Oscillation theorems for superlinear second-order damped differential equations

Applied Mathematics and Computation 189 (2007) 796–804

www.elsevier.com/locate/amc

Oscillation theorems for superlinear second-orderdamped differential equations q

Fang Lu, Fanwei Meng *

Department of Mathematics, Qufu Normal University, Qufu 273165, Shandong, People’s Republic of China

Abstract

In this paper, we concerned with the oscillation of the second-order superlinear differential equation of the form

0096-3

doi:10

q Th* Co

E-m

ðaðtÞy0ðtÞÞ0 þ pðtÞy0ðtÞ þ qðtÞf ðyðtÞÞ ¼ 0:

Several new oscillation criteria are established under quite general assumptions. Our methodology is somewhat differentfrom that of previous authors [Ch.G. Philos, Oscillation theorems for linear differential equations of second order, Arch.Math. (Basel) 53 (1989) 482–492; Ch.G. Philos, Oscillation criteria for second order superlinear differential equations, Ca-nad. J. Math. Anal. XLI (1989) 321–340; CH.G. Philos, I.K. Purnaras, Oscillations in superlinear differential equations ofsecond order. J. Math. Anal. Appl. 165 (1992) 1–11; Y.H. Yu, Oscillation criteria for second order nonlinear differentialequations with damping, Acta. Math. Appl. 16 (1993) 433–441 (in Chinese)]. Our results generalize and extend some earlierresults of Yu (1993). Example is also given to illustrate the results.� 2006 Published by Elsevier Inc.

Keywords: Superlinear differential equations; Superlinear damping; Oscillation

1. Introduction

We consider the oscillatory behavior of the second-order superlinear differential equation with damping

ðaðtÞy0ðtÞÞ0 þ pðtÞy0ðtÞ þ qðtÞf ðyðtÞÞ ¼ 0; ð1:1Þ
where a; p; q : ½t0;1Þ ! R ¼ ð�1;1Þ and f : R! R are continuous functions, and aðtÞ > 0; t P t0 > 0.
We recall that a function y : ½t0; t1Þ ! R, t1 > t0 is called a solution of Eq. (1.1) if yðtÞ satisfies Eq. (1.1) forall t 2 ½t0; t1Þ. In the sequel, it will be always assumed that solutions of Eq. (1.1) exist on some half-line½T ;1ÞðT P t0Þ. A solution yðtÞ of Eq. (1.1) is called oscillatory if it has arbitrarily large zeros, otherwise itis called nonoscillatory. Eq. (1.1) is called oscillatory if all its solutions are oscillatory.

003/$ - see front matter � 2006 Published by Elsevier Inc.

.1016/j.amc.2006.11.175

is research was partial supported by the NSF of shandong Province, China (Y2005A06).rresponding author.ail address: [email protected] (F. Meng).

mailto:[email protected]

F. Lu, F. Meng / Applied Mathematics and Computation 189 (2007) 796–804 797

Recently, Philos [2,3], Meng [4], Li and Yan [5], Yu [6] have studied the oscillatory behavior of superlineardifferential equations. The following oscillation criterions are their study results.

For the special case of Eq. (1.1), i.e., for the Emden–Fowler equation

y00ðtÞ þ qðtÞjyðtÞja sgnyðtÞ ¼ 0; a > 0: ð1:2Þ
Wong [1] established the following oscillation criterion.
Theorem A. If

limt!1

inf

Z t

t0

qðsÞds > 0 ð1:3Þ

and

limt!1

sup1

tn�1

Z t

t0

ðt � sÞn�1qðsÞds ¼ 1 for some integer n > 2; ð1:4Þ

Then, Eq. (1.2) is oscillatory for every a > 0.

Recently, Wong’s result has been extended in Philos [2] to more general equations of the form

y00ðtÞ þ qðtÞf ðyðtÞÞ ¼ 0; ð1:5Þ
where q : ½t0;1Þ ! R and f : R! R are continuous functions.
More precisely, Philos [2] presented the following oscillation criteria for differential Eq. (1.5).

Theorem B. Let the following conditions hold:

(i) yf ðyÞ > 0, f 0ðyÞP 0; when y 6¼ 0;

(ii)R1 du

f ðuÞ <1,R�1 du

f ðuÞ <1;

(iii)R1 ffiffiffiffiffiffiffi

f 0ðuÞp

f ðuÞ du <1,R�1 ffiffiffiffiffiffiffi

f 0ðuÞp

f ðuÞ du <1;

(iv) min infu>0

ffiffiffiffiffiffiffiffiffiffif 0ðuÞ

p R1u

ffiffiffiffiffiffiffif 0ðsÞp

f ðsÞ ds; infu<0


p R�1u

ffiffiffiffiffiffiffif 0ðsÞp

f ðsÞ ds� �

> 0

and suppose that there exists a continuously differentiable function u : ½t0;1Þ ! ð0;1Þ, which leads that u0 is

nonnegative and decreasing function, and we have

limt!1

inf

Z t

t0

uðsÞqðsÞds > �1; ð1:6Þ

limt!1

sup1

t2

Z t

t0

uðsÞZ s

t0

duuðuÞ

� �ds <1; ð1:7Þ

and

limt!1

sup1

tn�1

Z t

t0

ðt � sÞn�1uðsÞqðsÞds ¼ 1 for some integer n > 2: ð1:8Þ

Then Eq. (1.5) is oscillatory.

Several years ago, Yu [6] obtained a similar oscillation criterion to Theorem B for Eq. (1.1). Yu [6] extendthe main results of Philos [2] to Eq. (1.1) and obtained the more general results.

In this paper, we shall continue in this direction the study of oscillatory properties of Eq. (1.1). The purposeof this paper is to improve and extend the above-mentioned results. We shall further the investigation andoffer some new criteria for the oscillation of Eq. (1.1). Our results improve the main results of [2,6]. Exampleis given to illustrate the superiority of our results at the end of this paper.

It is no doubt that the Riccati substitution and its generalized forms play a very important role in the oscil-latory theory of superlinear differential equations. In this paper, we shall employ another method to derive

798 F. Lu, F. Meng / Applied Mathematics and Computation 189 (2007) 796–804

several oscillation criteria for Eq. (1.1), which are still new even in some particular cases. We believe that ourapproach is simpler and more general than the results of [2,6].

2. Preliminaries and a lemma

Let us begin with a lemma which will substantially simplify the proofs of our results. First we recall aclass functions defined on D ¼ fðt; sÞ : t P s P t0g. A function H 2 CðD;RÞ is said to belong to theclass P if

(I) Hðt; tÞ ¼ 0 for t P t0 and Hðt; sÞ > 0 when t 6¼ s;(II) Hðt; sÞ has partial derivatives on D such that

oHotðt; sÞ ¼ h1ðt; sÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðt; sÞ

p;

oHosðt; sÞ ¼ �h2ðt; sÞ


p
for some h1; h2 2 L1
locðD;RÞ.

Lemma 2.1. Let A0;A1;A2 2 Cð½t0;1Þ;RÞ with A2 > 0, and w 2 C1ð½t0;1Þ;RÞ. If there exists ða; bÞ � ½t0;1Þand c 2 ða; bÞ such that

w0 6 �A0ðsÞ þ A1ðsÞw� A2ðsÞw2; s 2 ða; bÞ; ð2:1Þ

then

1

Hðc; aÞ

Z c

aHðs; aÞA0ðsÞ �

1

4A2ðsÞU2

1ðs; aÞ� �

dsþ 1

Hðb; cÞ

Z b

cHðb; sÞA0ðsÞ �

1

4A2ðsÞU2

2ðb; sÞ� �

ds 6 0

ð2:2Þ

for every H 2 P, where

U1ðs; aÞ ¼ h1ðs; aÞ þ A1ðsÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðs; aÞ

p; U2ðb; sÞ ¼ h2ðb; sÞ � A1ðsÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðb; sÞ

p:

Proof. Multiplying (2.1) by Hðs; tÞ and integrating with respect to s from t to c for t 2 ða; c�, we have

Z c

tHðs; tÞA0ðsÞds 6 �

Z c

tHðs; tÞw0ðsÞdsþ

Z c

tHðs; tÞA1ðsÞwðsÞds�

Z c

tHðs; tÞA2ðsÞw2ðsÞds: ð2:3Þ

In view of (I) and (II), we see that

Z c

tHðs; tÞw0ðsÞds ¼ Hðc; tÞwðcÞ �

Z c

th1ðs; tÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðs; tÞ

pwðsÞds: ð2:4Þ

Using (2.4) in (2.3) leads to:

Z c

tHðs; tÞA0ðsÞds

6 �Hðc; tÞwðcÞ �Z c

t½A2ðsÞHðs; tÞw2ðsÞ � ðh1ðs; tÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðs; tÞ

pþ A1ðsÞHðs; tÞÞwðsÞ�ds

¼ �Hðc; tÞwðcÞ �Z c

t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2ðsÞHðs; tÞ

pwðsÞ � 1

2ffiffiffiffiffiffiffiffiffiffiffiA2ðsÞ

p U1ðs; tÞ !2

dsþZ c

t

1

4A2ðsÞU2

1ðs; tÞds

6 �Hðc; tÞwðcÞ þZ c

t

1

4A2ðsÞU2

1ðs; tÞds: ð2:5Þ


Similarly, if (2.1) is multiplied by Hðt; sÞ and then integrated from c to t for t 2 ½c; bÞ, then one gets
Z t
cHðt; sÞA0ðsÞds

6 Hðt; cÞwðcÞ �Z t

c½A2ðsÞHðt; sÞw2ðsÞ þ ðh2ðt; sÞ


p� A1ðsÞHðt; sÞÞwðsÞ�ds

¼ Hðt; cÞwðcÞ �Z t

c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2ðsÞHðt; sÞ

pwðsÞ þ 1

2ffiffiffiffiffiffiffiffiffiffiffiA2ðsÞ

p U2ðt; sÞ !2

dsþZ t

c

1

4A2ðsÞU2

2ðt; sÞds

6 Hðt; cÞwðcÞ þZ t

c

1

4A2ðsÞU2

2ðt; sÞds: ð2:6Þ

Letting t! aþ in (2.5) and t! b� in (2.6) and adding the resulting inequalities we have (2.2). h

3. Main results

In this section, we define the following functions that will be used in the proof of our results, suppose thatthere exists a function u 2 C1½½t0;1Þ; ð0;1Þ�, T P t0, let

nðtÞ ¼ aðtÞu0ðtÞ � pðtÞuðtÞ; gðtÞ ¼ 1

aðtÞuðtÞ ;

v½t; T � ¼ gðtÞZ t

TgðsÞds

� ��1

:

Theorem 3.1. Let conditions (i)–(iv) hold, suppose that there exists a function u 2 C1½½t0;1Þ; ð0;1Þ�, such that(1.6) satisfied, and

nðtÞP 0; n0ðtÞ 6 0; t P t0; ð3:1ÞZ 1gðsÞds ¼ 1: ð3:2Þ

If for every T P t0, there exists an interval ða; bÞ � ½T ;1Þ, and that there exists c 2 ða; bÞ, H 2 P, and for any

constant D > 0, such that

1

Hðc; aÞ

Z c

aHðs; aÞuðsÞqðsÞ � 1

4Dv½s; t0�U2

1ðs; aÞ� �

ds

þ 1

Hðb; cÞ

Z b

cHðb; sÞuðsÞqðsÞ � 1

4Dv½s; t0�U2

2ðb; sÞ� �

ds > 0; ð3:3Þ

where

U1ðs; aÞ ¼ h1ðs; aÞ þ nðsÞgðsÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðs; aÞ

p;

U2ðb; sÞ ¼ h2ðb; sÞ � nðsÞgðsÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðb; sÞ

p:

Then Eq. (1.1) is oscillatory.

Proof. Suppose to the contrary that there exists a nonoscillatory solution yðtÞ of Eq. (1.1). Without loss ofgenerality, we may assume that yðtÞ > 0 for all t P t0 > 0, because the similar argument holds also foryðtÞ < 0. Define the function wðtÞ by

wðtÞ ¼ uðtÞ aðtÞy0ðtÞ

f ðyðtÞÞ : ð3:4Þ


Differentiating (3.4) for t P t0, we have

w0ðtÞ ¼ �uðtÞqðtÞ þ nðtÞ y 0ðtÞf ðyðtÞÞ �

1

aðtÞuðtÞw2ðtÞf 0ðyðtÞÞ: ð3:5Þ

Integrating (3.5) from t0 to t ðt P t0 > 0Þ, after simple transformations, we conclude that

wðtÞ ¼ wðt0Þ �Z t

t0

uðsÞqðsÞdsþZ t

t0

nðsÞ y0ðsÞf ðyðsÞÞ ds�

Z t

t0

gðsÞw2ðsÞf 0ðyðsÞÞds: ð3:6Þ

By the Bonnet Theorem, there exists f 2 ½t0; t� for every t P t0, such that
Z t
t0

nðsÞ y0ðsÞf ðyðsÞÞ ds ¼ nðt0Þ

Z f

t0

y0ðsÞf ðyðsÞÞ ds ¼ nðt0Þ

Z yðfÞ

yðt0Þ

duf ðuÞ 6 nðt0Þ

Z 1

yðt0Þ

duf ðuÞ ¼ K; ð3:7Þ

where K > 0 is a constant. Hence, we have that for t P t0,

wðtÞ 6 L�Z t

t0

uðsÞqðsÞds�Z t

t0

gðsÞw2ðsÞf 0ðyðsÞÞds; ð3:8Þ

where L ¼ K þ wðt0Þ.We will discuss in three cases.Case 1. Suppose that y0ðtÞ is oscillatory. Then, there exists a sequence ftmgm¼1;2;... such that limm!1tm ¼ 1

and y0ðtmÞ ¼ 0 ðm ¼ 1; 2; . . .Þ of the interval ½t0;1Þ. So (3.8) give that
Z tm
t0

gðsÞw2ðsÞf 0ðyðsÞÞds 6 L�Z tm

t0

uðsÞqðsÞds; m ¼ 1; 2; . . .

Noting that (1.6), we obtain
Z 1
t0

gðsÞw2ðsÞf 0ðyðsÞÞds <1:

There exists a constant M > 0, such that
Z t
t0

gðsÞw2ðsÞf 0ðyðsÞÞds 6 M ; t P t0: ð3:9Þ

By Schwarz inequality and (3.9), we have

Z t

t0

y 0ðsÞf ðyðsÞÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 0ðyðsÞÞ

pds

2

¼Z t

t0

ffiffiffiffiffiffiffiffigðsÞ

p ffiffiffiffiffiffiffiffigðsÞ

pwðsÞ


p �ds

2

6

Z t

t0

gðsÞds� � Z t

t0

gðsÞw2ðsÞf 0ðyðsÞÞds� �

6 MZ t

t0

gðsÞds; t P t0: ð3:10Þ

Applying the condition (iv), we see that

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 0ðyðtÞÞ

p Z 1

yðtÞ


pf ðuÞ du P N 1; t P t0; ð3:11Þ

where N 1 is a positive constant.Let

N 2 ¼Z 1

yðt0Þ


pf ðuÞ du > 0:


Then, applying (3.11), we get

f 0ðyðtÞÞP N 21

Z 1

yðtÞ


pf ðuÞ du

" #�2

¼ N 21 N 2 �

Z yðtÞ

yðt0Þ


pf ðuÞ du

" #�2

¼ N 21 N 2 �

Z t

t0

y0ðsÞf ðyðsÞÞ


pds

� ��2

P N 21 N 2 þ

Z t

t0

y 0ðsÞf ðyðsÞÞ


pds

� ��2

:

Using (3.10) in the above inequality leads to:

f 0ðyðtÞÞP N 21 N 2 þ M

Z t

t0

gðsÞds� �1

2

" #�2

:

Hence, there exists a constant D > 0 and T 0 > t0, such that

f 0ðyðtÞÞP DZ t

t0

gðsÞds� ��1

; t P T 0: ð3:12Þ

Using (3.12) in (3.5) leads to:

w0ðtÞ 6 �uðtÞqðtÞ þ nðtÞgðtÞwðtÞ � Dv½t; t0�w2ðtÞ; t P T 0: ð3:13Þ

For a given T 0 P t0, there exist a, b and c such that T 0 6 a 6 b 6 c, Comparing inequalities (2.1) and (3.13),we identify that

A0ðtÞ ¼ uðtÞqðtÞ;A1ðtÞ ¼ nðtÞgðtÞ;A2ðtÞ ¼ Dv½t; t0�:

Applying Lemma 2.1 to (3.13), we have

1

Hðc; aÞ

Z c


4Dv½s; t0�U2

1ðs; aÞ� �

ds

þ 1

Hðb; cÞ

Z b


4Dv½s; t0�U2

2ðb; sÞ� �

ds 6 0; ð3:14Þ

which contradicts the assumption (3.3).Case 2. Suppose that y0ðtÞ > 0 for t P t1 P t0, then wðtÞ > 0, for t P t1, by (3.8) we obtain
Z t
t1

gðsÞw2ðsÞf 0ðyðsÞÞds 6 L�Z t

t1

uðsÞqðsÞds; t P t1:

Noting that (1.6), we see that
Z 1
t1

gðsÞw2ðsÞf 0ðyðsÞÞds <1: ð3:15Þ

The following proof is similar to case 1.Case 3. Suppose that y0ðtÞ < 0 for t P t1 P t0. If (3.15) holds, then, we have the discussion similar to

the case 2. If the integration in (3.15) is divergent, we can obtain the following inequality from (3.8) and(1.6)

L1 þZ t

t1

gðsÞw2ðsÞf 0ðyðsÞÞds 6 �wðtÞ; t P t1; ð3:16Þ


where L1 is a constant. By choosing t2 P t1, we can get

l � L1 þZ t2

t1

gðsÞw2ðsÞf 0ðyðsÞÞds > 1: ð3:17Þ

From (3.16) and (3.17), we see

wðtÞ < 0; t P t2:

Then, using (3.16), we find

gðtÞw2ðtÞf 0ðyðtÞÞL1 þ

R tt1

gðsÞw2ðsÞf 0ðyðsÞÞdsP � y0ðtÞf 0ðyðtÞÞ

f ðyðtÞÞ ; t P t2:

Integrating the above inequality, we get

ln L1 þZ t

t1

gðsÞw2ðsÞf 0ðyðsÞÞds� �

P lnf ðyðt2ÞÞf ðyðtÞÞ ; t P t2:

Hence

L1 þZ t

t1

gðsÞw2ðsÞf 0ðyðsÞÞds Pf ðyðt2ÞÞf ðyðtÞÞ ; t P t2: ð3:18Þ

By (3.16) and (3.18), we have

y 0ðtÞ 6 �f ðyðt2ÞÞgðtÞ < 0; t P t2:

Then

yðtÞ 6 yðt2Þ � f ðyðt2ÞÞZ t

t2

gðsÞds! �1; t!1:

The above inequality and the assumption yðtÞ > 0 are contradict. This completes the proof. h

The following theorem is a consequence of Theorem 3.1.

Theorem 3.2. Let conditions (i)–(iv) hold, suppose that there exists a function u 2 C1½½t0;1Þ; ð0;1Þ�, such that

(1.6), (3.1) and (3.2) satisfied, and there exists H 2 P, such that

lim supt!1

Z t

lHðs; lÞuðsÞqðsÞ � 1

4Dv½s; t0�U2

1ðs; lÞ� �

ds > 0 ð3:19Þ

and

lim supt!1

Z t

lHðt; sÞuðsÞqðsÞ � 1

4Dv½s; t0�U2

2ðt; sÞ� �

ds > 0 ð3:20Þ

hold for every l 2 ½t1;1Þ, where U1, U2, and D are the same as in Theorem 3.1. Then Eq. (1.1) is oscillatory.

Proof. Suppose that xðtÞ 6¼ 0 for all t 2 ½t2;1Þ for some t2 P t1. Set l ¼ a P t2 in (3.19). Clearly, we see from(3.19) that there exists c > a such that
Z c

4Dv½s; t0�U2

1ðs; aÞ� �

ds > 0: ð3:21Þ

Similarly, setting l ¼ c P t2 in (3.20), it follows that there exists b > c such that
Z b

4Dv½s; t0�U2

2ðb; sÞ� �

ds > 0: ð3:22Þ

From (3.21) and (3.22) we see that (3.3) is satisfied. Therefore, in view of Theorem 3.1, we may conclude thatEq. (1.1) is oscillatory.


If we choose Hðt; sÞ as follows:

Hðt; sÞ ¼ ðt � sÞk; t P s P t0;

where k > 1 is a constant, we have the following corollary. h

Corollary 3.2. Let conditions (i)–(iv) hold, suppose that there exists a function u 2 C1½½t0;1Þ; ð0;1Þ�, such that

(1.6), (3.1) and (3.2) satisfied, such that

lim supt!1

1

tk�1

Z t

lðs� lÞkuðsÞqðsÞ � 1

4Dv½s; t0�ðs� lÞk�2ðkþ nðsÞgðsÞðs� lÞÞ2

� �ds > 0 ð3:23Þ

and

lim supt!1

1

tk�1

Z t

lðt � sÞkuðsÞqðsÞ � 1

4Dv½s; t0�ðt � sÞk�2ðkþ nðsÞgðsÞðt � sÞÞ2

� �ds > 0; ð3:24Þ

for each l P t0, k > 1, then Eq. (1.1) is oscillatory.

Remark 3.1. If f ðyÞ ¼j yja sgny, when a > 1, then the conditions (i)–(iv) can be deleted in Theorems 3.1 and3.2.

Remark 3.2. Even if for the special case of Eq. (1.1), i.e., for Eq. (1.2) (a > 1) and Eq. (1.5), our results are alsonew.

Remark 3.3. In our results, the conditions that the integrateR1 ds

aðsÞ are either convergent or divergent, and the

damped coefficient pðtÞ is a ‘‘small’’ function are not necessary. Therefore, the restraint for aðtÞ and pðtÞ is

relaxed.

4. An example

In order to show the application of our results obtained in this paper, let us consider the following second-order differential equation with damping:

1

1þ t2y 0ðtÞ

� �0� 1

ty0ðtÞ þ qðtÞjyðtÞja sgnyðtÞ ¼ 0; a > 1; ð4:1Þ

where

qðtÞ ¼3þt2

42

t�ð6n�4Þpþ 1þt2

t

h i2

; ð6n� 4Þp 6 t 6 6n� 72

� p;

3þt2

42

ð6n�3Þp�t � 1þt2

t

h i2

; 6n� 72

� p < t 6 ð6n� 3Þp;

8>><>>:

for n 2 f1; 2; . . .g.Note that the function f ðyÞ ¼ jyja sgny satisfies the conditions (i)–(iv). From Eq. (4.1), we can see that

aðtÞ ¼ 1

1þ t2; pðtÞ ¼ � 1

t; when t P t0 ¼

p2:

Let uðtÞ � 1, we can see that

nðtÞ ¼ 1

t; gðtÞ ¼ 1þ t2;

v t;p2

h i¼ 1þ t2

t þ t33þ p

2þ p3

24

:

Moreover, nðtÞ and gðtÞ satisfy the conditions (3.1) and (3.2) of Theorem 3.1 respectively.


By taking Hðs; tÞ ¼ ðt � sÞ2, a ¼ ð6n� 4Þp, c ¼ 6n� 72

� p, b ¼ ð6n� 3Þp, we can easily obtain that the value

of integral in (3.3) is positive.Thus in view of Theorem 3.1, we may conclude that (4.1) is oscillatory. In fact, xðtÞ is a solution of (4.1)

then there exists a sequence ftng, ð6n� 4Þp < tn < ð6n� 3Þp such that xðtnÞ ¼ 0.The main results of [2,6] fails to Eq. (4.1). we see that our results are simpler and more general than [2,6].

References

[1] J.S.W. Wong, An oscillation criteria for second order nonlinear differential equations, Proc. Am. Math. Soc. 98 (1986) 109–112.[2] Ch.G. Philos, Oscillation criteria for second order superlinear differential equations, Canad. J. Math. Anal. XLI (1989) 321–340.[3] CH.G. Philos, I.K. Purnaras, Oscillations in superlinear differential equations of second order, J. Math. Anal. Appl. 165 (1992) 1–11.[4] F.W. Meng, An oscillation theorem for second order superlinear differential equations, Ind. J. Pure Appl. Math. 27 (1996) 651–658.[5] W.T. Li, J.R. Yan, An oscillation criteria for second order superlinear differential equations, Ind. J. Pure Appl. Math. 28 (1997) 735–

740.[6] Y.H. Yu, Oscillation criteria for second order nonlinear differential equations with damping, Acta Math. Appl. 16 (1993) 433–441 (in

Chinese).

Oscillation theorems for superlinear second-order damped differential equations

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