stability and the instability of the trivial solution of the neutral delay dierential equation (E) will be considered in the usual sense. For the sake of completeness, we will give here the de®nitions of these notions. The trivial solution of the dierential equation (E) is said to be stable (at 0) if for each > 0 there exists d d > 0 such that, for any / 2 C ÿr; 0; R with k/k < d, the solution x of (E)±(C) satis®es jx tj < for all t P ÿ r: Otherwise, the trivial solution of (E) is said to be unstable (at 0). Moreover, the trivial solution of (E) is called asymptotically stable (at 0) if it is stable (at 0) in the above sense and, in addition, there exists d0 > 0 such that, for any / 2 C ÿr; 0; R with k/k < d0 , the solution x of (E)±(C) satis®es lim x t 0:
t!1
In the particular case where ci 0 for all i 2 I (and the delays ri for i 2 I are arbitrary), the dierential equation (E) reduces to the (non-neutral) delay dierential equation X x0 t a tx t bj tx t ÿ sj : E0 j2J
As it concerns the (non-neutral) equation (E0 ), since the delays ri , i 2 I can be considered to be arbitrarily small so that r 6 s, we have the number s instead of r. The characteristic equation of (E0 ) is I0
kA
X
Bj eÿksj :
j2J
Before ending this section, let us especially consider the autonomous case, i.e. the special case of the neutral delay dierential equation
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
" E0
x t
X
#0 ci x t ÿ ri
ax t
i2I
X
207
bj x t ÿ sj ;
j2J
where I and J are initial segments of natural numbers and ci 2 R
for i 2 I;
ri 2 0; 1
a 2 R;
for i 2 I;
bj 2 R ÿ f0g
sj 2 0; 1
ri1 6 ri2
for i1 ; i2 2 I with i1 6 i2 ;
sj1 6 sj2
for j1 ; j2 2 J with j1 6 j2 :
for j 2 J ;
for j 2 J ;
The constant coecients a and bj for j 2 J of the diferential equation E0 can be considered to be T-periodic functions, for each real number T > 0. The characteristic equation of E0 is ! X X 0 ÿkri k 1 ci e bj eÿksj : a I i2I
j2J
Note that, as it is well-known, the trivial solution of the autonomous neutral delay dierential equation E0 is uniformly stable (respectively, uniformly asymptotically stable) if and only if it is stable (at 0) (resp., asymptotically stable (at 0)). The (non-neutral) delay dierential equation E0 has been studied in [11], while the autonomous neutral delay dierential equation E0 has been treated in [7]. A detailed analysis of the connection between our results and ones in [7,11] will be given at the end of the next section. 2. Statement of the main results and comments The main results of this paper are Theorems 1 and 2 below. Theorem 1 constitutes an asymptotic criterion for the solutions of the dierential equation (E). An estimate of the solutions of (E) and a stability criterion for the zero solution of (E) are established by Theorem 2. Theorem 1. Let k0 be a real root of the characteristic equation I and set X ci eÿk0 ri : qk0 1 i2I
Define fk0 t a~ t
X j2J
b~j teÿk0 sj
for t P ÿ r
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and set 1 F^k0 T
Z
T 0
fk t dt: 0
Assume that the root k0 has the following property: X X b j sj eÿk0 sj < qk : B jci j jqk0 j F^k0 ri eÿk0 ri qk0 P k0 0 i2I
j2J
Then, for any / 2 C ÿr; 0; R, the solution x of (E)±(C) satisfies Z t 1 Lk / fk u du 0 ; lim x t exp ÿ t!1 qk0 0 0 1 ck0 where
X ci / ÿ ri Lk0 / / 0 i2I
Z Z s 1 ÿk0 ri 0 1 e fk0 s/ sexp ÿ fk u du ds ÿ qk0 qk0 0 0 ÿri Z 0 Z s X 1 ÿk0 sj ~ e fk u du ds bj s/ s exp ÿ qk0 0 0 ÿsj j2J and ck0
X
ci 1 ÿ k0 ri eÿk0 ri
i2I
X
Bj sj eÿk0 sj :
j2J
Note. Property P k0 guarantees that qk0 > 0 and 1 ck0 > 0. Let us assume that X bj 0 on 0; 1 Q a j2J
and
X i2I
jci j
X
b j sj < 1: B
j2J
P From the ®rst assumption of (Q) it follows immediately that A j2J Bj 0 and hence k0 0 is a (real) root of the characteristic equation (I). By using again the ®rst assumption of (Q), we see that, for k0 0, we have fk0 0 on the interval ÿr; 1, and Fbk0 0. Furthermore, because of the second assumption of (Q), it is not dicult to verify that the root k0 0 of (I) has the property (P k0 ). Thus, an application of Theorem 1 with k0 0 leads to the following result: Let condition (Q) be satisfied. Then, for any / 2 C ÿr; 0; R, the solution x of (E)±(C) satisfies
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
lim x t
/ 0
P
209
P R0 ci / ÿri j2J ÿsj b~j s/ s ds P P : 1 i2I ci j2J Bj sj
i2I
t!1
Another immediate consequence of Theorem 1 is the following nonoscillation result: Let k0 be a real root of the characteristic equation (I) and let qk0 , fk0 and Fbk0 be as in Theorem 1. Assume that k0 has the property (P k0 ). Then, for any / 2 C ÿr; 0; R, the solution x of (E)±(C) will be non-oscillatory (i.e., either eventually positive or eventually negative), except possibly if / is such that Lk0 / 0, where Lk0 / is defined as in Theorem 1. As a complement of the last result, we can prove, by using the same method with that in [1] (see also [11]), that the set of all / 2 C ÿr; 0; R which satisfy Lk0 / 0 is a nowhere dense subset of the Banach space C ÿr; 0; R with the sup-norm. Theorem 2. Let k0 be a real root of the characteristic equation (I) and let qk0 , fk0 and Fbk0 be as in Theorem 1. Assume that the root k0 has the property (P k0 ). Consider ck0 as in Theorem 1 and set lk0
X i2I
! X Fbk0 b j sj eÿk0 sj : B jci j 1 ri eÿk0 ri qk0 j2J
Then, for any / 2 C ÿr; 0; R, the solution x of (E)±(C) satisfies jx tj 6 Nk0 k/k exp
1 qk0
Z 0
t
fk0 u du
for all t P 0;
where Nk0
Z s 1 lk0 1 lk0 1 1 fk0 u du : lk0 max exp ÿ s2ÿr;0 qk0 0 1 ck0 1 ck0
The constant Nk0 is greater than 1. Moreover, the trivial solution of the differential equation (E) is: (i) Stable (at 0) if Z t lim sup fk0 u du < 1 G1 k0 t!1
0
(and, in particular, if condition (Q) holds).
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Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
(ii) Asymptotically stable (at 0) if Z t lim fk0 u du ÿ1: G2 k0 t!1
0
(iii) Unstable (at 0) if Z t lim fk0 u du 1: G3 k0 t!1
0
Let us consider the particular case of the (non-neutral) delay dierential equation (E0 ). In this case, property (P(k0 )) of a real root k0 of the characteristic equation I0 becomes X b j sj eÿk0 sj < 1: B P0 k0 j2J
So, by applying our results to the dierential equation (E0 ), we are led to an improved version of the results of Philos [11]. The hypothesis that each one of the coefficients bj for j 2 J is of one sign on 0; 1 is posed in [11]; in this case, b j jBj j for j 2 J . However, this hypothesis is not needed for our we have B results (and those in [11]) to hold. Next, let us concentrate our interest to the special case of the autonomous neutral delay dierential equation (E0 ). Let k0 be a real root of the character0 istic equation I . We immediately have X bj eÿk0 sj k0 qk0 for t P ÿ r fk0 t a j2J
and Fbk0 jk0 jjqk0 j: Moreover, we see that the assumption that k0 has the property (P(k0 )) is equivalent to the assumption that k0 has the property: X X P0 k0 jci j 1 jk0 jri eÿk0 ri jbj jsj eÿk0 sj < 1: i2I
j2J
Thus, an application of the results given in this paper to the special case of the autonomous equation E0 leads to the results obtained by Kordonis et al. [7], under the following additional assumption: (Y) There exist a number T > 0 and positive integers li for i 2 I and mj for j 2 J so that r i li T
for i 2 I;
and
sj m j T
for j 2 J :
It must be noticed that the results given in [7] remain valid without the assumption (Y).
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211
It is an interesting problem to ®nd sucient conditions (on the coecients and the delays of the neutral delay dierential equation (E)) for the characteristic equation (I) to have a real root k0 with the property (P(k0 )). This problem has been investigated in detail by Philos [11] in the particular case of the characteristic equation I0 of the (non-neutral) delay dierential equation (E0 ) and by Kordonis et al. [7] in the special case of the characteristic equation 0 I of the autonomous neutral delay dierential equation E0 . 3. Proof of Theorem 1 From property (P(k0 )) it follows immediately that jqk0 j
X i2I
jci jeÿk0 ri < jqk0 j;
i:e: jqk0 j 1 ÿ
X
jci jeÿk0 ri
! > 0:
i2I
So, we always have X jci jeÿk0 ri > 0: 1ÿ i2I
But, by the de®nition of qk0 , we get X X ci eÿk0 ri P 1 ÿ jci jeÿk0 ri qk0 1 i2I
i2I
and consequently qk0 is necessarily positive. Hence, (P(k0 )) becomes ! X X Fbk0 b j sj eÿk0 sj < 1: B jci j 1 ri eÿk0 ri q k0 i2I j2J
3:1
Next, we will establish some equalities needed below. The T-periodicity of the functions a~ and b~j for j 2 J implies that the function fk0 is also T-periodic. So, by taking into account the fact that ri li T for i 2 I, we obtain for i 2 I and t P 0 Z ri Z ri Z t 1 fk0 u du fk0 u du fk0 u du ri ri 0 tÿri 0 Z T 1 fk0 u du ri T ( 0Z ) X Z T 1 T 1 ÿk0 sj a u du bj u du e ri T 0 T 0 j2J ! X ÿk0 sj A Bj e ri : j2J
212
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
Thus, since k0 is a root of (I), we have Z t 1 fk u du k0 ri for every t P 0 and all i 2 I: qk0 tÿri 0
3:2
In a similar way, by taking into account the fact that sj mj T for j 2 J and using again the hypothesis that k0 is a root of (I), we can obtain Z t 1 fk u du k0 sj for every t P 0 and all j 2 J : 3:3 qk0 tÿsj 0 Moreover, by taking again into account the fact that ri li T for i 2 I, we get for i 2 I and t P 0 Z ri Z ri Z t 1 jfk0 uj du jfk0 uj du jfk0 uj du ri ri 0 tÿri 0 Z T 1 jfk0 uj du ri : T 0 So, it holds Z t jfk0 uj du Fbk0 ri tÿri
for every t P 0 and all i 2 I:
3:4
Furthermore, for each index j 2 J , we can use the assumption that the function b~j is T-periodic and that sj mj T to obtain for t P 0 Z sj Z T Z sj Z t 1 1 ~ bj u du bj u du bj u du sj bj u du sj : sj 0 T 0 tÿsj 0 Hence, Z
t
tÿsj
b~j u du Bj sj
for every t P 0 and all j 2 J :
In a similar manner, one can verify that Z t b j sj for every t P 0 and all j 2 J : jb~j uj du B tÿsj
3:5
3:6
By using (3.2) and (3.4) for a point t t0 P 0 and an index i0 2 I, we obtain Z Z t0 Fbk 1 t0 1 fk0 u du 6 jfk0 uj du 0 ; jk0 j qk0 ri0 t0 ÿri qk0 qk0 ri0 t0 ÿri0 0
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
213
i.e. jk0 j 6
Fbk0 : qk0
Thus, by the de®nition of ck0 , we have jck0 j 6 6
X
jci j 1 jk0 jri eÿk0 ri
i2I
X i2I
!
jci j 1
X
jBj jsj eÿk0 sj
j2J
X Fbk0 b j sj eÿk0 sj lk ; B ri eÿk0 ri 0 qk0 j2J
where lk0 is de®ned as in Theorem 2. We have thus proved that jck0 j 6 lk0 . But, in view of (3.1), lk0 < 1 and so we always have 1 ck0 > 0. Now, consider an arbitrary function / 2 C ÿr; 0; R. Let x be the solution of (E)±(C) and de®ne Z t 1 fk0 u du y t x t exp ÿ qk0 0
for t P ÿ r:
Then, by using (3.2), we obtain for every t P 0 ( ) Z t X X 1 ci x t ÿ ri y t ci y t ÿ ri exp ÿ fk u du x t qk0 tÿri 0 i2I i2I Z t 1 fk u du exp qk0 0 0 # " Z t X 1 ÿk0 ri ci e y t ÿ ri exp fk0 u du : y t qk0 0 i2I Also, by virtue of (3.3), we get for any t P 0 ( " #) Z t X X 1 bj tx t ÿ sj bj ty t ÿ sj exp ÿ fk u du qk0 tÿsj 0 j2J j2J Z t 1 fk0 u du exp qk0 0 " # Z t X 1 ÿk0 sj bj te y t ÿ sj exp fk0 u du : qk0 0 j2J
214
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
So, for every t P 0, we have 8" 9 #0 < = X X ci x t ÿ ri ÿ a tx t ÿ bj tx t ÿ sj x t : ; i2I j2J #0 " Z t X 1 ÿk0 ri fk u du y t ci e y t ÿ ri exp ÿ qk0 0 0 i2I " # X 1 ÿk0 ri fk t y t ci e y t ÿ ri ÿ a ty t qk0 0 i2I " #0 X X ÿk0 sj ÿk0 ri bj te y t ÿ sj y t ci e y t ÿ ri ÿ j2J
"
1 fk t y t qk0 0 ÿ
X
X i2I
#i2I " ci e
ÿk0 ri
y t ÿ ri ÿ fk0 t ÿ "
bj teÿk0 sj y t ÿ sj y t
j2J
X
X
# bj te
j2J
ÿk0 sj
y t
#0
ci eÿk0 ri y t ÿ ri
i2I
X 1 1 fk0 t ci eÿk0 ri y t ÿ ri ÿ 1 fk0 ty t qk0 qk0 i2I " # X X bj teÿk0 sj y t ÿ bj teÿk0 sj y t ÿ sj j2J
j2J
#0
! 1 X ÿk0 ri ci e y t ÿ ri ÿ ci e fk0 ty t y t qk0 i2I i2I " # X X 1 ÿk0 ri ÿk0 sj fk t ci e y t ÿ ri bj te y t qk0 0 i2I j2J X bj teÿk0 sj y t ÿ sj ÿ "
"
X
j2J
ÿk0 ri
#0
X 1 ci eÿk0 ri y t ÿ ri ÿ fk0 t ci eÿk0 ri y t ÿ y t ÿ ri y t qk0 i2I i2I X bj teÿk0 sj y t ÿ y t ÿ sj : X
j2J
Thus, the fact that x satis®es (E) for all t P 0 is equivalent to #0 " X X 1 ci eÿk0 ri y t ÿ ri fk0 t ci eÿk0 ri y t ÿ y t ÿ ri y t qk0 i2I i2I X bj teÿk0 sj y t ÿ y t ÿ sj for t P 0: 3:7 ÿ j2J
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
215
Moreover, the initial condition (C) takes the following equivalent form
1 y t / t exp ÿ qk0
Z 0
t
fk0 u du
for t 2 ÿr; 0:
3:8
Furthermore, by taking into account the fact that the functions fk0 and b~j for j 2 J are T-periodic and that the delays ri , i 2 I and sj , j 2 J are multiples of T, we can verify that (3.7) is equivalent to Z 1 X ÿk0 ri t ci e y t ÿ ri ci e fk0 sy s ds y t qk0 i2I tÿri i2I Z t X eÿk0 sj b~j sy s ds K for t P 0; ÿ X
ÿk0 ri
tÿsj
j2J
3:9
where K y 0
X
ci e
ÿk0 ri
i2I
X
e
ÿk0 sj
X
0
ÿsj
j2J
y 0
Z
Z 1 X ÿk0 ri 0 y ÿri ÿ ci e fk0 sy s ds qk0 i2I ÿri
b~j sy s ds
Z X 1 ÿk0 ri 0 ci eÿk0 ri y ÿ ri ÿ e fk0 sy s ds qk0 ÿri i2I e
ÿk0 sj
Z
0
ÿsj
j2J
b~j sy s ds:
By (3.2) and (3.8) and the de®nition of Lk0 /, we can see that K Lk0 /. So, (3.9) can be written as follows: Z 1 X ÿk0 ri t ci e y t ÿ ri ci e fk0 sy s ds y t qk0 i2I tÿri i2I Z t X eÿk0 sj b~j sy s ds Lk0 / for t P 0: ÿ X
ÿk0 ri
j2J
tÿsj
Next, we de®ne z t y t ÿ
Lk0 / 1 ck 0
for t P 0:
3:10
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Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
Then, by using (3.2) and (3.5) and the way of de®nition of ck0 , it is not dicult to verify that (3.10) is equivalent to the following equation: z t
X
ci e
ÿk0 ri
i2I
ÿ
X
eÿk0 sj
Z 1 X ÿk0 ri t z t ÿ ri ci e fk0 sz s ds qk0 i2I tÿri
Z
t
tÿsj
j2J
b~j sz s ds for t P 0:
3:11
On the other hand, the initial condition (3.8) can be equivalently written
1 z t / t exp ÿ qk0
Z 0
t
fk0 u du ÿ
Lk0 / 1 ck0
for t 2 ÿr; 0:
3:12
By the de®nitions of y and z, what we have to prove is that lim z t 0:
3:13
t!1
In the rest of the proof we will establish (3.13). Set Z t 1 Lk / : fk0 u du ÿ 0 Mk0 / max / t exp ÿ t2ÿr;0 q 1c k0
0
k0
3:14
In view of (3.12), from (3.14) it follows that jz tj 6 Mk0 /
for t 2 ÿr; 0:
3:15
We will show that Mk0 / is a bound of z on the whole interval ÿr; 1, namely jz tj 6 Mk0 /
for all t P ÿ r:
3:16
To this end, let us consider an arbitrary number > 0. We claim that jz tj < Mk0 /
for every t P ÿ r:
3:17
Otherwise, because of (3.15), there exists a point t0 > 0 such that jz tj < Mk0 /
for t 2 ÿr; t0 ;
and
jz t0 j Mk0 / :
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
217
Then, by using (3.1), (3.4) and (3.6), from (3.11) we obtain
# Z t0 1 Mk0 / jz t0 j 6 jci j jz t0 ÿ ri j jfk sjjz sj ds eÿk0 ri qk0 t0 ÿri 0 i2I Z t0 X eÿk0 sj jb~j sjjz sj ds "
X
( 6
j2J
X i2I
"
"
1 jci j 1 qk0
X
ÿk0 sj
e
j2J
X i2I
t0 ÿsj
Z
Z
t0 t0 ÿsj
t0
t0 ÿri
# jfk0 sj ds eÿk0 ri
) ~ jbj sj ds Mk0 /
! # X F^k0 ÿk0 ri ÿk0 sj b jci j 1 ri e B j sj e Mk0 / q k0 j2J
< Mk0 / :
This is a contradiction and so (3.17) holds true. Since (3.17) is satis®ed for all numbers > 0, (3.16) is always ful®lled. Next, in view of (3.4), (3.6) and (3.16), from (3.11) we get for every t P 0 Z t X 1 jci j jz t ÿ ri j jfk0 sjjz sj ds eÿk0 ri jz tj 6 q tÿr k i 0 i2I Z t X eÿk0 sj jb~j sjjz sj ds ( 6
"
tÿsj
j2J
Z t 1 jci j 1 jfk0 sj ds eÿk0 ri qk0 tÿri i2I ) Z t X ÿk0 sj e jb~j sj ds Mk /
X
j2J
X i2I
tÿsj
0
! # X Fbk0 b j sj eÿk0 sj Mk /: B jci j 1 ri eÿk0 ri 0 qk0 j2J
Hence, we have jz tj 6 lk0 Mk0 /
for all t P 0;
3:18
where lk0 is de®ned as in Theorem 2. By using (3.4), (3.6) and (3.11) and taking into account (3.16) and (3.18), one can show, by an easy induction, that z satis®es
218
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
ÿ n jz tj 6 lk0 Mk0 /
for all t P nr ÿ r
n 0; 1; . . .:
3:19
But, (3.1) guarantees that 0 < lk0 < 1. Thus, from (3.19) it follows immediately that z tends to zero as t ! 1, i.e. (3.13) holds. The proof of the theorem is complete.
4. Proof of Theorem 2 Let / be an arbitrary function in C ÿr; 0; R and x be the solution of (E)± (C). Moreover, let Lk0 / be de®ned as in Theorem 1, and y and z be as in the proof of Theorem 1. Also, let Mk0 / be de®ned by (3.14) (in the proof of Theorem 1). As in the proof of Theorem 1 , we show that z satis®es (3.18). By the de®nition of z, from (3.18) it follows that jy tj 6
jLk0 /j lk0 Mk0 / for t P 0: 1 ck0
4:1
On the other hand, (3.14) gives Z s 1 jLk /j fk0 u du 0 : Mk0 / 6 k/k max exp ÿ s2ÿr;0 qk0 0 1 ck 0 For convenience, we put Rk0 max
s2ÿr;0
Z s 1 exp ÿ fk0 u du : qk0 0
Then the last equation can be written as follows: Mk0 / 6 k/kRk0
jLk0 /j : 1 ck0
4:2
Furthermore, a combination of (4.1) and (4.2) yields jy tj 6
1 lk0 jLk /j k/klk0 Rk0 1 ck 0 0
for t P 0:
4:3
But, by using (3.4) and (3.6) (which have been established in the proof of Theorem 1), we obtain
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
219
Z 1 ÿk0 ri 0 e jfk0 sjj/ sj qk0 ÿri i2I Z s 1 fk u du ds exp ÿ qk0 0 0 Z 0 Z s X 1 eÿk0 sj jb~j sjj/ sj exp ÿ fk0 u du ds qk0 0 ÿsj j2J
jLk0 /j 6 j/ 0j
X
jci j j/ ÿ ri j
Z 0 1 ÿk0 ri jci j 1 e jfk0 sj ds Rk0 qk0 ÿri i2I ( "Z #) 0 X eÿk0 sj jb~j sj ds Rk k/k
6 k/k k/k
X
ÿsj
j2J
k/k k/k
X i2I
k/k
X
Fbk jci j 1 0 ri eÿk0 ri Rk0 qk0 !
0
!
b j sj eÿk0 sj Rk : B 0
j2J
But, in view of (3.2), we have for any i 2 I Z s 1 fk0 u du eÿk0 ri Rk0 eÿk0 ri max exp ÿ s2ÿr;0 qk0 0 Z ÿri 1 ÿk0 ri exp ÿ fk0 u du Pe qk0 0 Z 0 1 ÿk0 ri exp fk u du e qk0 ÿri 0 eÿk0 ri exp k0 ri 1: Thus, we get
"
jLk0 /j 6 k/k k/k k/k
X
X j2J
i.e.
i2I
! # F^k0 ÿk0 ri jci j 1 ri e Rk 0 qk0 !
b j sj eÿk0 sj Rk ; B 0
ÿ jLk0 /j 6 k/k 1 lk0 Rk0 :
4:4
220
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
By combining (4.3) and (4.4), we take 1 lk0 ÿ 1 lk0 Rk0 lk0 Rk0 k/k jy tj 6 1 ck0
for every t P 0;
which, by the de®nitions of the function y and the constants Rk0 and Nk0 , gives Z t 1 fk u du for all t P 0: 4:5 jx tj 6 Nk0 k/k exp qk0 0 0 As in the proof of Theorem 1, we can see that jck0 j 6 lk0 and hence we immediately obtain Nk0 > 1. It remains to prove the stability criteria (i), (ii) and (iii). Assume ®rst that (G1 k0 )) holds and set Z t 1 fk0 u du : hk0 sup exp qk0 0 tP0 hk0 is obviously a real constant with hk0 P 1. Moreover, we put Hk0 hk0 Nk0 . Since Nk0 > 1, we also have Hk0 > 1. Let / be an arbitrary function in C ÿr; 0; R and x be the solution of (E)±(C). Then (4.5) gives jx tj 6 Hk0 k/k
for every t P 0:
Thus, since Hk0 > 1, we also have jx tj 6 Hk0 k/k
for all t P ÿ r:
This establishes that the trivial solution of (E) is stable (at 0). In particular, let us consider the case where condition (Q) holds. Then, as in Section 2, we see that k0 0, and fk0 0 on the interval ÿr; 1. In this case, (G1 k0 )) is always satis®ed. Next, let us suppose that (G2 k0 )) is ful®lled. Then, (G1 k0 )) is also satis®ed and hence the trivial solution of (E) is stable (at 0). Furthermore, let / 2 C ÿr; 0; R and x be the solution of (E)±(C). By (G2 k0 )), from (4.5) it follows that lim x t 0:
t!1
Hence, the trivial solution of (E) is asymptotically stable (at 0). Finally, let (G3 k0 )) be satis®ed. Then the trivial solution of (E) is unstable (at 0). Otherwise, there exists a number d d 1 > 0 such that, for any / 2 C ÿr; 0; R with k/k < d, the solution x of (E)±(C) satis®es jx tj < 1
for all t P ÿ r:
De®ne /0 t exp
1 qk0
Z
t 0
4:6
fk0 u du
for t 2 ÿr; 0:
Ch.G. Philos, I.K. Purnaras / Appl. Math. Comput. 117 (2001) 203±222
221
Clearly, /0 6 0. Furthermore, by the de®nition of Lk0 /, we have Z 0 Z X 1 1 ÿk0 ri 0 ci exp ÿ fk0 u du ÿ e fk0 s ds Lk0 /0 1 qk0 ÿri qk0 ÿri i2I Z 0 X eÿk0 sj b~j s ds ÿsj
j2J
1
X
ci 1 ÿ k0 ri eÿk0 ri
i2I
X
Bj sj eÿk0 sj
j2J
1 ck0 > 0: Let / 2 C ÿr; 0; R be de®ned by /
d1 /; k/0 k 0
where d1 is a number with 0 < d1 < d. Moreover, let x be the solution of (E)± (C). From Theorem 1 it follows that x satis®es Z t 1 Lk / d1 =k/0 kLk0 /0 fk0 u du 0 lim x t exp ÿ t!1 qk0 0 1 ck 0 1 ck0 d1 > 0: k/0 k But, we have k/k d1 < d and hence from (4.6) and condition (G3 k0 )) it follows that Z t 1 fk u du 0: lim x t exp ÿ t!1 qk0 0 0 This is a contradiction. The proof of our theorem is complete.
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