On the Oscillation of Second-Order Neutral Delay Differential Equations

Abstract

Some new oscillation criteria for the second-order neutral delay differential equation , are established, where , , , . These oscillation criteria extend and improve some known results. An example is considered to illustrate the main results.

1. Introduction

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 recent years, many studies have been made on the oscillatory behavior of solutions of neutral delay differential equations, and we refer to the recent papers [2–23] and the references cited therein.

This paper is concerned with the oscillatory behavior of the second-order neutral delay differential equation

(11)

where

In what follows we assume that

(I₁), ,

(I₂)

(I₃), , , , , where is a constant.

Some known results are established for (1.1) under the condition Grammatikopoulos et al. [6] obtained that if and, then the second-order neutral delay differential equation

(12)

oscillates. In [13], by employing Riccati technique and averaging functions method, Ruan established some general oscillation criteria for second-order neutral delay differential equation

(13)

Xu and Meng [18] as well as Zhuang and Li [23] studied the oscillation of the second-order neutral delay differential equation

(14)

Motivated by [11], we will further the investigation and offer some more general new oscillation criteria for (1.1), by employing a class of function operator and the Riccati technique and averaging technique.

Following [11], we say that a function belongs to the function class denoted by if where which satisfies 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

(15)

for and The function is defined by

(16)

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

(17)

2. Main Results

In this section, we give some new oscillation criteria for (1.1). We start with the following oscillation criteria.

Theorem.

If

(21)

where then (1.1) oscillates.

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 From (1.1), we have

(22)

Therefore is a decreasing function. We claim that for Otherwise, there exists such that Then from (2.2) we obtain

(23)

and hence,

(24)

Taking we get This contradiction proves that for Using definition of and applying (1.1), we get for sufficiently large

(25)

and thus,

(26)

Integrating (2.6) from to we obtain

(27)

Noting that we have

(28)

Since for we can find a constant such that for Then from (2.8) and the fact that is eventually decreasing, we have

(29)

which is a contradiction to (2.1). This completes the proof.

Theorem 2.2.

Assume that and there exist functions and such that

(210)

where is defined as in Theorem 2.1, the operator is defined by (1.5), and is defined by (1.6). 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 , , and for all Define

(211)

Then and

(212)

By (2.2) and the fact we get

(213)

From (2.11), (2.12), and (2.13), we have

(214)

Similarly, define

(215)

Then and

(216)

By (2.2) and the facting noting that we get

(217)

From (2.15), (2.16), and (2.17), we have

(218)

Therefore, from (2.14) and (2.18), we get

(219)

From (2.6), we obtain

(220)

Applying to (2.20), we get

(221)

By (1.7) and the above inequality, we obtain

(222)

Hence, from (2.22) we have

(223)

that is,

(224)

Taking the super limit in the above inequality, we get

(225)

which contradicts (2.10). This completes the proof.

Remark 2.3.

With the different choice of and Theorem 2.2 can be stated with different conditions for oscillation of (1.1). For example, if we choose for , , then

(226)

By Theorem 2.2 we can obtain the oscillation criterion for (1.1), the details are left to the reader.

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

Example 2.4.

Consider the following equation:

(227)

Let , , , and then by Theorem 2.1 every solution of (2.27) oscillates; for example, is an oscillatory solution of (2.27).

Remark 2.5.

The recent results cannot be applied in (2.27) since so our results are new ones.