- Email: [email protected]
Positive Solutions and Asymptotic Behavior of Delay Differential Equations with Nonlinear Impulses
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
207, 388]396 Ž1997.
AY975276
Positive Solutions and Asymptotic Behavior of Delay Differential Equations with Nonlinear Impulses* Yu Jiang and Yan Jurang Department of Mathematics, Shanxi Uni¨ ersity, Taiyuan 030006, Shanxi, People’s Republic of China Communicated by William F. Ames Received January 19, 1996
Consider the delay differential equation ŽDDE. with nonlinear impulses n
˙x Ž t . q
Ý pi Ž t . x Ž t y t i . s 0,
t / t j , t G t0 ,
is1
x Ž tq j . y x Ž t j . s Ij Ž x Ž t j . . ,
j s 1, 2, . . . ,
Ž).
q.
where pi g C Žw t 0 , `., R , t i G 0 for i s 1, 2, . . . , n, and I j g C ŽR, R. for j s 1, 2, . . . . The purpose of this paper is to obtain a necessary and sufficient condition for the existence of positive solutions of DDE without impulses and to establish a kind of order persistence of solutions of Eq. Ž). and criteria of the asymptotic behavior of Eq. Ž)., which can be used to improve and develop some of the known results in the literature. Q 1997 Academic Press
1. INTRODUCTION There have been many papers considering the delay differential equation n
˙x Ž t . q Ý pi Ž t . x Ž t y t i . s 0,
Ž 1.
is1
where pi g C Žw t 0 , `., Rq. , t i G 0 for i s 1, 2, . . . , n; see, for example, w1, 4, 5x and the references cited in w4x. There are only a few papers concerned with the impulsive delay differential equations, which is an important mathematical model of many evolution process; see w2, 3, 6, 7x. *This work was partially supported by the National Natural Science Foundation of China. 388 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
DELAY DIFFERENTIAL EQUATIONS
389
In this paper, we consider the delay differential equations ŽDDE. with nonlinear impulses, n
˙x Ž t . q Ý pi Ž t . x Ž t y t i . s 0,
t / t j , t G t0 ,
is1
x Ž tq j . y x Ž t j . s Ij Ž x Ž t j . . ,
j s 1, 2, . . . ,
Ž 2.
where pi g C Žw t 0 , `., Rq. , t i G 0 for i s 1, 2, . . . , n, and I j g C ŽR, R. for j s 1, 2, . . . . Let t s maxt 1 , t 2 , . . . , tn4 and t 0 - t 1 - t 2 - ??? , t j ª ` as j ª `. Set PC s f : w yt , 0 x ª R is piecewise left continuous 4 . With Eq. Ž2., one associates an initial condition of the form xs s f Ž s . ,
s g w yt , 0 x ,
Ž 3.
where xs s x Ž s q s . and f g PC. DEFINITION 1. x Ž t . is called a solution corresponding to t 0 of Eq. Ž2. if x: w t 0 y t , `. ª R is continuous for t / t j ; x is continuously differentiable . Ž y. exist and x Ž ty . s x Ž t j ., for t G t 0 , t / t j , and t / t j q t i , and x Ž tq j , x tj j Ž . j s 1, 2, . . . , and x satisfies Eq. 2 . DEFINITION 2. x Ž t . is called a solution corresponding to t 0 of the initial value problem Ž2. and Ž3. if x is a solution corresponding to t 0 of Eq. Ž2. and satisfies Ž3., denoted by x Ž t 0 , f .Ž t .. DEFINITION 3. x Ž t 0 , f .Ž t . is called a positive solution of Eq. Ž2. if f G 0, s g wyt , 0x, and x Ž t 0 , f .Ž t . ) 0, t G t 0 . DEFINITION 4. A solution x Ž t . of Eq. Ž2. is said to be nonoscillatory if it is eventually positive or eventually negative; otherwise it will be called oscillatory. Analogously, we can define the various solutions of Eq. Ž1.. Our aim in this paper is to obtain results on existence and asymptotic behavior of nonoscillatory solutions of Eq. Ž2.. This paper is organized as follows. In Section 2, we show a necessary and sufficient condition ensuring that there exist positive solutions in Eq. Ž1., the special case of which includes the corresponding results in w4x. Furthermore, under the conditions we establish in Section 3, several new results which assert that there exists a kind of order relation between the solutions of Eq. Ž1. and those of Eq. Ž2., that is, the differences between the solutions of Ž1. and the corresponding ones of Ž2., are always positive or negative. Finally, some criteria of asymptotic behavior of Eq. Ž2. are given in Section 4, which improve and generalize the known criteria w7x.
390
JIANG AND JURANG
2. POSITIVE SOLUTIONS OF EQ. Ž1. Let h i Ž t . s min t 0 , t y t i 4 , Hi Ž t . s max t 0 , t y t i 4 , and F s f g C Ž w yt , 0 x , Rq . < f Ž 0 . ) 0 and f Ž s . F f Ž 0 . , yt F s F 0 4 . Our aim in this section is to establish the following necessary and sufficient condition for the positive solutions of Eq. Ž1. For Eq. Ž1., the following statements are equi¨ alent:
THEOREM 1.
There exist k i Ž t . g C Žw t 0 , `., Rq. , i s 1, 2, . . . , n, such that
Ž a.
n
t
HH Ž t . Ý p Ž s . k Ž s . ds F ln k Ž t . , i
j
Ž b.
i
t G t 0 , j s 1, 2, . . . , n.
j
Ž 4.
is1
For ; f g F, x Ž t 0 , f .Ž t . is a positi¨ e solution of Eqs. Ž1. and Ž3..
Proof. Ž a. « Ž b . Assume that Ža. holds. Set n
b Ž t . s y Ý pi Ž t . k i Ž t . ,
g Ž t . s 0,
for t G t 0 .
is1
For all d g C Žw t 0 , `., R. such that b Ž t . F d Ž t . F g Ž t ., t G t 0 , it follows from Ž4. that t
n
t
HH Ž t . d Ž s . ds F HH Ž t . Ý p Ž s . k Ž s . ds F ln k Ž t . ,
y
i
j
j
i
j
is1
where t G t 0 , j s 1, 2, . . . , n. Hence, for ; f g F, we have n
g Ž t . s 0 G y Ý pi Ž t .
f Ž h i Ž t . y t0 . f Ž 0.
is1
ž
t
HH Ž t . d Ž s . ds
exp y
i
/
n
G y Ý pi Ž t . k i Ž t . s b Ž t . . is1
In view of Theorem 3.1.1 in w4x, x Ž t 0 , f .Ž t . is a positive solution corresponding to t 0 of Eqs. Ž1. and Ž3.. Žb. « Ža. Assume that y Ž t 0 , f .Ž t . is a positive solution of Eq. Ž1.. Let Ž y t . s y Ž t 0 , f .Ž t . and
a Ž t. s
˙y Ž t . , yŽ t.
t G t0 .
So y Ž t . s f Ž 0 . exp
t
žH
t0
a Ž s . ds ,
/
t G t0 .
391
DELAY DIFFERENTIAL EQUATIONS
Hence, it follows from Theorem 3.1.1 in w4x that n
a Ž t. q
Ý pi Ž t .
f Ž h i Ž t . y t0 . f Ž 0.
is1
exp
t
žH
H iŽ t .
a Ž s . ds s 0.
/
Ž 5.
Set t
HH Ž t . a Ž s . ds
k i Ž t . s exp y
ž
i
/
.
Integrating Ž5. from Hi Ž t . to t, we get n
t
HH Ž t . Ý p Ž s .
f Ž h i Ž s . y t0 . f Ž 0.
i
j
is1
s
k i Ž s . ds
n
t
t
HH Ž t . Ý p Ž s . k Ž s . ds s yHH Ž t . a Ž s . ds i
j
i
is1
j
s ln k j Ž t . ,
t G t 0 , j s 1, 2, . . . , n
which completes the proof of Theorem 1. Remark 1. If k i Ž t . ' e, it is easy from Ž4. to see that n
t
1
HH Ž t . Ý p Ž s . ds F e , i
j
t G t0 ,
is1
which is a well-known result.
3. SOME PROPERTIES OF SOLUTIONS OF EQ. Ž2. This section is devoted to the existence of positive and nonoscillatory solutions of Eq. Ž2., which are based on our comparison result established here. It is interesting in itself and needed in the following lemma, which is a clear extension of Lemma 1 in w5x. LEMMA 1. Assume that z Ž t . is a positi¨ e solution of Eq. Ž1. and x Ž t . is any solution of Eq. Ž2.. Set w Ž t . s x Ž t . rz Ž t . ,
t G t0 .
Then for t G t 0 , t / t j , t / t j q t i , j s 1, 2, . . . , and i s 1, 2, . . . , n, n
w ˙Ž t. s
Ý pi Ž t . is1
z Ž t y ti . zŽ t.
Ž w Ž t . y w Ž t y ti . . .
Ž 6.
392
JIANG AND JURANG
THEOREM 2.
Assume that n
pi Ž t . G 0
for i s 1, 2, . . . , n and
Ý pi Ž t . ) 0,
t G t0 , Ž 7.
is1
and Eq. Ž1. has a positi¨ e solution z Ž t 0 , w .Ž t . corresponding to t 0 . Then for f g PC such that
fs
½
0, b,
yt F s - 0, s s 0,
Ž 8.
where b / 0, bx Ž s , f . Ž t . ) 0,
tGs,
where x Ž s , f .Ž t . is a solution corresponding to s with s G t 0 of Eqs. Ž1. and Ž3.. Proof. By Theorem 1, without loss of generality, assume that z Ž t . ) 0 for t G t 0 y t . Consider the case of b ) 0 Žthe case of b - 0 can be treated in a similar way.. Since Eq. Ž1. is a linear equation, it is easy to see that there exists a constant C ) 0 such that Cz Ž t . is a positive solution of Eq. Ž1. and satisfies Cz Ž s . s b.
Ž 9.
We have to prove x Ž s , f .Ž t . ) 0 for t G s . Note that Cz Ž t . G f for t g w s y t , s x and it follows from Eq. Ž1. that there exists a constant T ) s such that x Ž s , f . Ž t . G Cz Ž t . ,
s F t F T.
Ž 10 .
We claim that T s `. Otherwise, there exists t ) s such that x Ž s , f . Ž t . s Cz Ž t . . In view of Ž9. and Ž10., we can choose t* s inf
sFtFt
w˙ Ž t . s 0 and w˙ Ž s . k 0, s F s F t 4
such that w ˙ Ž t* . s 0 and
w Ž t* . ) w Ž s . ,
s y t F s - t*,
where w Ž t . s x Ž t .rCz Ž t .. Then substituting t s t* into Ž6., Lemma 1 and Ž7. imply a contradiction which completes our proof.
DELAY DIFFERENTIAL EQUATIONS
393
Remark 2. If z Ž t 0 , f .Ž t . in the proof of Theorem 2 is a positive solution of Eqs. Ž1. and Ž3. with f g PC, the theorem is also true. In the following text, we will suppose that if x Ž t 0 , f .Ž t . is a solution of Eqs. Ž1. and Ž3., x*Ž t 0 , f .Ž t . is a solution of Eqs. Ž2. and Ž3.. THEOREM 3.
Suppose that the hypotheses of Theorem 2 hold and xIj Ž x . G 0,
j s 1, 2, . . . .
Ž 11.
Then z*Ž t 0 , w .Ž t . G z Ž t 0 , w .Ž t ., t G t 0 . Proof. Let z k Ž t 0 , w .Ž t ., k s 1, 2, . . . , be solutions of the equations n
˙x Ž t . q
Ý pi Ž t . x Ž t y t i . s 0,
t / t j , t G t0
is1
x Ž tq j . y x Ž t j . s Ij Ž x Ž t j . . ,
j s 1, 2, . . . , k ,
and w g PC. Since Eq. Ž1. and the first equation above are linear equations, z1Ž t 0 , w .Ž t . y z Ž t 0 , w .Ž t . is a solution corresponding to t 1 of the initial problem Eqs. Ž1. and Ž3., denoted by z1Ž t 1 , f .Ž t ., where
fs
½
0, I1 Ž x Ž t 1 . . ,
yt F s - 0, s s 0.
Hence, Theorem 2 and Ž11. imply z1 Ž t 1 , f . Ž t . ) 0,
t ) t1 .
So z1 Ž t 0 , w . Ž t . G z Ž t 0 , w . Ž t . ,
t G t0 .
By induction, for t G t 0 , 0 - z Ž t 0 , w . Ž t . F z1 Ž t 0 , w . Ž t . F ??? F z* Ž t 0 , w . Ž t . . The proof of Theorem 3 is completed. Applying Theorem 2 and the method used in Theorem 3, we can similarly obtain the following results. THEOREM 4. Assume that Ž4. and Ž7. are satisfied. Let x Ž t 0 , w .Ž t . be any solution of Eqs. Ž1. and Ž3.. Then for t G t 0 , Ž1. Ž2.
if I j Ž x . G 0, j s 1, 2, . . . , x*Ž t 0 , w .Ž t . G x Ž t 0 , w .Ž t .; if I j Ž x . F 0, j s 1, 2, . . . , x*Ž t 0 , w .Ž t . F x Ž t 0 , w .Ž t ..
394
JIANG AND JURANG
COROLLARY 1. In Theorem 4, if I j Ž x . G 0 w or I j Ž x . F 0x and x Ž t 0 , w .Ž t . is e¨ entually positi¨ e Ž or negati¨ e ., then x*Ž t 0 , w .Ž t . is also e¨ entually positi¨ e Ž or negati¨ e ..
4. ASYMPTOTIC BEHAVIOR OF EQ. Ž2. THEOREM 5.
Assume that Eq. Ž2. satisfies the conditions `
n
Ht Ý p Ž s . ds s `,
Ž 12 .
i
0
is1
`
I j Ž x . F bj < x < and
Ý bj - `,
j s 1, 2, . . . ,
Ž 13 .
js1
where bj G 0. Then e¨ ery nonoscillatory solution of Eq. Ž2. tends to zero as t ª `. Proof. Without loss of generality, assume that z Ž t . is an eventually positive solution of Eq. Ž2.. Take a sequence tUk 4 of t j 4`1 such that Ik Ž z Ž tUk .. ) 0 and choose a corresponding sequence bUk 4 from bj 4`1 . Therefore, there exists a sufficiently large T G t 0 , such that z Ž t . ) 0 and
˙z Ž t . F 0,
t / tj , t G T ,
Ž 14 .
that is, z Ž t . is decreasing in Ž t j , t jq1 x for t j G T, j s m, m q 1, . . . , with m g N. It is easy to see that z Ž t . is also decreasing in Ž tUk , tUkq1 x for tUk G T, k G m. Hence for t G tUk , U U U Uq z Ž t . F z Ž tUq k . F Ž 1 q b k . z Ž t k . F Ž 1 q b k . z Ž t ky1 .
F Ž 1 q bUk . Ž 1 q bUky1 . z Ž tUky1 . U F Ž 1 q bUk . Ž 1 q bUky1 . ??? Ž 1 q bm . z Ž tUm . .
Ž 15 .
In view of Ž13., 0 - ŁŽ1 q bUk . - `. Thus, from Ž15., there exists a constant M ) 0 such that zŽ t. - M
for t G T .
Now, we claim that lim inf t ª` z Ž t . s 0. Otherwise, set lim inf z Ž t . s l ) 0. tª`
Ž 16 .
395
DELAY DIFFERENTIAL EQUATIONS
Then there exists T1 G T, such that z Ž t . G lr2 for t y t ) T1. So n
0s˙ zŽ t. q
Ý pi Ž t . z Ž t y t i . is1
G˙ zŽ t. q
n
l 2
Ý pi Ž t . . is1
Integration from t to ` with t G T1 yields lyM
Ý
bUk y z Ž t . q
kGm
l
`
n
H Ý p Ž s . ds F 0, 2 t i
is1
which, in view of Ž12. and Ž13., implies a contradiction that complete our claim. Next, to prove that lim sup t ª` z Ž t . s 0, from Ž14. we can choose a subsequence j k 4`1 from tUk 4`m such that lim z Ž j k . s 0,
Ž 17 .
kª`
4` Uq 4`m between j k and and similarly find another subsequence hq k 1 of t k q j kq 1 , k s 1, 2, . . . , such that lim k ª` z Žhk . s lim sup t ª` z Ž t .. Assume that bUk , kU correspond to the moments j k , hk of impulsive effect, respectively. Then from Ž2. and Ž14., it follows from U U q 0 - z Ž hq k . F Ž 1 q b k . z Ž h k . F Ž 1 q b k . z Ž h ky1 .
F Ž 1 q bUk . Ž 1 q bUky1 . ??? Ž 1 q bUk . z Ž j k . ,
k s 1, 2, . . . ,
. lim k ª` z Žhq k
and Ž17. that s 0. Therefore, z Ž t . ª 0 as t ª `. The proof of Theorem 5 is completed. Theorem 1 in w7x is a special case of Theorem 5 here. Readers interested in it may combine it with Theorem 4 and obtain many results for themselves. LEMMA 2. Assume Ž7. and Ž11. hold. Let z Ž t . be a nonoscillatory solution of Eq. Ž1.. Let x Ž t . be any oscillatory solution of Eq. Ž2.. Then there exists K ) 0 such that e¨ entually xŽ t. F K zŽ t. . Proof. Without loss of generality, suppose that z Ž t . ) 0 for t G T. Hence, using the function w introduced in Lemma 1, we have to prove that w is bounded. Assume that it is not true. From the definition of w Ž t ., it is easy to see that Ž11. holds if and only if w Ž t j . Ž w Ž tq j . y w Ž t j . . G 0,
j s 1, 2, . . . .
396
JIANG AND JURANG
Moreover, noting that w is an oscillatory function, there exists t* G T q t such that either w Ž t* . F 0 and
w Ž t* . ) w Ž s .
for T F s - t*,
w ˙ Ž t* . G 0 and
w Ž t* . - w Ž s .
for T F s - t*,
or
4 Ž . in which t* may be an element of tq j . Substituting t* s t into 6 , Lemma 1 and Ž7. imply a contradiction. COROLLARY 2. Assume that Ž4., Ž7., and Ž11. ] Ž13. are satisfied. Then e¨ ery solution of Eq. Ž2. tends to zero as t ª `.
REFERENCES 1. O. Arino, G. Ladas, and Y. G. Sficas, On oscillations of some retarded differential equations, SIAM J. Math. Anal. 18 Ž1987., 64]73. 2. M. P. Chen, J. S. Yu, and J. H. Shen, The persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations, Computer Math. Anal. 27 Ž1994., 1]6. 3. K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl. 139 Ž1989., 110]122. 4. I. Gyori ¨ and G. Ladas, ‘‘Oscillation Theory of Delay Differential Equations with Applications,’’ Clarendon Press, Oxford, 1991. 5. M. R. S. Kulenovic, G. Ladas, and A. Meimaridou, Stability of solutions of linear delay differential equations, Proc. Amer. Math. Soc. 100 Ž1987., 433]441. 6. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, ‘‘Theory of Impulsive Differential Equations,’’ World Scientific, Singapore, 1989. 7. J. H. Shen and Z. C. Wang, Oscillation and asymptotic behavior of solutions of delay differential equations with impulses, Ann. Differential Equations 10 Ž1994., 61]68.
207, 388]396 Ž1997.
AY975276
Positive Solutions and Asymptotic Behavior of Delay Differential Equations with Nonlinear Impulses* Yu Jiang and Yan Jurang Department of Mathematics, Shanxi Uni¨ ersity, Taiyuan 030006, Shanxi, People’s Republic of China Communicated by William F. Ames Received January 19, 1996
Consider the delay differential equation ŽDDE. with nonlinear impulses n
˙x Ž t . q
Ý pi Ž t . x Ž t y t i . s 0,
t / t j , t G t0 ,
is1
x Ž tq j . y x Ž t j . s Ij Ž x Ž t j . . ,
j s 1, 2, . . . ,
Ž).
q.
where pi g C Žw t 0 , `., R , t i G 0 for i s 1, 2, . . . , n, and I j g C ŽR, R. for j s 1, 2, . . . . The purpose of this paper is to obtain a necessary and sufficient condition for the existence of positive solutions of DDE without impulses and to establish a kind of order persistence of solutions of Eq. Ž). and criteria of the asymptotic behavior of Eq. Ž)., which can be used to improve and develop some of the known results in the literature. Q 1997 Academic Press
1. INTRODUCTION There have been many papers considering the delay differential equation n
˙x Ž t . q Ý pi Ž t . x Ž t y t i . s 0,
Ž 1.
is1
where pi g C Žw t 0 , `., Rq. , t i G 0 for i s 1, 2, . . . , n; see, for example, w1, 4, 5x and the references cited in w4x. There are only a few papers concerned with the impulsive delay differential equations, which is an important mathematical model of many evolution process; see w2, 3, 6, 7x. *This work was partially supported by the National Natural Science Foundation of China. 388 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
DELAY DIFFERENTIAL EQUATIONS
389
In this paper, we consider the delay differential equations ŽDDE. with nonlinear impulses, n
˙x Ž t . q Ý pi Ž t . x Ž t y t i . s 0,
t / t j , t G t0 ,
is1
x Ž tq j . y x Ž t j . s Ij Ž x Ž t j . . ,
j s 1, 2, . . . ,
Ž 2.
where pi g C Žw t 0 , `., Rq. , t i G 0 for i s 1, 2, . . . , n, and I j g C ŽR, R. for j s 1, 2, . . . . Let t s maxt 1 , t 2 , . . . , tn4 and t 0 - t 1 - t 2 - ??? , t j ª ` as j ª `. Set PC s f : w yt , 0 x ª R is piecewise left continuous 4 . With Eq. Ž2., one associates an initial condition of the form xs s f Ž s . ,
s g w yt , 0 x ,
Ž 3.
where xs s x Ž s q s . and f g PC. DEFINITION 1. x Ž t . is called a solution corresponding to t 0 of Eq. Ž2. if x: w t 0 y t , `. ª R is continuous for t / t j ; x is continuously differentiable . Ž y. exist and x Ž ty . s x Ž t j ., for t G t 0 , t / t j , and t / t j q t i , and x Ž tq j , x tj j Ž . j s 1, 2, . . . , and x satisfies Eq. 2 . DEFINITION 2. x Ž t . is called a solution corresponding to t 0 of the initial value problem Ž2. and Ž3. if x is a solution corresponding to t 0 of Eq. Ž2. and satisfies Ž3., denoted by x Ž t 0 , f .Ž t .. DEFINITION 3. x Ž t 0 , f .Ž t . is called a positive solution of Eq. Ž2. if f G 0, s g wyt , 0x, and x Ž t 0 , f .Ž t . ) 0, t G t 0 . DEFINITION 4. A solution x Ž t . of Eq. Ž2. is said to be nonoscillatory if it is eventually positive or eventually negative; otherwise it will be called oscillatory. Analogously, we can define the various solutions of Eq. Ž1.. Our aim in this paper is to obtain results on existence and asymptotic behavior of nonoscillatory solutions of Eq. Ž2.. This paper is organized as follows. In Section 2, we show a necessary and sufficient condition ensuring that there exist positive solutions in Eq. Ž1., the special case of which includes the corresponding results in w4x. Furthermore, under the conditions we establish in Section 3, several new results which assert that there exists a kind of order relation between the solutions of Eq. Ž1. and those of Eq. Ž2., that is, the differences between the solutions of Ž1. and the corresponding ones of Ž2., are always positive or negative. Finally, some criteria of asymptotic behavior of Eq. Ž2. are given in Section 4, which improve and generalize the known criteria w7x.
390
JIANG AND JURANG
2. POSITIVE SOLUTIONS OF EQ. Ž1. Let h i Ž t . s min t 0 , t y t i 4 , Hi Ž t . s max t 0 , t y t i 4 , and F s f g C Ž w yt , 0 x , Rq . < f Ž 0 . ) 0 and f Ž s . F f Ž 0 . , yt F s F 0 4 . Our aim in this section is to establish the following necessary and sufficient condition for the positive solutions of Eq. Ž1. For Eq. Ž1., the following statements are equi¨ alent:
THEOREM 1.
There exist k i Ž t . g C Žw t 0 , `., Rq. , i s 1, 2, . . . , n, such that
Ž a.
n
t
HH Ž t . Ý p Ž s . k Ž s . ds F ln k Ž t . , i
j
Ž b.
i
t G t 0 , j s 1, 2, . . . , n.
j
Ž 4.
is1
For ; f g F, x Ž t 0 , f .Ž t . is a positi¨ e solution of Eqs. Ž1. and Ž3..
Proof. Ž a. « Ž b . Assume that Ža. holds. Set n
b Ž t . s y Ý pi Ž t . k i Ž t . ,
g Ž t . s 0,
for t G t 0 .
is1
For all d g C Žw t 0 , `., R. such that b Ž t . F d Ž t . F g Ž t ., t G t 0 , it follows from Ž4. that t
n
t
HH Ž t . d Ž s . ds F HH Ž t . Ý p Ž s . k Ž s . ds F ln k Ž t . ,
y
i
j
j
i
j
is1
where t G t 0 , j s 1, 2, . . . , n. Hence, for ; f g F, we have n
g Ž t . s 0 G y Ý pi Ž t .
f Ž h i Ž t . y t0 . f Ž 0.
is1
ž
t
HH Ž t . d Ž s . ds
exp y
i
/
n
G y Ý pi Ž t . k i Ž t . s b Ž t . . is1
In view of Theorem 3.1.1 in w4x, x Ž t 0 , f .Ž t . is a positive solution corresponding to t 0 of Eqs. Ž1. and Ž3.. Žb. « Ža. Assume that y Ž t 0 , f .Ž t . is a positive solution of Eq. Ž1.. Let Ž y t . s y Ž t 0 , f .Ž t . and
a Ž t. s
˙y Ž t . , yŽ t.
t G t0 .
So y Ž t . s f Ž 0 . exp
t
žH
t0
a Ž s . ds ,
/
t G t0 .
391
DELAY DIFFERENTIAL EQUATIONS
Hence, it follows from Theorem 3.1.1 in w4x that n
a Ž t. q
Ý pi Ž t .
f Ž h i Ž t . y t0 . f Ž 0.
is1
exp
t
žH
H iŽ t .
a Ž s . ds s 0.
/
Ž 5.
Set t
HH Ž t . a Ž s . ds
k i Ž t . s exp y
ž
i
/
.
Integrating Ž5. from Hi Ž t . to t, we get n
t
HH Ž t . Ý p Ž s .
f Ž h i Ž s . y t0 . f Ž 0.
i
j
is1
s
k i Ž s . ds
n
t
t
HH Ž t . Ý p Ž s . k Ž s . ds s yHH Ž t . a Ž s . ds i
j
i
is1
j
s ln k j Ž t . ,
t G t 0 , j s 1, 2, . . . , n
which completes the proof of Theorem 1. Remark 1. If k i Ž t . ' e, it is easy from Ž4. to see that n
t
1
HH Ž t . Ý p Ž s . ds F e , i
j
t G t0 ,
is1
which is a well-known result.
3. SOME PROPERTIES OF SOLUTIONS OF EQ. Ž2. This section is devoted to the existence of positive and nonoscillatory solutions of Eq. Ž2., which are based on our comparison result established here. It is interesting in itself and needed in the following lemma, which is a clear extension of Lemma 1 in w5x. LEMMA 1. Assume that z Ž t . is a positi¨ e solution of Eq. Ž1. and x Ž t . is any solution of Eq. Ž2.. Set w Ž t . s x Ž t . rz Ž t . ,
t G t0 .
Then for t G t 0 , t / t j , t / t j q t i , j s 1, 2, . . . , and i s 1, 2, . . . , n, n
w ˙Ž t. s
Ý pi Ž t . is1
z Ž t y ti . zŽ t.
Ž w Ž t . y w Ž t y ti . . .
Ž 6.
392
JIANG AND JURANG
THEOREM 2.
Assume that n
pi Ž t . G 0
for i s 1, 2, . . . , n and
Ý pi Ž t . ) 0,
t G t0 , Ž 7.
is1
and Eq. Ž1. has a positi¨ e solution z Ž t 0 , w .Ž t . corresponding to t 0 . Then for f g PC such that
fs
½
0, b,
yt F s - 0, s s 0,
Ž 8.
where b / 0, bx Ž s , f . Ž t . ) 0,
tGs,
where x Ž s , f .Ž t . is a solution corresponding to s with s G t 0 of Eqs. Ž1. and Ž3.. Proof. By Theorem 1, without loss of generality, assume that z Ž t . ) 0 for t G t 0 y t . Consider the case of b ) 0 Žthe case of b - 0 can be treated in a similar way.. Since Eq. Ž1. is a linear equation, it is easy to see that there exists a constant C ) 0 such that Cz Ž t . is a positive solution of Eq. Ž1. and satisfies Cz Ž s . s b.
Ž 9.
We have to prove x Ž s , f .Ž t . ) 0 for t G s . Note that Cz Ž t . G f for t g w s y t , s x and it follows from Eq. Ž1. that there exists a constant T ) s such that x Ž s , f . Ž t . G Cz Ž t . ,
s F t F T.
Ž 10 .
We claim that T s `. Otherwise, there exists t ) s such that x Ž s , f . Ž t . s Cz Ž t . . In view of Ž9. and Ž10., we can choose t* s inf
sFtFt
w˙ Ž t . s 0 and w˙ Ž s . k 0, s F s F t 4
such that w ˙ Ž t* . s 0 and
w Ž t* . ) w Ž s . ,
s y t F s - t*,
where w Ž t . s x Ž t .rCz Ž t .. Then substituting t s t* into Ž6., Lemma 1 and Ž7. imply a contradiction which completes our proof.
DELAY DIFFERENTIAL EQUATIONS
393
Remark 2. If z Ž t 0 , f .Ž t . in the proof of Theorem 2 is a positive solution of Eqs. Ž1. and Ž3. with f g PC, the theorem is also true. In the following text, we will suppose that if x Ž t 0 , f .Ž t . is a solution of Eqs. Ž1. and Ž3., x*Ž t 0 , f .Ž t . is a solution of Eqs. Ž2. and Ž3.. THEOREM 3.
Suppose that the hypotheses of Theorem 2 hold and xIj Ž x . G 0,
j s 1, 2, . . . .
Ž 11.
Then z*Ž t 0 , w .Ž t . G z Ž t 0 , w .Ž t ., t G t 0 . Proof. Let z k Ž t 0 , w .Ž t ., k s 1, 2, . . . , be solutions of the equations n
˙x Ž t . q
Ý pi Ž t . x Ž t y t i . s 0,
t / t j , t G t0
is1
x Ž tq j . y x Ž t j . s Ij Ž x Ž t j . . ,
j s 1, 2, . . . , k ,
and w g PC. Since Eq. Ž1. and the first equation above are linear equations, z1Ž t 0 , w .Ž t . y z Ž t 0 , w .Ž t . is a solution corresponding to t 1 of the initial problem Eqs. Ž1. and Ž3., denoted by z1Ž t 1 , f .Ž t ., where
fs
½
0, I1 Ž x Ž t 1 . . ,
yt F s - 0, s s 0.
Hence, Theorem 2 and Ž11. imply z1 Ž t 1 , f . Ž t . ) 0,
t ) t1 .
So z1 Ž t 0 , w . Ž t . G z Ž t 0 , w . Ž t . ,
t G t0 .
By induction, for t G t 0 , 0 - z Ž t 0 , w . Ž t . F z1 Ž t 0 , w . Ž t . F ??? F z* Ž t 0 , w . Ž t . . The proof of Theorem 3 is completed. Applying Theorem 2 and the method used in Theorem 3, we can similarly obtain the following results. THEOREM 4. Assume that Ž4. and Ž7. are satisfied. Let x Ž t 0 , w .Ž t . be any solution of Eqs. Ž1. and Ž3.. Then for t G t 0 , Ž1. Ž2.
if I j Ž x . G 0, j s 1, 2, . . . , x*Ž t 0 , w .Ž t . G x Ž t 0 , w .Ž t .; if I j Ž x . F 0, j s 1, 2, . . . , x*Ž t 0 , w .Ž t . F x Ž t 0 , w .Ž t ..
394
JIANG AND JURANG
COROLLARY 1. In Theorem 4, if I j Ž x . G 0 w or I j Ž x . F 0x and x Ž t 0 , w .Ž t . is e¨ entually positi¨ e Ž or negati¨ e ., then x*Ž t 0 , w .Ž t . is also e¨ entually positi¨ e Ž or negati¨ e ..
4. ASYMPTOTIC BEHAVIOR OF EQ. Ž2. THEOREM 5.
Assume that Eq. Ž2. satisfies the conditions `
n
Ht Ý p Ž s . ds s `,
Ž 12 .
i
0
is1
`
I j Ž x . F bj < x < and
Ý bj - `,
j s 1, 2, . . . ,
Ž 13 .
js1
where bj G 0. Then e¨ ery nonoscillatory solution of Eq. Ž2. tends to zero as t ª `. Proof. Without loss of generality, assume that z Ž t . is an eventually positive solution of Eq. Ž2.. Take a sequence tUk 4 of t j 4`1 such that Ik Ž z Ž tUk .. ) 0 and choose a corresponding sequence bUk 4 from bj 4`1 . Therefore, there exists a sufficiently large T G t 0 , such that z Ž t . ) 0 and
˙z Ž t . F 0,
t / tj , t G T ,
Ž 14 .
that is, z Ž t . is decreasing in Ž t j , t jq1 x for t j G T, j s m, m q 1, . . . , with m g N. It is easy to see that z Ž t . is also decreasing in Ž tUk , tUkq1 x for tUk G T, k G m. Hence for t G tUk , U U U Uq z Ž t . F z Ž tUq k . F Ž 1 q b k . z Ž t k . F Ž 1 q b k . z Ž t ky1 .
F Ž 1 q bUk . Ž 1 q bUky1 . z Ž tUky1 . U F Ž 1 q bUk . Ž 1 q bUky1 . ??? Ž 1 q bm . z Ž tUm . .
Ž 15 .
In view of Ž13., 0 - ŁŽ1 q bUk . - `. Thus, from Ž15., there exists a constant M ) 0 such that zŽ t. - M
for t G T .
Now, we claim that lim inf t ª` z Ž t . s 0. Otherwise, set lim inf z Ž t . s l ) 0. tª`
Ž 16 .
395
DELAY DIFFERENTIAL EQUATIONS
Then there exists T1 G T, such that z Ž t . G lr2 for t y t ) T1. So n
0s˙ zŽ t. q
Ý pi Ž t . z Ž t y t i . is1
G˙ zŽ t. q
n
l 2
Ý pi Ž t . . is1
Integration from t to ` with t G T1 yields lyM
Ý
bUk y z Ž t . q
kGm
l
`
n
H Ý p Ž s . ds F 0, 2 t i
is1
which, in view of Ž12. and Ž13., implies a contradiction that complete our claim. Next, to prove that lim sup t ª` z Ž t . s 0, from Ž14. we can choose a subsequence j k 4`1 from tUk 4`m such that lim z Ž j k . s 0,
Ž 17 .
kª`
4` Uq 4`m between j k and and similarly find another subsequence hq k 1 of t k q j kq 1 , k s 1, 2, . . . , such that lim k ª` z Žhk . s lim sup t ª` z Ž t .. Assume that bUk , kU correspond to the moments j k , hk of impulsive effect, respectively. Then from Ž2. and Ž14., it follows from U U q 0 - z Ž hq k . F Ž 1 q b k . z Ž h k . F Ž 1 q b k . z Ž h ky1 .
F Ž 1 q bUk . Ž 1 q bUky1 . ??? Ž 1 q bUk . z Ž j k . ,
k s 1, 2, . . . ,
. lim k ª` z Žhq k
and Ž17. that s 0. Therefore, z Ž t . ª 0 as t ª `. The proof of Theorem 5 is completed. Theorem 1 in w7x is a special case of Theorem 5 here. Readers interested in it may combine it with Theorem 4 and obtain many results for themselves. LEMMA 2. Assume Ž7. and Ž11. hold. Let z Ž t . be a nonoscillatory solution of Eq. Ž1.. Let x Ž t . be any oscillatory solution of Eq. Ž2.. Then there exists K ) 0 such that e¨ entually xŽ t. F K zŽ t. . Proof. Without loss of generality, suppose that z Ž t . ) 0 for t G T. Hence, using the function w introduced in Lemma 1, we have to prove that w is bounded. Assume that it is not true. From the definition of w Ž t ., it is easy to see that Ž11. holds if and only if w Ž t j . Ž w Ž tq j . y w Ž t j . . G 0,
j s 1, 2, . . . .
396
JIANG AND JURANG
Moreover, noting that w is an oscillatory function, there exists t* G T q t such that either w Ž t* . F 0 and
w Ž t* . ) w Ž s .
for T F s - t*,
w ˙ Ž t* . G 0 and
w Ž t* . - w Ž s .
for T F s - t*,
or
4 Ž . in which t* may be an element of tq j . Substituting t* s t into 6 , Lemma 1 and Ž7. imply a contradiction. COROLLARY 2. Assume that Ž4., Ž7., and Ž11. ] Ž13. are satisfied. Then e¨ ery solution of Eq. Ž2. tends to zero as t ª `.
REFERENCES 1. O. Arino, G. Ladas, and Y. G. Sficas, On oscillations of some retarded differential equations, SIAM J. Math. Anal. 18 Ž1987., 64]73. 2. M. P. Chen, J. S. Yu, and J. H. Shen, The persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations, Computer Math. Anal. 27 Ž1994., 1]6. 3. K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl. 139 Ž1989., 110]122. 4. I. Gyori ¨ and G. Ladas, ‘‘Oscillation Theory of Delay Differential Equations with Applications,’’ Clarendon Press, Oxford, 1991. 5. M. R. S. Kulenovic, G. Ladas, and A. Meimaridou, Stability of solutions of linear delay differential equations, Proc. Amer. Math. Soc. 100 Ž1987., 433]441. 6. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, ‘‘Theory of Impulsive Differential Equations,’’ World Scientific, Singapore, 1989. 7. J. H. Shen and Z. C. Wang, Oscillation and asymptotic behavior of solutions of delay differential equations with impulses, Ann. Differential Equations 10 Ž1994., 61]68.