Advances in Difference Equations

, 2010:763278 | Cite as

Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

  • Zhenlai Han
  • Tongxing Li
  • Shurong Sun
  • Weisong Chen
Open Access
Research Article
Part of the following topical collections:
  1. Recent Trends in Differential and Difference Equations

Abstract

Some sufficient conditions are established for the oscillation of second-order neutral differential equation Open image in new window , Open image in new window , where Open image in new window . 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).

Keywords

Function Class Nonoscillatory Solution Oscillation Criterion Oscillation Result Neutral Delay Differential Equation 

1. Introduction

In recent years, the oscillatory behavior of differential equations has been the subject of intensive study; we refer to the articles [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 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, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 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

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 Open image in new window such that Open image in new window , Open image in new window , Open image in new window and Open image in new window has a continuous and nonpositive partial derivative on Open image in new window with respect to the second variable. Moreover, let Open image in new window 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 Open image in new window such that
where Open image in new window and Open image in new window 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 Open image in new window Say a function Open image in new window belongs to a function class Open image in new window denoted by Open image in new window if Open image in new window where Open image in new window and Open image in new window which satisfies
In [10], Sun defined another type of function class Open image in new window and considered the oscillation of the second-order nonlinear damped differential equation

Say a function Open image in new window is said to belong to Open image in new window denoted by Open image in new window if Open image in new window where Open image in new window which satisfies Open image in new window for Open image in new window and has the partial derivative Open image in new window on Open image in new window such that Open image in new window is locally integrable with respect to Open image in new window in Open image in new window

In [8], by employing a class of function Open image in new window and a generalized Riccati transformation technique, Rogovchenko and Tuncay studied the oscillation of (1.10). Let Open image in new window Say a continuous function Open image in new window belongs to the class Open image in new window if:
Meng and Xu [22] considered the even-order neutral differential equations with deviating arguments

where Open image in new window , Open image in new window The authors introduced a class of functions Open image in new window Let Open image in new window and Open image in new window The function Open image in new window is said to belong to the class Open image in new window (defined by Open image in new window for short) if

Open image in new window , Open image in new window , Open image in new window for Open image in new window

Open image in new window has a continuous and nonpositive partial derivative on Open image in new window with respect to the second variable;

there exists a nondecreasing function Open image in new window such that

Xu and Meng [31] studied the oscillation of the second-order neutral delay differential equation
where Open image in new window by using the function class Open image in new window an operator Open image in new window and a Riccati transformation of the form
the authors established some oscillation criteria for (1.13). In [31], the operator Open image in new window is defined by
It is easy to verify that Open image in new window is a linear operator and that it satisfies
In 2009, by using the function class Open image in new window and defining a new operator Open image in new window , Liu and Bai [21] considered the oscillation of the second-order neutral delay differential equation
where Open image in new window The authors defined the operator Open image in new window by
It is easy to see that Open image in new window 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 Open image in new window and a generalized Riccati transformation of the form

Long and Wang [6] considered (1.22); by using the function class Open image in new window and the operator Open image in new window which is defined in [31], the authors established some oscillation results for (1.22).

In 1985, Grammatikopoulos et al. [16] obtained that if Open image in new window and Open image in new window then equation
is oscillatory. Li [18] studied (1.1) when Open image in new window 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 Open image in new window In 2002, Tanaka [27] studied the even-order neutral delay differential equation
where Open image in new window or Open image in new window 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 Open image in new window for Open image in new window and Open image in new window then (1.27) is oscillatory. We note that the result given in [28] fails to apply the cases Open image in new window or Open image in new window for Open image in new window To the best of our knowledge nothing is known regarding the qualitative behavior of (1.1) when Open image in new window Open image in new window

Motivated by [10, 21], for the sake of convenience, we give the following definitions.

Definition 1.1.

Assume that Open image in new window . The operator is defined by Open image in new window by

for Open image in new window and Open image in new window

Definition 1.2.

The function Open image in new window is defined by
It is easy to verify that Open image in new window 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 Open image in new window for Open image in new window Further, suppose that there exists a function Open image in new window such that for some Open image in new window and some Open image in new window one has

where Open image in new window Open image in new window Open image in new window Then every solution of (1.1) is oscillatory.

Proof.

Let Open image in new window be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists Open image in new window such that Open image in new window , Open image in new window , Open image in new window , for all Open image in new window Define Open image in new window for Open image in new window then Open image in new window for Open image in new window From (1.1), we have
We introduce a generalized Riccati transformation
Differentiating (2.4) from (2.2), we have Open image in new window Thus, there exists Open image in new window such that for all Open image in new window ,
Similarly, we introduce another generalized Riccati transformation
Differentiating (2.6), note that Open image in new window by (2.2), we have Open image in new window then for all sufficiently large Open image in new window one has
From (2.5) and (2.7), we have
By (2.3) and the above inequality, we obtain
From the above inequality and using monotonicity of Open image in new window for all Open image in new window we obtain
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 Open image in new window which ensures a certain flexibility in applications. Obviously, if (2.1) is satisfied for some Open image in new window it well also hold for any Open image in new window Parameter Open image in new window introduced in Theorem 2.1 plays an important role in the results that follow, and it is particularly important in the sequel that Open image in new window

With an appropriate choice of the functions Open image in new window and Open image in new window one can derive from Theorem 2.1 a number of oscillation criteria for (1.1). For example, consider a Kamenev-type function Open image in new window defined by
where Open image in new window is an integer. It is easy to see that Open image in new window and

As a consequence of Theorem 2.1, we have the following result.

Corollary 2.3.

Suppose that Open image in new window for Open image in new window Furthermore, assume that there exists a function Open image in new window such that for some integer Open image in new window and some Open image in new window

where Open image in new window and Open image in new window 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

for Open image in new window Hence, (2.17) is oscillatory for Open image in new window

Remark 2.5.

Corollary 2.3 can be applied to the second-order Euler differential equation

for Open image in new window Hence, (2.19) is oscillatory for Open image in new window

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.

Further, suppose that there exist functions Open image in new window and Open image in new window such that for all Open image in new window and for some Open image in new window
where Open image in new window , Open image in new window are as in Theorem 2.1. Suppose further that

where Open image in new window 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 Open image in new window of (1.1) such that Open image in new window , and Open image in new window for all Open image in new window We define the functions Open image in new window and Open image in new window as in Theorem 2.1; we arrive at inequality (2.10), which yields for Open image in new window sufficiently large
Therefore, for Open image in new window sufficiently large
It follows from (2.22) that
In order to prove that
suppose the contrary, that is,
Assumption (2.21) implies the existence of a Open image in new window such that
By (2.30), we have
and there exists a Open image in new window such that Open image in new window for all Open image in new window On the other hand, by virtue of (2.29), for any positive number Open image in new window there exists a Open image in new window such that, for all Open image in new window
Using integration by parts, we conclude that, for all Open image in new window
It follows from (2.33) that, for all Open image in new window
Since Open image in new window 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 Open image in new window as in Corollary 2.3, it is easy to verify that condition (2.21) is satisfied because, for any Open image in new window

Consequently, we have the following result.

Corollary 2.7.

where Open image in new window and Open image in new window are as in Theorem 2.1. Suppose further that (2.23) holds, where Open image in new window 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 Open image in new window for Open image in new window Further, suppose that Open image in new window such that (2.21) holds, there exist functions Open image in new window and Open image in new window such that for all Open image in new window and for some Open image in new window

where Open image in new window and Open image in new window are as in Theorem 2.1. Suppose further that (2.23) holds, where Open image in new window is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.

Theorem 2.9.

Assume that Open image in new window for Open image in new window Further, assume that there exists a function Open image in new window such that for each Open image in new window for some Open image in new window

where Open image in new window , Open image in new window are defined as in Theorem 2.1, the operator Open image in new window is defined by (1.28), and Open image in new window 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 Open image in new window of (1.1) such that Open image in new window , Open image in new window , and Open image in new window for all Open image in new window We define the functions Open image in new window and Open image in new window as in Theorem 2.1; we arrive at inequality (2.9). Applying Open image in new window 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

Thus by Theorem 2.9, we have the following oscillation result.

Corollary 2.10.

Suppose that Open image in new window for Open image in new window Further, assume that for each Open image in new window there exist a function Open image in new window and two constants Open image in new window such that for some Open image in new window

where Open image in new window , Open image in new window are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.

If we choose

According to Theorem 2.9, we have the following oscillation result.

Corollary 2.11.

Suppose that Open image in new window for Open image in new window Further, assume that for each Open image in new window there exist two functions Open image in new window such that for some Open image in new window

where Open image in new window , Open image in new window 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 Open image in new window for Open image in new window

Theorem 2.12.

Assume that Open image in new window for Open image in new window Suppose that there exists a function Open image in new window such that for some Open image in new window and for some Open image in new window one has

where Open image in new window Open image in new window and Open image in new window is as in Theorem 2.1. Then every solution of (1.1) is oscillatory.

Proof.

Let Open image in new window be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a solution Open image in new window of (1.1) such that Open image in new window , Open image in new window , and Open image in new window for all Open image in new window Proceeding as in the proof of Theorem 2.1, we obtain (2.2) and (2.3). In view of (2.2), we have Open image in new window for Open image in new window We introduce a generalized Riccati transformation
Differentiating (2.55) from (2.2), we have Open image in new window Thus, there exists Open image in new window such that for all Open image in new window ,
Similarly, we introduce another generalized Riccati transformation
Differentiating (2.57), then for all sufficiently large Open image in new window one has
From (2.56) and (2.58), we have
Note that Open image in new window then we have Open image in new window By (2.3) and the above inequality, we obtain

The rest of the proof is similar to that of Theorem 2.1, we omit the details. This completes the proof.

Take Open image in new window where Open image in new window is an integer. As a consequence of Theorem 2.12, we have the following result.

Corollary 2.13.

Suppose that Open image in new window for Open image in new window Furthermore, assume that there exists a function Open image in new window such that for some integer Open image in new window and some Open image in new window

where Open image in new window and Open image in new window 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

for Open image in new window Hence, (2.63) is oscillatory for Open image in new window

By (2.61), similar to the proof of Theorem 2.6, we have the following result.

Theorem 2.15.

Assume that Open image in new window for Open image in new window Assume also that Open image in new window such that (2.21) holds. Moreover, suppose that there exist functions Open image in new window and Open image in new window such that for all Open image in new window and for some Open image in new window
where Open image in new window and Open image in new window are as in Theorem 2.12. Suppose further that

where Open image in new window is defined as in Theorem 2.6. Then every solution of (1.1) is oscillatory.

Choosing Open image in new window , Open image in new window where Open image in new window is an integer. By Theorem 2.15, we have the following result.

Corollary 2.16.

where Open image in new window and Open image in new window are as in Theorem 2.12. Suppose further that (2.66) holds, where Open image in new window 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 Open image in new window for Open image in new window Assume also that Open image in new window such that (2.21) holds. Moreover, suppose that there exist functions Open image in new window and Open image in new window such that for all Open image in new window and for some Open image in new window

where Open image in new window and Open image in new window are as in Theorem 2.12. Suppose further that (2.66) holds, where Open image in new window 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 Open image in new window for Open image in new window Further, assume that there exists a function Open image in new window such that for each Open image in new window for some Open image in new window

where Open image in new window are defined as in Theorem 2.12, the operator Open image in new window is defined by (1.28), and Open image in new window is defined by (1.29). Then every solution of (1.1) is oscillatory.

If we choose Open image in new window as (2.47), then from Theorem 2.18, we have the following oscillation result.

Corollary 2.19.

Suppose that Open image in new window for Open image in new window Further, assume that for each Open image in new window there exist a function Open image in new window and two constants Open image in new window such that for some Open image in new window

where Open image in new window , Open image in new window are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.

If we choose Open image in new window as (2.50), then from Theorem 2.18, we have the following oscillation result.

Corollary 2.20.

Suppose that Open image in new window for Open image in new window Further, assume that for each Open image in new window there exist two functions Open image in new window such that for some Open image in new window

where Open image in new window , Open image in new window 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.

Notes

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).

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Copyright information

© Zhenlai Han et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Zhenlai Han
    • 1
    • 2
  • Tongxing Li
    • 1
    • 2
  • Shurong Sun
    • 1
    • 3
  • Weisong Chen
    • 1
  1. 1.School of ScienceUniversity of JinanJinanChina
  2. 2.School of Control Science and EngineeringShandong UniversityJinanChina
  3. 3.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaUSA

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