Oscillation of second-order nonlinear ODE with damping

Applied Mathematics and Computation 182 (2006) 1861–1871 www.elsevier.com/locate/amc

Oscillation of second-order nonlinear ODE with damping

q

Xueqin Zhao *, Fanwei Meng School of Mathematics Sciences, Qufu Normal University, Qufu 273165, Shandong, People’s Republic of China

Abstract We are concerned with a class of general type second-order differential equations with a nonlinear damping term. New oscillation criteria are established, which improve and extend some known results. Examples are also given to illustrate the results. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Oscillation; Nonlinear damping term; Riccati technique

1. Introduction Consider the second-order nonlinear damped differential equation 0

ðrðtÞk 1 ðxðtÞ; x0 ðtÞÞÞ þ pðtÞk 2 ðxðtÞ; x0 ðtÞÞx0 ðtÞ þ qðtÞf ðxðtÞÞ ¼ 0;

t P t0 ;

ð1:1Þ 2

where t0 is a fixed real number, r 2 C([t0, 1), (0, 1)), p; q 2 Cð½t0 ; 1Þ; RÞ, f 2 CðR; RÞ, and k 1 ; k 2 2 CðR ; RÞ. It is tacitly assumed that the functions r and k1 are continuously differentiable in their domain of definitions. Throughout this paper, we shall also assume that (A1) (A2) (A3) (A4)

p(t) P 0 for all t P t0, xf(x) > 0 for all x 5 0; aþ1 vk 1 ðu; vÞ P b1 jk 1 ðu; vÞj a for some b1 > 0 and all ðu; vÞ 2 R2 ; aþ1 1 vk 2 ðu; vÞf a ðuÞ P b2 jk 1 ðu; vÞj a for some b2 > 0, and all u; v 2 R2 ; f 0 (x) exists and f 0 ðxÞ ða1Þ=a

jf ðxÞj

P b3 > 0;

for some positive constant b3 and for all x 2 R=f0g; Or: (A5) q(t) P 0 for all t P t0, f satisfies f(x)/x P L for some positive constant L and for all x 5 0; q *

This research was supported by the NNSF of China and the NSF of Shangdong Province of China. Corresponding author. E-mail addresses: [email protected] (X. Zhao), [email protected] (F. Meng).

0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.06.022

1862

X. Zhao, F. Meng / Applied Mathematics and Computation 182 (2006) 1861–1871 aþ1

a1

(A6) vk 1 ðu; vÞ P b4 jk 1 ðu; vÞj a aþ1 u a for some b4 > 0, all v 2 R=f0g and all u 2 R; 1 (A7) vua k 2 ðu; vÞ P b5 jk 1 ðu; vÞj a for some b5 > 0, and all u; v 2 R2 ; By a solution of (1.1), we mean a function x 2 C1[Tx, 1), Tx P t0, which satisfies Eq. (1.1). In the sequel, we restrict our attention only to the nontrivial solutions of Eq. (1.1), that is, the solutions x(t) such that sup{jx(t)j : t P T} > 0 for all T P Tx. A nontrivial solution of Eq. (1.1) is called oscillatory if it has arbitrarily large zeros, otherwise, it is said to be non-oscillatory. Eq. (1.1) is called oscillatory if all its solutions are oscillatory. A close look at the form of Eq. (1.1) under the assumption (A2)–(A3) reveals that Eq. (1.1) can be considered as a natural generalization of the following equations 0

½rðtÞwðxðtÞÞx0 ðtÞ þ pðtÞx0 ðtÞ þ qðtÞf ðxðtÞÞ ¼ 0;

t P t0 :

ð1:2Þ

During the last two decades, the investigation of oscillatory solution related to Eq. (1.1) and its various particular case such as Eq. (1.2) has been attracting attention of numerous researchers and resulted in hundreds of papers with many interesting criteria proved [1–16]. Recently, Rogovchenko [9] introduced the basic assumptions, and main techniques used for proving oscillation criteria, meanwhile, stated some interesting results without proof and examples illustrating there relevance. Very recently, Mustafa [11] studied oscillation of a novel class of nonlinear differential equations with a damping term ½rðtÞy 0 ðtÞ0 þ pðtÞaðyðtÞÞy 0 ðtÞ þ qðtÞgðyðtÞÞ ¼ 0;

ð1:3Þ

and explained how Eq. (1.2) can be transformed to the form (1.3) by means of a simple integral transformation which preserves oscillatory character of solutions, and vice versa. The principal results of [11] solve the oscillation problem of some equations which known oscillation criteria fail to apply, such as the example in [11]. In 2004, Wong [16] researched the second-order nonlinear damped differential equations ½rðtÞWðxðtÞÞkðx0 ðtÞÞ0 þ pðtÞkðx0 ðtÞÞ þ qðtÞf ðxðtÞÞ ¼ 0;

t P t0 ;

ð1:4Þ

where k satisfies k2(y) 6 c1yk(y) for some constant c1 > 0 and for all x 2 R, via the following generalized Riccati type substitutions:   WðxðtÞÞkðx0 ðtÞÞ þ RðtÞ ; ð1:5Þ vðtÞ ¼ UðtÞaðtÞ f ðxðtÞÞ and the one with f(x(t)) replaced by x(t), where U 2 C1([t0, 1), R+) and R(t) 2 C([t0, 1), R) such that (aR) 2 C1[t0, 1). The results of [16] extend and improve results in Ayanlar [12]. Here, in the present paper, we shall establish several oscillation criteria for the general Eq. (1.1) by the generalized Riccati transform (1.5) with R(t)  0, and give some examples which other known oscillation criteria fail to apply. The results extend and improve some known results. Before giving the main results, we introduce some denotations. Let D = {(t, s) : t P s P t0} denote a subset in R2. We say that a function H = H(t, s) belong to a function class F, denoted by H 2 F. If H be continuous and satisfies the following conditions: (H1) H(t, t) = 0 and H(t, s) > 0 for t > s P t0; (H2) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi oH ðt; sÞ oH ðt; sÞ 6 0 and ¼ hðt; sÞ H ðt; sÞ os os where h 2 hC(D, R); i ðt;sÞ (H3) 0 < inf sPt0 lim inf t!1 HHðt;t < 1 < 1. 0Þ Specially, if we take, as in [13],

for ðt; sÞ 2 D;

X. Zhao, F. Meng / Applied Mathematics and Computation 182 (2006) 1861–1871

H ðt; sÞ ¼

Z

m ds ; nðsÞ

t

s

1863

m > a;

R1 1 for an arbitrary positive function n 2 C1[t0, 1) with t0 nðsÞ ds ¼ 1, we will find that the function H(t, s) satisfies conditions above. Consequently, when n(s) = 1 we have H(t, s) = (t  s)m, and when n(s) = s we have H(t, s) = (ln(t/s))m. 2. Oscillation results for f(x) with monotonicity Theorem 2.1. Suppose that assumptions (A1)–(A4) are satisfied. If there exist H 2 C(D) satisfying (H1)–(H2) and a positive function q 2 C1([t0, 1), (0, 1)) such that # Z t" 1 aa H ðt; sÞqðsÞraþ1 ðsÞhaþ1 1 ðt; sÞ lim sup H ðt; sÞqðsÞqðsÞ  ds ¼ 1; ð2:1Þ a H ðt; t0 Þ t0 ðb2 pðsÞ þ b1 b3 rðsÞÞ t!1 ða þ 1Þaþ1    0 ðsÞ  ffiffiffiffiffiffiffiffi; then Eq. (1.1) is oscillation. where h1 ðt; sÞ ¼ qqðsÞ  phðt;sÞ H ðt;sÞ

Proof. Otherwise, let x(t) be a non-oscillatory solution of Eq. (1.1). Without loss of generality, we may assume that x(t) > 0 on [T0, 1) for some sufficiently large T0 P t0. Define wðtÞ ¼ qðtÞ

rðtÞk 1 ðxðtÞ; x0 ðtÞÞ ; f ðxðtÞÞ

ð2:2Þ

t P T 0:

Then differentiating (2.2) and using Eq. (1.1), we obtain w0 ðtÞ 6

q0 ðtÞ pðtÞqðtÞk 2 ðxðtÞ; x0 ðtÞÞx0 ðtÞ rðtÞk 1 ðxðtÞ; x0 ðtÞÞ 0 wðtÞ  qðtÞqðtÞ   x ðtÞf 0 ðxðtÞÞqðtÞ: qðtÞ f ðxðtÞÞ f 2 ðxðtÞÞ

ð2:3Þ

By using assumptions (A1)–(A4), (2.3) implies that w(t) satisfies the differential inequality w0 ðtÞ 6

q0 ðtÞ b pðtÞ þ b1 b3 rðtÞ ðaþ1Þ=a wðtÞ  qðtÞqðtÞ  2 1 ; jwðtÞj aþ1 qðtÞ qa ðtÞr a ðtÞ

t P T 0:

ð2:4Þ

Multiplying (2.4) by H(t, s) (with t replaced by s), integrating with respect to s from s to t for t P s P T0, using integration by parts and (H2), we have Z t H ðt; sÞqðsÞqðsÞ ds s Z t Z t b pðsÞ þ b1 b3 rðsÞ ðaþ1Þ=a 6 H ðt; sÞwðsÞ þ H ðt; sÞh1 ðt; sÞjwðsÞj ds  H ðt; sÞ 2 1 ds jwðsÞj aþ1 qa ðsÞr a ðsÞ s s ( ) Z t b2 pðsÞ þ b1 b3 rðsÞ ðaþ1Þ=a ¼ H ðt; sÞwðsÞ þ H ðt; sÞ h1 ðt; sÞjwðsÞj  ds; ð2:5Þ jwðsÞj aþ1 1 qa ðsÞr a ðsÞ s where h1(t, s) is defined as the above. For given t and s, set F ðvÞ :¼ h1 v 

b 2 p þ b1 b 3 r

vðaþ1Þ=a ;

v > 0: qr a þ 1 b2 p þ b1 b3 r ð1Þ=a Because of F 0 ðvÞ ¼ h1  v , F(v) yields its maximum at 1 aþ1 a qa r a  a ah1 aþ1 v¼r q ; ða þ 1Þðb2 p þ b1 b3 rÞ 1 aþ1 a a

and F ðvÞ 6 F max ¼

aa raþ1 qhaþ1 1 ða þ 1Þ

aþ1

ðb2 p þ b1 b3 rÞ

a

:

ð2:6Þ

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X. Zhao, F. Meng / Applied Mathematics and Computation 182 (2006) 1861–1871

By using (2.6), we have Z t H ðt; sÞqðsÞqðsÞ ds 6 H ðt; sÞwðsÞ þ s

Hence, for all t P s P T0 Z " t

H ðt; sÞqðsÞqðsÞ 

s

aa ða þ 1Þaþ1

Z

aa ða þ 1Þ

aþ1

s

t

H ðt; sÞ

raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ a ds: ðb2 pðsÞ þ b1 b3 rðsÞÞ

# raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ a H ðt; sÞ ds 6 H ðt; sÞwðsÞ 6 H ðt; t 0 ÞjwðsÞj: ðb2 pðsÞ þ b1 b3 rðsÞÞ

ð2:7Þ

ð2:8Þ

Using the inequality (2.8) for s = T0, we get # Z t" aa raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ H ðt; sÞqðsÞqðsÞ  a H ðt; sÞ ds ða þ 1Þaþ1 ðb2 pðsÞ þ b1 b3 rðsÞÞ t0 # Z T0 " aa raþ1 ðsÞqðsÞhaþ1 1 ðt; sÞ ¼ H ðt; sÞqðsÞqðsÞ  a H ðt; sÞ ds ða þ 1Þaþ1 ðb2 pðsÞ þ b1 b3 rðsÞÞ t0 # Z t" aa raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ þ H ðt; sÞqðsÞqðsÞ  a H ðt; sÞ ds ða þ 1Þaþ1 ðb2 pðsÞ þ b1 b3 rðsÞÞ T0 Z T0 6 H ðt; t0 Þ jqðsÞjqðsÞ ds þ H ðt; t0 ÞjwðT 0 Þj: t0

It follows that # Z t" 1 aa raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ H ðt; sÞqðsÞqðsÞ  a H ðt; sÞ ds H ðt; t0 Þ t0 ða þ 1Þaþ1 ðb2 pðsÞ þ b1 b3 rðsÞÞ Z T0 6 jqðsÞjqðsÞ ds þ jwðT 0 Þj < 1:

ð2:9Þ

t0

Take lim sup in (2.9) as t ! 1, condition (2.1) gives the desired contradiction in (2.9), which completes the proof. h Investigating the condition (2.1) in Theorem 2.1, we can obtain the following corollary. Corollary 2.2. Suppose assumptions in Theorem 2.1 are valid and condition (2.1) is replaced by the conditions Z t 1 lim sup H ðt; sÞqðsÞqðsÞ ds ¼ 1; ð2:10Þ H ðt; t0 Þ t0 t!1 Z t 1 aa raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ lim sup H ðt; sÞ ð2:11Þ a ds < 1; aþ1 H ðt; t0 Þ t0 ðb2 pðsÞ þ b1 b3 rðsÞÞ t!1 ða þ 1Þ where h1(t, s) is defined as in Theorem 2.1. Then Eq. (1.1) is oscillatory. Theorem 2.3. Suppose assumptions (A1)–(A4) be valid. If there exist functions H 2 F, u 2 C([t0, 1), R) and a positive function q 2 C1([t0, 1), (0, 1)) such that Z t 1 raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ lim sup H ðt; sÞ ð2:12Þ a ds < 1; H ðt; t0 Þ t0 ðb2 pðsÞ þ b1 b3 rðsÞÞ t!1 Z 1 ðaþ1Þ=a uþ ðsÞðb2 pðsÞ þ b1 b3 rðsÞÞ ds ¼ 1; ð2:13Þ q1=a ðsÞraþ1=a ðsÞ t0

X. Zhao, F. Meng / Applied Mathematics and Computation 182 (2006) 1861–1871

1865

and for every s P t0 1 lim sup H ðt; sÞ t!1

Z t"

H ðt; sÞqðsÞqðsÞ 

s

# raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ H ðt; sÞ ds P uðsÞ; ðb2 pðsÞ þ b1 b3 rðsÞÞa

aa ða þ 1Þ

aþ1

ð2:14Þ

where h1(t, s) is defined as in Theorem 2.1, u+(s) = max{u(s), 0}. Then Eq. (1.1) is oscillatory. Proof. Otherwise, let x(t) be a non-oscillatory solution of Eq. (1.1). Without loss of generality, we may assume that x(t) > 0 on [T0, 1) for some sufficiently large T0 P t0. Define the function w(t) as in (2.2) and we get (2.5), (2.8). Dividing (2.8) through by H(t, s) and taking lim sup as t ! 1, we have # Z t" 1 aa raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ lim sup H ðt; sÞqðsÞqðsÞ  ð2:15Þ a H ðt; sÞ ds 6 wðsÞ; H ðt; sÞ s t!1 ða þ 1Þaþ1 ðb2 pðsÞ þ b1 b3 rðsÞÞ for t > s P T0. From condition (2.14), we have uðsÞ 6 wðsÞ;

ð2:16Þ

s P T 0;

and lim sup t!1

1 H ðt; T 0 Þ

Z

t

H ðt; sÞqðsÞqðsÞ ds P uðT 0 Þ:

ð2:17Þ

T0

Then, define functions Z t 1 ðb2 pðsÞ þ b1 b3 rðsÞÞH ðt; sÞ aðtÞ ¼ jwðsÞjðaþ1Þ=a ds; aþ1 1 H ðt; T 0 Þ T 0 r a ðsÞqa ðsÞ Z t 1 bðtÞ ¼ H ðt; sÞh1 ðt; sÞjwðsÞj ds; H ðt; T 0 Þ T 0 for t > T0. By (2.5) and (2.17), we obtain lim inf ½aðtÞ  bðtÞ 6 wðT 0 Þ  lim sup t!1

Now we claim that Z 1 ðb2 pðsÞ þ b1 b3 rðsÞÞ aþ1

T0

1

r a ðsÞqa ðsÞ

Suppose to the contrary that Z 1 ðb2 pðsÞ þ b1 b3 rðsÞÞ aþ1

T0

1

r a ðsÞqa ðsÞ

1 H ðt; T 0 Þ

Z

t

H ðt; sÞqðsÞqðsÞ ds 6 wðT 0 Þ  uðT 0 Þ < 1:

jwðsÞj

ðaþ1Þ=a

ds < 1:

ð2:19Þ

jwðsÞj

ðaþ1Þ=a

ds ¼ 1:

ð2:20Þ

In fact, by the assumption (H3), there exists a positive constant g such that  H ðt; sÞ inf lim inf > g > 0: sPt0 t!1 H ðt; t 0 Þ On the other hand, by (2.20) for any positive number l there exists a s1 > T0 such that Z s1 ðb2 pðsÞ þ b1 b3 rðsÞÞ l ðaþ1Þ=a ds P jwðsÞj aþ1 1 g r a ðsÞqa ðsÞ T0 for all t P s1.

ð2:18Þ

T0

ð2:21Þ

ð2:22Þ

1866

X. Zhao, F. Meng / Applied Mathematics and Computation 182 (2006) 1861–1871

By using the definition of a(t) and integration by parts, for all t P s1 Z t 1 H ðt; sÞðb2 pðsÞ þ b1 b3 rðsÞÞ ðaþ1Þ=a aðtÞ ¼ ds jwðsÞj aþ1 1 H ðt; T 0 Þ T 0 r a ðsÞqa ðsÞ ! Z t Z s 1 ðb2 pðsÞ þ b1 b3 rðsÞÞ ðaþ1Þ=a H ðt; sÞ d dl ds ¼ jwðsÞj aþ1 1 H ðt; T 0 Þ T 0 r a ðsÞqa ðsÞ T0 # "Z s Z t 1 oH ðt; sÞ ðb2 pðsÞ þ b1 b3 rðsÞÞ ðaþ1Þ=a ¼  dl ds jwðsÞj aþ1 1 H ðt; T 0 Þ T 0 os r a ðsÞqa ðsÞ T0 # "Z s Z t 1 oH ðt; sÞ ðb2 pðsÞ þ b1 b3 rðsÞÞ ðaþ1Þ=a P  dl ds jwðsÞj aþ1 1 H ðt; T 0 Þ s1 os r a ðsÞqa ðsÞ T0  Z t l 1 oH ðt; sÞ l H ðt; s1 Þ :  P ds ¼ g H ðt; T 0 Þ s1 os g H ðt; T 0 Þ ðt;s1 Þ 1Þ 1Þ From (2.21), for s1 we have lim inf t!1 HHðt;s > g > 0, and so there exists s2 P s1 such that HHðt;T P HHðt;s Pg ðt;t0 Þ ðt;t0 Þ 0Þ for all t P s2. And therefore we arrive at a(t) P l for all t P s2, and because of l be arbitrary positive constant, we conclude that

lim aðtÞ ¼ 1:

ð2:23Þ

t!1

1

From (2.18), consider a sequence ftn gn¼1  ½T 0 ; 1Þ such that limn!1tn = 1 and lim ½aðtn Þ  bðtn Þ ¼ lim inf ½aðtÞ  bðtÞ < 1:

n!1

t!1

Then there exists a constant M such that aðtn Þ  bðtn Þ 6 M;

n ¼ 1; 2; 3; . . .

ð2:24Þ

It follows from (2.23) and (2.24) that lim aðtn Þ ¼ 1;

ð2:25Þ

lim bðtn Þ ¼ 1:

ð2:26Þ

n!1 n!1

Then, by (2.24) and (2.25), for all n large enough 1

bðtn Þ M 1 6 < : aðtn Þ aðtn Þ 2

Thus for all n large enough, lim

n!1

baþ1 ðtn Þ ¼ 1: aa ðtn Þ

bðtn Þ aðtn Þ

> 12, and therefore ð2:27Þ

For b(tn), n 2 {1, 2, 3, . . .}, according to Ho¨lder inequality, we have Z tn 1 bðtn Þ ¼ H ðtn ; sÞh1 ðtn ; sÞjwðsÞj ds H ðtn ; T 0 Þ T 0 ! ! Z tn a=ðaþ1Þ ðb2 pðsÞ þ b1 b3 rðsÞÞ jwðsÞjH a=ðaþ1Þ ðtn ; sÞ H 1=ðaþ1Þ ðtn ; sÞrðsÞq1=ðaþ1Þ ðsÞh1 ðtn ; sÞ ¼ ds a=ðaþ1Þ 1=ðaþ1Þ rðsÞq1=ðaþ1Þ ðsÞ H a=ðaþ1Þ ðtn ; T 0 Þ ðb2 pðsÞ þ b1 b3 rðsÞÞ H ðtn ; T 0 Þ T0  a=ðaþ1Þ Z tn 1 ðb2 pðsÞ þ b1 b3 rðsÞÞH ðtn ; sÞ ðaþ1Þ=a jwðsÞj ds 6 H ðtn ; T 0 Þ T 0 raþ1=a ðsÞq1=a ðsÞ  1=ðaþ1Þ Z tn 1 H ðtn ; sÞraþ1 ðsÞqðsÞhaþ1 1 ðt n ; sÞ  ds ; a H ðtn ; T 0 Þ T 0 ðb2 pðsÞ þ b1 b3 rðsÞÞ

X. Zhao, F. Meng / Applied Mathematics and Computation 182 (2006) 1861–1871

1867

and therefore baþ1 ðtn Þ 1 6 a a ðtn Þ H ðtn ; T 0 Þ 6

1 gH ðtn ; t0 Þ

Z

tn T0

Z

t0

tn

H ðtn ; sÞraþ1 ðsÞqðsÞh1aþ1 ðtn ; sÞ ds a ðb2 pðsÞ þ b1 b3 rðsÞÞ H ðtn ; sÞraþ1 ðsÞqðsÞhaþ1 1 ðt n ; sÞ ds; ðb2 pðsÞ þ b1 b3 rðsÞÞa

for all large n. It follows from (2.27) that Z tn 1 H ðtn ; sÞraþ1 ðsÞqðsÞh1aþ1 ðtn ; sÞ lim ds ¼ 1; a n!1 H ðt n ; t 0 Þ t ðb2 pðsÞ þ b1 b3 rðsÞÞ 0

ð2:28Þ

so lim sup t!1

H ðt; sÞraþ1 ðsÞqðsÞh1aþ1 ðt; sÞ ds ¼ 1; a ðb2 pðsÞ þ b1 b3 rðsÞÞ

which contradicts with assumption (2.12). Thus, we have proved that (2.20) fails and the assertion (2.19) is valid. Then, it follows from (2.16) that u+(s) 6 jw(s)j for all s P T0, and Z 1 Z 1 ðaþ1Þ=a ðaþ1Þ=a uþ ðsÞðb2 pðsÞ þ b1 b3 rðsÞÞ jwðsÞj ðsÞðb2 pðsÞ þ b1 b3 rðsÞÞ ds 6 ds < 1; ð2:29Þ aþ1=a 1=a aþ1=a r r ðsÞq ðsÞÞ ðsÞq1=a ðsÞÞ T0 T0 which contradicts the condition (2.13). The proof is completed.

h

Theorem 2.4. Suppose assumptions (A1)–(A4) be valid. If there exist functions H 2 F, u 2 C([t0, 1), R) and a positive function q 2 C1([t0, 1), (0, 1)) such that (2.13) in Theorem 2.3 holds and Z t 1 H ðt; sÞqðsÞqðsÞ ds < 1; ð2:30Þ lim inf t!1 H ðt; t0 Þ t0 and for every s P t0 1 lim inf t!1 H ðt; sÞ

Z t" H ðt; sÞqðsÞqðsÞ  s

aa ða þ 1Þ

aþ1

# raþ1 ðsÞqðsÞhaþ1 1 ðt; sÞ a H ðt; sÞ ds P uðsÞ; ðb2 pðsÞ þ b1 b3 rðsÞÞ

ð2:31Þ

where h1(t, s) is defined as in Theorem 2.1. Then Eq. (1.1) is oscillatory. Proof. Otherwise, let x(t) be a non-oscillatory solution of Eq. (1.1). Without loss of generality, we may assume that x(t) > 0 on [T0, 1) for some sufficiently large T0 P t0. Define the function w(t) as in (2.2) and we get (2.5), (2.8). Following the proof of Theorem 2.3, we obtain (2.16) (just with lim sup replaced by lim inf in (2.15) for all s P T0). Then, by using condition (2.30), we get Z t 1 lim sup½aðtÞ  bðtÞ 6 wðT 0 Þ  lim inf H ðt; sÞqðsÞqðsÞ ds < 1; ð2:32Þ t!1 H ðt; T 0 Þ T 0 t!1 where a(t) and b(t) are defined as in the proof of Theorem 2.3. It follows from (2.31) and (2.30) that Z t 1 aa raþ1 ðsÞqðsÞhaþ1 1 ðt; sÞ lim inf ð2:33Þ a H ðt; sÞ ds < 1: t!1 H ðt; t0 Þ t0 ða þ 1Þaþ1 ðb2 pðsÞ þ b1 b3 rðsÞÞ 1

Then there exists a sequence ftn gn¼1 2 ðT 0 ; 1Þ such that limn!1tn = 1 and Z t 1 aa raþ1 ðsÞqðsÞhaþ1 1 ðt n ; sÞ lim a H ðt n ; sÞ ds n!1 H ðt n ; t 0 Þ t ða þ 1Þaþ1 ðb2 pðsÞ þ b1 b3 rðsÞÞ 0 Z t 1 aa raþ1 ðsÞqðsÞh1aþ1 ðtn ; sÞ ¼ lim inf a H ðt n ; sÞ ds < 1: tn !1 H ðt n ; t 0 Þ t ða þ 1Þaþ1 ðb2 pðsÞ þ b1 b3 rðsÞÞ 0

ð2:34Þ

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X. Zhao, F. Meng / Applied Mathematics and Computation 182 (2006) 1861–1871

Now, suppose that (2.20) holds. With the same argument as in Theorem 2.3, we conclude that (2.19) is satisfied. By (2.34), there exists a constant M such that (2.24) is fulfilled. Then following the procedure of the proof of Theorem 2.3, we obtain (2.28) which contradicts (2.34). Hence (2.19) holds. Making use of (2.16) and (2.19), we obtain (2.29), which contradicts condition (2.13). h The following theorem can be proved similarly to Theorem 2.4. Theorem 2.5. Suppose assumptions (A1)–(A4) be valid. If there exist functions H 2 F, u 2 C([t0, 1), R) and a positive function q 2 C1([t0, 1), (0, 1)), such that (2.13) and (2.31) hold, and Z t aþ1 1 r ðsÞqðsÞhaþ1 1 ðt; sÞ ð2:35Þ lim inf a H ðt; sÞ ds < 1; t!1 H ðt; t 0 Þ t ðb2 pðsÞ þ b1 b3 rðsÞÞ 0 where h1(t, s) is defined as in Theorem 2.1, then (1.1) is oscillatory. 3. Oscillation results for f(x) without monotonicity In this section, we shall consider the oscillation for Eq. (1.1) under the assumptions (A1) and (A5)–(A7). Theorem 3.1. Suppose that assumptions (A1) and (A5)–(A7) hold. If there exists H 2 C(D) satisfying (H1)–(H2) and a positive function q 2 C1([t0, 1), (0, 1)) such that # Z t" 1 aa H ðt; sÞqðsÞraþ1 ðsÞh1aþ1 ðt; sÞ LH ðt; sÞqðsÞqðsÞ  lim sup ds ¼ 1; ð3:1Þ a aþ1 H ðt; t0 Þ t0 ðb5 pðsÞ þ b4 rðsÞÞ t!1 ða þ 1Þ where h1(t, s) is defined as in Theorem 2.1. Then (1.1) is oscillation. Proof. Otherwise, let x(t) be a non-oscillatory solution of Eq. (1.1). Without loss of generality, we may assume that x(t) > 0 on [Tx, 1) for some sufficiently large Tx P t0. Define vðtÞ ¼ qðtÞ

rðtÞk 1 ðxðtÞ; x0 ðtÞÞ ; xðtÞ

ð3:2Þ

t P T 0:

Then differentiating (3.2) and using Eq. (1.1), we obtain v0 ðtÞ 6

q0 ðtÞ pðtÞqðtÞk 2 ðxðtÞ; x0 ðtÞÞx0 ðtÞ rðtÞk 1 ðxðtÞ; x0 ðtÞÞ 0 vðtÞ  LqðtÞqðtÞ   x ðtÞqðtÞ; qðtÞ xðtÞ x2 ðtÞ

ð3:3Þ

by using assumptions (A5)–(A7), (3.3) implies that v(t) satisfies the differential inequality v0 ðtÞ 6

q0 ðtÞ b pðtÞ þ b4 rðtÞ vðtÞ  LqðtÞqðtÞ  5 1 jvðtÞjðaþ1Þ=a ; aþ1 qðtÞ qa ðtÞr a ðtÞ

ð3:4Þ

t P T 0:

The rest proof is similar to that of Theorem 2.1 and hence omitted.

h

Corollary 3.2. Suppose assumptions in Theorem 3.1 are valid and condition (3.1) is replaced by the conditions Z t 1 lim sup H ðt; sÞqðsÞqðsÞ ds ¼ 1; ð3:5Þ H ðt; t0 Þ t0 t!1 Z t 1 aa raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ lim sup H ðt; sÞ ds < 1; ð3:6Þ aþ1 H ðt; t0 Þ t0 ðb5 pðsÞ þ b4 rðsÞÞa t!1 ða þ 1Þ where h1(t, s) is defined as in Theorem 2.1. Then Eq. (1.1) is oscillatory. Theorem 3.3. Suppose assumptions (A1) and (A5)–(A7) hold. If there exist functions H 2 F, u 2 C([t0, 1), R) and a positive function q 2 C1([t0, 1), (0, 1)) such that

X. Zhao, F. Meng / Applied Mathematics and Computation 182 (2006) 1861–1871

Z t 1 raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ lim sup H ðt; sÞ a ds < 1; H ðt; t0 Þ t0 ðb5 pðsÞ þ b4 rðsÞÞ t!1 Z 1 ðaþ1Þ=a uþ ðsÞðb5 pðsÞ þ b4 rðsÞÞ ds ¼ 1; 1=a ðsÞraþ1=a ðsÞ q t0

1869

ð3:7Þ ð3:8Þ

and for every s P t0 Z t"

1 lim sup H ðt; sÞ t!1

LH ðt; sÞqðsÞqðsÞ 

s

aa ða þ 1Þ

aþ1

# raþ1 ðsÞqðsÞhaþ1 1 ðt; sÞ a H ðt; sÞ ds P uðsÞ; ðb5 pðsÞ þ b4 rðsÞÞ

ð3:9Þ

where h1(t, s) is defined as in Theorem 2.1, u+(s) = max{u(s), 0}. Then Eq. (1.1) is oscillatory. Theorem 3.4. Suppose assumptions (A1) and (A5)–(A7) hold. If there exist functions H 2 F, u 2 C([t0, 1), R) and a positive function q 2 C1([t0, 1), (0, 1)) such that (3.8) in Theorem 3.3 holds and Z t 1 lim inf H ðt; sÞqðsÞqðsÞ ds < 1; ð3:10Þ t!1 H ðt; t0 Þ t0 and for every s P t0 1 lim inf t!1 H ðt; sÞ

Z t" LH ðt; sÞqðsÞqðsÞ  s

aa ða þ 1Þ

aþ1

# raþ1 ðsÞqðsÞh1aþ1 ðt; sÞ H ðt; sÞ ds P uðsÞ; ðb2 pðsÞ þ b1 b3 rðsÞÞa

ð3:11Þ

where h1(t, s) is defined as in Theorem 2.1. Then Eq. (1.1) is oscillatory. Theorem 3.5. Suppose assumptions (A1) and (A5)–(A7) hold. If there exist functions H 2 F, u 2 C([t0, 1), R) and a positive function q 2 C1([t0, 1), (0, 1)), such that (3.8) and (3.11) hold and Z t aþ1 1 r ðsÞqðsÞhaþ1 1 ðt; sÞ lim inf H ðt; sÞ ds < 1; ð3:12Þ t!1 H ðt; t0 Þ t0 ðb5 pðsÞ þ b4 rðsÞÞa where h1(t, s) is defined as in Theorem 2.1. Then Eq. (1.1) is oscillatory. 4. Some examples Example 4.1. Consider the nonlinear differential equation  0 2 ðpþ1Þ x0p ðtÞ tp ðx0 ðtÞÞ t þ 1 p1 ð1 þ x2 ðtÞÞð1 þ cx02 ðtÞÞ ðjxðtÞj xðtÞÞp ð1 þ x2 ðtÞÞð1 þ cx02 ðtÞÞ þ ½ktk1 ð2  sin tÞ  tk cos tjxðtÞjp1 xðtÞ ¼ 0;

ð4:1Þ

for t P t0 > 1, where 0 < p < 1 be any quotient of odd integer, c,k be arbitrary positive constants. Then if we take a = p, b1 = 1, b2 = 1, b3 = p, we will find assumptions (A1)–(A4) are satisfied. Moreover, taking H(t, s) = (t  s)2, h(t, s) = 2 and q(t)  1, we have Z t Z t Z t k1 k qðsÞ ds ¼ ½ks ð2  sin sÞ  s cos s ds ¼ dðsk ð2  sin sÞÞ ¼ tk ð2  sin tÞ  tk0 ð2  sin t0 Þ t0

t0 k

Pt  So, 1 t2

Z

t0

tk0 ð2

t 2

ðt  sÞ qðsÞ ds ¼ t0

P

k

 sin t0 Þ P t  1 t2 2 t2

Z

t 2

ðt  sÞ d

t0

Z

tk0 ð2

Z

 sin t0 Þ: 

t

qðsÞ ds t0

¼

2 t2

Z

t

Z

t0

t

s

 qðsÞ ds ds

t0 k

ðt  sÞðsk  tk0 ð2  sin t0 ÞÞ ds ¼ t0

ðt  sÞ

2t L1 L2 þ þ  L0 ! 1; ðk þ 1Þðk þ 2Þ t2 t

1870

X. Zhao, F. Meng / Applied Mathematics and Computation 182 (2006) 1861–1871 2tkþ1

2tk þ2

0 0 where L0 ¼ tk0 ð2  sin t0 Þ, L1 ¼ kþ2  L0 t0 , L2 ¼ 2L0 t0  kþ1 . On the other hand,   Z t Z aþ1 t 1 saþ1 2 1 ðt  sÞ2 2 ds ¼ ðt  sÞ1a saþ1 s2 2aþ1 ds 2 2 a t  s p Þ ðt  t0 Þ t0 ðt  t t0 ðs þ asÞ 0   1a Z t aþ1 t ðt  t0 Þ 2taþ1 1 1 2 aþ1 6 s 2 ds ¼  2 aþ1 t0 t ðt  t0 Þ ðt  t0 Þ t0

!

2aþ1 ; t0

t ! 1:

Consequently, by Corollary 2.2, Eq. (4.1) is oscillatory. Take account of the damped differential equation Example 4.2 h i0 t1p jx0 ðtÞjp1 ðx0 ðtÞÞ2 p1 2 p 2 sin2 ðtÞjx0 ðtÞj x0 ðtÞ þ 1 þ ð1 þ sin tÞx ðtÞð1 þ x ðtÞÞ ¼ 0; 2 p xðtÞð1 þ x ðtÞÞ

t P t0 ¼ 2;

ð4:2Þ

where 0 < p < 12 be arbitrary constant. Then if we take a = p, b1 = 1, b2 = 1, b3 = 1, we will find assumptions (A1)–(A4) are satisfied. Moreover, taking H(t, s) = (t  s)2, h(t, s) = 2, q(t) = t2 and making use of L 0 Houspital rule, we have Z t Z t Z t 1 2 2 2 2 2 lim ðt  sÞ s ð1 þ sin sÞ ds ¼ lim s ð1 þ sin sÞ ds 6 lim 2s2 ds ¼ 1 < 1: t!1 ðt  t Þ2 t t!1 t t!1 t 0 0 0 0 On the other hand for t P s P 2 and every s P 2,    2 2 2 h1 ðt; sÞ ¼  þ  ¼ þ 2; ts s ts #  aþ1 Z " aþ1 1 t aa 2 ðsin2 sÞ 2 2 2 2 2 lim inf 2 þ2 ðt  sÞ ð1 þ sin sÞs  ðt  sÞ s ds 1 a t!1 t ts ða þ 1Þaþ1 s ðsp þ sin2 sÞ Z Z 1 t 1 t 2aþ1 aa 2 2 1a aþ1 ðt  sÞ s ds  lim sup 2 ðt  sÞ s3 ðt  s þ 1Þ ds P lim 2 aþ1 t!1 t t t!1 s s ða þ 1Þ Z t aþ1 Z t a a a ðt þ 1Þ 1 ð2aÞ 1 2 3  P lim s ds  lim sup s ds ¼ : aþ1 aþ1 2 aþ1 t!1 s s s t t!1 ða þ 1Þ ða þ 1Þ s 1 2aa 1 ð2aÞa Setting uðsÞ ¼  , s0 ¼ ðaþ1Þ aþ1 , it follows that aþ1 2 s ða þ 1Þ s #ðaþ1Þ=a Z 1 Z 1" 1 a sp þ sin2 s 1 ð2aÞ 1 ðaþ1Þ=a  Qþ ðsÞ 2 s3=a ds aþ1 ds P aþ1 2 2  s s a ða þ 1Þ t0 t0 s a ðsin sÞ þ ðaþ1Þ=a ðaþ1Þ=a Z 1 Z s0  1 1 1 1  s0 2  s0 2 s3=a ds þ s3=a ds ¼ s s þ s s þ t0 s0 ðaþ1Þ=a Z s0  Z 1 1 1  s0 2 P s3=a ds þ ðs  s0 Þðaþ1Þ=a ds ¼ 1: s s þ t0 s0 that is because of 3 > 2(a + 1) while 0 < a < 1/2. Thus by Theorem 2.4, Eq. (4.2) is oscillatory. References [1] S.R. Grace, Oscillation theorems for nonlinear differential equations of second order, J. Math. Anal. Appl. 171 (1992) 220–241. [2] S.R. Grace, B.S. Lalli, Integral averaging techniques for the oscillation of second order nonlinear differential equations, J. Math. Anal. Appl. 149 (1990) 277–311.

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