Abstract and Applied Analysis

Volume 2013 (2013), Article ID 281581, 5 pages

http://dx.doi.org/10.1155/2013/281581

## Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 10 May 2013; Revised 11 August 2013; Accepted 12 August 2013

Academic Editor: Aref Jeribi

Copyright © 2013 Fatima N. Ahmed et al. 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.

#### Abstract

Some new sufficient conditions for oscillation of all solutions of the first-order linear neutral delay differential equations are obtained. Our new results improve many well-known results in the literature. Some examples are inserted to illustrate our results.

#### 1. Introduction

A neutral delay differential equation (NDDE) is a differential equation in which the highest-order derivative of the unknown function is evaluated both at the present state at time and at the past state at time for some positive constant .

In the last two decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions of neutral delay differential equations. Particularly, we mention the papers by Ladas and Sficas [1], Chuanxi and Ladas [2], Ruan [3], Elabbasy and Saker [4], Kulenović et al. [5], and Karpuz and Öcalan [6] who investigated NDDEs with variable coefficients. To a large extent, this is due to its theoretical interest as well as to its importance in applications. It suffices to note that NDDEs appear in the study of networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits) in population dynamics and also in many applications in epidemics and infection diseases. We refer reader to [1–18] for relevant studies on this subject.

In this paper, we consider the linear first-order NDDE of the type where and . When and is a constant, Jaroš [9] established some new oscillation conditions for all solutions of (1), and his technique was based on the study of the characteristic equation

Zhang [19], Ladas and Sficas [1], Grammatikopoulos et al. [10], and Yu et al. [8] considered (1) when , and they obtained some sufficient conditions for oscillation of (1). The purpose of this work is to present some new sufficient conditions under which all solutions of (1) are oscillatory. In order to achieve this object, we are first concerned with NDDE (1) with constant coefficients (when is a constant). That is,

Some illustrating examples are given. In some sense, the established results extend and improve some previous investigations such as [1, 8–10, 19].

As usual, a solution of (1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory if it is eventually positive or eventually negative. A function is called eventually positive (or negative) if there exists such that (or ) for all. Equation (1) is called oscillatory if all its solutions are oscillatory; otherwise, it is called nonoscillatory.

#### 2. Main Results

In this section, we give some new sufficient conditions for the oscillation of all solutions of (1) and (3). This is done by using the following well-known lemmas which are from [11, 12].

Lemma 1. *Consider the NDDE
**
where , and .**Let be a positive solution of (4). Set
**
If , then is a positive and decreasing solution of (4); that is,
*

Lemma 2. *Let and be positive constants. Let be an eventually positive solution of the delay differential inequality
**
Then for sufficiently large,
**
where
*

Our main results can now be given as follows.

Theorem 3. *Consider NDDE (3). Assume that *(i)*, and*(ii)*,
**where is the unique real root of the equation
**
Then all solutions of (3) are oscillatory.*

*Proof. *Assume, for the sake of a contradiction, that (3) has a nonoscillatory solution . Without loss of generality, assume that . Let
So that is also a positive solution of (3).

That is,
where

Set for

Thus it follows from Lemma 1 that is a positive and decreasing solution of
and in particular (as implies that .), it follows that

But we have

This implies that

Applying Lemma 2 with (18) we get

Then is bounded.

Dividing (16) by and integrating from to , we get

Let .

Then, it follows from (20) that for and sufficiently small,

As is arbitrary, so we have

Let

Then

Let be the unique real root of the equation

Then

Hence

This contradicts condition (ii) and then completes the proof.

*Example 4. *Consider the NDDE

We note that

Then we have (i) ,
(ii) where is the unique real root of the equation
Then all the hypotheses of Theorem 3 are satisfied, and therefore every solution of (28) oscillates. (Indeed is such a solution.)

Theorem 5. *Consider the NDDE (1). Assume that* (iii)* , and is periodic with period ,* (iv)

*,*

*where is defined as in Theorem 3. Then all solutions of (1) are oscillatory.*

*Proof. *Assume, for the sake of contradiction, that (1) has a nonoscillatory solution . Without loss of generality, assume that . Let
which is oscillation invariant transformation. Then is a positive solution of the equation
where is periodic with period .

Let

Then is decreasing positive solution of the equation

Set

This implies that , since .

Dividing both sides of (33) by and then integrating from to , we obtain that

Hence

Since is periodic with period , then we obtain

Substituting in (38) we find, for all ,

Now, we want to prove that is bounded.

Applying the assumption (iv), we can find such that
where is similar as in the proof of Theorem 3.

Integrating (33) from to we obtain

Using Bonnet’s Theorem and in particular (as ), we get

Integrating (33) from to , we get

Using Bonnet’s Theorem and in particular (as ), we get

Combining (43) and (45), we conclude
or

Then is bounded.

Now, let

But we have proved that is bounded; that is, is finite.

From (40), we obtain

Therefore, we get

Hence

This contradicts our assumption (iv) and then completes the proof.

*Example 6. *Consider the NDDE
where

Then we have(1);
(2) is periodic with period and satisfies

where is the unique real root of the equation

Therefore (52) satisfies all the hypotheses of Theorem 5. Hence every solution of this equation is oscillatory.

Theorem 7. *Suppose that condition (iii) holds. If* (v)* ,**
then every solution of (1) is oscillatory.*

*Proof. *Proceeding as in the proof of Theorem 5, we get (49) which implies that

Hence

But this is a contradiction of assumption (v), and then the proof is complete.

*Example 8. *Consider the NDDE

Here we have

Note that is positive and periodic with period , and also(1),
(2)

Then (58) satisfies hypotheses of Theorem 7, and so all its solutions are oscillatory.

#### Funding

This research has been completed with the support of these Grants: ukm-DLP-2011-049, DIP-2012-31 and FRGS/1/2012/SG04/ukm/01/1.

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