Research Article | Open Access

# Oscillation and Asymptotic Behaviour of a Higher-Order Nonlinear Neutral-Type Functional Differential Equation with Oscillating Coefficients

**Academic Editor:**J. C. Butcher

#### 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.

#### References

- L. H. Erbe, Q. Kong, and B. G. Zhang,
*Oscillation Theory for Functional-Differential Equations*, vol. 190 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1995. - I. Györi and G. Ladas,
*Oscillation Theory of Delay Differential Equations with Applications*, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1991. - A. Zein and T. Abu-Kaff, “Bounded oscillation of higher order neutral differential equations with oscillating coefficients,”
*Applied Mathematics E-Notes*, vol. 6, pp. 126–131, 2006. View at: Google Scholar | Zentralblatt MATH - Y. Bolat and Ö. Akin, “Oscillatory behaviour of higher order neutral type nonlinear forced differential equation with oscillating coefficients,”
*Journal of Mathematical Analysis and Applications*, vol. 290, no. 1, pp. 302–309, 2004. View at: Publisher Site | Google Scholar - R. P. Agarwal, S. R. Grace, and D. O'Regan,
*Oscillation Theory for Difference and Functional Differential Equations*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at: Zentralblatt MATH - R. P. Agarwal and S. R. Grace, “The oscillation of higher-order differential equations with deviating arguments,”
*Computers & Mathematics with Applications*, vol. 38, no. 3-4, pp. 185–199, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. P. Agarwal and S. R. Grace, “Oscillation of certain functional-differential equations,”
*Computers & Mathematics with Applications*, vol. 38, no. 5-6, pp. 143–153, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. R. Grace and B. S. Lalli, “Oscillation theorems for ${n}^{th}$
-order delay differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 91, no. 2, pp. 352–366, 1983. View at: Publisher Site | Google Scholar - N. Parhi, “Oscillations of higher order differential equations of neutral type,”
*Czechoslovak Mathematical Journal*, vol. 50, no. 1, pp. 155–173, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - F. Yuecai, “Oscillatory behaviour of higher order neutral nonlinear neutral functional differential equation with oscillating coefficients,”
*Journal of South China Normal University*, no. 3, pp. 6–11, 1999. View at: Google Scholar - M. Bohner, B. Karpuz, and Ö. Öcalan, “Iterated oscillation criteria for delay dynamic equations of first order,”
*Advances in Difference Equations*, Article ID 458687, 12 pages, 2008. View at: Google Scholar | Zentralblatt MATH - D. Bainov and D. P. Mishev,
*Oscillation Theory of Operator-Differential Equations*, World Scientific, Singapore, 1995. - G. S. Ladde, V. Lakshmikantham, and B. G. Zhang,
*Oscillation Theory of Differential Equations with Deviating Arguments*, vol. 110 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1987. - D. Bainov and D. P. Mishev,
*Oscillation Theory of Delay Differential Equations with Applications*, Clarendon Press, Oxford, UK, 1991. - S. R. Grace and B. S. Lalli, “Oscillation theorems for certain neutral differential equations,”
*Czechoslovak Mathematical Journal*, vol. 38, no. 4, pp. 745–753, 1988. View at: Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2011 Mustafa Kemal Yildiz 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.