- Email: [email protected]
Oscillation of second order delay differential equations
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Applied Mathematics and Computation 198 (2008) 930–935 www.elsevier.com/locate/amc
Oscillation of second order delay differential equations Aizhi Weng, Jitao Sun
*
Department of Mathematics, Tongji University, 200092, PR China
Abstract We"establish the oscillation# criteria for the second order functional equation 00 l m n X X X xðtÞ þ ci ðtÞxðt si Þ þ pi ðtÞxðt di Þ qi ðtÞxðt ri Þ ¼ 0 i¼1
i¼1
i¼1
and " xðtÞ þ
l X
#00 ci ðtÞxðt si Þ
þ
i¼1
m X
pi ðtÞxðt di Þ
i¼1
n X
qi ðtÞxðt ri Þ ¼ f ðtÞ:
i¼1
New oscillation criteria improve the one recently established by Manojlovic et al. [J. Manojlovic, Y. Shoukaku, T. Tanigawa, N. Yoshida, Oscillation criteria for second order differential equations with positive and negative coefficients, Appl. Math. Comp. 181 (2006) 853–863]. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Oscillation; Second order; Delay; Differential equation
1. Introduction In the present paper, we consider the oscillation of the second order functional equation " #00 l m n X X X xðtÞ þ ci ðtÞxðt si Þ þ pi ðtÞxðt di Þ qi ðtÞxðt ri Þ ¼ 0; t P t0 i¼1
i¼1
ð1:1Þ
i¼1
and " xðtÞ þ
l X i¼1
*
#00 ci ðtÞxðt si Þ
þ
m X i¼1
pi ðtÞxðt di Þ
n X
qi ðtÞxðt ri Þ ¼ f ðtÞ;
i¼1
Corresponding author. E-mail address: [email protected] (J. Sun).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.09.016
t P t0
ð1:2Þ
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
931
where m P n, si ði ¼ 1; 2; . . . ; lÞ, di ði ¼ 1; 2; . . . ; mÞ, ri ði ¼ 1; 2; . . . ; nÞ are non-negative constants, ri 6 di ði ¼ 1; 2; . . . ; nÞ and ci 2 Cð½0; 1Þ; ½0; 1ÞÞ ði ¼ 1; 2; . . . ; lÞ; pi 2 Cð½0; 1Þ; ½0; 1ÞÞ ði ¼ 1; 2; . . . ; mÞ; qi 2 Cð½0; 1Þ; ½0; 1ÞÞ ði ¼ 1; 2; . . . ; nÞ: We make the following assumptions: (H1) (H2) (H3) (H4)
0 6 s1 < s2 < < sl 6 ri ; i ¼ 1; 2; . . . ; n. qi ðtÞ 6 qi ðt ri Þ; pi ðtÞ P qi ðt di Þ; i ¼ 1; 2; . . . ; n. 9j0 2Pf1; 2; . . . ; ng; pj0 ðtÞ qj0 ðt dj0 Þ P a; a ¼ const. l 0 6 i¼1 ci ðtÞ 6 c; c ¼ const.
Manojlovic et al. [1], Shi et al. [2] and Wang [3] and many others studied the oscillation character of this type or its generalized form [4–8]. Ref. [1] proved the following Theorems A and B: Theorem A. Assume that (H2)–(H4) hold. Eq. (1.1) is oscillatory, if n Z 1 Z sri X qi ðnÞ dn ds 6 1: i¼1
0
ð1:3Þ
sdi
Theorem B. Assume that (H2)–(H4) and (1.3) hold and there exists a function F ðtÞ 2 C 2 ð½0; 1Þ; RÞ such that F 00 ðtÞ ¼ f ðtÞ;
lim F ðtÞ ¼ 0:
ð1:4Þ
t!1
Then, any solution xðtÞ of Eq. (1.2) either oscillates or satisfies limt!1 xðtÞ ¼ 0. Our main results are as follows: Theorem 1.1. Assume that (H1)–(H4) hold. Eq. (1.1) is oscillatory, if there exists j1 2 f1; 2; . . . ; lg such that n Z 1 Z sri X qi ðnÞ dn ds 6 cj1 ðtÞ: ð1:5Þ i¼1
0
sdi
Theorem 1.2. Assume that (H1)–(H4) and (1.5) hold and there exists a function F ðtÞ 2 C 2 ð½0; 1Þ; RÞ such that F 00 ðtÞ ¼ f ðtÞ and limt!1 F ðtÞ is finite. Then, any solution xðtÞ of Eq. (1.2) either oscillates or satisfies limt!1 xðtÞ ¼ 0. 2. Proof of Theorem 1.1 Proof. Denote T ¼ maxfsi ; dj ; rk : 1 6 i 6 l; 1 6 j 6 m; 1 6 k 6 ng: Suppose that xðtÞ is a non-oscillatory solution of (1.1). Without any loss of generality, we assume that xðtÞ > 0 for t P t0 . Denote wðtÞ ¼ xðtÞ þ
l X i¼1
ci ðtÞxðt si Þ
n Z X i¼1
By (2.1) and (H1)–(H3), it follows that
t0
t
Z
sri
sdi
qi ðnÞxðnÞ dn ds:
ð2:1Þ
932
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
" w00 ðtÞ ¼ xðtÞ þ
l X
#00 ci ðtÞxðt si Þ
þ
i¼1
¼ 6
m X i¼1 m X
n X
qi ðt di Þxðt di Þ
i¼1
pi ðtÞxðt di Þ þ
n X
qi ðt ri Þxðt ri Þ
i¼1
qi ðtÞxðt ri Þ þ
i¼1
n X
qi ðt di Þxðt di Þ
n X
i¼1
½pi ðtÞ qi ðt di Þxðt di Þ þ
i¼1
6
n X
qi ðt ri Þxðt ri Þ
i¼1
n X ½qi ðtÞ qi ðt ri Þxðt ri Þ i¼1
m X
½pi ðtÞ qi ðt di Þxðt di Þ 6 ½pj0 ðtÞ qj0 ðt dj0 Þxðt dj0 Þ;
t P t0 þ T
i¼1
which yields that w00 ðtÞ 6 axðt dj0 Þ < 0;
t P t0 þ T :
ð2:2Þ
By (2.2), we conclude that there exists t1 P t0 þ T such that w0 ðtÞ P 0 or w0 ðtÞ < 0 for t P t1 . (i) If w0 ðtÞ P 0 for t P t1 . Integrating (2.2), we have 0
0
0
1 > w ðt1 Þ P w ðt1 Þ w ðtÞ P a
Z
t
xðs dj0 Þ ds
t1
which implies that xðtÞ is integrable in ½t1 ; 1Þ. Hence, zðtÞ ¼ xðtÞ þ ½t1 ; 1Þ since ci ðtÞ is bounded. On the other hand, we have n Z tri X z0 ðtÞ ¼ w0 ðtÞ þ qi ðnÞxðnÞ dn P 0; t P t1 : i¼1
Pl
i¼1 ci ðtÞxðt
si Þ is integrable too in
tdi
zðtÞ is non-decreasing, that is, zðtÞ P zðt1 Þ > 0 for t P t1 . Therefore, zðtÞ is non-integrable in ½t1 ; 1Þ. We obtain a contradiction. (ii) If w0 ðtÞ < 0 for t P t1 . We define l n Z t Z sri X X yðtÞ ¼ xðtÞ þ ci ðtÞxðt si Þ qi ðnÞxðnÞ dn ds: i¼1 0
0
i¼1
00
sdi
t1
00
y ðtÞ ¼ w ðtÞ; y ðtÞ ¼ w ðtÞ for t P t1 , together with (2.2), yielding that y 0 ðtÞ is decreasing for t P t1 . The inequality y 0 ðtÞ 6 y 0 ðt1 Þ < 0 implies that lim yðtÞ ¼ 1:
ð2:3Þ
t!1
We claim that xðtÞ is bounded in ½t0 ; 1Þ. If not, then limt!1 xðtÞ ¼ 1 so that there exists t2 P t1 satisfying max xðtÞ ¼ xðt2 sj1 Þ; yðt2 Þ < 0:
t1 T 6t6t2 sj1
Then, we have 0 > yðt2 Þ ¼ xðt2 Þ þ
l X
ci ðt2 Þxðt2 si Þ
i¼1
n Z X i¼1
P xðt2 Þ þ cj1 ðt2 Þxðt2 sj1 Þ xðt2 sj1 Þ " P cj1 ðt2 Þ
n Z X i¼1
P 0:
t1
t2
Z
sri
sdi
t2 t1
n Z X
#
i¼1
t1
Z
sri
qi ðnÞxðnÞ dn ds
sdi
t2
Z
sri
sdi
qi ðnÞ dn ds "
qi ðnÞ dn ds xðt2 sj1 Þ P cj1 ðt2 Þ
n Z X i¼1
0
1
Z
sri
sdi
# qi ðnÞ dn ds xðt2 sj1 Þ
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
933
The contradiction shows that xðtÞ must be bounded. Then, there exists a constant M such that xðtÞ 6 M for t P t1 . By (H4) and (1.5), it follows that n Z t Z sri n Z t2 Z sri n Z 1 Z sri X X X qi ðnÞxðnÞ dn ds P M qi ðnÞ dn ds P M qi ðnÞ dn ds yðtÞ P i¼1
sdi
t1
i¼1
t1
sdi
i¼1
0
sdi
P Mc > 1 which is contradictory to (2.3). This completes the proof. h Example 2.1. Let us consider the equation 00
½xðtÞ þ 3xðt pÞ þ ðt þ 1Þxðt 2pÞ 3et2p xðt pÞ ¼ 0;
ð2:4Þ
t > 0:
We have here l ¼ 1; c1 ðtÞ ¼ 3; s1 ¼ p; m ¼ 1; p1 ðtÞ ¼ t þ 1; d1 ¼ 2p; n ¼ 1;
q1 ðtÞ ¼ 3et2p ;
r1 ¼ p:
Straightforward calculations yield q1 ðtÞ 6 q1 ðt r1 Þ; p1 ðtÞ q1 ðt d1 Þ ¼ t þ 1 3et ep P ð1 3ep Þ > 0; Z 1 Z sr1 q1 ðnÞ dn ds ¼ 3ð1 ep Þ < c1 ðtÞ: 0
t > 0;
sd1
Therefore, according to Theorem 1.1, Eq. (2.4) is oscillatory. But, the theorem in Ref. [1] is invalid to this example since Z 1 Z sr1 q1 ðnÞ dn ds ¼ 3ð1 ep Þ > 1: 0
sd1
3. Proof of Theorem 1.2 Proof. Suppose that xðtÞ is a non-oscillatory solution of (1.2). Without any loss of generality, we assume that xðtÞ > 0 for t P t0 . Denote limt!1 F ðtÞ ¼ A. We define l n Z t Z sri X X W ðtÞ ¼ xðtÞ þ ci ðtÞxðt si Þ qi ðnÞxðnÞ dn ds F ðtÞ þ A þ 1: ð3:1Þ i¼1
It follows that " 00
W ðtÞ ¼ xðtÞ þ
l X
#00 ci ðtÞxðt si Þ
þ
i¼1
" ¼ f ðtÞ
m X
sdi
t0
i¼1
n X i¼1
pi ðtÞxðt di Þ þ
i¼1
n X
qi ðt di Þxðt di Þ
n X
qi ðt ri Þxðt ri Þ f ðtÞ
i¼1
#
qi ðtÞxðt ri Þ þ
n X
i¼1
i¼1
qi ðt di Þxðt di Þ
n X
qi ðt ri Þxðt ri Þ
i¼1
f ðtÞ m n n X X X ¼ pi ðtÞxðt di Þ þ qi ðt di Þxðt di Þ þ ½qi ðtÞ qi ðt ri Þxðt ri Þ i¼1
6
m X i¼1
i¼1
i¼1
½pi ðtÞ qi ðt di Þxðt di Þ 6 ½pj0 ðtÞ qj0 ðt dj0 Þxðt dj0 Þ; t P t0 þ T
934
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
which yields that W 00 ðtÞ 6 axðt dj0 Þ < 0;
t P t0 þ T :
ð3:2Þ
By (3.2) there exists n1 P t0 þ T such that W 0 ðtÞ P 0 or W 0 ðtÞ < 0 for t P n1 . By conditions of Theorem 1.2, there exists sufficiently large g1 such that F ðtÞ þ A þ 1 > 0 for t P g1 . Let t1 ¼ maxfn1 ; g1 g. (i) If W 0 ðtÞ P 0 for t P t1 , Integrating (3.2), we have 1 > W 0 ðt1 Þ P W 0 ðt1 Þ W 0 ðtÞ P a
Z
t
xðs dj0 Þ ds t1
which implies that xðtÞ is integrable and bounded in ½t1 ; 1Þ so that U ðtÞ ¼ xðtÞ þ integrable and bounded in ½t1 ; 1Þ. Denoting UðtÞ ¼ U ðtÞ F ðtÞ;
Pl
i¼1 ci ðtÞxðt
si Þ is ð3:3Þ
t P t1 ;
then, UðtÞ is bounded in ½t1 ; 1Þ since limt!1 F ðtÞ exists and U ðtÞ is bounded. By (3.1), we have that n Z tri X 0 0 qi ðsÞxðsÞ ds P 0; U ðtÞ ¼ W ðtÞ þ tdi
i¼1
hence UðtÞ is bounded and monotonous, limt!1 UðtÞ exists. By (3.3), limt!1 U ðtÞ exists, let limt!1 U ðtÞ ¼ l, where l 2 ½0; 1Þ. We claim that l ¼ 0. If l 2 ð0; 1Þ, then there exists t2 P t1 such that l U ðtÞ > ; t P t2 ; 2 which contradicts the fact that U ðtÞ is integrable in ½t1 ; 1Þ. Therefore, limt!1 U ðtÞ ¼ 0. Since 0 < xðtÞ 6 U ðtÞ for t P t1 , we have that limt!1 xðtÞ ¼ 0. (ii) If W 0 ðtÞ < 0 for t P t1 , We define l n Z t Z sri X X Y ðtÞ ¼ xðtÞ þ ci ðtÞxðt si Þ qi ðnÞxðnÞ dn ds F ðtÞ þ A þ 1: ð3:4Þ i¼1 0
i¼1 0
00
sdi
t1
00
Then, Y ðtÞ ¼ W ðtÞ; Y ðtÞ ¼ W ðtÞ for t P t1 , together with (3.2), yield that Y 0 ðtÞ is decreasing for t P t1 . The equality Y 0 ðtÞ 6 Y 0 ðt1 Þ < 0 implies that lim Y ðtÞ ¼ 1:
ð3:5Þ
t!1
We claim that xðtÞ is bounded in ½t0 ; 1Þ. If not, then limt!1 xðtÞ ¼ 1 so that there exists t2 P t1 satisfying max
t1 T 6t6t2 sj1
xðtÞ ¼ xðt2 sj1 Þ;
Y ðt2 Þ < 0:
Then, we have l X
0 > Y ðt2 Þ ¼ xðt2 Þ þ
ci ðt2 Þxðt2 si Þ
n Z X
i¼1
i¼1
P xðt2 Þ þ cj1 ðt2 Þxðt2 sj1 Þ xðt2 sj1 Þ P ½cj1 ðt2 Þ
n Z X i¼1
t1
t2
Z
t2
t1
n Z X i¼1
t1
Z
sri
qi ðnÞxðnÞ dn ds F ðtÞ þ A þ 1
sdi
t2
Z
sri
qi ðnÞ dn ds F ðtÞ þ A þ 1
sdi
sri
sdi
qi ðnÞ dn dsxðt2 sj1 Þ P ½cj1 ðt2 Þ
n Z X i¼1
0
1
Z
sri
qi ðnÞ dn dsxðt2 sj1 Þ P 0
sdi
The contradiction shows that xðtÞ must be bounded. Then, there exists a constant L > 0 such that xðtÞ 6 L for t P t1 . It follows that
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
Y ðtÞ P L
n Z X i¼1
t1
t2
Z
sri
qi ðnÞ dn ds P L sdi
n Z X i¼1
0
1
Z
935
sri
qi ðnÞ dn ds P Lc > 1
sdi
which contradicts (3.5). We conclude that if xðtÞ is a non-oscillatory solution of (1.2), then limt!1 xðtÞ ¼ 0. This completes the proof. h Example 3.1. Consider the equation 00
½xðtÞ þ 3xðt pÞ þ ðt þ 1Þxðt 2pÞ 3et2p xðt pÞ ¼ et þ
1 1 sin ; 3 t t
t > 0:
ð3:6Þ
Here F ðtÞ ¼ et þ t sin 1t ! 1 as t ! 1. The left of equality (3.6) is exactly the left of equality (2.4). All conditions of Theorem 1.2 are satisfied, thus any solution xðtÞ of Eq. (3.6) is either oscillatory or satisfies limt!1 xðtÞ ¼ 0: Acknowledgement This work was supported by the NNSF of China under Grant 60474008. References [1] J. Manojlovic, Y. Shoukaku, T. Tanigawa, N. Yoshida, Oscillation criteria for second order differential equations with positive and negative coefficients, Appl. Math. Comp. 181 (2006) 853–863. [2] W. Shi, P. Wang, Oscillatory criteria of a class of second-order neutral functional differential equations, Applied. Math. Comp. 146 (2003) 211–226. [3] P. Wang, Oscillation criteria for second-order neutral equations with distributed deviating arguments, J. Comp. Math. Appl. 47 (2004) 1935–1946. [4] C. Lee, C. Yeh, An oscillation theorem, Appl. Math. Lett. 20 (2007) 238–240. [5] Y. Sun, Oscillation of second order functional differential equations with distributed damping, Appl. Math. Comp. 178 (2006) 519–526. [6] P. Wang, Y. Wu, Oscillation of certain second-order functional differential equations with damping, J. Comp. Appl. Math. 157 (2003) 49–56. [7] L. Erbe, Q. Kong, B. Zheng, Oscillation Theory for Functional Differential Equations, Dekker, New York, 1995. [8] R. Agarwal, S. Grace, D. Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Netherland, 2000.
Applied Mathematics and Computation 198 (2008) 930–935 www.elsevier.com/locate/amc
Oscillation of second order delay differential equations Aizhi Weng, Jitao Sun
*
Department of Mathematics, Tongji University, 200092, PR China
Abstract We"establish the oscillation# criteria for the second order functional equation 00 l m n X X X xðtÞ þ ci ðtÞxðt si Þ þ pi ðtÞxðt di Þ qi ðtÞxðt ri Þ ¼ 0 i¼1
i¼1
i¼1
and " xðtÞ þ
l X
#00 ci ðtÞxðt si Þ
þ
i¼1
m X
pi ðtÞxðt di Þ
i¼1
n X
qi ðtÞxðt ri Þ ¼ f ðtÞ:
i¼1
New oscillation criteria improve the one recently established by Manojlovic et al. [J. Manojlovic, Y. Shoukaku, T. Tanigawa, N. Yoshida, Oscillation criteria for second order differential equations with positive and negative coefficients, Appl. Math. Comp. 181 (2006) 853–863]. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Oscillation; Second order; Delay; Differential equation
1. Introduction In the present paper, we consider the oscillation of the second order functional equation " #00 l m n X X X xðtÞ þ ci ðtÞxðt si Þ þ pi ðtÞxðt di Þ qi ðtÞxðt ri Þ ¼ 0; t P t0 i¼1
i¼1
ð1:1Þ
i¼1
and " xðtÞ þ
l X i¼1
*
#00 ci ðtÞxðt si Þ
þ
m X i¼1
pi ðtÞxðt di Þ
n X
qi ðtÞxðt ri Þ ¼ f ðtÞ;
i¼1
Corresponding author. E-mail address: [email protected] (J. Sun).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.09.016
t P t0
ð1:2Þ
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
931
where m P n, si ði ¼ 1; 2; . . . ; lÞ, di ði ¼ 1; 2; . . . ; mÞ, ri ði ¼ 1; 2; . . . ; nÞ are non-negative constants, ri 6 di ði ¼ 1; 2; . . . ; nÞ and ci 2 Cð½0; 1Þ; ½0; 1ÞÞ ði ¼ 1; 2; . . . ; lÞ; pi 2 Cð½0; 1Þ; ½0; 1ÞÞ ði ¼ 1; 2; . . . ; mÞ; qi 2 Cð½0; 1Þ; ½0; 1ÞÞ ði ¼ 1; 2; . . . ; nÞ: We make the following assumptions: (H1) (H2) (H3) (H4)
0 6 s1 < s2 < < sl 6 ri ; i ¼ 1; 2; . . . ; n. qi ðtÞ 6 qi ðt ri Þ; pi ðtÞ P qi ðt di Þ; i ¼ 1; 2; . . . ; n. 9j0 2Pf1; 2; . . . ; ng; pj0 ðtÞ qj0 ðt dj0 Þ P a; a ¼ const. l 0 6 i¼1 ci ðtÞ 6 c; c ¼ const.
Manojlovic et al. [1], Shi et al. [2] and Wang [3] and many others studied the oscillation character of this type or its generalized form [4–8]. Ref. [1] proved the following Theorems A and B: Theorem A. Assume that (H2)–(H4) hold. Eq. (1.1) is oscillatory, if n Z 1 Z sri X qi ðnÞ dn ds 6 1: i¼1
0
ð1:3Þ
sdi
Theorem B. Assume that (H2)–(H4) and (1.3) hold and there exists a function F ðtÞ 2 C 2 ð½0; 1Þ; RÞ such that F 00 ðtÞ ¼ f ðtÞ;
lim F ðtÞ ¼ 0:
ð1:4Þ
t!1
Then, any solution xðtÞ of Eq. (1.2) either oscillates or satisfies limt!1 xðtÞ ¼ 0. Our main results are as follows: Theorem 1.1. Assume that (H1)–(H4) hold. Eq. (1.1) is oscillatory, if there exists j1 2 f1; 2; . . . ; lg such that n Z 1 Z sri X qi ðnÞ dn ds 6 cj1 ðtÞ: ð1:5Þ i¼1
0
sdi
Theorem 1.2. Assume that (H1)–(H4) and (1.5) hold and there exists a function F ðtÞ 2 C 2 ð½0; 1Þ; RÞ such that F 00 ðtÞ ¼ f ðtÞ and limt!1 F ðtÞ is finite. Then, any solution xðtÞ of Eq. (1.2) either oscillates or satisfies limt!1 xðtÞ ¼ 0. 2. Proof of Theorem 1.1 Proof. Denote T ¼ maxfsi ; dj ; rk : 1 6 i 6 l; 1 6 j 6 m; 1 6 k 6 ng: Suppose that xðtÞ is a non-oscillatory solution of (1.1). Without any loss of generality, we assume that xðtÞ > 0 for t P t0 . Denote wðtÞ ¼ xðtÞ þ
l X i¼1
ci ðtÞxðt si Þ
n Z X i¼1
By (2.1) and (H1)–(H3), it follows that
t0
t
Z
sri
sdi
qi ðnÞxðnÞ dn ds:
ð2:1Þ
932
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
" w00 ðtÞ ¼ xðtÞ þ
l X
#00 ci ðtÞxðt si Þ
þ
i¼1
¼ 6
m X i¼1 m X
n X
qi ðt di Þxðt di Þ
i¼1
pi ðtÞxðt di Þ þ
n X
qi ðt ri Þxðt ri Þ
i¼1
qi ðtÞxðt ri Þ þ
i¼1
n X
qi ðt di Þxðt di Þ
n X
i¼1
½pi ðtÞ qi ðt di Þxðt di Þ þ
i¼1
6
n X
qi ðt ri Þxðt ri Þ
i¼1
n X ½qi ðtÞ qi ðt ri Þxðt ri Þ i¼1
m X
½pi ðtÞ qi ðt di Þxðt di Þ 6 ½pj0 ðtÞ qj0 ðt dj0 Þxðt dj0 Þ;
t P t0 þ T
i¼1
which yields that w00 ðtÞ 6 axðt dj0 Þ < 0;
t P t0 þ T :
ð2:2Þ
By (2.2), we conclude that there exists t1 P t0 þ T such that w0 ðtÞ P 0 or w0 ðtÞ < 0 for t P t1 . (i) If w0 ðtÞ P 0 for t P t1 . Integrating (2.2), we have 0
0
0
1 > w ðt1 Þ P w ðt1 Þ w ðtÞ P a
Z
t
xðs dj0 Þ ds
t1
which implies that xðtÞ is integrable in ½t1 ; 1Þ. Hence, zðtÞ ¼ xðtÞ þ ½t1 ; 1Þ since ci ðtÞ is bounded. On the other hand, we have n Z tri X z0 ðtÞ ¼ w0 ðtÞ þ qi ðnÞxðnÞ dn P 0; t P t1 : i¼1
Pl
i¼1 ci ðtÞxðt
si Þ is integrable too in
tdi
zðtÞ is non-decreasing, that is, zðtÞ P zðt1 Þ > 0 for t P t1 . Therefore, zðtÞ is non-integrable in ½t1 ; 1Þ. We obtain a contradiction. (ii) If w0 ðtÞ < 0 for t P t1 . We define l n Z t Z sri X X yðtÞ ¼ xðtÞ þ ci ðtÞxðt si Þ qi ðnÞxðnÞ dn ds: i¼1 0
0
i¼1
00
sdi
t1
00
y ðtÞ ¼ w ðtÞ; y ðtÞ ¼ w ðtÞ for t P t1 , together with (2.2), yielding that y 0 ðtÞ is decreasing for t P t1 . The inequality y 0 ðtÞ 6 y 0 ðt1 Þ < 0 implies that lim yðtÞ ¼ 1:
ð2:3Þ
t!1
We claim that xðtÞ is bounded in ½t0 ; 1Þ. If not, then limt!1 xðtÞ ¼ 1 so that there exists t2 P t1 satisfying max xðtÞ ¼ xðt2 sj1 Þ; yðt2 Þ < 0:
t1 T 6t6t2 sj1
Then, we have 0 > yðt2 Þ ¼ xðt2 Þ þ
l X
ci ðt2 Þxðt2 si Þ
i¼1
n Z X i¼1
P xðt2 Þ þ cj1 ðt2 Þxðt2 sj1 Þ xðt2 sj1 Þ " P cj1 ðt2 Þ
n Z X i¼1
P 0:
t1
t2
Z
sri
sdi
t2 t1
n Z X
#
i¼1
t1
Z
sri
qi ðnÞxðnÞ dn ds
sdi
t2
Z
sri
sdi
qi ðnÞ dn ds "
qi ðnÞ dn ds xðt2 sj1 Þ P cj1 ðt2 Þ
n Z X i¼1
0
1
Z
sri
sdi
# qi ðnÞ dn ds xðt2 sj1 Þ
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
933
The contradiction shows that xðtÞ must be bounded. Then, there exists a constant M such that xðtÞ 6 M for t P t1 . By (H4) and (1.5), it follows that n Z t Z sri n Z t2 Z sri n Z 1 Z sri X X X qi ðnÞxðnÞ dn ds P M qi ðnÞ dn ds P M qi ðnÞ dn ds yðtÞ P i¼1
sdi
t1
i¼1
t1
sdi
i¼1
0
sdi
P Mc > 1 which is contradictory to (2.3). This completes the proof. h Example 2.1. Let us consider the equation 00
½xðtÞ þ 3xðt pÞ þ ðt þ 1Þxðt 2pÞ 3et2p xðt pÞ ¼ 0;
ð2:4Þ
t > 0:
We have here l ¼ 1; c1 ðtÞ ¼ 3; s1 ¼ p; m ¼ 1; p1 ðtÞ ¼ t þ 1; d1 ¼ 2p; n ¼ 1;
q1 ðtÞ ¼ 3et2p ;
r1 ¼ p:
Straightforward calculations yield q1 ðtÞ 6 q1 ðt r1 Þ; p1 ðtÞ q1 ðt d1 Þ ¼ t þ 1 3et ep P ð1 3ep Þ > 0; Z 1 Z sr1 q1 ðnÞ dn ds ¼ 3ð1 ep Þ < c1 ðtÞ: 0
t > 0;
sd1
Therefore, according to Theorem 1.1, Eq. (2.4) is oscillatory. But, the theorem in Ref. [1] is invalid to this example since Z 1 Z sr1 q1 ðnÞ dn ds ¼ 3ð1 ep Þ > 1: 0
sd1
3. Proof of Theorem 1.2 Proof. Suppose that xðtÞ is a non-oscillatory solution of (1.2). Without any loss of generality, we assume that xðtÞ > 0 for t P t0 . Denote limt!1 F ðtÞ ¼ A. We define l n Z t Z sri X X W ðtÞ ¼ xðtÞ þ ci ðtÞxðt si Þ qi ðnÞxðnÞ dn ds F ðtÞ þ A þ 1: ð3:1Þ i¼1
It follows that " 00
W ðtÞ ¼ xðtÞ þ
l X
#00 ci ðtÞxðt si Þ
þ
i¼1
" ¼ f ðtÞ
m X
sdi
t0
i¼1
n X i¼1
pi ðtÞxðt di Þ þ
i¼1
n X
qi ðt di Þxðt di Þ
n X
qi ðt ri Þxðt ri Þ f ðtÞ
i¼1
#
qi ðtÞxðt ri Þ þ
n X
i¼1
i¼1
qi ðt di Þxðt di Þ
n X
qi ðt ri Þxðt ri Þ
i¼1
f ðtÞ m n n X X X ¼ pi ðtÞxðt di Þ þ qi ðt di Þxðt di Þ þ ½qi ðtÞ qi ðt ri Þxðt ri Þ i¼1
6
m X i¼1
i¼1
i¼1
½pi ðtÞ qi ðt di Þxðt di Þ 6 ½pj0 ðtÞ qj0 ðt dj0 Þxðt dj0 Þ; t P t0 þ T
934
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
which yields that W 00 ðtÞ 6 axðt dj0 Þ < 0;
t P t0 þ T :
ð3:2Þ
By (3.2) there exists n1 P t0 þ T such that W 0 ðtÞ P 0 or W 0 ðtÞ < 0 for t P n1 . By conditions of Theorem 1.2, there exists sufficiently large g1 such that F ðtÞ þ A þ 1 > 0 for t P g1 . Let t1 ¼ maxfn1 ; g1 g. (i) If W 0 ðtÞ P 0 for t P t1 , Integrating (3.2), we have 1 > W 0 ðt1 Þ P W 0 ðt1 Þ W 0 ðtÞ P a
Z
t
xðs dj0 Þ ds t1
which implies that xðtÞ is integrable and bounded in ½t1 ; 1Þ so that U ðtÞ ¼ xðtÞ þ integrable and bounded in ½t1 ; 1Þ. Denoting UðtÞ ¼ U ðtÞ F ðtÞ;
Pl
i¼1 ci ðtÞxðt
si Þ is ð3:3Þ
t P t1 ;
then, UðtÞ is bounded in ½t1 ; 1Þ since limt!1 F ðtÞ exists and U ðtÞ is bounded. By (3.1), we have that n Z tri X 0 0 qi ðsÞxðsÞ ds P 0; U ðtÞ ¼ W ðtÞ þ tdi
i¼1
hence UðtÞ is bounded and monotonous, limt!1 UðtÞ exists. By (3.3), limt!1 U ðtÞ exists, let limt!1 U ðtÞ ¼ l, where l 2 ½0; 1Þ. We claim that l ¼ 0. If l 2 ð0; 1Þ, then there exists t2 P t1 such that l U ðtÞ > ; t P t2 ; 2 which contradicts the fact that U ðtÞ is integrable in ½t1 ; 1Þ. Therefore, limt!1 U ðtÞ ¼ 0. Since 0 < xðtÞ 6 U ðtÞ for t P t1 , we have that limt!1 xðtÞ ¼ 0. (ii) If W 0 ðtÞ < 0 for t P t1 , We define l n Z t Z sri X X Y ðtÞ ¼ xðtÞ þ ci ðtÞxðt si Þ qi ðnÞxðnÞ dn ds F ðtÞ þ A þ 1: ð3:4Þ i¼1 0
i¼1 0
00
sdi
t1
00
Then, Y ðtÞ ¼ W ðtÞ; Y ðtÞ ¼ W ðtÞ for t P t1 , together with (3.2), yield that Y 0 ðtÞ is decreasing for t P t1 . The equality Y 0 ðtÞ 6 Y 0 ðt1 Þ < 0 implies that lim Y ðtÞ ¼ 1:
ð3:5Þ
t!1
We claim that xðtÞ is bounded in ½t0 ; 1Þ. If not, then limt!1 xðtÞ ¼ 1 so that there exists t2 P t1 satisfying max
t1 T 6t6t2 sj1
xðtÞ ¼ xðt2 sj1 Þ;
Y ðt2 Þ < 0:
Then, we have l X
0 > Y ðt2 Þ ¼ xðt2 Þ þ
ci ðt2 Þxðt2 si Þ
n Z X
i¼1
i¼1
P xðt2 Þ þ cj1 ðt2 Þxðt2 sj1 Þ xðt2 sj1 Þ P ½cj1 ðt2 Þ
n Z X i¼1
t1
t2
Z
t2
t1
n Z X i¼1
t1
Z
sri
qi ðnÞxðnÞ dn ds F ðtÞ þ A þ 1
sdi
t2
Z
sri
qi ðnÞ dn ds F ðtÞ þ A þ 1
sdi
sri
sdi
qi ðnÞ dn dsxðt2 sj1 Þ P ½cj1 ðt2 Þ
n Z X i¼1
0
1
Z
sri
qi ðnÞ dn dsxðt2 sj1 Þ P 0
sdi
The contradiction shows that xðtÞ must be bounded. Then, there exists a constant L > 0 such that xðtÞ 6 L for t P t1 . It follows that
A. Weng, J. Sun / Applied Mathematics and Computation 198 (2008) 930–935
Y ðtÞ P L
n Z X i¼1
t1
t2
Z
sri
qi ðnÞ dn ds P L sdi
n Z X i¼1
0
1
Z
935
sri
qi ðnÞ dn ds P Lc > 1
sdi
which contradicts (3.5). We conclude that if xðtÞ is a non-oscillatory solution of (1.2), then limt!1 xðtÞ ¼ 0. This completes the proof. h Example 3.1. Consider the equation 00
½xðtÞ þ 3xðt pÞ þ ðt þ 1Þxðt 2pÞ 3et2p xðt pÞ ¼ et þ
1 1 sin ; 3 t t
t > 0:
ð3:6Þ
Here F ðtÞ ¼ et þ t sin 1t ! 1 as t ! 1. The left of equality (3.6) is exactly the left of equality (2.4). All conditions of Theorem 1.2 are satisfied, thus any solution xðtÞ of Eq. (3.6) is either oscillatory or satisfies limt!1 xðtÞ ¼ 0: Acknowledgement This work was supported by the NNSF of China under Grant 60474008. References [1] J. Manojlovic, Y. Shoukaku, T. Tanigawa, N. Yoshida, Oscillation criteria for second order differential equations with positive and negative coefficients, Appl. Math. Comp. 181 (2006) 853–863. [2] W. Shi, P. Wang, Oscillatory criteria of a class of second-order neutral functional differential equations, Applied. Math. Comp. 146 (2003) 211–226. [3] P. Wang, Oscillation criteria for second-order neutral equations with distributed deviating arguments, J. Comp. Math. Appl. 47 (2004) 1935–1946. [4] C. Lee, C. Yeh, An oscillation theorem, Appl. Math. Lett. 20 (2007) 238–240. [5] Y. Sun, Oscillation of second order functional differential equations with distributed damping, Appl. Math. Comp. 178 (2006) 519–526. [6] P. Wang, Y. Wu, Oscillation of certain second-order functional differential equations with damping, J. Comp. Appl. Math. 157 (2003) 49–56. [7] L. Erbe, Q. Kong, B. Zheng, Oscillation Theory for Functional Differential Equations, Dekker, New York, 1995. [8] R. Agarwal, S. Grace, D. Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Netherland, 2000.