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Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

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 [113]; Especially, the study of the oscillation of neutral delay differential equations is of great interest in the last three decades; see for example [1438] 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

(1.1)

where Throughout this paper, we assume that

  1. (a)

    , and is not identically zero on any ray of the form for any where is a constant;

  2. (b)

    for is a constant;

  3. (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

(1.2)

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

(1.3)

The author obtained that if

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

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

(1.8)

and has partial derivatives and on such that

(1.9)

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

(1.10)

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:

  1. (i)

    and for

  2. (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

(1.11)

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

(1.12)

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

(1.13)

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

(1.14)

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

(1.15)

for and The function is defined by

(1.16)

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

(1.17)

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

(1.18)

where The authors defined the operator by

(1.19)

for and The function is defined by

(1.20)

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

(1.21)

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

(1.22)

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

(1.23)

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

(1.24)

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

(1.25)

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

(1.26)

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

(1.27)

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

(1.28)

for and

Definition 1.2.

The function is defined by

(1.29)

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

(1.30)

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

(2.1)

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

(2.2)

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

(2.3)

We introduce a generalized Riccati transformation

(2.4)

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

(2.5)

Similarly, we introduce another generalized Riccati transformation

(2.6)

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

(2.7)

From (2.5) and (2.7), we have

(2.8)

By (2.3) and the above inequality, we obtain

(2.9)

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

(2.10)

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

(2.11)

and, for all

(2.12)

By (2.12),

(2.13)

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

(2.14)

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

(2.15)

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

(2.16)

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

(2.17)

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

(2.18)

for Hence, (2.17) is oscillatory for

Remark 2.5.

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

(2.19)

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

(2.20)

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

(2.21)

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

(2.22)

where , are as in Theorem 2.1. Suppose further that

(2.23)

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

(2.24)

Therefore, for sufficiently large

(2.25)

It follows from (2.22) that

(2.26)

for all and for any Consequently, for all we obtain

(2.27)

In order to prove that

(2.28)

suppose the contrary, that is,

(2.29)

Assumption (2.21) implies the existence of a such that

(2.30)

By (2.30), we have

(2.31)

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

(2.32)

Using integration by parts, we conclude that, for all

(2.33)

It follows from (2.33) that, for all

(2.34)

Since is an arbitrary positive constant, we get

(2.35)

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

(2.36)

and, by virtue of (2.27),

(2.37)

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

(2.38)

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

(2.39)

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

(2.40)

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

(2.41)

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

(2.42)

By (1.30) and the above inequality, we obtain

(2.43)

Hence, from (2.43) we have

(2.44)

that is,

(2.45)

Taking the super limit in the above inequality, we get

(2.46)

which contradicts (2.41). This completes the proof.

If we choose

(2.47)

for and then we have

(2.48)

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

(2.49)

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

If we choose

(2.50)

where then we have

(2.51)

where , are defined as the following:

(2.52)

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

(2.53)

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

(2.54)

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

(2.55)

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

(2.56)

Similarly, we introduce another generalized Riccati transformation

(2.57)

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

(2.58)

From (2.56) and (2.58), we have

(2.59)

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

(2.60)

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

(2.61)

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

(2.62)

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

(2.63)

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

(2.64)

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

(2.65)

where and are as in Theorem 2.12. Suppose further that

(2.66)

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

(2.67)

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

(2.68)

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

(2.69)

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

(2.70)

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

(2.71)

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

(2.72)

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.

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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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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

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Keywords

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