- Research Article
- Open access
- Published:
Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations
Advances in Difference Equations volume 2010, Article number: 763278 (2010)
Abstract
Some sufficient conditions are established for the oscillation of second-order neutral differential equation ,
, where
. The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of second-order neutral delay differential equations, Rat. Mat. 1 (1985), Grace and Lalli (1987), Ruan (1993), H. J. Li (1996), H. J. Li (1997), Xu and Xia (2008).
1. Introduction
In recent years, the oscillatory behavior of differential equations has been the subject of intensive study; we refer to the articles [1–13]; Especially, the study of the oscillation of neutral delay differential equations is of great interest in the last three decades; see for example [14–38] and references cited therein. Second-order neutral delay differential equations have applications in problems dealing with vibrating masses attached to an elastic bar and in some variational problems (see [39]).
This paper is concerned with the oscillatory behavior of the second-order neutral delay differential equation

where Throughout this paper, we assume that
-
(a)
,
and
is not identically zero on any ray of the form
for any
where
is a constant;
-
(b)
for
is a constant;
-
(c)
,
,
,
,
,
where
is a constant.
In the study of oscillation of differential equations, there are two techniques which are used to reduce the higher-order equations to the first-order Riccati equation (or inequality). One of them is the Riccati transformation technique. The other one is called the generalized Riccati technique. This technique can introduce some new sufficient conditions for oscillation and can be applied to different equations which cannot be covered by the results established by the Riccati technique.
Philos [7] examined the oscillation of the second-order linear ordinary differential equation

and used the class of functions as follows. Suppose there exist continuous functions such that
,
,
and
has a continuous and nonpositive partial derivative on
with respect to the second variable. Moreover, let
be a continuous function with

The author obtained that if

then every solution of (1.2) oscillates. Li [4] studied the equation

used the generalized Riccati substitution, and established some new sufficient conditions for oscillation. Li utilized the class of functions as in [7] and proved that if there exists a positive function such that

where and
then every solution of (1.5) oscillates. Yan [13] used Riccati technique to obtain necessary and sufficient conditions for nonoscillation of (1.5). Applying the results given in [4, 13], every solution of the equation

is oscillatory.
An important tool in the study of oscillation is the integral averaging technique. Just as we can see, most oscillation results in [1, 3, 5, 7, 11, 12] involved the function class Say a function
belongs to a function class
denoted by
if
where
and
which satisfies

and has partial derivatives and
on
such that

In [10], Sun defined another type of function class and considered the oscillation of the second-order nonlinear damped differential equation

Say a function is said to belong to
denoted by
if
where
which satisfies
for
and has the partial derivative
on
such that
is locally integrable with respect to
in
In [8], by employing a class of function and a generalized Riccati transformation technique, Rogovchenko and Tuncay studied the oscillation of (1.10). Let
Say a continuous function
belongs to the class
if:
-
(i)
and
for
-
(ii)
has a continuous and nonpositive partial derivative
satisfying, for some
,
where
is nonnegative.
Meng and Xu [22] considered the even-order neutral differential equations with deviating arguments

where ,
The authors introduced a class of functions
Let
and
The function
is said to belong to the class
(defined by
for short) if
,
,
for
has a continuous and nonpositive partial derivative on
with respect to the second variable;
there exists a nondecreasing function such that

Xu and Meng [31] studied the oscillation of the second-order neutral delay differential equation

where by using the function class
an operator
and a Riccati transformation of the form

the authors established some oscillation criteria for (1.13). In [31], the operator is defined by

for and
The function
is defined by

It is easy to verify that is a linear operator and that it satisfies

In 2009, by using the function class and defining a new operator
, Liu and Bai [21] considered the oscillation of the second-order neutral delay differential equation

where The authors defined the operator
by

for and
The function
is defined by

It is easy to see that is a linear operator and that it satisfies

Wang [11] established some results for the oscillation of the second-order differential equation

by using the function class and a generalized Riccati transformation of the form

Long and Wang [6] considered (1.22); by using the function class and the operator
which is defined in [31], the authors established some oscillation results for (1.22).
In 1985, Grammatikopoulos et al. [16] obtained that if and
then equation

is oscillatory. Li [18] studied (1.1) when and established some oscillation criteria for (1.1). In [15, 19, 25], the authors established some general oscillation criteria for second-order neutral delay differential equation

where In 2002, Tanaka [27] studied the even-order neutral delay differential equation

where or
The author established some comparison theorems for the oscillation of (1.26). Xu and Xia [28] investigated the second-order neutral differential equation

and obtained that if for
and
then (1.27) is oscillatory. We note that the result given in [28] fails to apply the cases
or
for
To the best of our knowledge nothing is known regarding the qualitative behavior of (1.1) when
Motivated by [10, 21], for the sake of convenience, we give the following definitions.
Definition 1.1.
Assume that . The operator is defined by
by

for and
Definition 1.2.
The function is defined by

It is easy to verify that is a linear operator and that it satisfies

In this paper, we obtain some new oscillation criteria for (1.1). The paper is organized as follows. In the next section, we will use the generalized Riccati transformation technique to give some sufficient conditions for the oscillation of (1.1), and we will give two examples to illustrate the main results. The key idea in the proofs makes use of the idea used in [23]. The method used in this paper is different from that of [27].
2. Main Results
In this section, we give some new oscillation criteria for (1.1). We start with the following oscillation result.
Theorem 2.1.
Assume that for
Further, suppose that there exists a function
such that for some
and some
one has

where Then every solution of (1.1) is oscillatory.
Proof.
Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists
such that
,
,
, for all
Define
for
then
for
From (1.1), we have

It is obvious that and
for
imply
for
Using (2.2) and condition
there exists
such that for
we get

We introduce a generalized Riccati transformation

Differentiating (2.4) from (2.2), we have Thus, there exists
such that for all
,

Similarly, we introduce another generalized Riccati transformation

Differentiating (2.6), note that by (2.2), we have
then for all sufficiently large
one has

From (2.5) and (2.7), we have

By (2.3) and the above inequality, we obtain

Multiplying (2.9) by and integrating from
to
we have, for any
and for all

From the above inequality and using monotonicity of for all
we obtain

and, for all

By (2.12),

which contradicts (2.1). This completes the proof.
Remark 2.2.
We note that it suffices to satisfy (2.1) in Theorem 2.1 for any which ensures a certain flexibility in applications. Obviously, if (2.1) is satisfied for some
it well also hold for any
Parameter
introduced in Theorem 2.1 plays an important role in the results that follow, and it is particularly important in the sequel that
With an appropriate choice of the functions and
one can derive from Theorem 2.1 a number of oscillation criteria for (1.1). For example, consider a Kamenev-type function
defined by

where is an integer. It is easy to see that
and

As a consequence of Theorem 2.1, we have the following result.
Corollary 2.3.
Suppose that for
Furthermore, assume that there exists a function
such that for some integer
and some

where and
are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.
For an application of Corollary 2.3, we give the following example.
Example 2.4.
Consider the second-order neutral differential equation

where ,
Let
,
, and
Then
,
Take
Applying Corollary 2.3 with
for any

for Hence, (2.17) is oscillatory for
Remark 2.5.
Corollary 2.3 can be applied to the second-order Euler differential equation

where Let
, and
Then
,
Take
,
Applying Corollary 2.3 with
for any

for Hence, (2.19) is oscillatory for
It may happen that assumption (2.1) is not satisfied, or it is not easy to verify, consequently, that Theorem 2.1 does not apply or is difficult to apply. The following results provide some essentially new oscillation criteria for (1.1).
Theorem 2.6.
Assume that for
and for some

Further, suppose that there exist functions and
such that for all
and for some

where ,
are as in Theorem 2.1. Suppose further that

where Then every solution of (1.1) is oscillatory.
Proof.
We proceed as in the proof of Theorem 2.1, assuming, without loss of generality, that there exists a solution of (1.1) such that
, and
for all
We define the functions
and
as in Theorem 2.1; we arrive at inequality (2.10), which yields for
sufficiently large

Therefore, for sufficiently large

It follows from (2.22) that

for all and for any
Consequently, for all
we obtain

In order to prove that

suppose the contrary, that is,

Assumption (2.21) implies the existence of a such that

By (2.30), we have

and there exists a such that
for all
On the other hand, by virtue of (2.29), for any positive number
there exists a
such that, for all

Using integration by parts, we conclude that, for all

It follows from (2.33) that, for all

Since is an arbitrary positive constant, we get

which contradicts (2.17). Consequently, (2.28) holds, so

and, by virtue of (2.27),

which contradicts (2.23). This completes the proof.
Choosing as in Corollary 2.3, it is easy to verify that condition (2.21) is satisfied because, for any

Consequently, we have the following result.
Corollary 2.7.
Suppose that for
Furthermore, assume that there exist functions
and
such that for all
for some integer
and some

where and
are as in Theorem 2.1. Suppose further that (2.23) holds, where
is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
From Theorem 2.6, we have the following result.
Theorem 2.8.
Assume that for
Further, suppose that
such that (2.21) holds, there exist functions
and
such that for all
and for some

where and
are as in Theorem 2.1. Suppose further that (2.23) holds, where
is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
Theorem 2.9.
Assume that for
Further, assume that there exists a function
such that for each
for some

where ,
are defined as in Theorem 2.1, the operator
is defined by (1.28), and
is defined by (1.29). Then every solution of (1.1) is oscillatory.
Proof.
We proceed as in the proof of Theorem 2.1, assuming, without loss of generality, that there exists a solution of (1.1) such that
,
, and
for all
We define the functions
and
as in Theorem 2.1; we arrive at inequality (2.9). Applying
to (2.9), we get

By (1.30) and the above inequality, we obtain

Hence, from (2.43) we have

that is,

Taking the super limit in the above inequality, we get

which contradicts (2.41). This completes the proof.
If we choose

for and
then we have

Thus by Theorem 2.9, we have the following oscillation result.
Corollary 2.10.
Suppose that for
Further, assume that for each
there exist a function
and two constants
such that for some

where ,
are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.
If we choose

where then we have

where ,
are defined as the following:

According to Theorem 2.9, we have the following oscillation result.
Corollary 2.11.
Suppose that for
Further, assume that for each
there exist two functions
such that for some

where ,
are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.
In the following, we give some new oscillation results for (1.1) when for
Theorem 2.12.
Assume that for
Suppose that there exists a function
such that for some
and for some
one has

where and
is as in Theorem 2.1. Then every solution of (1.1) is oscillatory.
Proof.
Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a solution
of (1.1) such that
,
, and
for all
Proceeding as in the proof of Theorem 2.1, we obtain (2.2) and (2.3). In view of (2.2), we have
for
We introduce a generalized Riccati transformation

Differentiating (2.55) from (2.2), we have Thus, there exists
such that for all
,

Similarly, we introduce another generalized Riccati transformation

Differentiating (2.57), then for all sufficiently large one has

From (2.56) and (2.58), we have

Note that then we have
By (2.3) and the above inequality, we obtain

Multiplying (2.60) by and integrating from
to
we have, for any
and for all

The rest of the proof is similar to that of Theorem 2.1, we omit the details. This completes the proof.
Take where
is an integer. As a consequence of Theorem 2.12, we have the following result.
Corollary 2.13.
Suppose that for
Furthermore, assume that there exists a function
such that for some integer
and some

where and
are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.
For an application of Corollary 2.13, we give the following example.
Example 2.14.
Consider the second-order neutral differential equation

where ,
,
,
,
,
, and
, for
Let
,
, and
. Then
,
Applying Corollary 2.13 with
for any

for Hence, (2.63) is oscillatory for
By (2.61), similar to the proof of Theorem 2.6, we have the following result.
Theorem 2.15.
Assume that for
Assume also that
such that (2.21) holds. Moreover, suppose that there exist functions
and
such that for all
and for some

where and
are as in Theorem 2.12. Suppose further that

where is defined as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
Choosing ,
where
is an integer. By Theorem 2.15, we have the following result.
Corollary 2.16.
Suppose that for
Furthermore, assume that there exist functions
and
such that for all
some integer
and some

where and
are as in Theorem 2.12. Suppose further that (2.66) holds, where
is defined as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
From Theorem 2.15, we have the following result.
Theorem 2.17.
Assume that for
Assume also that
such that (2.21) holds. Moreover, suppose that there exist functions
and
such that for all
and for some

where and
are as in Theorem 2.12. Suppose further that (2.66) holds, where
is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
Next, by (2.60), similar to the proof of Theorem 2.9, we have the following result.
Theorem 2.18.
Assume that for
Further, assume that there exists a function
such that for each
for some

where are defined as in Theorem 2.12, the operator
is defined by (1.28), and
is defined by (1.29). Then every solution of (1.1) is oscillatory.
If we choose as (2.47), then from Theorem 2.18, we have the following oscillation result.
Corollary 2.19.
Suppose that for
Further, assume that for each
there exist a function
and two constants
such that for some

where ,
are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.
If we choose as (2.50), then from Theorem 2.18, we have the following oscillation result.
Corollary 2.20.
Suppose that for
Further, assume that for each
there exist two functions
such that for some

where ,
are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.
Remark 2.21.
The results of this paper can be extended to the more general equation of the form

The statement and the formulation of the results are left to the interested reader.
Remark 2.22.
One can easily see that the results obtained in [15, 16, 18, 19, 25, 28] cannot be applied to (2.17), (2.63), so our results are new.
References
Agarwal RP, Wang Q-R: Oscillation and asymptotic behavior for second-order nonlinear perturbed differential equations. Mathematical and Computer Modelling 2004,39(13):1477-1490. 10.1016/j.mcm.2004.07.007
Džurina J, Stavroulakis IP: Oscillation criteria for second-order delay differential equations. Applied Mathematics and Computation 2003,140(2-3):445-453. 10.1016/S0096-3003(02)00243-6
Kamenev IV: An integral test for conjugacy for second order linear differential equations. Matematicheskie Zametki 1978,23(2):249-251.
Li H: Oscillation criteria for second order linear differential equations. Journal of Mathematical Analysis and Applications 1995,194(1):217-234. 10.1006/jmaa.1995.1295
Li W-T, Agarwal RP: Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations. Journal of Mathematical Analysis and Applications 2000,245(1):171-188. 10.1006/jmaa.2000.6749
Long Q, Wang Q-R: New oscillation criteria of second-order nonlinear differential equations. Applied Mathematics and Computation 2009,212(2):357-365. 10.1016/j.amc.2009.02.040
Philos ChG: Oscillation theorems for linear differential equations of second order. Archiv der Mathematik 1989,53(5):482-492. 10.1007/BF01324723
Rogovchenko YV, Tuncay F: Oscillation criteria for second-order nonlinear differential equations with damping. Nonlinear Analysis: Theory, Methods & Applications 2008,69(1):208-221. 10.1016/j.na.2007.05.012
Sun YG, Meng FW: Note on the paper of Džurina and Stavroulakis. Applied Mathematics and Computation 2006,174(2):1634-1641. 10.1016/j.amc.2005.07.008
Sun YG: New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping. Journal of Mathematical Analysis and Applications 2004,291(1):341-351. 10.1016/j.jmaa.2003.11.008
Wang Q-R: Interval criteria for oscillation of certain second order nonlinear differential equations. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2005,12(6):769-781.
Wang Q-R: Oscillation criteria for nonlinear second order damped differential equations. Acta Mathematica Hungarica 2004,102(1-2):117-139. 10.1023/B:AMHU.0000023211.53752.03
Yan JR: Oscillatory properties of second-order differential equations with an "integralwise small" coefficient. Acta Mathematica Sinica 1987,30(2):206-215.
Agarwal RP, Grace SR: Oscillation theorems for certain neutral functional-differential equations. Computers & Mathematics with Applications 1999,38(11-12):1-11. 10.1016/S0898-1221(99)00280-1
Grace SR, Lalli BS: Oscillation of nonlinear second order neutral differential equations. Radovi Matematički 1987, 3: 77-84.
Grammatikopoulos MK, Ladas G, Meimaridou A: Oscillations of second order neutral delay differential equations. Radovi Matematički 1985,1(2):267-274.
Karpuz B, Manojlović JV, Öcalan Ö, Shoukaku Y: Oscillation criteria for a class of second-order neutral delay differential equations. Applied Mathematics and Computation 2009,210(2):303-312. 10.1016/j.amc.2008.12.075
Li H: Oscillatory theorems for second-order neutral delay differential equations. Nonlinear Analysis: Theory, Methods & Applications 1996,26(8):1397-1409. 10.1016/0362-546X(94)00346-J
Li H: Oscillation of solutions of second-order neutral delay differential equations with integrable coefficients. Mathematical and Computer Modelling 1997,25(3):69-79. 10.1016/S0895-7177(97)00016-2
Lin X, Tang XH: Oscillation of solutions of neutral differential equations with a superlinear neutral term. Applied Mathematics Letters 2007,20(9):1016-1022. 10.1016/j.aml.2006.11.006
Liu L, Bai Y: New oscillation criteria for second-order nonlinear neutral delay differential equations. Journal of Computational and Applied Mathematics 2009,231(2):657-663. 10.1016/j.cam.2009.04.009
Meng F, Xu R: Kamenev-type oscillation criteria for even order neutral differential equations with deviating arguments. Applied Mathematics and Computation 2007,190(2):1402-1408. 10.1016/j.amc.2007.02.017
Parhi N, Rath RN: On oscillation of solutions of forced nonlinear neutral differential equations of higher order. Czechoslovak Mathematical Journal 2003,53(128)(4):805-825. 10.1007/s10587-004-0805-8
Rath RN, Misra N, Padhy LN: Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation. Mathematica Slovaca 2007,57(2):157-170. 10.2478/s12175-007-0006-7
Ruan SG: Oscillations of second order neutral differential equations. Canadian Mathematical Bulletin 1993,36(4):485-496. 10.4153/CMB-1993-064-4
Şahiner Y: On oscillation of second order neutral type delay differential equations. Applied Mathematics and Computation 2004,150(3):697-706. 10.1016/S0096-3003(03)00300-X
Tanaka S: A oscillation theorem for a class of even order neutral differential equations. Journal of Mathematical Analysis and Applications 2002,273(1):172-189. 10.1016/S0022-247X(02)00235-4
Xu R, Xia Y: A note on the oscillation of second-order nonlinear neutral functional differential equations. International Journal of Contemporary Mathematical Sciences 2008,3(29–32):1441-1450.
Xu R, Meng F: Some new oscillation criteria for second order quasi-linear neutral delay differential equations. Applied Mathematics and Computation 2006,182(1):797-803. 10.1016/j.amc.2006.04.042
Xu R, Meng F: Oscillation criteria for second order quasi-linear neutral delay differential equations. Applied Mathematics and Computation 2007,192(1):216-222. 10.1016/j.amc.2007.01.108
Xu R, Meng F: New Kamenev-type oscillation criteria for second order neutral nonlinear differential equations. Applied Mathematics and Computation 2007,188(2):1364-1370. 10.1016/j.amc.2006.11.004
Xu Z, Liu X: Philos-type oscillation criteria for Emden-Fowler neutral delay differential equations. Journal of Computational and Applied Mathematics 2007,206(2):1116-1126. 10.1016/j.cam.2006.09.012
Ye L, Xu Z: Oscillation criteria for second order quasilinear neutral delay differential equations. Applied Mathematics and Computation 2009,207(2):388-396. 10.1016/j.amc.2008.10.051
Zafer A: Oscillation criteria for even order neutral differential equations. Applied Mathematics Letters 1998,11(3):21-25. 10.1016/S0893-9659(98)00028-7
Zhang Q, Yan J, Gao L: Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients. Computers & Mathematics with Applications 2010,59(1):426-430. 10.1016/j.camwa.2009.06.027
Han Z, Li T, Sun S, Sun Y: Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396]. Applied Mathematics and Computation 2010,215(11):3998-4007. 10.1016/j.amc.2009.12.006
Han Z, Li T, Sun S, Chen W: On the oscillation of second-order neutral delay differential equations. Advances in Difference Equations 2010, 2010:-8.
Li T, Han Z, Zhao P, Sun S: Oscillation of even-order neutral delay differential equations. Advances in Difference Equations 2010, 2010:-7.
Hale J: Theory of Functional Differential Equations, Applied Mathematical Sciences. 2nd edition. Springer, New York, NY, USA; 1977:x+365.
Acknowledgment
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), Shandong Postdoctoral Funded Project (200802018) and supported by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), also supported by University of Jinan Research Funds for Doctors (XBS0843).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Han, Z., Li, T., Sun, S. et al. Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations. Adv Differ Equ 2010, 763278 (2010). https://doi.org/10.1155/2010/763278
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/763278