#### Abstract

New oscillation criteria are established for the second-order nonlinear neutral functional differential equations of the form , , where , , and . Our results improve and extend some known results in the literature. Some examples are also provided to show the importance of these results.

#### 1. Introduction

This paper is concerned with the oscillation problem of the second-order nonlinear functional differential equation of the following form: where is a constant, .

Throughout this paper, we will assume the following hypotheses: for ,, , ,;, and there exists a function such that

By a solution of (1.1), we mean a function for some which has the property that and satisfies (1.1) on . As is customary, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on ; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all of its nonconstant solutions are oscillatory.

We note that neutral delay differential equations find numerous applications in electric networks. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines which rise in high-speed computers where the lossless transmission lines are used to interconnect switching circuits; see [1]. Therefore, there is constant interest in obtaining new sufficient conditions for the oscillation or nonoscillation of the solutions of varietal types of the second-order equations, see, e.g., papers [2â€“17].

Known oscillation criteria require various restrictions on the coefficients of the studied neutral differential equations.

Agarwal et al. [2], Chern et al. [3], DÅ¾urina and Stavroulakis [4], Kusano et al. [5, 6], Mirzov [7], and Sun and Meng [8] observed some similar properties between and the corresponding linear equation Liu and Bai [10], Xu and Meng [11, 12], and Dong [13] established some oscillation criteria for (1.3) with neutral term under the assumption that

Han et al. [14] examined the oscillation of second-order linear neutral differential equation where , and obtained some oscillation criteria for (1.6) when Han et al. [15] studied the oscillation of (1.6) under the case and

Tripathy [16] considered the nonlinear dynamic equation of the form where is a the ratios of two positive odd integers, and obtained some oscillation criteria under the following conditions:

DÅ¾urina [17] was concerned with the oscillation behavior of the solutions of the second-order neutral differential equations as follows where is a the ratios of two positive odd integers, and obtained some new results under the following conditions

Our purpose of this paper is to establish some new oscillation criteria for (1.1), and we will also consider the cases (1.5) and

To the best of my knowledge, there is no result for the oscillation of (1.1) under the conditions both and (1.13).

In this paper, we will use a new inequality to establish some oscillation criteria for (1.1) for the first time. Some examples will be given to show the importance of these results. In Sections 3 and 4, for the sake of convenience, we denote that

#### 2. Lemma

In this section, we give the following lemma, which we will use in the proofs of our main results.

Lemma 2.1. *Assume that . If , then one has
*

*Proof. * Suppose that or . Then we have (2.1). Suppose that . Define the function by . Then for . Thus, is a convex function. By the definition of convex function, for , we have
that is,
This completes the proof.

#### 3. Oscillation Criteria for the Case (1.5)

In this section, we will establish some oscillation criteria for (1.1) under the case (1.5).

Theorem 3.1. *Suppose that (1.5) holds, for . Furthermore, assume that there exists a function such that
**
Then (1.1) is oscillatory. *

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that and for all . By applying (1.1), for all sufficiently large , we obtain that
Using (2.1) and the definition of , we conclude that
In view of (1.1), we obtain that
Thus, is decreasing function. Now we have two possible cases for eventually and eventually.

(i) Suppose that for . Then, from (3.4), we get
which implies that
Letting , by (1.5), we find , which is a contradiction.

(ii) Suppose that for . We define a Riccati substitution
Then . From (3.4), we have
Differentiating (3.7), we find that
Therefore, by (3.7), (3.8), and (3.9), we see that
Similarly, we introduce a Riccati substitution
Then . From (3.4), we have
Differentiating (3.11), we find that
Therefore, by (3.11), (3.12), and (3.13), we see that
Thus, from (3.10) and (3.14), we have
It is follows from (3.3) that
Integrating the above inequality from to , we obtain that
Define
Using the following inequality:
we have
On the other hand, define
So we have
Thus, from (3.17), we get
which contradicts (3.1). This completes the proof.

When , where are constants, we obtain the following result.

Theorem 3.2. * Suppose that (1.5) holds, , for . Further, assume that there exists a function such that
**
Then (1.1) is oscillatory. *

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that and for all . Using (1.1), for all sufficiently large , we obtain that
By applying (2.1) and the definition of , we conclude that
The remainder of the proof is similar to that of Theorem 3.1 and hence is omitted.

Theorem 3.3. *Suppose that (1.5) holds, for . Furthermore, assume that there exists a function such that
**
Then (1.1) is oscillatory. *

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that and for all . Proceeding as in the proof of Theorem 3.1, we get (3.3) and (3.4). In view of (3.4), is decreasing function. Now we have two possible cases for eventually and eventually.

(i) Suppose that for . Then, similar to the proof of case of Theorem 3.1, we obtain a contradiction.

(ii) Suppose that for . We define a Riccati substitution
Then . From (3.4), we have
Differentiating (3.28), we find that
Therefore, by (3.28), (3.29), and (3.30), we see that

Similarly, we introduce a Riccati substitution
Then . Differentiating (3.32), we find that
Therefore, by (3.32) and (3.33), we see that
Thus, from (3.31) and (3.33), we have
It follows from (3.3) that
Integrating the above inequality from to , we obtain that
Define
Using (3.19), we have
On the other hand, define
So we have
Thus, from (3.37), we get
which contradicts (3.27). This completes the proof.

When , where are constants, we obtain the following result.

Theorem 3.4. * Suppose that (1.5) holds, for . Furthermore, assume that there exists a function such that
**
Then (1.1) is oscillatory. *

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that and for all . Using (1.1) and the definition of , we obtain (3.26) for all sufficiently large . The remainder of the proof is similar to that of Theorem 3.3 and hence is omitted.

#### 4. Oscillation Criteria for the Case (1.13)

In this section, we will establish some oscillation criteria for (1.1) under the case (1.13).

In the following, we assume that are constants.

Theorem 4.1. * Suppose that (1.13) holds, ,? ?for . Further, assume that there exists a function such that (3.24) holds. If there exists a function for such that
**
then (1.1) is oscillatory. *

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that and for all . Proceeding as in the proof of Theorem 3.2, we get (3.26). In view of (1.1), we have (3.4). Thus, is decreasing function. Now we have two possible cases for eventually and eventually.

(i) Suppose that for . Then, by Theorem 3.2, we obtain a contradiction with (3.24).

(ii) Suppose that for . We define the function by
Then . Noting that is increasing, we get
Dividing the above inequality by , and integrating it from to , we obtain that
Letting , we have
that is,
Hence, by (4.2), we get

Similarly, we define the function by
Then . Noting that is increasing, we get the following:
Thus . So by (4.7), we see that
Differentiating (4.2), we obtain that
by (3.4), and we have , so

Similarly, we see that
Therefore, by (4.12) and (4.13), we get the following:
Using (3.26) and (4.14), we obtain that
Multiplying (4.15) by , and integrating it from to , we have
Using (3.19), (4.7), and (4.10), we find that
Letting , we obtain a contradiction with (4.1). This completes the proof.

From Theorems 3.4 and 4.1, we have the following result.

Theorem 4.2. *Suppose that (1.13) holds, for . Further, assume that there exists a function such that (3.43) holds. If there exists a function for such that (4.1) holds, then (1.1) is oscillatory. *

#### 5. Examples

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

*Example 5.1. * Study the second-order neutral differential equation
where are constants.

Let . It is easy to see that all the conditions of Theorem 3.1 hold. Hence, (5.1) is oscillatory.

*Example 5.2. *Consider the second-order quasilinear neutral differential equation
where are constants, .

Let . Then, we have if . Hence, by Theorem 3.2, (5.2) is oscillatory if