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Interval oscillation criteria for second order neutral nonlinear differential equations
Applied Mathematics and Computation 157 (2004) 39–51 www.elsevier.com/locate/amc
Interval oscillation criteria for second order neutral nonlinear differential equations Rong-Kun Zhuang a
a,1
, Wan-Ton Li
b,*,2
Department of Mathematics, Huizhou University, Huizhou 516015, People’s Republic of China b Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
Abstract Some oscillation criteria for the second order neutral nonlinear differential equation n X qi ðtÞfi ðyðsi ðtÞÞÞ ¼ 0; t P t0 ½yðtÞ þ pðtÞyðrðtÞÞ00 þ i¼1
are established. New oscillation criteria are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of ½t0 ; 1Þ, rather than on the whole half-line. Our results are more natural according to the Sturm Separation Theorem. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Oscillation; Nonlinear differential equation; Neutral type; Riccati transformation; Interval criteria
1. Introduction In this paper we consider the oscillation behavior of solutions of the second order neutral nonlinear differential equation *
Corresponding author. E-mail addresses: [email protected] (R.-K. Zhuang), [email protected] (W.-T. Li). 1 Supported by the NSF of Educational Department of Guangdong Province (0176). 2 Supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011A25-007-Z), the Foundation for University Key Teacher by the Ministry of Education of China, and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China. 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.06.016
40
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51 00
½yðtÞ þ pðtÞyðrðtÞÞ þ
n X
qi ðtÞfi ðyðsi ðtÞÞÞ ¼ 0;
ð1Þ
i¼1
where t P t0 , and the functions p, r, si , qi , fi , (i ¼ 1; 2; . . . ; n) are to be specified in the following text. By a solution of (1), we mean a continuously differentiable function yðtÞ which is defined for t P minfrðt0 Þ; si ðt0 Þ; i ¼ 1; 2; . . . ; ng such that yðtÞ ¼ /ðtÞ for minfrðt0 Þ; si ðt0 Þ; i ¼ 1; 2; . . . ; ng 6 t0 and satisfies (1) for all t P t0 . We restrict our attention only to the nontrivial solution yðtÞ of (1), i.e., to the solution yðtÞ such that supfjyðtÞj: t P T g > 0 for all T P t0 . A nontrivial solution of (1) is called oscillatory if it has arbitrary large zeros, otherwise, it is called nonoscillatory. Eq. (1) is called oscillatory if all its solutions are oscillatory. The oscillation problem for nonlinear delay equation such as ½pðtÞy 0 ðtÞ0 þ qðtÞf ðyðsðtÞÞÞgðy 0 ðtÞÞ ¼ 0;
t P t0 ;
ð2Þ
as well as for the nonlinear ordinary differential equation 0
½pðtÞy 0 ðtÞ þ qðtÞf ðyðtÞÞgðy 0 ðtÞÞ ¼ 0;
t P t0 ;
ð3Þ
and the linear ordinary equation 0
½pðtÞy 0 ðtÞ þ qðtÞyðtÞ ¼ 0;
t P t0
ð4Þ
has been studied by many authors with different methods. Some result can be found in [1–9] and references therein. On the one hand, the recent paper by Rogovechenko [2] contains various conditions for nonlinear delay equations obtained by using an integral averaging technique similar to that exploited in [8]. On the other hand, most known oscillation criteria involve the integrals of p and q and hence require the information of p and q on the entire half-line ½t0 ; 1Þ. However, from the Sturm Separation Theorem, we see that oscillation is only an interval property, i.e., if there exists a sequence of subintervals ½ai ; bi of ½t0 ; 1Þ, as ai ! 1, such that for each i there exists a solution of Eq. (4) that has at least two zeros in ½ai ; bi , then every solution of Eq. (4) is oscillatory, no matter how ‘‘bad’’ Eq. (4) is (or, p and q are) on the remaining part of ½t0 :1Þ. Kong [10] applied this idea to oscillation and established an interval criterion for oscillation of the second order linear differential equation (4). However, these results do not apply to nonlinear ODEs [3]. Recently, Li and Agarwal [11–15], Li and Huo [16,17], Li and Cheng [18], Li [19–21], Zhuang and Li [22] and Huang [23] further studied interval oscillation criteria for nonlinear ODEs. But these results cannot be applied to the neutral nonlinear equation (1). Motivated by the idea of Kong [10] and Rogovchenko [2], and Li and Agarwal [11], in this paper we use of the generalized Riccati techinque and averaging technique and by considering the function H ðt; sÞkðsÞ which may not
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
41
have a nonpositive partial derivative on D0 with the second variable, we relax the usual assumption ðoH ðt; sÞ=osÞ 6 0 on D0 ¼ fðt; sÞ: t > s P t0 g in [8], and we extend to nonlinear neutral differential equation an idea of Li and Agarwal [11,12] on interval criteria for oscillation of second order nonlinear ODEÕs, and obtain several new interval criteria for oscillation, this is, criteria given by the behavior of Eq. (1) only on a sequence of subintervals of ½t0 ; 1Þ. Hereafter, we always assume without mentioning that: ðC1 Þ The function p : ½t0 ; 1Þ ! ½0; 1Þ is a continuous function. ðC2 Þ The function qi : ½t0 ; 1Þ ! ½0; 1Þ, i ¼ 1; 2; . . . ; n, are continuous and qi 6 0 on any ½T ; 1Þ for some T P t0 . ðC3 Þ The functions fi : R ! R are continuous and fi ðyÞ=y P li > 0 or fi0 ðyÞ P li > 0 for y 6¼ 0, i ¼ 1; 2; . . . ; n, li are constants. ðC4 Þ The function r : ½t0 ; 1Þ ! R is continuous and nondecreasing, rðtÞ 6 t for t P t0 and limt!1 rðtÞ ¼ 1. ðC5 Þ The functions si : ½t0 ; 1Þ ! R are continuous, si 6 t for t P t0 and limt!1 si ðtÞ ¼ 1, i ¼ 1; 2; . . . ; n.
2. Main results In the sequel, we use the notation D0 ¼ fðt; sÞ: t > s P t0 g;
D ¼ fðt:sÞ: t P s P t0 g:
First we give two lemmas which will be used in the following results. Lemma 2.1. If yðtÞ is a nonoscillation solution of Eq. (1), then zðtÞz0 ðtÞ is eventually positive, where zðtÞ ¼ yðtÞ þ pðtÞyðrðtÞÞ for t P t0 . Proof. Without loss of generality we may assume that yðrðtÞÞ > 0 for t P T1 P t0 , where T1 is a positive number. Then zðtÞ > 0 and z00 ðtÞ ¼
n X
qi ðtÞfi ðyðsi ðtÞÞÞ 6 0;
t P T1 :
ð5Þ
i¼1
Therefore, z0 ðtÞ is decreasing. We claim that z0 ðtÞ > 0 for t P t0 . If it is not the case, we suppose that there exists a real number T2 P T1 such that z0 ðT2 Þ 6 0. It follows from ðC3 Þ, ðC4 Þ and (5) that there exists a real number T20 P T1 such that z0 ðT20 Þ < 0 and z0 ðtÞ 6 z0 ðT20 Þ for t P T20 . This implies that zðtÞ 6 zðT20 Þ þ z0 ðT20 Þðt T20 Þ;
t P T20 ;
42
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
which implies that zðtÞ ! 1 as t ! 1. This contradicts to zðtÞ > 0 for t P T1 . This completes our proof. h Lemma 2.2. Assume that yðtÞ 2 C 2 ½t0 ; 1Þ satisfies y 0 ðtÞ > 0;
yðtÞ > 0;
y 00 ðtÞ 6 0;
t P t0 ;
then for each 0 < li < 1, there exists a T2 P t0 such that yðsi ðtÞÞ P li yðtÞ
si ðtÞ ; t
t P T2 ; i ¼ 1; 2; . . . ; n:
The proof is similar to that of [1] (Lemma 2.1); here we omit it. Theorem 2.1. Assume that the functions H 2 CðD; RÞ, h1 ; h2 2 CðD0 ; RÞ, k; v 2 C 0 ð½t0 ; 1Þ; ð0; 1ÞÞ satisfy the following conditions: ðH1 Þ H ðt; tÞ ¼ 0 for t P t0 ; H ðt; sÞ > 0 on D0 ; 0
ðH2 Þ
o ðH ðt; sÞkðtÞÞ ot
ðtÞ þ H ðt; sÞkðtÞ vvðtÞ ¼ h1 ðt; sÞ; 8ðt; sÞ 2 D0 ;
ðH3 Þ
o ðH ðt; sÞkðsÞÞ os
ðsÞ þ H ðt; sÞkðsÞ vvðsÞ ¼ h2 ðt; sÞ; 8ðt; sÞ 2 D0 .
0
Assume also that for each sufficiently large T0 P t0 , there exists increasing divergent sequences of positive number fan g, fbn g, fcn g with T0 6 an < cn < bn such that Z
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i an Z bn n 1 vðsÞ X þ H ðbn ; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds H ðbn ; cn Þ cn s i¼1 i Z cn Z bn 1 vðsÞh1 ðs; an Þ 1 vðsÞh2 ðbn ; sÞ þ : ð6Þ > 4H ðcn ; an Þ an H ðs; an ÞkðsÞ 4H ðbn ; cn Þ cn H ðbn ; sÞkðsÞ
1 H ðcn ; an Þ
cn
H ðs; an ÞkðsÞ
Then Eq. (1) is oscillatory. Proof. Let yðtÞ be a nonoscillatory solution of Eq. (1). By Lemma 2.1, zðtÞz0 ðtÞ is eventually positive. Without loss of generality, we assume that yðtÞ > 0, zðsi ðtÞÞ > 0. z0 ðtÞ > 0 and zðrðsi ðtÞÞÞ > 0, i ¼ 1; 2; . . . ; n, for t P T1 P t0 , where T1 is a positive number. Since the case when yðtÞ is eventually negative can be treated analogously. Hence by Lemma 2.2, for any 0 < li < 1 there exists a T2 P T1 such that
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
zðsi ðtÞÞ P li zðtÞ
si ðtÞ ; t
i ¼ 1; 2; . . . ; n; t P T2 :
43
ð7Þ
Now, observing Eq. (1), we have
z00 ðtÞ þ
n X
qi ðtÞfi ðyðsi ðtÞÞÞ ¼ 0:
ð8Þ
i¼1
Using ðC1 Þ, ðC2 Þ, ðC3 Þ and (8), we get
z00 ðtÞ þ
n X
li qi ðtÞ½zðsi ðtÞÞ pðsi ðtÞÞzðrðsi ðtÞÞÞ 6 0:
ð9Þ
i¼1
In view of the fact that zðtÞ P yðtÞ and z0 ðtÞ > 0, yield
z00 ðtÞ þ
n X
li qi ðtÞ½1 pðsi ðtÞÞzðsi ðtÞÞ 6 0:
ð10Þ
i¼1
Define
wðtÞ ¼
vðtÞz0 ðtÞ ; zðtÞ
t P T ¼ maxfT0 ; T1 ; T2 g:
ð11Þ
Then 0 2 v0 ðtÞ z00 ðtÞ z ðtÞ wðtÞ þ vðtÞ vðtÞ w ðtÞ ¼ vðtÞ zðtÞ zðtÞ n X v0 ðtÞ si ðtÞ 1 2 wðtÞ vðtÞ ½1 pðsi ðtÞÞ w ðtÞ: 6 li li qi ðtÞ vðtÞ t vðtÞ i¼1 0
ð12Þ
Next we multiply (12), with t replaced by s, by H ðt; sÞkðsÞ and integrate from t1 to t ðbn P t2 > t1 P cn P T Þ. After simple computations, we get
44
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
Z
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i t1 Z t Z t v0 ðsÞ wðsÞ ds w0 ðsÞH ðt; sÞkðsÞ ds þ H ðt; sÞkðsÞ 6 vðsÞ t1 t1 Z t H ðt; sÞkðsÞ 2 w ðsÞ ds vðsÞ t1 Z t o v0 ðsÞ ðH ðt; sÞkðsÞÞ þ H ðt; sÞkðsÞ ¼ H ðt; t1 Þkðt1 Þwðt1 Þ þ wðsÞ ds os vðsÞ t1 Z t H ðt; sÞkðsÞ 2 w ðsÞ ds vðsÞ t1 Z t Z t H ðt; sÞkðsÞ 2 w ðsÞ ds ¼ H ðt; t1 Þkðt1 Þwðt1 Þ h2 ðt; sÞwðsÞ ds vðsÞ t1 t1 Z 1 t vðsÞh22 ðt; sÞ ds ¼ H ðt; t1 Þkðt1 Þwðt1 Þ þ 4 t1 H ðt; sÞkðsÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )2 Z t (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ðt; sÞkðsÞ 1 vðsÞ wðsÞ þ h2 ðt; sÞ ds: vðsÞ 2 H ðt; sÞkðsÞ t1 t
H ðt; sÞkðsÞ
Hence, we get Z t n vðsÞ X H ðt; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i t1 Z 1 t vðsÞh22 ðt; sÞ ds: 6 H ðt; t1 Þkðt1 Þwðt1 Þ þ 4 t1 H ðt; sÞkðsÞ
ð13Þ
Now put t1 ¼ cn and let t ¼ t2 ! b n in (13). Dividing both sides by H ðbn ; cn Þ, we obtain Z bn n 1 vðsÞ X H ðbn ; sÞkðsÞ l li ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds H ðbn ; cn Þ cn s i¼1 i Z bn 1 vðsÞh22 ðbn ; sÞ ds: ð14Þ 6 kðcn Þwðcn Þ þ 4H ðbn ; cn Þ cn H ðbn ; sÞkðsÞ Next go back to (12), and repeat the calculations multiplying first by H ðs; tÞkðsÞ instead of by H ðt; sÞkðsÞ and then integrating from t to t2 ðbn P t2 >t1 P cn P T Þ. We get Z t2 n vðsÞ X H ðs; tÞkðsÞ l li ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i t Z 1 t2 vðsÞh21 ðs; tÞ ds: ð15Þ 6 H ðt2 ; tÞkðt2 Þwðt2 Þ þ 4 t H ðs; tÞkðsÞ
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
45
Let t ¼ t1 ! aþ n and put t2 ¼ cn . Then divide both side in (15) by H ðcn ; an Þ, we get Z cn n 1 vðsÞ X H ðs; an ÞkðsÞ l li ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds H ðcn ; an Þ an s i¼1 i Z cn 1 vðsÞh21 ðs; an Þ ds: ð16Þ 6 kðcn Þwðcn Þ þ 4H ðcn ; an Þ an H ðs; an ÞkðsÞ Now we claim that every nontrivial solution of Eq. (1) has at least one zero tn 2 ðan ; bn Þ. Suppose the contrary. Then by the assumption that yðtÞ > 0 for t 2 ðan ; bn Þ and (14), (16) hold. Adding (14) and (16), we have the inequality Z
1 H ðcn ; an Þ þ
6
cn
H ðs; an ÞkðsÞ
an
1 H ðbn ; cn Þ
1 4H ðcn ; an Þ
Z
Z
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i
bn
H ðbn ; sÞkðsÞ
cn cn an
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i
vðsÞh1 ðs; an Þ 1 ds þ H ðs; an ÞkðsÞ 4H ðbn ; cn Þ
Z
bn cn
vðsÞh2 ðbn ; sÞ : H ðbn ; sÞkðsÞ
ð17Þ
which contradicts the assumption (6). Thus, the claim holds, i.e., every nontrivial solution of Eq. (1) has at least one zero ti 2 ðai ; bi Þ. Noting that bi P ti > ai P T0 , i 2 N , we see that every solution has arbitrarily large zeros. Hence Eq. (1) is oscillatory. h As immediate consequences of Theorem 2.1, the following result is true. Theorem 2.2. Let condition (6) in Theorem 2.1 be replaced by ) Z t( n vðsÞ X vðsÞh21 ðs; lÞ ds lim sup H ðs; lÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ t!1 s i¼1 i 4H ðs; lÞkðsÞ l ð18Þ
> 0;
and Z t( lim sup
t!1
l
) n vðsÞ X vðsÞh22 ðt; sÞ ds H ðt; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4H ðt; sÞkðsÞ
>0
for each sufficient large l P T0 P t0 . Then Eq. (1) is oscillatory.
ð19Þ
46
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
Proof. For any T P T0 P t0 , let an ¼ T . In (18) we choose l ¼ an . Then there exists cn > an such that Z
cn
an
(
) n vðsÞ X vðsÞh21 ðs; an Þ ds > 0: H ðs; an ÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4H ðs; an ÞkðsÞ
ð20Þ In (19) we choose l ¼ cn . Then there exists bn > cn such that ) Z bn ( n vðsÞ X vðsÞh22 ðbn ; sÞ ds > 0: H ðbn ; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4H ðbn ; sÞkðsÞ cn ð21Þ Combining (20) and (21) we obtain (6). The conclusion thus comes from Theorem 2.1. The proof is complete. h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If p h1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt; sÞ and h2 ðt; sÞ are replaced by h1 ðt; sÞ H ðt; sÞkðsÞ and h2 ðt; sÞ H ðt; sÞkðsÞ, respectively, in Theorems 2.1 and 2.2, we have the following results. The proof are quite similar, so we omit the details. Theorem 2.3. Assume that the functions H 2 CðD; RÞ, h1 ; h2 2 CðD0 ; RÞ, k; v 2 C 0 ð½t0 ; 1Þ; ð0; 1ÞÞ satisfy the following conditions: ðH1 Þ H ðt; tÞ ¼ 0 for t P t0 ; H ðt; sÞ > 0 on D0 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ðtÞ ðH2 Þ oto ðH ðt; sÞkðtÞÞ þ H ðt; sÞkðtÞ vvðtÞ ¼ h1 ðt; sÞ H ðt; sÞkðtÞ; 8ðt; sÞ 2 D0 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ðsÞ ¼ h2 ðt; sÞ H ðt; sÞkðsÞ; 8ðt; sÞ 2 D0 . ðH3 Þ oso ðH ðt; sÞkðsÞÞ þ H ðt; sÞkðsÞ vvðsÞ Assume also that for each sufficiently large T0 P t0 , there exists increasing divergent sequences of positive number fan g, fbn g, fcn g with T0 6 an < cn < bn such that Z
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i an Z bn n 1 vðsÞ X þ H ðbn ; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds H ðbn ; cn Þ cn s i¼1 i Z cn Z bn 1 1 > vðsÞh21 ðs; an Þ þ vðsÞh22 ðbn ; sÞ: ð22Þ 4H ðcn ; an Þ an 4H ðbn ; cn Þ cn
1 H ðcn ; an Þ
cn
H ðs; an ÞkðsÞ
Then Eq. (1) is oscillatory.
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
47
Theorem 2.4. Let condition (22) in Theorem 2.3 be replaced by Z t( lim sup
t!1
l
) n vðsÞ X vðsÞh21 ðs; lÞ ds > 0; H ðs; lÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4
ð23Þ and Z t( lim sup
t!1
l
) n vðsÞ X vðsÞh22 ðt; sÞ ds > 0 H ðt; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4
ð24Þ for each sufficient large l P T0 P t0 . Then Eq. (1) is oscillatory. We choose kðsÞ 1 and H ðt; sÞ ¼ H ðt sÞ, in Theorem 2.3, we have that oðH ðt sÞ oðH ðt sÞÞ ¼ ; ot os and denote them by hðt sÞ. Then hðt sÞ v0 ðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 ðt; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ H ðt sÞ; H ðt sÞ vðtÞ and hðt sÞ v0 ðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 ðt; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ðt sÞ: H ðt sÞ vðsÞ Applying Theorem 2.3 we have Theorem 2.5. Assume that for any T P t0 there exist T 6 an < cn such that Z cn Z cn n vðsÞ X H ðs an ÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds þ H ðs an Þ s i¼1 i an an kð2cn sÞ
1 > 4
Z
cn
an
1 þ 4
Z
n vð2cn sÞ X l li qi ð2cn sÞ½1 pðsi ð2cn sÞÞsi ð2cn sÞ ds 2cn s i¼1 i
#2 hðs an Þ v0 ðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ H ðs an Þ ds H ðs an Þ vðsÞ "
cn
an
#2 hðs an Þ v0 ð2cn sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vð2cn sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ðs an Þ ds: H ðs an Þ vð2cn sÞ
Then Eq. (1) is oscillatory.
"
ð25Þ
48
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
Proof. Let bn ¼ 2cn an . Then H ðbn cn Þ ¼ H ðcn an Þ ¼ H
bn an 2
and for any u 2 L½a; b, we have Z
bn
uðsÞ ds ¼
Z
cn
Hence Z
cn
uð2cn sÞ ds:
an
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i cn Z cn n vð2cn sÞ X ¼ H ðs an Þ l li qi ð2cn sÞ 2cn s i¼1 i an bn
H ðbn sÞ
½1 pðsi ð2cn sÞÞsi ð2cn sÞ ds; and Z
bn
vðsÞh22 ðbn sÞ ¼
cn
Z
cn
vð2cn sÞh22 ðs an Þ ds
an
¼
Z
cn
an
#2 hðs an Þ v0 ð2cn sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hðs an Þ ds: vð2cn sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hðs an Þ vð2cn sÞ "
Thus that (25) holds implies that (22) holds, and therefore Eq. (1) is oscillatory by Theorem 2.3. The proof is complete. h From above oscillation criteria, we can obtain different sufficient conditions for oscillation of Eq. (1) by different choices of H ðt; sÞ and kðtÞ. We choose H ðt sÞ ¼ ðt sÞk ;
t P s P t0 ;
k > 1 is a constant. Let vðtÞ 1, kðtÞ 1. Based on the above results we obtain the following oscillation criteria of KamenevÕs type. Theorem 2.6. Eq. (1) is oscillatory provided that for each l P t0 , and there exists k > 1 such that the following inequalities are satisfied: Z t n 1 1X k2 ðs lÞk li li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds > lim sup k1 t!1 t s i¼1 4ðk 1Þ l ð26Þ
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
49
and lim sup
t!1
Z
1 tk1
t
ðt sÞ
l
k
n 1X k2 : li li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds > s i¼1 4ðk 1Þ
ð27Þ Then Eq. (1) is oscillatory. The proof is similar to that of Theorem 2.5 of Li and Agarwal [12] and omitted it here. Theorem 2.7. Eq. (1) is oscillatory provided that for each l P t0 , and there exists k > 1 such that the following inequality is satisfied: ( Z t n 1 1X k lim sup k1 ðs lÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ t!1 t s i¼1 i l ) n 1 X k2 þ : li li qi ð2t sÞ½1 pðsi ð2t sÞÞsi ð2t sÞ ds > 2t s i¼1 2ðk 1Þ ð28Þ Then Eq. (1) is oscillatory. Proof. Noting that lim
t!1
k2 2tk1
Z l
t
ðs lÞk2 ds ¼
k2 : 2ðk 1Þ
ð29Þ
From (28) and (29), let l ¼ an , we have " Z t( n X 1 1 k 1 lim sup k1 ðs an Þ li li qi ðsÞ½1 pðsi ðsÞÞ þ t!1 t s 2t s an i¼1 ) # n X k2 k2 ds > 0: li li qi ð2t sÞ½1 pðsi ð2t sÞÞsi ð2t sÞ ðs an Þ 2 i¼1 Hence " Z t( n n X 1 X k 1 lim sup ðs an Þ li li qi ðsÞ½1 pðsi ðsÞÞ þ l li qi ð2t sÞ t!1 s i¼1 2t s i¼1 i an ) # k2 k2 ½1 pðsi ð2t sÞÞsi ð2t sÞ ðs an Þ ds > 0: 2
50
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
Then, there exist cn > an such that " Z cn ( n X k 1 ðs an Þ l li qi ðsÞ½1 pðsi ðsÞÞ s i¼1 i an n X 1 l li qi ð2cn sÞ½1 pðsi ð2cn sÞÞsi ð2cn sÞ þ 2cn s i¼1 i Z k2 c n > ðs an Þk2 ds; 2 an
#)
i.e., (25) holds. By Theorem 2.5, Eq. (1) is oscillatory. The proof is complete. h
References [1] L.H. Erbe, Oscillation criteria for second order nonlinear delay equation, Can. Math. Bull. 16 (1972) 49–59. [2] Y.V. Rogovchenko, On oscillation of a second-order nonlinear delay differential equation, Funkcial. Ekvac. 43 (2000) 1–29. [3] Y.V. Rogovchkno, Oscillation criteria for certain nonlinear differential equation, J. Math. Anal. Appl. 229 (1999) 399–416. [4] H.J. Li, C.C. Yeh, Oscillation criteria for second-order neutral delay equations, Comput. Math. Appl. 36 (10–12) (1998) 123–132. [5] S.R. Grace, Oscillation theorems for nonlinear differential equations of second order, J. Math. Anal. Appl. 171 (1992) 220–241. [6] M.A. El-Sayed, An oscillation criteria for a forced second linear differential equation, Proc. Amer. Math. Soc. 24 (1982) 169–182. [7] I.V. Kamenev, An integral criterion of oscillation for linear differential equations of second order, Mat. Zametki. 23 (1978) 249–251. [8] GH.G. Philos, Oscillation theorems linear differential equations of second order, Arch. Math. 53 (1989) 483–492. [9] W.T. Li, Oscillation of certain second-order nonlinear differential equations, J. Math. Anal. Appl. 217 (1998) 1–14. [10] Q.K. Kong, Interval criterion for oscillation of second order linear differential equations, J. Math. Anal. Appl. 229 (1999) 258–270. [11] W.T. Li, R.P. Agarwal, Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equation, J. Math. Anal. Appl. 245 (2000) 171–188. [12] W.T. Li, R.P. Agarwal, Interval oscillation criteria for second order nonlinear differential equations with damping, Comput. Math. Appl. 40 (2000) 217–230. [13] W.T. Li, R.P. Agarwal, Interval oscillation criteria for a forced nonlinear ordinary differential equations, Appl. Anal. 75 (3/4) (2000) 341–347. [14] W.T. Li, R.P. Agarwal, Interval oscillation criteria for second order forced nonlinear differential equations with damping, Panamerican Math. J. 11 (3) (2001) 109–117. [15] W.T. Li, R.P. Agarwal, Interval oscillation criteria for second-order nonlinear differential equations, Ukrainian Math. J. 53 (9) (2001) 1391–1406. [16] W.T. Li, H.F. Huo, Interval oscillation criteria for nonlinear second order differential equations, Indian J. Pure Appl. Math. 32 (7) (2001) 1003–1014.
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[17] W.T. Li, H.F. Huo, Interval oscillation criteria for second order forced nonlinear differential equations with damping, Annles Univ. Sci. Budapest 44 (2001) 83–93. [18] W.T. Li, S.S. Cheng, An oscillation criteria for nonhomogeneous half-linear differential equations, Appl. Math. Lett. 15 (2002) 259–263. [19] W.T. Li, Oscillation criteria for second-order half-linear ordinary differential equations with damping, Rocky Mountain J. Math., in press. [20] W.T. Li, Interval oscillation criteria for second order nonlinear differential equations with damping, Taiwanese J. Math., in press. [21] W.T. Li, Interval oscillation criteria for second order half-linear differential equations, Acta Math. Sinica 35 (3) (2002) 509–516. [22] R.K. Zhuang, W.T. Li, Interval oscillation criteria for second order quasilinear perturbed differential equations, Int. J. Diff. Equat. Appl. 7 (2003) 165–179. [23] C.C. Hung, Oscillation and nonoscillation for linear second order differential equation, J. Math. Anal. Appl. 210 (1997) 712–723.
Interval oscillation criteria for second order neutral nonlinear differential equations Rong-Kun Zhuang a
a,1
, Wan-Ton Li
b,*,2
Department of Mathematics, Huizhou University, Huizhou 516015, People’s Republic of China b Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
Abstract Some oscillation criteria for the second order neutral nonlinear differential equation n X qi ðtÞfi ðyðsi ðtÞÞÞ ¼ 0; t P t0 ½yðtÞ þ pðtÞyðrðtÞÞ00 þ i¼1
are established. New oscillation criteria are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of ½t0 ; 1Þ, rather than on the whole half-line. Our results are more natural according to the Sturm Separation Theorem. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Oscillation; Nonlinear differential equation; Neutral type; Riccati transformation; Interval criteria
1. Introduction In this paper we consider the oscillation behavior of solutions of the second order neutral nonlinear differential equation *
Corresponding author. E-mail addresses: [email protected] (R.-K. Zhuang), [email protected] (W.-T. Li). 1 Supported by the NSF of Educational Department of Guangdong Province (0176). 2 Supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011A25-007-Z), the Foundation for University Key Teacher by the Ministry of Education of China, and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China. 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.06.016
40
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51 00
½yðtÞ þ pðtÞyðrðtÞÞ þ
n X
qi ðtÞfi ðyðsi ðtÞÞÞ ¼ 0;
ð1Þ
i¼1
where t P t0 , and the functions p, r, si , qi , fi , (i ¼ 1; 2; . . . ; n) are to be specified in the following text. By a solution of (1), we mean a continuously differentiable function yðtÞ which is defined for t P minfrðt0 Þ; si ðt0 Þ; i ¼ 1; 2; . . . ; ng such that yðtÞ ¼ /ðtÞ for minfrðt0 Þ; si ðt0 Þ; i ¼ 1; 2; . . . ; ng 6 t0 and satisfies (1) for all t P t0 . We restrict our attention only to the nontrivial solution yðtÞ of (1), i.e., to the solution yðtÞ such that supfjyðtÞj: t P T g > 0 for all T P t0 . A nontrivial solution of (1) is called oscillatory if it has arbitrary large zeros, otherwise, it is called nonoscillatory. Eq. (1) is called oscillatory if all its solutions are oscillatory. The oscillation problem for nonlinear delay equation such as ½pðtÞy 0 ðtÞ0 þ qðtÞf ðyðsðtÞÞÞgðy 0 ðtÞÞ ¼ 0;
t P t0 ;
ð2Þ
as well as for the nonlinear ordinary differential equation 0
½pðtÞy 0 ðtÞ þ qðtÞf ðyðtÞÞgðy 0 ðtÞÞ ¼ 0;
t P t0 ;
ð3Þ
and the linear ordinary equation 0
½pðtÞy 0 ðtÞ þ qðtÞyðtÞ ¼ 0;
t P t0
ð4Þ
has been studied by many authors with different methods. Some result can be found in [1–9] and references therein. On the one hand, the recent paper by Rogovechenko [2] contains various conditions for nonlinear delay equations obtained by using an integral averaging technique similar to that exploited in [8]. On the other hand, most known oscillation criteria involve the integrals of p and q and hence require the information of p and q on the entire half-line ½t0 ; 1Þ. However, from the Sturm Separation Theorem, we see that oscillation is only an interval property, i.e., if there exists a sequence of subintervals ½ai ; bi of ½t0 ; 1Þ, as ai ! 1, such that for each i there exists a solution of Eq. (4) that has at least two zeros in ½ai ; bi , then every solution of Eq. (4) is oscillatory, no matter how ‘‘bad’’ Eq. (4) is (or, p and q are) on the remaining part of ½t0 :1Þ. Kong [10] applied this idea to oscillation and established an interval criterion for oscillation of the second order linear differential equation (4). However, these results do not apply to nonlinear ODEs [3]. Recently, Li and Agarwal [11–15], Li and Huo [16,17], Li and Cheng [18], Li [19–21], Zhuang and Li [22] and Huang [23] further studied interval oscillation criteria for nonlinear ODEs. But these results cannot be applied to the neutral nonlinear equation (1). Motivated by the idea of Kong [10] and Rogovchenko [2], and Li and Agarwal [11], in this paper we use of the generalized Riccati techinque and averaging technique and by considering the function H ðt; sÞkðsÞ which may not
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
41
have a nonpositive partial derivative on D0 with the second variable, we relax the usual assumption ðoH ðt; sÞ=osÞ 6 0 on D0 ¼ fðt; sÞ: t > s P t0 g in [8], and we extend to nonlinear neutral differential equation an idea of Li and Agarwal [11,12] on interval criteria for oscillation of second order nonlinear ODEÕs, and obtain several new interval criteria for oscillation, this is, criteria given by the behavior of Eq. (1) only on a sequence of subintervals of ½t0 ; 1Þ. Hereafter, we always assume without mentioning that: ðC1 Þ The function p : ½t0 ; 1Þ ! ½0; 1Þ is a continuous function. ðC2 Þ The function qi : ½t0 ; 1Þ ! ½0; 1Þ, i ¼ 1; 2; . . . ; n, are continuous and qi 6 0 on any ½T ; 1Þ for some T P t0 . ðC3 Þ The functions fi : R ! R are continuous and fi ðyÞ=y P li > 0 or fi0 ðyÞ P li > 0 for y 6¼ 0, i ¼ 1; 2; . . . ; n, li are constants. ðC4 Þ The function r : ½t0 ; 1Þ ! R is continuous and nondecreasing, rðtÞ 6 t for t P t0 and limt!1 rðtÞ ¼ 1. ðC5 Þ The functions si : ½t0 ; 1Þ ! R are continuous, si 6 t for t P t0 and limt!1 si ðtÞ ¼ 1, i ¼ 1; 2; . . . ; n.
2. Main results In the sequel, we use the notation D0 ¼ fðt; sÞ: t > s P t0 g;
D ¼ fðt:sÞ: t P s P t0 g:
First we give two lemmas which will be used in the following results. Lemma 2.1. If yðtÞ is a nonoscillation solution of Eq. (1), then zðtÞz0 ðtÞ is eventually positive, where zðtÞ ¼ yðtÞ þ pðtÞyðrðtÞÞ for t P t0 . Proof. Without loss of generality we may assume that yðrðtÞÞ > 0 for t P T1 P t0 , where T1 is a positive number. Then zðtÞ > 0 and z00 ðtÞ ¼
n X
qi ðtÞfi ðyðsi ðtÞÞÞ 6 0;
t P T1 :
ð5Þ
i¼1
Therefore, z0 ðtÞ is decreasing. We claim that z0 ðtÞ > 0 for t P t0 . If it is not the case, we suppose that there exists a real number T2 P T1 such that z0 ðT2 Þ 6 0. It follows from ðC3 Þ, ðC4 Þ and (5) that there exists a real number T20 P T1 such that z0 ðT20 Þ < 0 and z0 ðtÞ 6 z0 ðT20 Þ for t P T20 . This implies that zðtÞ 6 zðT20 Þ þ z0 ðT20 Þðt T20 Þ;
t P T20 ;
42
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
which implies that zðtÞ ! 1 as t ! 1. This contradicts to zðtÞ > 0 for t P T1 . This completes our proof. h Lemma 2.2. Assume that yðtÞ 2 C 2 ½t0 ; 1Þ satisfies y 0 ðtÞ > 0;
yðtÞ > 0;
y 00 ðtÞ 6 0;
t P t0 ;
then for each 0 < li < 1, there exists a T2 P t0 such that yðsi ðtÞÞ P li yðtÞ
si ðtÞ ; t
t P T2 ; i ¼ 1; 2; . . . ; n:
The proof is similar to that of [1] (Lemma 2.1); here we omit it. Theorem 2.1. Assume that the functions H 2 CðD; RÞ, h1 ; h2 2 CðD0 ; RÞ, k; v 2 C 0 ð½t0 ; 1Þ; ð0; 1ÞÞ satisfy the following conditions: ðH1 Þ H ðt; tÞ ¼ 0 for t P t0 ; H ðt; sÞ > 0 on D0 ; 0
ðH2 Þ
o ðH ðt; sÞkðtÞÞ ot
ðtÞ þ H ðt; sÞkðtÞ vvðtÞ ¼ h1 ðt; sÞ; 8ðt; sÞ 2 D0 ;
ðH3 Þ
o ðH ðt; sÞkðsÞÞ os
ðsÞ þ H ðt; sÞkðsÞ vvðsÞ ¼ h2 ðt; sÞ; 8ðt; sÞ 2 D0 .
0
Assume also that for each sufficiently large T0 P t0 , there exists increasing divergent sequences of positive number fan g, fbn g, fcn g with T0 6 an < cn < bn such that Z
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i an Z bn n 1 vðsÞ X þ H ðbn ; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds H ðbn ; cn Þ cn s i¼1 i Z cn Z bn 1 vðsÞh1 ðs; an Þ 1 vðsÞh2 ðbn ; sÞ þ : ð6Þ > 4H ðcn ; an Þ an H ðs; an ÞkðsÞ 4H ðbn ; cn Þ cn H ðbn ; sÞkðsÞ
1 H ðcn ; an Þ
cn
H ðs; an ÞkðsÞ
Then Eq. (1) is oscillatory. Proof. Let yðtÞ be a nonoscillatory solution of Eq. (1). By Lemma 2.1, zðtÞz0 ðtÞ is eventually positive. Without loss of generality, we assume that yðtÞ > 0, zðsi ðtÞÞ > 0. z0 ðtÞ > 0 and zðrðsi ðtÞÞÞ > 0, i ¼ 1; 2; . . . ; n, for t P T1 P t0 , where T1 is a positive number. Since the case when yðtÞ is eventually negative can be treated analogously. Hence by Lemma 2.2, for any 0 < li < 1 there exists a T2 P T1 such that
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
zðsi ðtÞÞ P li zðtÞ
si ðtÞ ; t
i ¼ 1; 2; . . . ; n; t P T2 :
43
ð7Þ
Now, observing Eq. (1), we have
z00 ðtÞ þ
n X
qi ðtÞfi ðyðsi ðtÞÞÞ ¼ 0:
ð8Þ
i¼1
Using ðC1 Þ, ðC2 Þ, ðC3 Þ and (8), we get
z00 ðtÞ þ
n X
li qi ðtÞ½zðsi ðtÞÞ pðsi ðtÞÞzðrðsi ðtÞÞÞ 6 0:
ð9Þ
i¼1
In view of the fact that zðtÞ P yðtÞ and z0 ðtÞ > 0, yield
z00 ðtÞ þ
n X
li qi ðtÞ½1 pðsi ðtÞÞzðsi ðtÞÞ 6 0:
ð10Þ
i¼1
Define
wðtÞ ¼
vðtÞz0 ðtÞ ; zðtÞ
t P T ¼ maxfT0 ; T1 ; T2 g:
ð11Þ
Then 0 2 v0 ðtÞ z00 ðtÞ z ðtÞ wðtÞ þ vðtÞ vðtÞ w ðtÞ ¼ vðtÞ zðtÞ zðtÞ n X v0 ðtÞ si ðtÞ 1 2 wðtÞ vðtÞ ½1 pðsi ðtÞÞ w ðtÞ: 6 li li qi ðtÞ vðtÞ t vðtÞ i¼1 0
ð12Þ
Next we multiply (12), with t replaced by s, by H ðt; sÞkðsÞ and integrate from t1 to t ðbn P t2 > t1 P cn P T Þ. After simple computations, we get
44
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
Z
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i t1 Z t Z t v0 ðsÞ wðsÞ ds w0 ðsÞH ðt; sÞkðsÞ ds þ H ðt; sÞkðsÞ 6 vðsÞ t1 t1 Z t H ðt; sÞkðsÞ 2 w ðsÞ ds vðsÞ t1 Z t o v0 ðsÞ ðH ðt; sÞkðsÞÞ þ H ðt; sÞkðsÞ ¼ H ðt; t1 Þkðt1 Þwðt1 Þ þ wðsÞ ds os vðsÞ t1 Z t H ðt; sÞkðsÞ 2 w ðsÞ ds vðsÞ t1 Z t Z t H ðt; sÞkðsÞ 2 w ðsÞ ds ¼ H ðt; t1 Þkðt1 Þwðt1 Þ h2 ðt; sÞwðsÞ ds vðsÞ t1 t1 Z 1 t vðsÞh22 ðt; sÞ ds ¼ H ðt; t1 Þkðt1 Þwðt1 Þ þ 4 t1 H ðt; sÞkðsÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )2 Z t (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ðt; sÞkðsÞ 1 vðsÞ wðsÞ þ h2 ðt; sÞ ds: vðsÞ 2 H ðt; sÞkðsÞ t1 t
H ðt; sÞkðsÞ
Hence, we get Z t n vðsÞ X H ðt; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i t1 Z 1 t vðsÞh22 ðt; sÞ ds: 6 H ðt; t1 Þkðt1 Þwðt1 Þ þ 4 t1 H ðt; sÞkðsÞ
ð13Þ
Now put t1 ¼ cn and let t ¼ t2 ! b n in (13). Dividing both sides by H ðbn ; cn Þ, we obtain Z bn n 1 vðsÞ X H ðbn ; sÞkðsÞ l li ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds H ðbn ; cn Þ cn s i¼1 i Z bn 1 vðsÞh22 ðbn ; sÞ ds: ð14Þ 6 kðcn Þwðcn Þ þ 4H ðbn ; cn Þ cn H ðbn ; sÞkðsÞ Next go back to (12), and repeat the calculations multiplying first by H ðs; tÞkðsÞ instead of by H ðt; sÞkðsÞ and then integrating from t to t2 ðbn P t2 >t1 P cn P T Þ. We get Z t2 n vðsÞ X H ðs; tÞkðsÞ l li ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i t Z 1 t2 vðsÞh21 ðs; tÞ ds: ð15Þ 6 H ðt2 ; tÞkðt2 Þwðt2 Þ þ 4 t H ðs; tÞkðsÞ
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
45
Let t ¼ t1 ! aþ n and put t2 ¼ cn . Then divide both side in (15) by H ðcn ; an Þ, we get Z cn n 1 vðsÞ X H ðs; an ÞkðsÞ l li ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds H ðcn ; an Þ an s i¼1 i Z cn 1 vðsÞh21 ðs; an Þ ds: ð16Þ 6 kðcn Þwðcn Þ þ 4H ðcn ; an Þ an H ðs; an ÞkðsÞ Now we claim that every nontrivial solution of Eq. (1) has at least one zero tn 2 ðan ; bn Þ. Suppose the contrary. Then by the assumption that yðtÞ > 0 for t 2 ðan ; bn Þ and (14), (16) hold. Adding (14) and (16), we have the inequality Z
1 H ðcn ; an Þ þ
6
cn
H ðs; an ÞkðsÞ
an
1 H ðbn ; cn Þ
1 4H ðcn ; an Þ
Z
Z
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i
bn
H ðbn ; sÞkðsÞ
cn cn an
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i
vðsÞh1 ðs; an Þ 1 ds þ H ðs; an ÞkðsÞ 4H ðbn ; cn Þ
Z
bn cn
vðsÞh2 ðbn ; sÞ : H ðbn ; sÞkðsÞ
ð17Þ
which contradicts the assumption (6). Thus, the claim holds, i.e., every nontrivial solution of Eq. (1) has at least one zero ti 2 ðai ; bi Þ. Noting that bi P ti > ai P T0 , i 2 N , we see that every solution has arbitrarily large zeros. Hence Eq. (1) is oscillatory. h As immediate consequences of Theorem 2.1, the following result is true. Theorem 2.2. Let condition (6) in Theorem 2.1 be replaced by ) Z t( n vðsÞ X vðsÞh21 ðs; lÞ ds lim sup H ðs; lÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ t!1 s i¼1 i 4H ðs; lÞkðsÞ l ð18Þ
> 0;
and Z t( lim sup
t!1
l
) n vðsÞ X vðsÞh22 ðt; sÞ ds H ðt; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4H ðt; sÞkðsÞ
>0
for each sufficient large l P T0 P t0 . Then Eq. (1) is oscillatory.
ð19Þ
46
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
Proof. For any T P T0 P t0 , let an ¼ T . In (18) we choose l ¼ an . Then there exists cn > an such that Z
cn
an
(
) n vðsÞ X vðsÞh21 ðs; an Þ ds > 0: H ðs; an ÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4H ðs; an ÞkðsÞ
ð20Þ In (19) we choose l ¼ cn . Then there exists bn > cn such that ) Z bn ( n vðsÞ X vðsÞh22 ðbn ; sÞ ds > 0: H ðbn ; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4H ðbn ; sÞkðsÞ cn ð21Þ Combining (20) and (21) we obtain (6). The conclusion thus comes from Theorem 2.1. The proof is complete. h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If p h1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt; sÞ and h2 ðt; sÞ are replaced by h1 ðt; sÞ H ðt; sÞkðsÞ and h2 ðt; sÞ H ðt; sÞkðsÞ, respectively, in Theorems 2.1 and 2.2, we have the following results. The proof are quite similar, so we omit the details. Theorem 2.3. Assume that the functions H 2 CðD; RÞ, h1 ; h2 2 CðD0 ; RÞ, k; v 2 C 0 ð½t0 ; 1Þ; ð0; 1ÞÞ satisfy the following conditions: ðH1 Þ H ðt; tÞ ¼ 0 for t P t0 ; H ðt; sÞ > 0 on D0 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ðtÞ ðH2 Þ oto ðH ðt; sÞkðtÞÞ þ H ðt; sÞkðtÞ vvðtÞ ¼ h1 ðt; sÞ H ðt; sÞkðtÞ; 8ðt; sÞ 2 D0 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ðsÞ ¼ h2 ðt; sÞ H ðt; sÞkðsÞ; 8ðt; sÞ 2 D0 . ðH3 Þ oso ðH ðt; sÞkðsÞÞ þ H ðt; sÞkðsÞ vvðsÞ Assume also that for each sufficiently large T0 P t0 , there exists increasing divergent sequences of positive number fan g, fbn g, fcn g with T0 6 an < cn < bn such that Z
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i an Z bn n 1 vðsÞ X þ H ðbn ; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds H ðbn ; cn Þ cn s i¼1 i Z cn Z bn 1 1 > vðsÞh21 ðs; an Þ þ vðsÞh22 ðbn ; sÞ: ð22Þ 4H ðcn ; an Þ an 4H ðbn ; cn Þ cn
1 H ðcn ; an Þ
cn
H ðs; an ÞkðsÞ
Then Eq. (1) is oscillatory.
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
47
Theorem 2.4. Let condition (22) in Theorem 2.3 be replaced by Z t( lim sup
t!1
l
) n vðsÞ X vðsÞh21 ðs; lÞ ds > 0; H ðs; lÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4
ð23Þ and Z t( lim sup
t!1
l
) n vðsÞ X vðsÞh22 ðt; sÞ ds > 0 H ðt; sÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ s i¼1 i 4
ð24Þ for each sufficient large l P T0 P t0 . Then Eq. (1) is oscillatory. We choose kðsÞ 1 and H ðt; sÞ ¼ H ðt sÞ, in Theorem 2.3, we have that oðH ðt sÞ oðH ðt sÞÞ ¼ ; ot os and denote them by hðt sÞ. Then hðt sÞ v0 ðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 ðt; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ H ðt sÞ; H ðt sÞ vðtÞ and hðt sÞ v0 ðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 ðt; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ðt sÞ: H ðt sÞ vðsÞ Applying Theorem 2.3 we have Theorem 2.5. Assume that for any T P t0 there exist T 6 an < cn such that Z cn Z cn n vðsÞ X H ðs an ÞkðsÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds þ H ðs an Þ s i¼1 i an an kð2cn sÞ
1 > 4
Z
cn
an
1 þ 4
Z
n vð2cn sÞ X l li qi ð2cn sÞ½1 pðsi ð2cn sÞÞsi ð2cn sÞ ds 2cn s i¼1 i
#2 hðs an Þ v0 ðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ H ðs an Þ ds H ðs an Þ vðsÞ "
cn
an
#2 hðs an Þ v0 ð2cn sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vð2cn sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ðs an Þ ds: H ðs an Þ vð2cn sÞ
Then Eq. (1) is oscillatory.
"
ð25Þ
48
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
Proof. Let bn ¼ 2cn an . Then H ðbn cn Þ ¼ H ðcn an Þ ¼ H
bn an 2
and for any u 2 L½a; b, we have Z
bn
uðsÞ ds ¼
Z
cn
Hence Z
cn
uð2cn sÞ ds:
an
n vðsÞ X l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds s i¼1 i cn Z cn n vð2cn sÞ X ¼ H ðs an Þ l li qi ð2cn sÞ 2cn s i¼1 i an bn
H ðbn sÞ
½1 pðsi ð2cn sÞÞsi ð2cn sÞ ds; and Z
bn
vðsÞh22 ðbn sÞ ¼
cn
Z
cn
vð2cn sÞh22 ðs an Þ ds
an
¼
Z
cn
an
#2 hðs an Þ v0 ð2cn sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hðs an Þ ds: vð2cn sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hðs an Þ vð2cn sÞ "
Thus that (25) holds implies that (22) holds, and therefore Eq. (1) is oscillatory by Theorem 2.3. The proof is complete. h From above oscillation criteria, we can obtain different sufficient conditions for oscillation of Eq. (1) by different choices of H ðt; sÞ and kðtÞ. We choose H ðt sÞ ¼ ðt sÞk ;
t P s P t0 ;
k > 1 is a constant. Let vðtÞ 1, kðtÞ 1. Based on the above results we obtain the following oscillation criteria of KamenevÕs type. Theorem 2.6. Eq. (1) is oscillatory provided that for each l P t0 , and there exists k > 1 such that the following inequalities are satisfied: Z t n 1 1X k2 ðs lÞk li li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds > lim sup k1 t!1 t s i¼1 4ðk 1Þ l ð26Þ
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
49
and lim sup
t!1
Z
1 tk1
t
ðt sÞ
l
k
n 1X k2 : li li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ ds > s i¼1 4ðk 1Þ
ð27Þ Then Eq. (1) is oscillatory. The proof is similar to that of Theorem 2.5 of Li and Agarwal [12] and omitted it here. Theorem 2.7. Eq. (1) is oscillatory provided that for each l P t0 , and there exists k > 1 such that the following inequality is satisfied: ( Z t n 1 1X k lim sup k1 ðs lÞ l li qi ðsÞ½1 pðsi ðsÞÞsi ðsÞ t!1 t s i¼1 i l ) n 1 X k2 þ : li li qi ð2t sÞ½1 pðsi ð2t sÞÞsi ð2t sÞ ds > 2t s i¼1 2ðk 1Þ ð28Þ Then Eq. (1) is oscillatory. Proof. Noting that lim
t!1
k2 2tk1
Z l
t
ðs lÞk2 ds ¼
k2 : 2ðk 1Þ
ð29Þ
From (28) and (29), let l ¼ an , we have " Z t( n X 1 1 k 1 lim sup k1 ðs an Þ li li qi ðsÞ½1 pðsi ðsÞÞ þ t!1 t s 2t s an i¼1 ) # n X k2 k2 ds > 0: li li qi ð2t sÞ½1 pðsi ð2t sÞÞsi ð2t sÞ ðs an Þ 2 i¼1 Hence " Z t( n n X 1 X k 1 lim sup ðs an Þ li li qi ðsÞ½1 pðsi ðsÞÞ þ l li qi ð2t sÞ t!1 s i¼1 2t s i¼1 i an ) # k2 k2 ½1 pðsi ð2t sÞÞsi ð2t sÞ ðs an Þ ds > 0: 2
50
R.-K. Zhuang, W.-T. Li / Appl. Math. Comput. 157 (2004) 39–51
Then, there exist cn > an such that " Z cn ( n X k 1 ðs an Þ l li qi ðsÞ½1 pðsi ðsÞÞ s i¼1 i an n X 1 l li qi ð2cn sÞ½1 pðsi ð2cn sÞÞsi ð2cn sÞ þ 2cn s i¼1 i Z k2 c n > ðs an Þk2 ds; 2 an
#)
i.e., (25) holds. By Theorem 2.5, Eq. (1) is oscillatory. The proof is complete. h
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