Oscillation of Even-Order Neutral Delay Differential Equations

This paper is concerned with the oscillatory behavior of the even-order neutral delay differential equations

(1.1)

where is even

In what follows we assume that

(I₁) ,

(I₂) , , , , , , where is a constant,

(I₃) and for is a constant.

Neutral differential equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale [1].

In the last decades, there are many studies that have been made on the oscillatory behavior of solutions of differential equations [2–6] and neutral delay differential equations [7–23].

For instance, Grammatikopoulos et al. [10] examined the oscillation of second-order neutral delay differential equations

(1.2)

where

Liu and Bai [13] investigated the second-order neutral differential equations

(1.3)

where

Meng and Xu [14] studied the oscillation of even-order neutral differential equations

(1.4)

where

Ye and Xu [21] considered the second-order quasilinear neutral delay differential equations

(1.5)

where

Zafer [22] discussed oscillation criteria for the equations

(1.6)

where

In 2009, Zhang et al. [23] considered the oscillation of the even-order nonlinear neutral differential equation (1.1) when

To the best of our knowledge, the above oscillation results cannot be applied when and it seems to have few oscillation results for (1.1) when

Xu and Xia [17] established a new oscillation criteria for the second-order neutral differential equations

(1.7)

Motivated by Liu and Bai [13], we will further the investigation and offer some more general new oscillation criteria for (1.1), by employing a class of function and operator The method used in this paper is different from [17].

Following [13], we say that a function belongs to the function class if where which satisfies

(1.8)

for and has the partial derivative on such that is locally integrable with respect to in

By choosing the special function it is possible to derive several oscillation criteria for a wide range of differential equations.

Define the operator by

(1.9)

for and The function is defined by

(1.10)

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

(1.11)

In this section, we give some new oscillation criteria for (1.1). In order to prove our theorems we will need the following lemmas.

Lemma 2.1 (see [5]).

Let If is eventually of one sign for all large say then there exist a and an integer with even for or odd for such that implies that for and implies that for

Lemma 2.2 (see [5]).

If the function is as in Lemma 2.1 and for then for every there exists a constant such that

(2.1)

Lemma 2.3 (see [14]).

Suppose that is an eventually positive solution of (1.1). Then there exists a number such that for

(2.2)

Theorem 2.4.

Assume that .

Further, there exist functions and , such that for some and for every

(2.3)

where the operator is defined by (1.9), and is defined by (1.10). Then every solution of (1.1) is oscillatory.

Proof.

Let be a nonoscillatory solution of (1.1). Then there exists such that for all Without loss of generality, we assume that for all

By Lemma 2.3, there exists such like that (2.2) for Using definition of and applying (1.1), we get for sufficiently large

(2.4)

thus

(2.5)

where

It is easy to check that we can apply Lemma 2.2 for and conclude that there exist and such that

(2.6)

Next, define

(2.7)

Then, and

(2.8)

From (2.6), (2.7), and (2.8), we have

(2.9)

Similarly, define

(2.10)

Then, and

(2.11)

From (2.6), (2.10), and (2.11), we have

(2.12)

Therefore, from (2.9) and (2.12), we get

(2.13)

From (2.5), note that , and then we obtain

(2.14)

Applying to (2.14), we get

(2.15)

By (1.11) and the above inequality, we obtain

(2.16)

Hence, from (2.16), we have

(2.17)

that is,

(2.18)

Taking the super limit in the above inequality, we get

(2.19)

which contradicts (2.3). This completes the proof.

We can apply Theorem 2.4 to the second-order neutral delay differential equations

(2.20)

We get the following new result.

Theorem 2.5.

Assume that Further, there exist functions and such that

(2.21)

where is defined as in Theorem 2.4, the operator is defined by (1.9), and is defined by (1.10). Then every solution of (2.20) is oscillatory.

Proof.

Let be a nonoscillatory solution of (2.20). Then there exists such that for all

Without loss of generality, we assume that for all Proceeding as in the proof of Theorem 2.4, we have (2.2) and (2.5) Next, define

(2.22)

Then and

(2.23)

From (2.22) and (2.23), we have

(2.24)

Similarly, define

(2.25)

Then The rest of the proof is similar to that of the proof of Theorem 2.4, hence we omit the details.

Remark 2.6.

With the different choice of and Theorem 2.4 (or Theorem 2.5) can be stated with different conditions for oscillation of (1.1) (or (2.20)). For example, if we choose for then

(2.26)

By Theorem 2.4 (or Theorem 2.5) we can obtain the oscillation criterion for (1.1) (or (2.20)); the details are left to the reader.

For an application, we give the following example to illustrate the main results.

Example 2.7.

Consider the following equations:

(2.27)

Let Take it is easy to verify that

(2.28)

By Theorem 2.5, let one has (2.21). Hence, every solution of (2.27) oscillates. For example, is an oscillatory solution of (2.27).

Remark 2.8.

The recent results cannot be applied in (1.1) and (2.20) when for Therefore, our results are new.

Remark 2.9.

It would be interesting to find another method to study (1.1) and (2.20) when , or for

Remark 2.10.

It would be more interesting to find another method to study (1.1) when is odd.