Journal of Applied MathematicsVolume 2011 (2011), Article ID 158060, 8 pagesdoi:10.1155/2011/158060

Research Article

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

Mustafa Kemal Yildiz,¹ Emrah Karaman,² and Hülya Durur³

¹Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, ANS Campus, 03200 Afyon, Turkey²Department of Mathematics, Faculty of Science and Arts, Karabük University, 78050 Karabük, Turkey³Department of Technical Programs, Vocational High School of Ardahan, Ardahan University, 75000 Ardahan, Turkey

Received 6 August 2010; Accepted 16 April 2011

Academic Editor: J. C. Butcher

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.

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, (C₁)there exists a function such that is continuous and nondecreasing and satisfies the inequality where is a positive constant, and (C₂), (C₃) 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 (C₂), . 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 (C₂). 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 (C₁) 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 (C₂), (C₃) 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 (C₂) 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 (C₁). 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 (C₃), we obtain as . This is a contradiction. So, is impossible. Therefore, is the only possible case. That is, . Since is bounded, by virtue of (C₂) 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 (C₂) 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 (C₂) and (C₃) of Theorem 3.3 are satisfied, every bounded solution of (3.24) oscillates or tends to zero at infinity.

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