#### Abstract

We will study oscillation of bounded solutions of higher-order nonlinear neutral delay differential equations of the following type: , , , where , , , , , , , , , and . We obtain sufficient conditions for the oscillation of all solutions of this equation.

#### 1. Introduction

In this paper, we are concerned with the oscillation of the solutions of a certain more general higher-order nonlinear neutral-type functional differential equation with an oscillating coefficient of the form where is oscillatory and , , , , , , , , and . As it is customary, a solution is said to be oscillatory if is not eventually positive or not eventually negative. Otherwise, the solution is called nonoscillatory. A differential equation is called oscillatory if all of its solutions oscillate. Otherwise, it is nonoscillatory. In this paper, we restrict our attention to real-valued solutions .

In [1, 2], several authors have investigated the linear delay differential equation where and . A classical result is that every solution of (1.2) oscillates if

In [3], Zein and Abu-Kaff have investigated the higher-order nonlinear delay differential equation where , , , , , , , , is continuous, for , there exists an oscillatory function , such that , .

In [4], Bolat and Akin have investigated the higher-order nonlinear differential equation where , , , for , and are oscillating functions, for , , , , for , , is nondecreasing function, for , and . If is odd, , , and for , then every bounded solution of (1.5) is either oscillatory or tends to zero as . If is even, , and , there exists a continuously differentiable function then every bounded solution of (1.5) is either oscillatory or tends to zero as .

Recently, many studies have been made on the oscillatory and asymptotic behaviour of solutions of higher-order neutral-type functional differential equations. Most of the known results which were studied are the cases when , where is the identity function; see, for example, [1–15] and references cited there in.

The purpose of this paper is to study oscillatory behaviour of solutions of (1.1). For the general theory of differential equations, one can refer to [5, 6, 12–14]. Many references to some applications of the differential equations can be found in [2].

In this paper, the function is defined by

#### 2. Some Auxiliary Lemmas

Lemma 2.1 (see [5]). *Let be a positive and -times differentiable function on . If is of constant sign and not identically zero in any interval , then there exist a and an integer , such that is even, if is nonnegative, or odd, if is nonpositive, and that, as , if , for , and if , for . *

Lemma 2.2 (see [5]). *Let be as in Lemma 2.1. In addition and for every ; then for every , , the following hold:
*

#### 3. Main Results

Theorem 3.1. *Assume that is even,*(C_{1})*there exists a function such that is continuous and nondecreasing and satisfies the inequality
where is a positive constant, and
*(C_{2})*,
*(C_{3})*and every solution of the first-order delay differential equation
**
is oscillatory. Then every bounded solution of (1.1) is either oscillates or tends to zero as .*

*Proof. *Assume that (1.1) has a bounded nonoscillatory solution . Without loss of generality, assume that is eventually positive (the proof is similar when is eventually negative). That is, , , and for . Further, suppose that does not tend to zero as . By (1.1) and (1.7), we have
It follows that is strictly monotone and eventually of constant sign. Since is bounded and does not tend to zero as , by virtue of (C_{2}), . Then we can find a such that eventually and is also bounded for sufficiently large . Because is even and odd for and is bounded, by Lemma 2.1, since (otherwise, is not bounded), there exists a such that for
In particular, since for , is increasing. Since is bounded, by (C_{2}). Then, there exists a by (1.7),
for . We may find a such that for , we have
From (3.4) and (3.7), we can obtain the result of
for . Since is defined for , and with for and not identically zero, applying directly Lemma 2.2 (second part, since is positive and increasing), it follows from Lemma 2.2 that
Using (C_{1}) and (3.7), we find for ,
It follows from (3.4) and the above inequality that is an eventually positive solution of
By a well-known result (see [14, Theorem 3.1]), the differential equation
has an eventually positive solution. This contradicts the fact that (1.1) is oscillatory, and the proof is completed.

Thus, from Theorem 3.1 and [11, Theorem 2.3] (see also [11, Example 3.1]), we can obtain the following corollary.

Corollary 3.2. *If
**
then every bounded solution of (1.1) is either oscillatory or tends to zero as .*

Theorem 3.3. *Assume that is odd and (C _{2}), (C_{3}) hold. Then, every bounded solution of (1.1) either oscillates or tends to zero as .*

*Proof. *Assume that (1.1) has a bounded nonoscillatory solution . Without loss of generality, assume that is eventually positive (the proof is similar when is eventually negative). That is, , , and for . Further, we assume that does not tend to zero as . By (1.1) and (1.7), we have for
That is, . It follows that is strictly monotone and eventually of constant sign. Since , there exists a , such that for , we have . Since is bounded, by virtue of (C_{2}) and (1.7), there is a such that is also bounded, for . Because is odd and is bounded, by Lemma 2.1, since (otherwise, is not bounded), there exists , such that for , we have . In particular, since for , is decreasing. Since is bounded, we may write , . Assume that . Let . Then, there exist a constant and a with , such that for . Since is bounded, by (C_{1}). Therefore, there exists a constant and a with , such that for . So, we may find with , such that for . From (3.14), we have
If we multiply (3.15) by and integrate from to , then we obtain
where
Since , for and , we have for . From (3.16), we have
By (C_{3}), we obtain
as . This is a contradiction. So, is impossible. Therefore, is the only possible case. That is, . Since is bounded, by virtue of (C_{2}) and (1.7), we obtain
Now, let us consider the case of for . By (1.1) and (1.7),
That is, . It follow that is strictly monotone and eventually of constant sign. Since , there exists a , such that for , we have . Since is bounded, by virtue of (C_{2}) and (1.7), there is a such that is also bounded, for . Assume that . Then, . Therefore, and for . From this, we observe that is bounded. Because is odd and is bounded, by Lemma 2.1, since (otherwise, is not bounded), there exists a , such that for and . That is, for and . In particular, for , we have . Therefore, is increasing. So, we can assume that . As in the proof of , we may prove that . As for the rest, it is similar to the case . That is, . This contradicts our assumption. Hence, the proof is completed.

*Example 3.4. *We consider difference equation of the form
where , , , , , , and . By taking ,
we check that all the conditions of Theorem 3.1 are satisfied and that every bounded solution of (3.22) oscillates or tends to zero at infinity.

*Example 3.5. *We consider difference equation of the form
where , , , , and , , . Hence, we have
Since Conditions (C_{2}) and (C_{3}) of Theorem 3.3 are satisfied, every bounded solution of (3.24) oscillates or tends to zero at infinity.