#### Abstract

The purpose of this paper is to examine oscillatory properties of the third-order neutral delay differential equation . Some oscillatory and asymptotic criteria are presented. These criteria improve and complement those results in the literature. Moreover, some examples are given to illustrate the main results.

#### 1. Introduction

This paper is concerned with the oscillation and asymptotic behavior of the third-order neutral differential equation We always assume that (H1),(H2).

We set . By a solution of , we mean a nontrivial function , which has the properties , , and satisfies on . We consider only those solutions of which satisfy for all . We assume that possesses such a solution. A solution of is called oscillatory if it has arbitrarily large zeros on ; otherwise, it is called nonoscillatory. Equation is said to be almost oscillatory if all its solutions are oscillatory or convergent to zero asymptotically.

Recently, great attention has been devoted to the oscillation of differential equations; see, for example, the papers [1–30]. Hartman and Wintner [9], Hanan [10], and Erbe [8] studied a particular case of , namely, the third-order differential equation Equation with plays an important role in the study of the oscillation of third-order trinomial delay differential equation see [6, 12, 27]. Baculíková and Džurina [21, 22], Candan and Dahiya [25], Grace et al. [28], and Saker and Džurina [30] examined the oscillation behavior of with . It seems that there are few results on the oscillation of with a neutral term. Baculíková and Džurina [23, 24] and Thandapani and Li [17] investigated the oscillation of under the assumption Graef et al. [13] and Candan and Dahiya [26] considered the oscillation of

In this paper, we shall further the investigation of the oscillations of and . Three cases: are studied.

In the following, all functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all large enough. Without loss of generality, we can deal only with the positive solutions of .

#### 2. Main Results

In this section, we will give the main results.

Theorem 2.1. *Assume that (1.4) holds, . If for some function , for all sufficiently large and for , one has
**
then is almost oscillatory.*

*Proof. *Assume that is a positive solution of . Based on the condition (1.4), there exist two possible cases: (1),
(2) for , is large enough.Assume that case (1) holds. We define the function by
Then, for . Using , we have
Since
we have that
Thus, we get
for . Differentiating (2.3), we obtain
It follows from , (2.3), and (2.4) that
that is,
which follows from (2.6) and (2.7) that
Hence, we have
Integrating the last inequality from to , we get
which contradicts (2.1).

Assume that case (2) holds. Using the similar proof of [23, Lemma 2], we can get due to condition (2.2). This completes the proof.

Theorem 2.2. *Assume that (1.5) holds, . Further, assume that for some function , for all sufficiently large and for , one has (2.1) and (2.2). If
**
where
**
then is almost oscillatory.*

*Proof. *Assume that is a positive solution of . Based on the condition (1.5), there exist three possible cases (1), (2) (as those of Theorem 2.1), and (3), for , is large enough.

Assume that case (1) and case (2) hold, respectively. We can obtain the conclusion of Theorem 2.2 by applying the proof of Theorem 2.1.

Assume that case (3) holds. From , is decreasing. Thus, we get
Dividing the above inequality by and integrating it from to , we obtain
Letting , we have
that is,
Define function by
Then, for . Hence, by (2.19) and (2.20), we get
Differentiating (2.20), we obtain
Using , we have (2.4). From and (2.4), we have
In view of (3), we see that
Hence,
which implies that
By (2.20) and (2.23), (2.24), and (2.26), we obtain
Multiplying the last inequality by and integrating it from to , we have
which follows that
due to (2.21), which contradicts (2.14). This completes the proof.

Theorem 2.3. *Assume that (1.6) holds, . Further, assume that for some function , for all sufficiently large and for , one has (2.1), (2.2), and (2.14). If
**
where
**
then is almost oscillatory.*

*Proof. *Assume that is a positive solution of . Based on the condition (1.6), there exist four possible cases (1), (2), (3) (as those of Theorem 2.2), and (4),
for , is large enough.

Assume that case (1), case (2), and case (3) hold, respectively. We can obtain the conclusion of Theorem 2.3 by using the proof of Theorem 2.2.

Assume that case (4) holds. Since , we get
which implies that
for some constant . By (2.33), we obtain
Using (2.34), we see that
From , (2.33), and (2.35), we have
Integrating the last inequality from to , we get
Integrating again, we have
Integrating again, we obtain
which contradicts (2.30). This completes the proof.

Theorem 2.4. *Assume that (1.6) holds, . Further, assume that for some function , for all sufficiently large and for , one has (2.1), (2.2) and (2.14). If
**
then is almost oscillatory.*

*Proof. *Assume that is a positive solution of . Based on the condition (1.6), there exist four possible cases (1), (2), (3), and (4) (as those of Theorem 2.3).

Assume that case (1), case (2), and case (3) hold, respectively. We can obtain the conclusion of Theorem 2.4 by using the proof of Theorem 2.2.

Assume that case (4) holds. Then, . Assume that . Then, from the proof of [23, Lemma 2], we see that there exists a constant such that
The rest of the proof is similar to that of Theorem 2.3 and hence is omitted.

#### 3. Examples

In this section, we will present some examples to illustrate the main results.

*Example 3.1. *Consider the third-order neutral delay differential equation
where .

Let . It follows from Theorem 2.1 that every solution of (3.1) is either oscillatory or , if for some .

Note that (3.1) is almost oscillatory, if due to [23, Corollary 3].

*Example 3.2. *Consider the third-order neutral delay differential equation
.

Let . It follows from Theorem 2.1 that every solution of (3.2) is almost oscillatory. One such solution is .

*Example 3.3. *Consider the third-order neutral delay differential equation
where .

Let . It follows from Theorem 2.2 that every solution of (3.3) is either oscillatory or , if for some .

Note that [22, Theorem 1] cannot be applied to (3.3) when .

*Example 3.4. *Consider the third-order neutral delay differential equation

Let . It follows from Theorem 2.3 that every solution of (3.4) is either oscillatory or , if .

#### 4. Remarks

*Remark 4.1. *In [3], Agarwal et al. established a well-known result; see [4, Lemma 6.1]. Using [4, Lemma 6.1] and defining the function as in Theorem 2.1 with , we can replace condition (2.1) with
that is oscillatory. Similarly, we can replace condition (2.14) by
that is oscillatory.

*Remark 4.2. *The results for can be extended to the nonlinear differential equations.

*Remark 4.3. *It is interesting to find a method to study for the case when

*Remark 4.4. *It is interesting to find other methods to present some sufficient conditions which guarantee that every solution of is oscillatory.

#### Acknowledgments

The authors would like to thank the referees and Professor P. J. Y. Wong for the comments which helped to improve the paper. This research is supported by NNSF of PR China (Grant nos. 61034007, 60874016, and 50977054). The second author would like to express his gratitude to Professors Ravi P. Agarwal and Martin Bohner for their selfless guidance.