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Oscillation Caused By Impulses
Journal of Mathematical Analysis and Applications 255, 163᎐176 Ž2001. doi:10.1006rjmaa.2000.7218, available online at http:rrwww.idealibrary.com on
Oscillation Caused By Impulses Mingshu Peng Department of Mathematics, Beijing Normal Uni¨ ersity, Beijing 100875, People’s Republic of China; and Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China Submitted by William F. Ames Received December 1, 1998
The present paper is devoted to the investigation of the oscillation of a kind of very extensively studied second order nonlinear delay differential equations with impulses, some interesting results are obtained, which illustrate that impulses play a very important role in giving rise to the oscillations of equations. 䊚 2001 Academic Press
Key Words: oscillation; impulse; delay differential equation; nonlinearity.
Consider the impulsive delay differential equation
Ž aŽ t .
xX Ž t .
␣ y1 X
X
x Ž t . . q f Ž t , x Ž t . , x Ž t y . . s 0,
ˆ X xX Ž tq k . s Ik Ž x Ž t k . .
x Ž tq k . s Ik Ž x Ž t k . . ,
t / tk
Ž 1.
where ␣ , ) 0, 0 - t 1 - t 2 - ⭈⭈⭈ - t k - ⭈⭈⭈ and lim t ª⬁ t k s ⬁. Suppose that xX Ž t k . s xX Ž ty k . s lim
x Ž tk q h . y x Ž tk . h
hªy0
and xX Ž tq k . s lim
x Ž t k q h . y x Ž tq k . h
hªq0
.
163 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
164
MINGSHU PENG
Throughout the paper, assume that the following conditions hold: Ži. f Ž t, u, ¨ . is continuous in w t 0 y , q⬁. = Žy⬁, q⬁. = Žy⬁, q⬁., where t 0 G 0, uf Ž t, u, ¨ . ) 0 Ž u¨ ) 0. and f Žt ,Žu¨,.¨ . G pŽ t . Ž ¨ / 0., where pŽ t . is continuous in w t 0 y , q⬁., pŽ t . G 0, and x Ž x . ) 0 Ž x / 0., X Ž x . G 0; Žii. Ik Ž x . and Iˆk Ž x . are continuous in Žy⬁, q⬁.; there exist positive numbers c k , cUk , d k , dUk such that cUk F Ik Ž x .rx F c k , d k F Iˆk Ž x .rx F dUk ; Žiii. aŽ t . is a positive continuous function in w t 0 y , q⬁. and AŽ t . t s Ht 0 dsra1r ␣ Ž s .. Recently, there has been increasing interest on the oscillationrnonoscillation of the first order linear delay differential equations with impulses Žsee paper w1᎐6x., and good results have been obtained. But few are on the second order nonlinear delay differential equations with impulses, e.g., w7, 8x, etc. The present paper is devoted to the study of the oscillation of a type of very extensive second order nonlinear delay differential equations with impulses. Some interesting results are gained here. In addition, some examples show that, though some delay differential equations without impulses are non-oscillatory, they may become oscillatory if some impulses are added to them. That is, in some cases, impulses play a dominating role in the oscillations of equations. As to some related results about the oscillation of some second-order nonlinear ODE with impulses, we refer to paper w9x by Chen and Feng. For the theory of delay differential equations and impulsive differential equations, please see the recent books by Gyori ¨ and Ladas w10x and Lakshmikantham et al. w11x, respectively. We introduce the notation as follows: PC s x: w  y ,  x ª R ¬ x Ž t . is twice continuously differentiable for t g w  y ,  x_ t k , k s 1, 2, . . . 4 , X . Ž y. X Ž q. X Ž y. exist and x Ž ty . Ž . X Ž y. Ž . x Ž tq k , x tk , x tk , x tk k s x t k , x t k s x t k for t k g w  y ,  x4 . ⍀ s x: w  y , q⬁. ª R ¬ x Ž t . is continuous first for t / t k , x Ž tq k , y y x Ž t k . exist and x Ž t k . s x Ž t k ., x Ž t . is continuously differentiable for t G  , X q . X Ž y. Ž . XŽ y . exist and t / t k , t / t k q and xX Ž tq k , x tk , x tk q , x tk q X X y x Ž t k . s x Ž t k .4 . For any  G 0, g PC , a function x: w  y , q⬁. ª R is called a solution of Eq. Ž1. satisfying the initial value condition xŽ t. s Ž t. , if x g ⍀ and satisfies Ž1. and Ž2..
tg w y, x
Ž 2.
165
OSCILLATION CAUSED BY IMPULSES
Using the method of steps, one can show that for any ) 0 and g PC , the initial value problem Ž1., Ž2. has a unique solution x g ⍀ . A solution of Ž1. is said to be non-oscillatory if this solution is eventually positive or eventually negative. Otherwise, this solution is said to be oscillatory. This paper is organized as follows. In Section 1 we shall offer two interesting lemmas, which will be used in Section 2 to prove our main theorems. To illustrate our results, three examples are also included in Section 3.
1. SOME LEMMAS LEMMA 1 w9, Theorem 1.4.1x. ŽA 0 . ŽA 1 .
Assume that
m g PC 1w Rq, R x and mŽ t . is left-continuous at t k , k s 1, 2, . . . For k s 1, 2, . . . , t G t 0 , mX Ž t . F p Ž t . m Ž t . q q Ž t . , mŽ
tq k
t / tk
. F d k m Ž t k . q bk
where q, p g PC 1 w Rq, R x, d k G 0 and bk are constants. Then mŽ t . F mŽ t0 . q
Ł
t 0-t k -t
Ý
t 0 -t k -t
q
ž
d k exp
Ł
t k-t j-t
t
Ht s-tŁ-t d 0
k
exp
t
žH
d j exp t
žH s
k
p Ž s . ds
t0
ž
/
t
Ht p Ž s . ds k
//
bk
p Ž . d q Ž s . ds,
/
t G t0 .
LEMMA 2. Let x Ž t . be a solution of Eq. Ž1.. Suppose that there exists some T G t 0 such that x Ž t . ) 0 for t G T. If 1
q⬁
Ht
j
1r ␣
a
Ł Ž s . t -t -s j
l
di ci
ds s q⬁
for some t j ŽG t 1 ., then xX Ž tq k . G0 for t g Ž t k , t kq1 x, where t k G T.
and
xX Ž t . G 0
Ž 3.
166
MINGSHU PENG
Proof. At first, we prove that xX Ž t k . G 0 for any t k G T. If not, then . s Iˆj Ž xX Ž t j .. F there exists some j such that t j G T, xX Ž t j . - 0, and xX Ž tq j X d j x Ž t j . - 0. Let X q a Ž tq j . x Ž tj .
␣y1 X
␣ x Ž tq j . s y Ž  ) 0 .
S Ž t . s a Ž t . xX Ž t .
and
␣y1 X
x Ž t. .
By Ž1., for t g Ž t jqiy1 , t jqi x, i s 1, 2, . . . , we have
Ž aŽ t .
X
␣ y1 X
xX Ž t .
x Ž t . . s yf Ž t , x Ž t . , x Ž t y . . F yp Ž t . Ž x Ž t y . . F 0.
Ž 4.
Hence, SŽ t . is monotonically decreasing in Ž t jqiy1 , t jqi x. So we have a Ž t jq1 . xX Ž t jq1 .
␣ y1 X
X q x Ž t jq1 . F a Ž tq j . x Ž tj .
␣y1 X
␣ x Ž tq j . s y - 0
and a Ž t jq2 . xX Ž t jq2 .
␣ y1 X
X q x Ž t jq2 . F a Ž tq jq1 . x Ž t jq1 .
␣y1 X
x Ž tq jq1 .
s a Ž t jq1 . Iˆjq1 Ž xX Ž t jq1 . . ␣
␣y1
F Ž d jq1 . a Ž t jq1 . xX Ž t jq1 .
Iˆjq1 Ž xX Ž t jq1 . .
␣ y1 X
x Ž t jq1 .
␣
F y Ž d jq1 .  ␣ - 0. By induction, we obtain a Ž t . xX Ž t .
␣
␣ y1 X
x Ž t . F y Ž d jq1 d jq2 ⭈⭈⭈ d jqn .  ␣ s y ␣
Ł
t j-t k-t
d k␣ - 0,
Ž 5. for t g Ž t jqn , t jqnq1 x. Therefore
 X
x Ž t. F y
Ł
t j-t k-t
dk
a1r ␣ Ž t .
.
167
OSCILLATION CAUSED BY IMPULSES
. Ž . In view of condition Žii., we have x Ž tq k F c k x t k . Applying Lemma 1, we obtain x Ž t . F x Ž tq j .
Ł
t j-t k-t
ck
t
q
Ht s-tŁ-t c t -tŁ-s d a k
j
i
k
j
1
1r ␣
1
Ž s.
t ) tj .
ds,
Ž 6.
In view of the fact that Ł t j - t k - t c k s Ł t j - t l - s c i Ł s - t l - t c l , we have xŽ t. F
Ł
t j-t k-t
½
c k x Ž tq k . y
1
t
Ht
j
1r ␣
a
Ł Ž s . t -t -s j
l
di
5
ds ,
ci
t ) t j . Ž 7.
Since x Ž t k . ) 0 Ž t k G T ., one can find that Ž7. contradicts Ž3. as t ª ⬁. Therefore xX Ž t k . G 0
Ž tk G T . .
X Ž . . By condition Žii., we have, for any t k G T, xX Ž tq k G d k x t k G 0. Because SŽ t . is decreasing in Ž t jqiy1 , t jqi x, we get, for t g Ž t jqiy1 , t jqi x, SŽ t . G 0, which implies xX Ž t . G 0. The proof of this lemma is complete.
Remark 1. In the case that x Ž t . is eventually negative, if Ž3. holds true, X . Ž . Ž x then xX Ž tq k F 0 and x t F 0, for t g t jqiy1 , t jqi where t k G T. 2. MAIN RESULTS THEOREM 1. Assume that Ž3. holds and there exists a positi¨ e integer k 0 such that cUk G 1 for k G k 0 . If 1
q⬁
Ht
Ł
t 0-t 0, n -s
0
0, n
p Ž s . ds s q⬁,
Ž 8.
where
¡1
0, n s
~Ž
¢Ž
t 0, n s t k q / t m dUk
. .
␣
␣ dUm
t 0, n s t k t 0, n s t k q s t m
Ž m ) k. Ž 9.
168
MINGSHU PENG
and t 0, n s t k or t k q Ž t 1 s t 0, 1 - t 0, 2 - ⭈⭈⭈ - t 0, n - t 0, nq1 - ⭈⭈⭈ ., then e¨ ery solution of Ž1. is oscillatory. Proof. Without loss of generality, we can assume k 0 s 1. If Ž1. has a non-oscillatory solution x Ž t ., we might as well assume that x Ž t . ) 0 Ž t G t 0 .. It follows from Lemma 2 that xX Ž t . G 0 for t g Ž t k , t kq1 x, where k s 1, 2, . . . . Let
wŽ t. s
a Ž t . xX Ž t .
␣y1 X
x Ž t.
Ž xŽ t y . .
.
Ž 10 .
. Ž . Ž . Ž. Then w Ž tq k G 0 k s 1, 2, . . . , w t G 0 for t G t 0 . Using condition i and Ž . Eq. 1 , we get
wX Ž t . s y
y
f Ž t, xŽ t. , xŽ t y . .
Ž xŽ t y . . a Ž t . xX Ž t .
␣y1 X
x Ž t . X Ž x Ž t y . . xX Ž t y .
2 Ž xŽ t y . .
F yp Ž t . . Ž 11 .
It follows from the continuity of aŽ t ., condition Žii., cUk G 1, and X Ž x . G 0 that
w Ž tq k . s
X q a Ž tq k . x Ž tk .
¡Ž d F
~ Žd
␣y1 X
x Ž tq k .
Ž x Ž tq k y.. ␣
␣ y1
␣
␣ y1 X
U k
xX Ž t k . . a Ž t k . xX Ž t k . ␣ s Ž dUk . w Ž t k . Ž x Ž tk y . . tk y / tm Ž 0 - m - k .
U k
. a Ž t k . xX Ž t k . Ž cUm x Ž t m . . ␣
¢
x Ž tk . ␣ y1
xX Ž t k . Ž dUk . aŽ t k . xX Ž t k . ␣ F s Ž dUk . w Ž t k . Ž x Ž tk y . . tk y s tm Ž 0 - m - k .
Ž 12 .
169
OSCILLATION CAUSED BY IMPULSES
and ␣y1 X
X q a Ž tq k q . x Ž tk q .
w Ž tq k q. s
¡aŽ t
x Ž tq k q.
Ž x Ž tq k .. ␣y1 X
q . xX Ž tq k q.
k
x Ž tq k q.
Ž cUk x Ž t k . . F
a Ž t k q . xX Ž t k q .
␣ y1 X
x Ž tk q .
Ž x Ž tk . .
s w Ž tk q . tk q / tm F
~ aŽ t
Ž m ) k.
␣ y1
m
Iˆm Ž xX Ž t m . . . Iˆm Ž xX Ž t m . . Ž cUk x Ž t k . . ␣
␣ y1 X
Ž dUm . aŽ t m . xX Ž t m . F Ž cUk x Ž t k . .
x Ž tm .
␣
F
¢
Ž 13 .
Ž dUm . aŽ t k q . xX Ž t k q . Ž x Ž tk . .
␣ y1 X
x Ž tk q .
␣
s Ž dUm . w Ž t k q . tk q s tm
Ž m ) k.
It follows from inequalities Ž11. ᎐ Ž13. that wX Ž t . F yp Ž t . ,
t / t 0, n
w Ž tq 0, n . F 0, n w Ž t 0, n . , where t 0, n s t k or t k q Ž t 1 s t 0, 1 - t 0, 2 - ⭈⭈⭈ - t 0, n - t 0, nq1 - ⭈⭈⭈ . and 0, n is defined by Ž9.. Then, applying Lemma 1, we obtain wŽ t. F
Ł
t 0-t 0, n -t
½
0, n w Ž tq 0 . y
t
Ht t -tŁ-s 0
0
0, n
1 0, n
5
p Ž s . ds ,
t G t 0 Ž 14 .
In view of Ž8., Ž14., and w Ž t . G 0, we get a contradiction as t ª ⬁. Hence, every solution of Ž1. is oscillatory. The proof of Theorem 1 is complete.
170
MINGSHU PENG
Assume that Ž3. holds and Ž ab. G Ž a. Ž b . for any
THEOREM 2. ab ) 0. If
1
q⬁
Ht
Ł
t 0-t 0, n -s
0
0, n
p Ž s . ds s q⬁,
Ž 15 .
where
¡Ž d
U ␣ m cUk ␣ dUk
.
Ž
0, n
s~
t 0, n s t k q s t m
.
Ž
t 0, n s t k
.
1
Ž
cUk
dUk
¢ Ž c Ž
.
U m
and t k y / t m
t 0, n s t k q / t m
.
Ž m ) k.
Ž 16 .
Ž m ) k.
␣
t 0, n s t k
.
and t k y s t m
Ž0 - m - k. ,
then e¨ ery solution of Ž1. is oscillatory. Proof. If Ž1. has a non-oscillatory solution x Ž t ., without loss of generality, we can assume x Ž t . ) 0 Ž t G t 0 .. Let w Ž t . be defined by Ž10.. Then w Ž tq k . G0
Ž k s 1, 2, . . . . , w Ž t . G 0 Ž t G t 0 . .
It is easy to see that
wŽ
tq k
.s
X q a Ž tq k . x Ž tk .
¡Ž d F
~ Žd
␣y1 X
x Ž tq k .
Ž x Ž tq k y.. ␣
␣ y1
␣
␣ y1
U k
xX Ž t k . . a Ž t k . xX Ž t k . ␣ s Ž dUk . w Ž t k . Ž x Ž tk y . . tk y / tm Ž 0 - m - k .
U k
xX Ž t k . . a Ž t k . xX Ž t k . Ž cUm . Ž x Ž t m . . ␣
¢
F
␣ y1
Ž 17 . ␣
xX Ž t k . Ž dUk . aŽ t k . xX Ž t k . Ž dUk . s w Ž tk . Ž cUm . Ž x Ž t k y . . Ž cUm . tk y s tm Ž 0 - m - k .
171
OSCILLATION CAUSED BY IMPULSES
and wŽ
tq k
q. s
␣y1 X
X q a Ž tq k q . x Ž tk q .
¡aŽ t
x Ž tq k q.
Ž x Ž tq k .. ␣y1 X
q . xX Ž tq k q.
k
x Ž tq k q.
Ž cUk x Ž t k . . F
a Ž t k q . xX Ž t k q .
Ž
sy
1
Ž cUk .
cUk
~ aŽ t
x Ž tk q .
. Ž x Ž tk . .
w Ž tk q .
tk q / tm F
␣ y1 X
Ž m ) k.
␣ y1
m
Iˆm Ž xX Ž t m . . . Iˆm Ž xX Ž t m . . Ž cUk x Ž t k . . ␣
Ž 18 .
␣ y1
F
xX Ž t m . Ž dUm . aŽ t m . xX Ž t m . Ž cUk . Ž x Ž t k . .
F
Ž dUm . aŽ t k q . xX Ž t k q . Ž cUk . Ž x Ž t k . .
␣
␣ y1 X
x Ž tk q .
␣
¢
Ž dUm . s w Ž tk q . Ž cUk . tk q s tm Ž m ) k . .
It follows from inequalities Ž10., Ž17., and Ž18. that wX Ž t . F yp Ž t . ,
t / t 0, n
w Ž tq 0, n . F 0, n w Ž t 0, n . , where t 0, n s t k or t k q Ž t 1 s t 0, 1 - t 0, 2 - ⭈⭈⭈ - t 0, n - t 0, nq1 - ⭈⭈⭈ . and 0, n is defined by Ž16.. Then, applying Lemma 1, we obtain wŽ t. F
Ł
t 0-t 0, n -t
½
0, n w Ž tq 0 . y
1
t
Ht t -tŁ-s 0
0
0, n
0, n
5
p Ž s . ds ,
t G t0 .
Ž 19 . In view of Ž15., Ž19., and w Ž t . G 0, we get a contradiction as t ª ⬁. Hence, every solution of Ž1. is oscillatory. The proof of Theorem 2 is complete.
172
MINGSHU PENG
Using Theorems 1 and 2, we can obtain some corollaries as follows: COROLLARY 1. Assume that Ž3. holds and there exists a positi¨ e integer k 0 such that cUk G 1, dUk F 1 for k G k 0 . If q⬁
H
p Ž s . ds s q⬁,
Ž 20 .
then e¨ ery solution of Ž1. is oscillatory. Proof. Without loss of generality, let k 0 s 1. By cUk G 1, dUk F 1, we know that 1r 0, n G 1. Therefore t
Ht t -tŁ-s 0
0
0, n
1
p Ž s . ds G
0, n
t
Ht p Ž s . ds.
Ž 21 .
0
Let t ª ⬁; it follows from Ž20. and Ž21. that Ž8. holds. By Theorem 1, we get that all solutions of Ž1. are oscillatory. COROLLARY 2. Assume that Ž3. holds and there exist a positi¨ e integer k 0 and a constant ␥ ) 0 such that cUk
1
G 1,
G
␣
Ž dUk .
␥
t kq 1
for k G k 0
ž / tk
Ž 22 .
and q⬁
H
t r p Ž t . dt s q⬁,
Ž 23 .
then e¨ ery solution of Ž1. is oscillatory. Proof. Without loss of generality, let k 0 s 1. Then we have t
Ht t -tŁ-s 0
0
0, n
s
t1
Ht
1
p Ž s . ds
0, n
p Ž s . ds q
0
q G
Ž
dUa
.
1
Ž
dU1 dU2
1
Ž
1
dU1
.
␣
t2
Ht
⭈⭈⭈
.
␣
Ž
1
Ž
Ht p Ž s . ds
p Ž s . ds q
⭈⭈⭈
1 dU1 dU2
n
1
dU1 dU2
p Ž s . ds q
t
dUn
1
q
t2
Ht
␣
1
Ž t
dUn
.

dU1 dU2
.
␣
H p Ž s . ds a t n
t3
Ht
2
p Ž s . ds
.
␣
t3
Ht
2
p Ž s . ds q ⭈⭈⭈
173
OSCILLATION CAUSED BY IMPULSES
G G s
1 t 1␥ 1 t 1␥ 1 t 1␥
t2 ␥
t 2 p Ž s . ds q
Ht
1
s p Ž s . ds q
0
nq1
p Ž s . ds
n
t2 ␥
Ht
t ␥
Ht t
s p Ž s . ds q ⭈⭈⭈ q
t ␥
Ht s
p Ž s . ds
n
1
t ␥
Ht s
t 3 p Ž s . ds q ⭈⭈⭈ q
2
t1 ␥
Ht
t3 ␥
Ht
p Ž s . ds
Ž 24 .
0
for t g Ž t n , t nq1 x. Let t ª ⬁; it follows from Ž23. and Ž24. that Ž8. holds. According to Theorem 1, we get a conclusion that Eq. Ž1. is oscillatory. COROLLARY 3. Assume that Ž3. and Ž23. hold, and Ž ab. G Ž a. Ž b . for any ab ) 0. Furthermore, suppose that there exist a positi¨ e integer k 0 and a constant ␥ ) 0 such that
Ž cUk .
t kq1 y t k ) ,
Ž dUk .
␣
G
t kq1
␥
ž / tk
for k G k 0 .
Ž 25 .
Then e¨ ery solution of Eq. Ž1. is oscillatory. Corollary 3 can be deduced from Theorem 2. Its proof is similar to that of Corollary 2 and it is omitted. Remark 2. Using the same technique and the same argument as above, one also can obtain new criteria about the oscillation of the advanced differential equation with impulses
Ž aŽ t .
xX Ž t .
X
␣ y1 X
x Ž t . . q f Ž t , x Ž t . , x Ž t q . . s 0,
x Ž tq k . s Ik Ž x Ž t k . . ,
t / tk
ˆ X xX Ž tq k . s Ik Ž x Ž t k . . .
Ž 26 .
3. EXAMPLES EXAMPLE 1. Consider the impulsive delay differential equation xY q
1
4t q xŽ k . s xŽ k. ,
2
x ty
ž
1 5
/
s 0,
t / k, k s 1, 2, 3, . . .
xX Ž kq . s Ž kr Ž k q 1 . . xX Ž k . ,
Ž 27 .
k s 1, 2, . . . ,
where d k s dUk s krŽ k q 1., c k s cUk s 1, pŽ t . s 1r4t 2 , t k s k, and Ž x . s x.
174
MINGSHU PENG
Obviously, ␣ s 1, the conditions Ži. and Žii. are satisfied and di
1
q⬁
Ht
1r ␣
a
j
Ł Ž s . t -t -s j
ci
l
ds d jq1
s A Ž t jq1 . y A Ž t j . y q ⭈⭈⭈ q s1q
c jq1
Ž aŽ t jq2 . y AŽ t jq1 . .
d jq1 d jq2 ⭈⭈⭈ d jqn c jq1 c jq2 ⭈⭈⭈ c jqn
jq1 jq2
q
jq1 jq3
Ž AŽ t jqnq1 . y AŽ t jqn . . q ⭈⭈⭈
q ⭈⭈⭈ q
jq1 jqn
q ⭈⭈⭈
s q⬁. Let k 0 s 1, ␥ s 1. Then 1
Ž
dUk
t ␥ p Ž t . dt s
H
.
s
␣
kq1 k
s
t kq 1 tk
and q⬁
H
q⬁
tp Ž t . dt s
q⬁
H
t=
1 2t2
dt s q⬁.
By Corollary 2, we know that every solution of Eq. Ž27. is oscillatory. EXAMPLE 2. Consider the super-linear impulsive equation xY q
1 3
x 2 ny1 t y
ž
1
/
3 t x Ž kq . s Ž k q 1rk . x Ž k . ,
s 0,
t / k,
k s 1, 2, . . .
xX Ž kq . s xX Ž k . ,
Ž 28 .
k s 1, 2, . . . ,
where n ) 1 is a natural number, and c k s cUk s Ž k q 1.rk, d k s dUk s 1, pŽ t . s 1rt 3 , t k s k, and Ž x . s x 2 ny1. Let k 0 s 1, ␥ s 3. Obviously, the conditions Ži., Žii., and Ž3. are satisfied and
Ž cUk .
Ž dUk .
␣
s
ž
kq1 k
2 ny1
/
G
t kq1
ž / tk
3
175
OSCILLATION CAUSED BY IMPULSES
and q⬁
H
t ␥ p Ž t . dt s
q⬁
H
t 3 p Ž t . dt s
q⬁
H
t3 =
1 t3
dt s q⬁.
By Corollary 3, we know that every solution of Eq. Ž28. is oscillatory. But the delay differential equation xY q Ž1rt 3 . x 2 ny1 Ž t y 13 . s 0, by paper w12x, is non-oscillatory. EXAMPLE 3. Consider the sub-linear impulsive equation xY q
1 2
1
x 1r3 t y
t q xŽ k . s xŽ k. ,
s 0, t / k, k s 1, 2, . . . 12 Ž 29 . X X q x Ž k . s Ž kr Ž k q 1 . . x Ž k . , k s 1, 2, . . . ,
ž
/
where c k s cUk s 1, d k s dUk s krŽ k q 1., pŽ t . s 1rt 2 , t k s k, and Ž x . s x 1r3. Let k 0 s 1, ␥ s 1. Obviously, the conditions Ži., Žii., and Ž3. are satisfied and Ž ab. s Ž a. Ž b . for any ab ) 0,
Ž cUk .
Ž
dUk
.
␣
s
kq1 k
t kq1
G
tk
,
and q⬁
H
t ␥ p Ž t . dt s
q⬁
H
tp Ž t . dt s
q⬁
H
t=
1 t2
dt s q⬁.
By Corollary 3, we know that every solution of Eq. Ž29. is oscillatory. But the delay differential equation xY q Ž1rt 2 . x 1r3 Ž t y 121 . s 0, by paper w12x, is non-oscillatory.
REFERENCES 1. A. Zhao and J. Yan, Existence of positive solutions for delay differential equations with impulses, J. Math. Anal. Appl. 210 Ž1997., 667᎐678. 2. K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl. 139 Ž1989., 110᎐122. 3. M.-P. Chen, J. S. Yu, and J. H. Shen, The persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations, Comput. Math. Appl. 27 Ž1994., 1᎐6. 4. Y. Jiang and Y. Jurang, Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses, J. Math. Anal. Appl. 207 Ž1997., 388᎐396. 5. A. Domoshnitsky and M. Drakhlin, Nonoscillation of first order impulsive differential equations with delay, J. Math. Anal. Appl. 206 Ž1997., 254᎐269.
176
MINGSHU PENG
6. Y. Zhang, A. Zhao, and J. Yan, Oscillation criteria for impulsive delay differential equations, J. Math. Anal. Appl. 205 Ž1997., 461᎐470. 7. M. Peng and W. Ge, Oscillation criteria for second order nonlinear delay differential equations with impulses, Comput. Math. Appl. 39 Ž2000., 217᎐225. 8. L. Berezansky and E. Braverman, On oscillation of a second order impulsive delay differential equation, J. Math. Anal. Appl. 233 Ž1999., 276᎐300. 9. C. Yong-shao and F. Wei-zhen, Oscillation of second order nonlinear ODE with impulses, J. Math. Anal. Appl. 210 Ž1997., 150᎐169. 10. I. Gyori ¨ and G. Ladas, ‘‘Oscillation Theory of Delay Differential Equations with Applications,’’ Clarendon Press, Oxford, 1991. 11. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, ‘‘Theory of Impulsive Differential Equations,’’ World Scientific, Singapore, 1989. 12. A. H. Nasr, Necessary and sufficient conditions for the oscillation of forced nonlinear second order differential equations with delayed arguments, J. Math. Anal. Appl. 212 Ž1997., 51᎐59.
Oscillation Caused By Impulses Mingshu Peng Department of Mathematics, Beijing Normal Uni¨ ersity, Beijing 100875, People’s Republic of China; and Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China Submitted by William F. Ames Received December 1, 1998
The present paper is devoted to the investigation of the oscillation of a kind of very extensively studied second order nonlinear delay differential equations with impulses, some interesting results are obtained, which illustrate that impulses play a very important role in giving rise to the oscillations of equations. 䊚 2001 Academic Press
Key Words: oscillation; impulse; delay differential equation; nonlinearity.
Consider the impulsive delay differential equation
Ž aŽ t .
xX Ž t .
␣ y1 X
X
x Ž t . . q f Ž t , x Ž t . , x Ž t y . . s 0,
ˆ X xX Ž tq k . s Ik Ž x Ž t k . .
x Ž tq k . s Ik Ž x Ž t k . . ,
t / tk
Ž 1.
where ␣ , ) 0, 0 - t 1 - t 2 - ⭈⭈⭈ - t k - ⭈⭈⭈ and lim t ª⬁ t k s ⬁. Suppose that xX Ž t k . s xX Ž ty k . s lim
x Ž tk q h . y x Ž tk . h
hªy0
and xX Ž tq k . s lim
x Ž t k q h . y x Ž tq k . h
hªq0
.
163 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
164
MINGSHU PENG
Throughout the paper, assume that the following conditions hold: Ži. f Ž t, u, ¨ . is continuous in w t 0 y , q⬁. = Žy⬁, q⬁. = Žy⬁, q⬁., where t 0 G 0, uf Ž t, u, ¨ . ) 0 Ž u¨ ) 0. and f Žt ,Žu¨,.¨ . G pŽ t . Ž ¨ / 0., where pŽ t . is continuous in w t 0 y , q⬁., pŽ t . G 0, and x Ž x . ) 0 Ž x / 0., X Ž x . G 0; Žii. Ik Ž x . and Iˆk Ž x . are continuous in Žy⬁, q⬁.; there exist positive numbers c k , cUk , d k , dUk such that cUk F Ik Ž x .rx F c k , d k F Iˆk Ž x .rx F dUk ; Žiii. aŽ t . is a positive continuous function in w t 0 y , q⬁. and AŽ t . t s Ht 0 dsra1r ␣ Ž s .. Recently, there has been increasing interest on the oscillationrnonoscillation of the first order linear delay differential equations with impulses Žsee paper w1᎐6x., and good results have been obtained. But few are on the second order nonlinear delay differential equations with impulses, e.g., w7, 8x, etc. The present paper is devoted to the study of the oscillation of a type of very extensive second order nonlinear delay differential equations with impulses. Some interesting results are gained here. In addition, some examples show that, though some delay differential equations without impulses are non-oscillatory, they may become oscillatory if some impulses are added to them. That is, in some cases, impulses play a dominating role in the oscillations of equations. As to some related results about the oscillation of some second-order nonlinear ODE with impulses, we refer to paper w9x by Chen and Feng. For the theory of delay differential equations and impulsive differential equations, please see the recent books by Gyori ¨ and Ladas w10x and Lakshmikantham et al. w11x, respectively. We introduce the notation as follows: PC s x: w  y ,  x ª R ¬ x Ž t . is twice continuously differentiable for t g w  y ,  x_ t k , k s 1, 2, . . . 4 , X . Ž y. X Ž q. X Ž y. exist and x Ž ty . Ž . X Ž y. Ž . x Ž tq k , x tk , x tk , x tk k s x t k , x t k s x t k for t k g w  y ,  x4 . ⍀ s x: w  y , q⬁. ª R ¬ x Ž t . is continuous first for t / t k , x Ž tq k , y y x Ž t k . exist and x Ž t k . s x Ž t k ., x Ž t . is continuously differentiable for t G  , X q . X Ž y. Ž . XŽ y . exist and t / t k , t / t k q and xX Ž tq k , x tk , x tk q , x tk q X X y x Ž t k . s x Ž t k .4 . For any  G 0, g PC , a function x: w  y , q⬁. ª R is called a solution of Eq. Ž1. satisfying the initial value condition xŽ t. s Ž t. , if x g ⍀ and satisfies Ž1. and Ž2..
tg w y, x
Ž 2.
165
OSCILLATION CAUSED BY IMPULSES
Using the method of steps, one can show that for any ) 0 and g PC , the initial value problem Ž1., Ž2. has a unique solution x g ⍀ . A solution of Ž1. is said to be non-oscillatory if this solution is eventually positive or eventually negative. Otherwise, this solution is said to be oscillatory. This paper is organized as follows. In Section 1 we shall offer two interesting lemmas, which will be used in Section 2 to prove our main theorems. To illustrate our results, three examples are also included in Section 3.
1. SOME LEMMAS LEMMA 1 w9, Theorem 1.4.1x. ŽA 0 . ŽA 1 .
Assume that
m g PC 1w Rq, R x and mŽ t . is left-continuous at t k , k s 1, 2, . . . For k s 1, 2, . . . , t G t 0 , mX Ž t . F p Ž t . m Ž t . q q Ž t . , mŽ
tq k
t / tk
. F d k m Ž t k . q bk
where q, p g PC 1 w Rq, R x, d k G 0 and bk are constants. Then mŽ t . F mŽ t0 . q
Ł
t 0-t k -t
Ý
t 0 -t k -t
q
ž
d k exp
Ł
t k-t j-t
t
Ht s-tŁ-t d 0
k
exp
t
žH
d j exp t
žH s
k
p Ž s . ds
t0
ž
/
t
Ht p Ž s . ds k
//
bk
p Ž . d q Ž s . ds,
/
t G t0 .
LEMMA 2. Let x Ž t . be a solution of Eq. Ž1.. Suppose that there exists some T G t 0 such that x Ž t . ) 0 for t G T. If 1
q⬁
Ht
j
1r ␣
a
Ł Ž s . t -t -s j
l
di ci
ds s q⬁
for some t j ŽG t 1 ., then xX Ž tq k . G0 for t g Ž t k , t kq1 x, where t k G T.
and
xX Ž t . G 0
Ž 3.
166
MINGSHU PENG
Proof. At first, we prove that xX Ž t k . G 0 for any t k G T. If not, then . s Iˆj Ž xX Ž t j .. F there exists some j such that t j G T, xX Ž t j . - 0, and xX Ž tq j X d j x Ž t j . - 0. Let X q a Ž tq j . x Ž tj .
␣y1 X
␣ x Ž tq j . s y Ž  ) 0 .
S Ž t . s a Ž t . xX Ž t .
and
␣y1 X
x Ž t. .
By Ž1., for t g Ž t jqiy1 , t jqi x, i s 1, 2, . . . , we have
Ž aŽ t .
X
␣ y1 X
xX Ž t .
x Ž t . . s yf Ž t , x Ž t . , x Ž t y . . F yp Ž t . Ž x Ž t y . . F 0.
Ž 4.
Hence, SŽ t . is monotonically decreasing in Ž t jqiy1 , t jqi x. So we have a Ž t jq1 . xX Ž t jq1 .
␣ y1 X
X q x Ž t jq1 . F a Ž tq j . x Ž tj .
␣y1 X
␣ x Ž tq j . s y - 0
and a Ž t jq2 . xX Ž t jq2 .
␣ y1 X
X q x Ž t jq2 . F a Ž tq jq1 . x Ž t jq1 .
␣y1 X
x Ž tq jq1 .
s a Ž t jq1 . Iˆjq1 Ž xX Ž t jq1 . . ␣
␣y1
F Ž d jq1 . a Ž t jq1 . xX Ž t jq1 .
Iˆjq1 Ž xX Ž t jq1 . .
␣ y1 X
x Ž t jq1 .
␣
F y Ž d jq1 .  ␣ - 0. By induction, we obtain a Ž t . xX Ž t .
␣
␣ y1 X
x Ž t . F y Ž d jq1 d jq2 ⭈⭈⭈ d jqn .  ␣ s y ␣
Ł
t j-t k-t
d k␣ - 0,
Ž 5. for t g Ž t jqn , t jqnq1 x. Therefore
 X
x Ž t. F y
Ł
t j-t k-t
dk
a1r ␣ Ž t .
.
167
OSCILLATION CAUSED BY IMPULSES
. Ž . In view of condition Žii., we have x Ž tq k F c k x t k . Applying Lemma 1, we obtain x Ž t . F x Ž tq j .
Ł
t j-t k-t
ck
t
q
Ht s-tŁ-t c t -tŁ-s d a k
j
i
k
j
1
1r ␣
1
Ž s.
t ) tj .
ds,
Ž 6.
In view of the fact that Ł t j - t k - t c k s Ł t j - t l - s c i Ł s - t l - t c l , we have xŽ t. F
Ł
t j-t k-t
½
c k x Ž tq k . y
1
t
Ht
j
1r ␣
a
Ł Ž s . t -t -s j
l
di
5
ds ,
ci
t ) t j . Ž 7.
Since x Ž t k . ) 0 Ž t k G T ., one can find that Ž7. contradicts Ž3. as t ª ⬁. Therefore xX Ž t k . G 0
Ž tk G T . .
X Ž . . By condition Žii., we have, for any t k G T, xX Ž tq k G d k x t k G 0. Because SŽ t . is decreasing in Ž t jqiy1 , t jqi x, we get, for t g Ž t jqiy1 , t jqi x, SŽ t . G 0, which implies xX Ž t . G 0. The proof of this lemma is complete.
Remark 1. In the case that x Ž t . is eventually negative, if Ž3. holds true, X . Ž . Ž x then xX Ž tq k F 0 and x t F 0, for t g t jqiy1 , t jqi where t k G T. 2. MAIN RESULTS THEOREM 1. Assume that Ž3. holds and there exists a positi¨ e integer k 0 such that cUk G 1 for k G k 0 . If 1
q⬁
Ht
Ł
t 0-t 0, n -s
0
0, n
p Ž s . ds s q⬁,
Ž 8.
where
¡1
0, n s
~Ž
¢Ž
t 0, n s t k q / t m dUk
. .
␣
␣ dUm
t 0, n s t k t 0, n s t k q s t m
Ž m ) k. Ž 9.
168
MINGSHU PENG
and t 0, n s t k or t k q Ž t 1 s t 0, 1 - t 0, 2 - ⭈⭈⭈ - t 0, n - t 0, nq1 - ⭈⭈⭈ ., then e¨ ery solution of Ž1. is oscillatory. Proof. Without loss of generality, we can assume k 0 s 1. If Ž1. has a non-oscillatory solution x Ž t ., we might as well assume that x Ž t . ) 0 Ž t G t 0 .. It follows from Lemma 2 that xX Ž t . G 0 for t g Ž t k , t kq1 x, where k s 1, 2, . . . . Let
wŽ t. s
a Ž t . xX Ž t .
␣y1 X
x Ž t.
Ž xŽ t y . .
.
Ž 10 .
. Ž . Ž . Ž. Then w Ž tq k G 0 k s 1, 2, . . . , w t G 0 for t G t 0 . Using condition i and Ž . Eq. 1 , we get
wX Ž t . s y
y
f Ž t, xŽ t. , xŽ t y . .
Ž xŽ t y . . a Ž t . xX Ž t .
␣y1 X
x Ž t . X Ž x Ž t y . . xX Ž t y .
2 Ž xŽ t y . .
F yp Ž t . . Ž 11 .
It follows from the continuity of aŽ t ., condition Žii., cUk G 1, and X Ž x . G 0 that
w Ž tq k . s
X q a Ž tq k . x Ž tk .
¡Ž d F
~ Žd
␣y1 X
x Ž tq k .
Ž x Ž tq k y.. ␣
␣ y1
␣
␣ y1 X
U k
xX Ž t k . . a Ž t k . xX Ž t k . ␣ s Ž dUk . w Ž t k . Ž x Ž tk y . . tk y / tm Ž 0 - m - k .
U k
. a Ž t k . xX Ž t k . Ž cUm x Ž t m . . ␣
¢
x Ž tk . ␣ y1
xX Ž t k . Ž dUk . aŽ t k . xX Ž t k . ␣ F s Ž dUk . w Ž t k . Ž x Ž tk y . . tk y s tm Ž 0 - m - k .
Ž 12 .
169
OSCILLATION CAUSED BY IMPULSES
and ␣y1 X
X q a Ž tq k q . x Ž tk q .
w Ž tq k q. s
¡aŽ t
x Ž tq k q.
Ž x Ž tq k .. ␣y1 X
q . xX Ž tq k q.
k
x Ž tq k q.
Ž cUk x Ž t k . . F
a Ž t k q . xX Ž t k q .
␣ y1 X
x Ž tk q .
Ž x Ž tk . .
s w Ž tk q . tk q / tm F
~ aŽ t
Ž m ) k.
␣ y1
m
Iˆm Ž xX Ž t m . . . Iˆm Ž xX Ž t m . . Ž cUk x Ž t k . . ␣
␣ y1 X
Ž dUm . aŽ t m . xX Ž t m . F Ž cUk x Ž t k . .
x Ž tm .
␣
F
¢
Ž 13 .
Ž dUm . aŽ t k q . xX Ž t k q . Ž x Ž tk . .
␣ y1 X
x Ž tk q .
␣
s Ž dUm . w Ž t k q . tk q s tm
Ž m ) k.
It follows from inequalities Ž11. ᎐ Ž13. that wX Ž t . F yp Ž t . ,
t / t 0, n
w Ž tq 0, n . F 0, n w Ž t 0, n . , where t 0, n s t k or t k q Ž t 1 s t 0, 1 - t 0, 2 - ⭈⭈⭈ - t 0, n - t 0, nq1 - ⭈⭈⭈ . and 0, n is defined by Ž9.. Then, applying Lemma 1, we obtain wŽ t. F
Ł
t 0-t 0, n -t
½
0, n w Ž tq 0 . y
t
Ht t -tŁ-s 0
0
0, n
1 0, n
5
p Ž s . ds ,
t G t 0 Ž 14 .
In view of Ž8., Ž14., and w Ž t . G 0, we get a contradiction as t ª ⬁. Hence, every solution of Ž1. is oscillatory. The proof of Theorem 1 is complete.
170
MINGSHU PENG
Assume that Ž3. holds and Ž ab. G Ž a. Ž b . for any
THEOREM 2. ab ) 0. If
1
q⬁
Ht
Ł
t 0-t 0, n -s
0
0, n
p Ž s . ds s q⬁,
Ž 15 .
where
¡Ž d
U ␣ m cUk ␣ dUk
.
Ž
0, n
s~
t 0, n s t k q s t m
.
Ž
t 0, n s t k
.
1
Ž
cUk
dUk
¢ Ž c Ž
.
U m
and t k y / t m
t 0, n s t k q / t m
.
Ž m ) k.
Ž 16 .
Ž m ) k.
␣
t 0, n s t k
.
and t k y s t m
Ž0 - m - k. ,
then e¨ ery solution of Ž1. is oscillatory. Proof. If Ž1. has a non-oscillatory solution x Ž t ., without loss of generality, we can assume x Ž t . ) 0 Ž t G t 0 .. Let w Ž t . be defined by Ž10.. Then w Ž tq k . G0
Ž k s 1, 2, . . . . , w Ž t . G 0 Ž t G t 0 . .
It is easy to see that
wŽ
tq k
.s
X q a Ž tq k . x Ž tk .
¡Ž d F
~ Žd
␣y1 X
x Ž tq k .
Ž x Ž tq k y.. ␣
␣ y1
␣
␣ y1
U k
xX Ž t k . . a Ž t k . xX Ž t k . ␣ s Ž dUk . w Ž t k . Ž x Ž tk y . . tk y / tm Ž 0 - m - k .
U k
xX Ž t k . . a Ž t k . xX Ž t k . Ž cUm . Ž x Ž t m . . ␣
¢
F
␣ y1
Ž 17 . ␣
xX Ž t k . Ž dUk . aŽ t k . xX Ž t k . Ž dUk . s w Ž tk . Ž cUm . Ž x Ž t k y . . Ž cUm . tk y s tm Ž 0 - m - k .
171
OSCILLATION CAUSED BY IMPULSES
and wŽ
tq k
q. s
␣y1 X
X q a Ž tq k q . x Ž tk q .
¡aŽ t
x Ž tq k q.
Ž x Ž tq k .. ␣y1 X
q . xX Ž tq k q.
k
x Ž tq k q.
Ž cUk x Ž t k . . F
a Ž t k q . xX Ž t k q .
Ž
sy
1
Ž cUk .
cUk
~ aŽ t
x Ž tk q .
. Ž x Ž tk . .
w Ž tk q .
tk q / tm F
␣ y1 X
Ž m ) k.
␣ y1
m
Iˆm Ž xX Ž t m . . . Iˆm Ž xX Ž t m . . Ž cUk x Ž t k . . ␣
Ž 18 .
␣ y1
F
xX Ž t m . Ž dUm . aŽ t m . xX Ž t m . Ž cUk . Ž x Ž t k . .
F
Ž dUm . aŽ t k q . xX Ž t k q . Ž cUk . Ž x Ž t k . .
␣
␣ y1 X
x Ž tk q .
␣
¢
Ž dUm . s w Ž tk q . Ž cUk . tk q s tm Ž m ) k . .
It follows from inequalities Ž10., Ž17., and Ž18. that wX Ž t . F yp Ž t . ,
t / t 0, n
w Ž tq 0, n . F 0, n w Ž t 0, n . , where t 0, n s t k or t k q Ž t 1 s t 0, 1 - t 0, 2 - ⭈⭈⭈ - t 0, n - t 0, nq1 - ⭈⭈⭈ . and 0, n is defined by Ž16.. Then, applying Lemma 1, we obtain wŽ t. F
Ł
t 0-t 0, n -t
½
0, n w Ž tq 0 . y
1
t
Ht t -tŁ-s 0
0
0, n
0, n
5
p Ž s . ds ,
t G t0 .
Ž 19 . In view of Ž15., Ž19., and w Ž t . G 0, we get a contradiction as t ª ⬁. Hence, every solution of Ž1. is oscillatory. The proof of Theorem 2 is complete.
172
MINGSHU PENG
Using Theorems 1 and 2, we can obtain some corollaries as follows: COROLLARY 1. Assume that Ž3. holds and there exists a positi¨ e integer k 0 such that cUk G 1, dUk F 1 for k G k 0 . If q⬁
H
p Ž s . ds s q⬁,
Ž 20 .
then e¨ ery solution of Ž1. is oscillatory. Proof. Without loss of generality, let k 0 s 1. By cUk G 1, dUk F 1, we know that 1r 0, n G 1. Therefore t
Ht t -tŁ-s 0
0
0, n
1
p Ž s . ds G
0, n
t
Ht p Ž s . ds.
Ž 21 .
0
Let t ª ⬁; it follows from Ž20. and Ž21. that Ž8. holds. By Theorem 1, we get that all solutions of Ž1. are oscillatory. COROLLARY 2. Assume that Ž3. holds and there exist a positi¨ e integer k 0 and a constant ␥ ) 0 such that cUk
1
G 1,
G
␣
Ž dUk .
␥
t kq 1
for k G k 0
ž / tk
Ž 22 .
and q⬁
H
t r p Ž t . dt s q⬁,
Ž 23 .
then e¨ ery solution of Ž1. is oscillatory. Proof. Without loss of generality, let k 0 s 1. Then we have t
Ht t -tŁ-s 0
0
0, n
s
t1
Ht
1
p Ž s . ds
0, n
p Ž s . ds q
0
q G
Ž
dUa
.
1
Ž
dU1 dU2
1
Ž
1
dU1
.
␣
t2
Ht
⭈⭈⭈
.
␣
Ž
1
Ž
Ht p Ž s . ds
p Ž s . ds q
⭈⭈⭈
1 dU1 dU2
n
1
dU1 dU2
p Ž s . ds q
t
dUn
1
q
t2
Ht
␣
1
Ž t
dUn
.

dU1 dU2
.
␣
H p Ž s . ds a t n
t3
Ht
2
p Ž s . ds
.
␣
t3
Ht
2
p Ž s . ds q ⭈⭈⭈
173
OSCILLATION CAUSED BY IMPULSES
G G s
1 t 1␥ 1 t 1␥ 1 t 1␥
t2 ␥
t 2 p Ž s . ds q
Ht
1
s p Ž s . ds q
0
nq1
p Ž s . ds
n
t2 ␥
Ht
t ␥
Ht t
s p Ž s . ds q ⭈⭈⭈ q
t ␥
Ht s
p Ž s . ds
n
1
t ␥
Ht s
t 3 p Ž s . ds q ⭈⭈⭈ q
2
t1 ␥
Ht
t3 ␥
Ht
p Ž s . ds
Ž 24 .
0
for t g Ž t n , t nq1 x. Let t ª ⬁; it follows from Ž23. and Ž24. that Ž8. holds. According to Theorem 1, we get a conclusion that Eq. Ž1. is oscillatory. COROLLARY 3. Assume that Ž3. and Ž23. hold, and Ž ab. G Ž a. Ž b . for any ab ) 0. Furthermore, suppose that there exist a positi¨ e integer k 0 and a constant ␥ ) 0 such that
Ž cUk .
t kq1 y t k ) ,
Ž dUk .
␣
G
t kq1
␥
ž / tk
for k G k 0 .
Ž 25 .
Then e¨ ery solution of Eq. Ž1. is oscillatory. Corollary 3 can be deduced from Theorem 2. Its proof is similar to that of Corollary 2 and it is omitted. Remark 2. Using the same technique and the same argument as above, one also can obtain new criteria about the oscillation of the advanced differential equation with impulses
Ž aŽ t .
xX Ž t .
X
␣ y1 X
x Ž t . . q f Ž t , x Ž t . , x Ž t q . . s 0,
x Ž tq k . s Ik Ž x Ž t k . . ,
t / tk
ˆ X xX Ž tq k . s Ik Ž x Ž t k . . .
Ž 26 .
3. EXAMPLES EXAMPLE 1. Consider the impulsive delay differential equation xY q
1
4t q xŽ k . s xŽ k. ,
2
x ty
ž
1 5
/
s 0,
t / k, k s 1, 2, 3, . . .
xX Ž kq . s Ž kr Ž k q 1 . . xX Ž k . ,
Ž 27 .
k s 1, 2, . . . ,
where d k s dUk s krŽ k q 1., c k s cUk s 1, pŽ t . s 1r4t 2 , t k s k, and Ž x . s x.
174
MINGSHU PENG
Obviously, ␣ s 1, the conditions Ži. and Žii. are satisfied and di
1
q⬁
Ht
1r ␣
a
j
Ł Ž s . t -t -s j
ci
l
ds d jq1
s A Ž t jq1 . y A Ž t j . y q ⭈⭈⭈ q s1q
c jq1
Ž aŽ t jq2 . y AŽ t jq1 . .
d jq1 d jq2 ⭈⭈⭈ d jqn c jq1 c jq2 ⭈⭈⭈ c jqn
jq1 jq2
q
jq1 jq3
Ž AŽ t jqnq1 . y AŽ t jqn . . q ⭈⭈⭈
q ⭈⭈⭈ q
jq1 jqn
q ⭈⭈⭈
s q⬁. Let k 0 s 1, ␥ s 1. Then 1
Ž
dUk
t ␥ p Ž t . dt s
H
.
s
␣
kq1 k
s
t kq 1 tk
and q⬁
H
q⬁
tp Ž t . dt s
q⬁
H
t=
1 2t2
dt s q⬁.
By Corollary 2, we know that every solution of Eq. Ž27. is oscillatory. EXAMPLE 2. Consider the super-linear impulsive equation xY q
1 3
x 2 ny1 t y
ž
1
/
3 t x Ž kq . s Ž k q 1rk . x Ž k . ,
s 0,
t / k,
k s 1, 2, . . .
xX Ž kq . s xX Ž k . ,
Ž 28 .
k s 1, 2, . . . ,
where n ) 1 is a natural number, and c k s cUk s Ž k q 1.rk, d k s dUk s 1, pŽ t . s 1rt 3 , t k s k, and Ž x . s x 2 ny1. Let k 0 s 1, ␥ s 3. Obviously, the conditions Ži., Žii., and Ž3. are satisfied and
Ž cUk .
Ž dUk .
␣
s
ž
kq1 k
2 ny1
/
G
t kq1
ž / tk
3
175
OSCILLATION CAUSED BY IMPULSES
and q⬁
H
t ␥ p Ž t . dt s
q⬁
H
t 3 p Ž t . dt s
q⬁
H
t3 =
1 t3
dt s q⬁.
By Corollary 3, we know that every solution of Eq. Ž28. is oscillatory. But the delay differential equation xY q Ž1rt 3 . x 2 ny1 Ž t y 13 . s 0, by paper w12x, is non-oscillatory. EXAMPLE 3. Consider the sub-linear impulsive equation xY q
1 2
1
x 1r3 t y
t q xŽ k . s xŽ k. ,
s 0, t / k, k s 1, 2, . . . 12 Ž 29 . X X q x Ž k . s Ž kr Ž k q 1 . . x Ž k . , k s 1, 2, . . . ,
ž
/
where c k s cUk s 1, d k s dUk s krŽ k q 1., pŽ t . s 1rt 2 , t k s k, and Ž x . s x 1r3. Let k 0 s 1, ␥ s 1. Obviously, the conditions Ži., Žii., and Ž3. are satisfied and Ž ab. s Ž a. Ž b . for any ab ) 0,
Ž cUk .
Ž
dUk
.
␣
s
kq1 k
t kq1
G
tk
,
and q⬁
H
t ␥ p Ž t . dt s
q⬁
H
tp Ž t . dt s
q⬁
H
t=
1 t2
dt s q⬁.
By Corollary 3, we know that every solution of Eq. Ž29. is oscillatory. But the delay differential equation xY q Ž1rt 2 . x 1r3 Ž t y 121 . s 0, by paper w12x, is non-oscillatory.
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6. Y. Zhang, A. Zhao, and J. Yan, Oscillation criteria for impulsive delay differential equations, J. Math. Anal. Appl. 205 Ž1997., 461᎐470. 7. M. Peng and W. Ge, Oscillation criteria for second order nonlinear delay differential equations with impulses, Comput. Math. Appl. 39 Ž2000., 217᎐225. 8. L. Berezansky and E. Braverman, On oscillation of a second order impulsive delay differential equation, J. Math. Anal. Appl. 233 Ž1999., 276᎐300. 9. C. Yong-shao and F. Wei-zhen, Oscillation of second order nonlinear ODE with impulses, J. Math. Anal. Appl. 210 Ž1997., 150᎐169. 10. I. Gyori ¨ and G. Ladas, ‘‘Oscillation Theory of Delay Differential Equations with Applications,’’ Clarendon Press, Oxford, 1991. 11. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, ‘‘Theory of Impulsive Differential Equations,’’ World Scientific, Singapore, 1989. 12. A. H. Nasr, Necessary and sufficient conditions for the oscillation of forced nonlinear second order differential equations with delayed arguments, J. Math. Anal. Appl. 212 Ž1997., 51᎐59.