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Oscillation criteria for second-order impulsive delay difference equations
Applied Mathematics and Computation 146 (2003) 227–235 www.elsevier.com/locate/amc
Oscillation criteria for second-order impulsive delay difference equations q Mingshu Peng Department of Applied Mathematics, Northern Jiao Tong University, Beijing 100044, PR China
Abstract We investigate the oscillation of a kind of very extensively studied second-order nonlinear delay difference equations under impulsive perturbations, some interesting results are obtained, and some examples which illustrate that impulses play a very important role in giving rise to the oscillations of equations are also included. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Oscillation; Impulse; Delay difference equation; Nonlinearity
1. Introduction Consider the impulsive delay difference equation 8 < Dðan1 jDxðn 1Þja1 Dxðn 1ÞÞ þ f ðn; xðnÞ; xðn lÞÞ ¼ 0; n 6¼ nk ; k 2 N; : a1 a1 ank jDxðnk Þj Dxðnk Þ ¼ Nk ðank 1 jDxðnk 1Þj Dxðnk 1ÞÞ;
ð1Þ
where D denotes the forward difference operator, i.e., DxðnÞ ¼ xðn þ 1Þ xðnÞ, a > 0, l 2 N, N is the natural number set, 0 6 n0 < n1 < < nk < , and limn!1 nk ¼ 1. Throughout this paper, assume that the following conditions hold: (i) uf ðn; u; vÞ > 0 (uv > 0) and there exists a nonnegative sequence fpn g and a function / such that
q
This work is partially supported by NNSF of China.
0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00539-8
228
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
f ðn; u; vÞ P pn /ðcÞ
v 6¼ 0;
where / satisfies x/ðxÞ > 0 (x 6¼ 0), and /ðuÞ /ðvÞ ¼ gðu; vÞðu vÞ
for uv 6¼ 0
and gðu; vÞ is a nonnegative function; (ii) there exist positive numbers dk , dk such that k ðxÞ=x 6 dk ; Pn dk 6 N1=a 1 (iii) fan gn0 is a positive sequence and AðnÞ ¼ s¼n0 1=as . It is well known that ordinary differential equations with impulses and delay differential equations have been considered by many authors. The theory of impulsive differential equations is emerging as an important area of investigation, since it is much richer than the corresponding theory of differential equations without impulse effects. Moreover, such equations may exhibit several real world phenomena, such as rhythmical beating, merging of solutions, and noncontinuity of solutions. In the recent years, there is increasing interest on the oscillation/nonoscillation of impulsive delay differential equations, and numerous papers have been published on this class of equations and good results were obtained (see Refs. [1–7] etc. and the references therein). But fewer papers are on impulsive difference equations (for more details, please see paper [7]). The aim of this paper is devoted to the study of the oscillation/nonoscillation of a type of very extensively studied second-order nonlinear delay difference equations under impulsive perturbations and the methods and techniques employed here, via the impulsive difference inequalities, come from those employed in papers [1,2]. Some interesting results are gained here. In addition, some examples show that, though some delay differential equations without impulses are nonoscillatory, they may become oscillatory if some impulses are added to them. That is, in some cases, impulses play a dominating part in causing the oscillations of equations. For notation convince, let N ½n1 ; n2 ¼ fnjn 2 N; n1 6 n 6 n2 g; N ½n1 ; n2 Þ ¼ fnjn 2 N; n1 6 n < n2 g and N ½n1 ; 1Þ ¼ fnjn 2 N; n P n1 g: By a solution of Eq. (1) we mean a real valued sequence fxðnÞg defined on N ½n0 l; 1Þ which satisfies (1) for n P n0 . It is clear that Eq. (1) has a unique 1 solution fxðnÞgn0 l , under the initial conditions: xi ¼ yi ; i ¼ n0 l; . . . ; n0 ; where yi (i ¼ n0 l; . . . ; n0 ) are given real constants.
ð2Þ
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
229
A solution of (1) is said to be nonoscillatory 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 2 we shall offer two interesting Lemmas, which will be used in Section 3 to prove our main oscillation theorems. To illustrate our results, three examples are also included in Section 4.
2. Some lemmas Lemma 1. Let xðnÞ be a solution of Eq. (1). Suppose that there exists some N P n0 such that xðnÞ > 0 for n P N . If þ1 X 1
Y
1=a
s¼nj
as
1=a
¼ þ1
di
ð3Þ
nj 6 ni 6 s
holds for all sufficiently large nj ( P n1 ). Then Dxðnk 1Þ P 0
DxðnÞ P 0
and
for n 2 N ½nk ; nkþ1 Þ, where nk 1 P N . Proof. At first, we prove that Dxðnk 1Þ P 0 for any nk P N . If not, then there a1 exists some j such that nj 1 P N , Dxðnj 1Þ < 0 and anj jDxðnj Þ a1 a1 Dxðnj Þ ¼ Nj ðanj 1 jDxðnj 1Þj Dxðnj 1ÞÞ 6 dj anj 1 jDxðnj 1Þj Dxðnj 1Þ < 0. Let anj 1 jDxðnj 1Þj
a1
Dxðnj 1Þ ¼ ba
ðb > 0Þ
and a1
SðnÞ ¼ an1 jDxðn 1Þj
Dxðn 1Þ:
By (1), for n 2 N ðnjþi1 ; njþi Þ, i ¼ 1; 2; . . . , we have DSðnÞ ¼ Dðan1 jDxðn 1Þja1 Dxðn 1ÞÞ ¼ f ðn; xðnÞ; xðn lÞÞ 6 pn /ðxðn lÞÞ 6 0:
ð4Þ
Hence, SðnÞ is monotonically decreasing in N ðnjþi1 ; njþi . So a1
anjþ1 1 jDxðnjþ1 1Þj
a1
Dxðnjþ1 1ÞÞ 6 atj jDxðnj Þj
Dxðnj Þ 6 dj ba < 0
230
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
and anjþ2 1 jDxðnjþ2 1Þja1 Dxðnjþ2 1Þ 6 anjþ1 jDxðnjþ1 Þja1 Dxðnjþ1 Þ ¼ Njþ1 ðanjþ1 1 jDxðnjþ1 1ÞÞj a1
6 djþ1 anjþ1 1 jDxðnjþ1 1Þj
a1
Dxðnjþ1 1ÞÞ
Dxðnjþ1 1Þ ¼ djþ1 dj ba < 0:
By induction, we obtain an jDxðnÞj
a1
Y
DxðnÞ 6 ðdj djþ1 djþ2 djþl Þba ¼ ba
dk < 0:
ð5Þ
nj 6 nk 6 n
Therefore DxðnÞ 6
b
Q
1=a
nj 6 nk 6 n 1=a an
dk
ð6Þ
;
which, by summing from nj to n, provide xðn þ 1Þ 6 xðnj Þ b
n X 1 i¼nj
Y
ds1=a 1=a ai nj 6 ns 6 i
ð7Þ
for n P nj
in view of xðnÞ > 0, one can find that the right side of (7) converges to 1 as n ! 1, whereas the left side is eventually positive, which is a contradiction. Therefore Dxðnk 1Þ P 0
nk 1 P N : a1
By condition (ii), we have, for any nk P N , ank jDxðnk Þj Dxðnk Þ ¼ a1 Nk ðank 1 jDxðnk 1Þj Dxðnk 1ÞÞ P 0. Because SðnÞ is decreasing in N ðnjþi1 ; njþi , we get, for n 2 N ½njþi1 ; njþi Þ, SðnÞ P 0, which implies DxðnÞ P 0. The proof of this lemma is complete. Remark 1. In the case that xðnÞ is eventually negative, if (3) holds true, then Dxðnk 1Þ 6 0 and DxðnÞ 6 0, for n 2 N ½njþi1 ; njþi Þ, where nk P N . To establish our main results, we also need the following lemma, which is a discrete version of Theorem 1.4.1 in [6] by Lakshmikantham, Bainov and Simeonov: Lemma 2. Assume that DmðnÞ 6 ln mðnÞ þ qn ;
n 6¼ nk ;
mðnk þ 1Þ 6 bk mðnk Þ þ ek ; where fln g and fqn g are two real valued consequences and ln > 1, bk , ck are constants and bk P 0. Then
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
Y
mðnÞ 6 mðn0 Þ
bk
n0
Y
Y
ð1 þ li Þ þ
n0
ð1 þ li Þ þ
nk
n1 X
X
ek
n0
Y
i¼n0 ; i6¼nk i
Y
bk
Y
231
bj
nk
ð1 þ ls Þqi ;
n P n0 :
i
The proof can be followed from mathematical induction and direct analysis and it is omitted. 3. Oscillation criteria Theorem 1. Assume that (3) holds. If X Y 1 pi ¼ þ1 d nj 6 ns 6 i s i¼nj þ1; i6¼n
ð8Þ
k
holds for all sufficiently large nj . Then every solution of (1) is oscillatory. Proof. Without loss of generality, we can assume k0 ¼ 1. If (1) has a nonoscillatory solution xðnÞ, we might assume that xðnÞ > 0 (n P n0 ). It follows from Lemma 2 that DxðnÞ P 0 for n 2 N ½nk ; nkþ1 , k ¼ 1; 2; . . . Let a1
wðnÞ ¼
an1 jDxðn 1Þj Dxðn 1Þ : /ðxðn lÞÞ
ð9Þ
Then wðnk Þ P 0 (k ¼ 1; 2; . . .), wðnÞ P 0 for n P n0 . Using condition (i) and Eq. (1), we get a1
DwðnÞ ¼
a1
Dðan1 jDxðn 1Þj Dxðn 1ÞÞ an jDxðnÞj DxðnÞD/ðxðn lÞÞ /ðxðn lÞÞ /ðxðn lÞÞ/ðxðn l þ 1ÞÞ a1
¼
f ðn; xðnÞ; xðn lÞÞ an jDxðnÞj DxðnÞD/ðxðn lÞÞ 6 pn : /ðxðn lÞÞ /ðxðn lÞÞ/ðxðn l þ 1ÞÞ
Then a1
wðnk þ 1Þ ¼
a1
ank jDxðnk Þj Dxðnk Þ dk ank 1 jDxðnk 1Þj Dxðnk 1Þ 6 /ðxðnk l þ 1ÞÞ /ðxðnk lÞÞ
¼ dk wðnk Þ: It follows from the above inequalities that wðnÞ satisfied the following difference inequalities: DwðnÞ 6 pn ;
n 6¼ nk ;
wðnk þ 1Þ 6 dk wðnk Þ:
232
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
Then, applying Lemma 2, we obtain ( Y wðn þ 1Þ 6 dk wðnj Þ
n X
pi
i¼nj þ1; i6¼nk ; k2N
n0
Y nj 6 nk 6 i
1 dk
) ;
n P nj : ð10Þ
In view of (8), (10) and wðnÞ P 0, we get a contradiction as n ! 1. Hence, every solution of (1) is oscillatory. The proof of Theorem 1 is complete. Using Theorem 1, we can obtain some corollaries as follows: Corollary 1. Assume that (3) holds and there exists a positive integer k0 such that dk 6 1 for k P k0 . If þ1 X
pn ¼ þ1:
ð11Þ
n6¼nk ; k2N
Then every solution of (1) is oscillatory. Proof. Without loss of generality, let k0 ¼ 1. By dk 6 1, we know that 1=dk P 1. Therefore n X
pi
i¼nj þ1; i6¼nk ; k2N
Y nj 6 nl 6 i
1 P dl
n X
pi :
ð12Þ
i¼nj þ1; i6¼nk ; k2N
Let n ! 1, it follows from (11) and (12) that (8) holds. By Theorem 1, we get that all solutions of (1) are oscillatory. Corollary 2. Assume that (3) holds and there exist a positive integer k0 and a constant c > 0 such that 1 P dk
nk þ 1 nk
c for k P k0
ð13Þ
and þ1 X
nc pn ¼ þ1:
n6¼nk ; k2N
Then every solution of (1) is oscillatory.
ð14Þ
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
233
Proof. Without loss of generality, Let k0 ¼ 1. Then we have n X
Y
pi
n0 6 nl 6 i i¼n0 ; i6¼nk ; k2N nX 2 1
P
1 d1
1 P c n1 1 P c n1
njþ1 1 nX n2 1 1 1 X 1 1 X 1 ¼ p þ p þ þ pi i i dl d1 i¼n þ1 d1 d2 dj i¼nj þ1 i¼n0
ps þ þ
"s¼n1 þ1 nX 2 1
nc2 pi
" n1 þ1 nX 2 1 n1 þ1
1
1 d1 d2 dn
þ þ
n X
c
i pi þ þ
nl þ1
ps
s¼nl þ1
#
nclþ1 pi
nl þ1 n X
n X
# c
i pi ¼
1 nc1
n X
ic pi
i¼n1 þ1; i6¼nk ; k2N
for n 2 N ðnlþ1 ; nlþ2 Þ. Let n ! 1, one can obtain that (8) holds. It follows from Theorem 1 that Eq. (1) is oscillatory. Remark 2. Using the same technique and the same argument as above, one also can obtain new criteria about the oscillation of the advanced difference equation with impulses ( a1 Dðan1 jDxðn 1Þj Dxðn 1ÞÞ þ f ðn; xðnÞ; xðn þ lÞÞ ¼ 0; n 6¼ nk ; ank jDxðnk Þja1 Dxðnk Þ ¼ Nk ðank 1 jDxðnk 1Þja1 Dxðnk 1ÞÞ: ð15Þ Remark 3. When l ¼ 0, under the same assumption as in Theorem 1 or Corollaries 1 and 2, the conclusions of Theorem 1 or Corollaries 1 and 2 also follow. 4. Examples Example 1. Consider the impulsive delay difference equation 8 2 ln n lnðn2 1Þ > 2 > xðn lÞ ¼ 0; n 6¼ 3k; k ¼ 1; 2; . . . ; < D xðn 1Þ þ lnðn lÞ k > > : Dxðnk Þ ¼ Dxðnk 1Þ; kþ1 ð16Þ where l 2 N, dk ¼ dk ¼ k=ðk þ 1Þ, pn ¼ ð2 ln n lnðn2 1ÞÞ= lnðn lÞ, nk ¼ 3k, and /ðxÞ ¼ x. Obviously, a ¼ 1, the conditions (i)–(iii) are satisfied and þ1 Y X 1 di ¼ dj ðAðnjþ1 1Þ Aðnj 1ÞÞ 1=a s¼nj as nj 6 ni 6 s þ þ dj djþ1 djþl ðAðnjþlþ1 Þ Aðnjþl ÞÞ þ 3j 3j 3j þ þ þ þ ¼ 1: ¼ jþ1 jþ2 jþl
234
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
Let k0 ¼ 1, c ¼ 1. Then 1 k þ 1 nkþ1 ¼ ¼ dk k nk and þ1 X
nc p n ¼
n6¼nk
þ1 X
npn ¼ þ1:
n6¼nk
By Corollary 2, we know that every solution of Eq. (16) is oscillatory. But the delay difference equation D2 xðn 1Þ þ
2 ln n lnðn2 1Þ xðn lÞ ¼ 0 lnðn lÞ
has a nonoscillatory solution xn ¼ ln n. Example 2. Consider the sub-linear impulsive equation 8 1 2s1 > 2 > < D xðn 1Þ þ 3 x ðn lÞ ¼ 0; n 6¼ 3k; k ¼ 1; 2; . . . ; n 2s1 k > > Dxð3kÞ; : Dxð3k þ 1Þ ¼ kþ1
ð17Þ
where s > 1, l 6 0, s and l are integers, and dk ¼ dk ¼ k=ðk þ 1Þ, pn ¼ 1=n3 , nk ¼ 3k and /ðxÞ ¼ x2n1 . Let k0 ¼ 1, c ¼ 3. Obviously, the conditions (i)–(iii) and (3) are satisfied and 2s1 3 1 nkþ1 nkþ1 ¼ P dk nk nk and þ1 X n6¼nk ; k2N
nc p n ¼
þ1 X
n3 pn ¼ þ1:
n6¼nk ; k2N
By Corollary 2, we know that every solution of Eq. (17) is oscillatory. Example 3. Consider the sub-linear impulsive equation 8 1 > < D2 xðn 1Þ þ 2 x1=3 ðn lÞ ¼ 0; n 6¼ 3k; k ¼ 1; 2; . . . ; n k > : Dxð3k þ 1Þ ¼ Dxð3kÞ; kþ1
ð18Þ
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
235
where dk ¼ dk ¼ k=ðk þ 1Þ, pn ¼ 1=n2 , nk ¼ 3k and /ðxÞ ¼ x1=3 . Let k0 ¼ 1, c ¼ 1. Obviously, the conditions (i)–(iii) and (3) are satisfied, and 1 k þ 1 nkþ1 ¼ ¼ dk k nk and þ1 X n6¼nk ; k2N
nc p n ¼
þ1 X
npn ¼ þ1:
n6¼nk ; k2N
By Corollary 2, we know that every solution of Eq. (18) is oscillatory.
References [1] M. Peng, W. Ge, Oscillation criteria for second order nonlinear delay differential equations with impulses, Comput. Math. Appl. 39 (2000) 217–225. [2] M. Peng, Oscillation caused by impulses, J. Math. Anal. Appl. 255 (2001) 163–176. [3] M. Peng, A comparison theorem of second-order impulsive delay differential equations, J. Beijing Normal Univ. 36 (2000) 746–747 (Natural Ed., in Chinese). [4] M. Peng, W. Ge, Q. Xu, Preservation of nonoscillatory behavior of solutions second-order differential equations under impulsive perturbations, Appl. Math. Lett. 15 (2) (2002) 203–210. [5] M. Peng, W. Ge, Q. Xu, Preservation of nonoscillation for second-order delay differential equations under impulsive perturbations, Acta Math. Sinica 45 (5) (2002) 935–940 (in Chinese). [6] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of impulsive differential equations, World Scientific, Singapore, 1989. [7] M. Peng, Oscillation theorems of second-order nonlinear neutral delay difference equations with impulses, Comput. Math. Appl. 44 (5,6) (2002) 741–748.
Oscillation criteria for second-order impulsive delay difference equations q Mingshu Peng Department of Applied Mathematics, Northern Jiao Tong University, Beijing 100044, PR China
Abstract We investigate the oscillation of a kind of very extensively studied second-order nonlinear delay difference equations under impulsive perturbations, some interesting results are obtained, and some examples which illustrate that impulses play a very important role in giving rise to the oscillations of equations are also included. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Oscillation; Impulse; Delay difference equation; Nonlinearity
1. Introduction Consider the impulsive delay difference equation 8 < Dðan1 jDxðn 1Þja1 Dxðn 1ÞÞ þ f ðn; xðnÞ; xðn lÞÞ ¼ 0; n 6¼ nk ; k 2 N; : a1 a1 ank jDxðnk Þj Dxðnk Þ ¼ Nk ðank 1 jDxðnk 1Þj Dxðnk 1ÞÞ;
ð1Þ
where D denotes the forward difference operator, i.e., DxðnÞ ¼ xðn þ 1Þ xðnÞ, a > 0, l 2 N, N is the natural number set, 0 6 n0 < n1 < < nk < , and limn!1 nk ¼ 1. Throughout this paper, assume that the following conditions hold: (i) uf ðn; u; vÞ > 0 (uv > 0) and there exists a nonnegative sequence fpn g and a function / such that
q
This work is partially supported by NNSF of China.
0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00539-8
228
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
f ðn; u; vÞ P pn /ðcÞ
v 6¼ 0;
where / satisfies x/ðxÞ > 0 (x 6¼ 0), and /ðuÞ /ðvÞ ¼ gðu; vÞðu vÞ
for uv 6¼ 0
and gðu; vÞ is a nonnegative function; (ii) there exist positive numbers dk , dk such that k ðxÞ=x 6 dk ; Pn dk 6 N1=a 1 (iii) fan gn0 is a positive sequence and AðnÞ ¼ s¼n0 1=as . It is well known that ordinary differential equations with impulses and delay differential equations have been considered by many authors. The theory of impulsive differential equations is emerging as an important area of investigation, since it is much richer than the corresponding theory of differential equations without impulse effects. Moreover, such equations may exhibit several real world phenomena, such as rhythmical beating, merging of solutions, and noncontinuity of solutions. In the recent years, there is increasing interest on the oscillation/nonoscillation of impulsive delay differential equations, and numerous papers have been published on this class of equations and good results were obtained (see Refs. [1–7] etc. and the references therein). But fewer papers are on impulsive difference equations (for more details, please see paper [7]). The aim of this paper is devoted to the study of the oscillation/nonoscillation of a type of very extensively studied second-order nonlinear delay difference equations under impulsive perturbations and the methods and techniques employed here, via the impulsive difference inequalities, come from those employed in papers [1,2]. Some interesting results are gained here. In addition, some examples show that, though some delay differential equations without impulses are nonoscillatory, they may become oscillatory if some impulses are added to them. That is, in some cases, impulses play a dominating part in causing the oscillations of equations. For notation convince, let N ½n1 ; n2 ¼ fnjn 2 N; n1 6 n 6 n2 g; N ½n1 ; n2 Þ ¼ fnjn 2 N; n1 6 n < n2 g and N ½n1 ; 1Þ ¼ fnjn 2 N; n P n1 g: By a solution of Eq. (1) we mean a real valued sequence fxðnÞg defined on N ½n0 l; 1Þ which satisfies (1) for n P n0 . It is clear that Eq. (1) has a unique 1 solution fxðnÞgn0 l , under the initial conditions: xi ¼ yi ; i ¼ n0 l; . . . ; n0 ; where yi (i ¼ n0 l; . . . ; n0 ) are given real constants.
ð2Þ
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
229
A solution of (1) is said to be nonoscillatory 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 2 we shall offer two interesting Lemmas, which will be used in Section 3 to prove our main oscillation theorems. To illustrate our results, three examples are also included in Section 4.
2. Some lemmas Lemma 1. Let xðnÞ be a solution of Eq. (1). Suppose that there exists some N P n0 such that xðnÞ > 0 for n P N . If þ1 X 1
Y
1=a
s¼nj
as
1=a
¼ þ1
di
ð3Þ
nj 6 ni 6 s
holds for all sufficiently large nj ( P n1 ). Then Dxðnk 1Þ P 0
DxðnÞ P 0
and
for n 2 N ½nk ; nkþ1 Þ, where nk 1 P N . Proof. At first, we prove that Dxðnk 1Þ P 0 for any nk P N . If not, then there a1 exists some j such that nj 1 P N , Dxðnj 1Þ < 0 and anj jDxðnj Þ a1 a1 Dxðnj Þ ¼ Nj ðanj 1 jDxðnj 1Þj Dxðnj 1ÞÞ 6 dj anj 1 jDxðnj 1Þj Dxðnj 1Þ < 0. Let anj 1 jDxðnj 1Þj
a1
Dxðnj 1Þ ¼ ba
ðb > 0Þ
and a1
SðnÞ ¼ an1 jDxðn 1Þj
Dxðn 1Þ:
By (1), for n 2 N ðnjþi1 ; njþi Þ, i ¼ 1; 2; . . . , we have DSðnÞ ¼ Dðan1 jDxðn 1Þja1 Dxðn 1ÞÞ ¼ f ðn; xðnÞ; xðn lÞÞ 6 pn /ðxðn lÞÞ 6 0:
ð4Þ
Hence, SðnÞ is monotonically decreasing in N ðnjþi1 ; njþi . So a1
anjþ1 1 jDxðnjþ1 1Þj
a1
Dxðnjþ1 1ÞÞ 6 atj jDxðnj Þj
Dxðnj Þ 6 dj ba < 0
230
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
and anjþ2 1 jDxðnjþ2 1Þja1 Dxðnjþ2 1Þ 6 anjþ1 jDxðnjþ1 Þja1 Dxðnjþ1 Þ ¼ Njþ1 ðanjþ1 1 jDxðnjþ1 1ÞÞj a1
6 djþ1 anjþ1 1 jDxðnjþ1 1Þj
a1
Dxðnjþ1 1ÞÞ
Dxðnjþ1 1Þ ¼ djþ1 dj ba < 0:
By induction, we obtain an jDxðnÞj
a1
Y
DxðnÞ 6 ðdj djþ1 djþ2 djþl Þba ¼ ba
dk < 0:
ð5Þ
nj 6 nk 6 n
Therefore DxðnÞ 6
b
Q
1=a
nj 6 nk 6 n 1=a an
dk
ð6Þ
;
which, by summing from nj to n, provide xðn þ 1Þ 6 xðnj Þ b
n X 1 i¼nj
Y
ds1=a 1=a ai nj 6 ns 6 i
ð7Þ
for n P nj
in view of xðnÞ > 0, one can find that the right side of (7) converges to 1 as n ! 1, whereas the left side is eventually positive, which is a contradiction. Therefore Dxðnk 1Þ P 0
nk 1 P N : a1
By condition (ii), we have, for any nk P N , ank jDxðnk Þj Dxðnk Þ ¼ a1 Nk ðank 1 jDxðnk 1Þj Dxðnk 1ÞÞ P 0. Because SðnÞ is decreasing in N ðnjþi1 ; njþi , we get, for n 2 N ½njþi1 ; njþi Þ, SðnÞ P 0, which implies DxðnÞ P 0. The proof of this lemma is complete. Remark 1. In the case that xðnÞ is eventually negative, if (3) holds true, then Dxðnk 1Þ 6 0 and DxðnÞ 6 0, for n 2 N ½njþi1 ; njþi Þ, where nk P N . To establish our main results, we also need the following lemma, which is a discrete version of Theorem 1.4.1 in [6] by Lakshmikantham, Bainov and Simeonov: Lemma 2. Assume that DmðnÞ 6 ln mðnÞ þ qn ;
n 6¼ nk ;
mðnk þ 1Þ 6 bk mðnk Þ þ ek ; where fln g and fqn g are two real valued consequences and ln > 1, bk , ck are constants and bk P 0. Then
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
Y
mðnÞ 6 mðn0 Þ
bk
n0
Y
Y
ð1 þ li Þ þ
n0
ð1 þ li Þ þ
nk
n1 X
X
ek
n0
Y
i¼n0 ; i6¼nk i
Y
bk
Y
231
bj
nk
ð1 þ ls Þqi ;
n P n0 :
i
The proof can be followed from mathematical induction and direct analysis and it is omitted. 3. Oscillation criteria Theorem 1. Assume that (3) holds. If X Y 1 pi ¼ þ1 d nj 6 ns 6 i s i¼nj þ1; i6¼n
ð8Þ
k
holds for all sufficiently large nj . Then every solution of (1) is oscillatory. Proof. Without loss of generality, we can assume k0 ¼ 1. If (1) has a nonoscillatory solution xðnÞ, we might assume that xðnÞ > 0 (n P n0 ). It follows from Lemma 2 that DxðnÞ P 0 for n 2 N ½nk ; nkþ1 , k ¼ 1; 2; . . . Let a1
wðnÞ ¼
an1 jDxðn 1Þj Dxðn 1Þ : /ðxðn lÞÞ
ð9Þ
Then wðnk Þ P 0 (k ¼ 1; 2; . . .), wðnÞ P 0 for n P n0 . Using condition (i) and Eq. (1), we get a1
DwðnÞ ¼
a1
Dðan1 jDxðn 1Þj Dxðn 1ÞÞ an jDxðnÞj DxðnÞD/ðxðn lÞÞ /ðxðn lÞÞ /ðxðn lÞÞ/ðxðn l þ 1ÞÞ a1
¼
f ðn; xðnÞ; xðn lÞÞ an jDxðnÞj DxðnÞD/ðxðn lÞÞ 6 pn : /ðxðn lÞÞ /ðxðn lÞÞ/ðxðn l þ 1ÞÞ
Then a1
wðnk þ 1Þ ¼
a1
ank jDxðnk Þj Dxðnk Þ dk ank 1 jDxðnk 1Þj Dxðnk 1Þ 6 /ðxðnk l þ 1ÞÞ /ðxðnk lÞÞ
¼ dk wðnk Þ: It follows from the above inequalities that wðnÞ satisfied the following difference inequalities: DwðnÞ 6 pn ;
n 6¼ nk ;
wðnk þ 1Þ 6 dk wðnk Þ:
232
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
Then, applying Lemma 2, we obtain ( Y wðn þ 1Þ 6 dk wðnj Þ
n X
pi
i¼nj þ1; i6¼nk ; k2N
n0
Y nj 6 nk 6 i
1 dk
) ;
n P nj : ð10Þ
In view of (8), (10) and wðnÞ P 0, we get a contradiction as n ! 1. Hence, every solution of (1) is oscillatory. The proof of Theorem 1 is complete. Using Theorem 1, we can obtain some corollaries as follows: Corollary 1. Assume that (3) holds and there exists a positive integer k0 such that dk 6 1 for k P k0 . If þ1 X
pn ¼ þ1:
ð11Þ
n6¼nk ; k2N
Then every solution of (1) is oscillatory. Proof. Without loss of generality, let k0 ¼ 1. By dk 6 1, we know that 1=dk P 1. Therefore n X
pi
i¼nj þ1; i6¼nk ; k2N
Y nj 6 nl 6 i
1 P dl
n X
pi :
ð12Þ
i¼nj þ1; i6¼nk ; k2N
Let n ! 1, it follows from (11) and (12) that (8) holds. By Theorem 1, we get that all solutions of (1) are oscillatory. Corollary 2. Assume that (3) holds and there exist a positive integer k0 and a constant c > 0 such that 1 P dk
nk þ 1 nk
c for k P k0
ð13Þ
and þ1 X
nc pn ¼ þ1:
n6¼nk ; k2N
Then every solution of (1) is oscillatory.
ð14Þ
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
233
Proof. Without loss of generality, Let k0 ¼ 1. Then we have n X
Y
pi
n0 6 nl 6 i i¼n0 ; i6¼nk ; k2N nX 2 1
P
1 d1
1 P c n1 1 P c n1
njþ1 1 nX n2 1 1 1 X 1 1 X 1 ¼ p þ p þ þ pi i i dl d1 i¼n þ1 d1 d2 dj i¼nj þ1 i¼n0
ps þ þ
"s¼n1 þ1 nX 2 1
nc2 pi
" n1 þ1 nX 2 1 n1 þ1
1
1 d1 d2 dn
þ þ
n X
c
i pi þ þ
nl þ1
ps
s¼nl þ1
#
nclþ1 pi
nl þ1 n X
n X
# c
i pi ¼
1 nc1
n X
ic pi
i¼n1 þ1; i6¼nk ; k2N
for n 2 N ðnlþ1 ; nlþ2 Þ. Let n ! 1, one can obtain that (8) holds. It follows from Theorem 1 that Eq. (1) is oscillatory. Remark 2. Using the same technique and the same argument as above, one also can obtain new criteria about the oscillation of the advanced difference equation with impulses ( a1 Dðan1 jDxðn 1Þj Dxðn 1ÞÞ þ f ðn; xðnÞ; xðn þ lÞÞ ¼ 0; n 6¼ nk ; ank jDxðnk Þja1 Dxðnk Þ ¼ Nk ðank 1 jDxðnk 1Þja1 Dxðnk 1ÞÞ: ð15Þ Remark 3. When l ¼ 0, under the same assumption as in Theorem 1 or Corollaries 1 and 2, the conclusions of Theorem 1 or Corollaries 1 and 2 also follow. 4. Examples Example 1. Consider the impulsive delay difference equation 8 2 ln n lnðn2 1Þ > 2 > xðn lÞ ¼ 0; n 6¼ 3k; k ¼ 1; 2; . . . ; < D xðn 1Þ þ lnðn lÞ k > > : Dxðnk Þ ¼ Dxðnk 1Þ; kþ1 ð16Þ where l 2 N, dk ¼ dk ¼ k=ðk þ 1Þ, pn ¼ ð2 ln n lnðn2 1ÞÞ= lnðn lÞ, nk ¼ 3k, and /ðxÞ ¼ x. Obviously, a ¼ 1, the conditions (i)–(iii) are satisfied and þ1 Y X 1 di ¼ dj ðAðnjþ1 1Þ Aðnj 1ÞÞ 1=a s¼nj as nj 6 ni 6 s þ þ dj djþ1 djþl ðAðnjþlþ1 Þ Aðnjþl ÞÞ þ 3j 3j 3j þ þ þ þ ¼ 1: ¼ jþ1 jþ2 jþl
234
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
Let k0 ¼ 1, c ¼ 1. Then 1 k þ 1 nkþ1 ¼ ¼ dk k nk and þ1 X
nc p n ¼
n6¼nk
þ1 X
npn ¼ þ1:
n6¼nk
By Corollary 2, we know that every solution of Eq. (16) is oscillatory. But the delay difference equation D2 xðn 1Þ þ
2 ln n lnðn2 1Þ xðn lÞ ¼ 0 lnðn lÞ
has a nonoscillatory solution xn ¼ ln n. Example 2. Consider the sub-linear impulsive equation 8 1 2s1 > 2 > < D xðn 1Þ þ 3 x ðn lÞ ¼ 0; n 6¼ 3k; k ¼ 1; 2; . . . ; n 2s1 k > > Dxð3kÞ; : Dxð3k þ 1Þ ¼ kþ1
ð17Þ
where s > 1, l 6 0, s and l are integers, and dk ¼ dk ¼ k=ðk þ 1Þ, pn ¼ 1=n3 , nk ¼ 3k and /ðxÞ ¼ x2n1 . Let k0 ¼ 1, c ¼ 3. Obviously, the conditions (i)–(iii) and (3) are satisfied and 2s1 3 1 nkþ1 nkþ1 ¼ P dk nk nk and þ1 X n6¼nk ; k2N
nc p n ¼
þ1 X
n3 pn ¼ þ1:
n6¼nk ; k2N
By Corollary 2, we know that every solution of Eq. (17) is oscillatory. Example 3. Consider the sub-linear impulsive equation 8 1 > < D2 xðn 1Þ þ 2 x1=3 ðn lÞ ¼ 0; n 6¼ 3k; k ¼ 1; 2; . . . ; n k > : Dxð3k þ 1Þ ¼ Dxð3kÞ; kþ1
ð18Þ
M. Peng / Appl. Math. Comput. 146 (2003) 227–235
235
where dk ¼ dk ¼ k=ðk þ 1Þ, pn ¼ 1=n2 , nk ¼ 3k and /ðxÞ ¼ x1=3 . Let k0 ¼ 1, c ¼ 1. Obviously, the conditions (i)–(iii) and (3) are satisfied, and 1 k þ 1 nkþ1 ¼ ¼ dk k nk and þ1 X n6¼nk ; k2N
nc p n ¼
þ1 X
npn ¼ þ1:
n6¼nk ; k2N
By Corollary 2, we know that every solution of Eq. (18) is oscillatory.
References [1] M. Peng, W. Ge, Oscillation criteria for second order nonlinear delay differential equations with impulses, Comput. Math. Appl. 39 (2000) 217–225. [2] M. Peng, Oscillation caused by impulses, J. Math. Anal. Appl. 255 (2001) 163–176. [3] M. Peng, A comparison theorem of second-order impulsive delay differential equations, J. Beijing Normal Univ. 36 (2000) 746–747 (Natural Ed., in Chinese). [4] M. Peng, W. Ge, Q. Xu, Preservation of nonoscillatory behavior of solutions second-order differential equations under impulsive perturbations, Appl. Math. Lett. 15 (2) (2002) 203–210. [5] M. Peng, W. Ge, Q. Xu, Preservation of nonoscillation for second-order delay differential equations under impulsive perturbations, Acta Math. Sinica 45 (5) (2002) 935–940 (in Chinese). [6] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of impulsive differential equations, World Scientific, Singapore, 1989. [7] M. Peng, Oscillation theorems of second-order nonlinear neutral delay difference equations with impulses, Comput. Math. Appl. 44 (5,6) (2002) 741–748.