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International Journal of Differential Equations
Volume 2010 (2010), Article ID 354726, 7 pages
http://dx.doi.org/10.1155/2010/354726
Research Article

Conditions for Oscillation of a Neutral Differential Equation

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 18 November 2009; Revised 2 April 2010; Accepted 13 May 2010

Academic Editor: Josef Diblík

Copyright © 2010 Weiping Yan and Jurang Yan. 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

For a neutral differential equation with positive and changeable sign coefficients , oscillation criteria are established, where q(t) is not required as nonnegative. Several new results are obtained.

1. Introduction

Consider the delay neutral differential equation with positive and changeable sign coefficients where the following conditions are assumed to hold throughout this paper:; and and ; and for all .

Recently, oscillation of first-order differential equations and difference equations with positive and negative coefficients has been investigated by many authors. Several interesting results have been obtained. We refer to [112] and the references cited therein. However, to the best of our knowledge, up to now, there are not works on oscillation of solutions of (1.1) with able to change sign. The purpose of this paper is to study oscillation properties of (1.1) by some new technique. Our results improve and extend several known results in the literature. In particular, our results can be applied to linear neutral differential equation

By a solution of (1.1) we mean a function for some such that is continuously differentiable on and satisfies (1.1) for , where . As is customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise the solution is called nonoscillatory.

In the sequel, unless otherwise specified, when we write a functional inequality on it will hold for all sufficiently large .

First, we establish the following lemma. It extends and improves in [3, Lemma ], [4, Lemma ], [7, Lemma ], and [9, Lemma ].

Lemma 1.1. Assume that where and is the inverse function of . Let be a nonoscillatory solution of (1.1) and Then

Proof. Let be an eventually positive solution. The case when is an eventually negative solution is similar and its proof is omitted. Thus we have By , (1.4), and (1.2), we obtain which implies that is decreasing. Hence, if (1.5) does not hold, then eventually and there exist and positive constant such that for all , that is,
We consider the following two possible cases.
The first case. is unbounded, that is, . Thus there exists a sequence of points such that and . From (1.8) we have This is a contradiction.
The second case. is bounded, that is, . Choose a sequence of points such that and as . Let , , and . Then . Thus, in view of (1.8) we obtain Therefore which is also a contradiction. Hence (1.5) holds. The proof of Lemma 1.1 is complete.

2. Main Results

In this section, we will prove a comparison theorem on oscillation for (1.1). For convenience of discussions, in the rest of this paper we will use the following notations: where is the inverse function of .

The following comparison theorem is the main result of this paper.

Theorem 2.1. Assume that (1.2) and (1.3) hold and there exists a nonnegative integer such that all solutions of the following delay differential equation: are oscillatory. Then all solutions of (1.1) are also oscillatory.

Proof. Suppose that is an eventually positive solution of (1.1). The proof of the case where is eventually negative is similar and will be omitted. By Lemma 1.1, we have where is given by (1.4). Thus By induction, we see that From (1.4), (2.3), and (2.5), we obtain It follows from (1.7) that By a well-known result (see, e.g., [4, Corollary ]) we can conclude that (2.2) has also an eventually positive solution. This is a contradiction. The proof of Theorem 2.1 is complete.

Following the proof of Theorem 2.1 and taking into account in (2.7) the positivity of the functions and and the properties of the delay functions and , we state as corollary the following claim.

Corollary 2.2. Assume that (1.2) and (1.3) hold and there exists a nonnegative integer such that all solutions of the delay differential equation are oscillatory. Then all solutions of (1.1) are also oscillatory.

Corollary 2.3. Consider (1.1) with . Assume that there exists a nonnegative integer such that all solutions of the delay differential equation are oscillatory. Then all solutions of (1.1) are also oscillatory.

Remark 2.4. Corollaries 2.2 and 2.3 extend and improve [3, Theorems and ].

3. Explicit Oscillation Conditions

In this section, we will give several explicit oscillation conditions for (1.1). Let where and .

By [11, Corollary and Theorem ] with a well-known oscillation criterion for first-order linear delay differential equations, we have the following result.

Theorem 3.1. Assume that (1.2) and (1.3) hold and there exists a nonnegative integer and such that or there exists a nonnegative integer such that Then all solutions of (1.1) are oscillatory.

Remark 3.2. Theorem 3.1 extends and improves [4, Theorem ], [2, Theorem ], and the relative results in [12].

Consider the autonomous neutral differential equation where , and are positive constants, and are real number, and

Theorem 3.3. Assume that (3.5) holds and there exists a nonnegative integer satisfying Then all solutions of (3.4) are oscillatory.

Proof. By (3.2), we find that for (3.4) which implies that (3.6) holds. By Theorem 3.1, (3.6) leads to that all solutions of (3.4) are oscillatory. The proof of Theorem 3.3 is complete.

Corollary 3.4. Assume that (3.5) with holds and Then all solutions of (3.4) are oscillatory.

Remark 3.5. When , (3.8) reduces to Equation (3.9) improves conditions (2) and (4) in [1, Corollary ] where the following two oscillation criteria for the solutions of (3.4) with and are obtained: Obviously, since , it follows that (3.9) is respectively better than (3.10).

Acknowledgments

The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the paper. This work was supported by Tianyuan Mathematics Fund of China (no. 10826080) and Youth Science Foundation of Shanxi Province (no. 2009021001-1).

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