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Abstract and Applied Analysis
Volume 2010, Article ID 564068, 11 pages
http://dx.doi.org/10.1155/2010/564068
Research Article

New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received 23 November 2009; Revised 8 May 2010; Accepted 11 June 2010

Academic Editor: Ağacık Zafer

Copyright © 2010 Yuzhen Bai and Lihua Liu. 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 establish the oscillation and asymptotic criteria for the second-order neutral delay differential equations with positive and negative coefficients having the forms and . The obtained new oscillation criteria extend and improve the recent results given in the paperof B. Karpuz et al. (2009).

1. Introduction

In this paper, we consider the oscillation of all solutions of the second-order neutral delay differential equations with positive and negative coefficients having the forms

We introduce the following class of functions equipped with the functions satisfying the following properties: is strictly increasing and holds, holds for all

In this paper, we make the following assumptions: are bounded starting segments of positive integers; that is, for all for all and for all with for all for all and for all and and that there exists a function which satisfies and

In order to establish our main results, we will assume that there exists a mapping satisfying the following conditions: for all and for all where and for all and there exists such that and

A function is called a solution of (1.1) (or (1.2)) provided that satisfies (1.1)(or (1.2)) identically on and where , and We restrict our attention only to the nontrivial solution that is, to the solution such that for all A nontrivial solution of (1.1) (or (1.2)) is called oscillatory if it has arbitrary large zeros, otherwise, it is called nonoscillatory.

The oscillation and nonoscillation of solutions of second-order neutral delay differential equations have been studied by many authors; see [110]. However, to the best of our knowledge,there seem to be fewoscillation results for (1.1) and (1.2).

Recently,Manojlovićet al. [4] andWeng and Sun[10] have studied oscillation and asymptotic behavior of all solutions of the following equations: and several well-known results have been obtained.

By using weaker conditions than in [4, 10], Karpuz et al. [1] have established oscillation criteria for differential equation

In this paper, we shall continue in the direction to study the oscillatory properties of (1.1) and (1.2). We establish new oscillation criteria for (1.1) and (1.2), which extend and improve the corresponding results in [1, 4, 10]. We also give two examples to illustrate our main results.

2. Main Results

The following properties of the set in [1] are needed for our subsequent discussion.

Property 1. If and then

Corollary 2.1. Suppose that and exists; then

Property 2. If and where then we have .

Property 3. Let be such that Suppose with and If holds, then

Property 4. Let be such that Suppose with , and If holds, then holds.

For simplicity, we denote the set of bounded functions by where For an arbitrary function which satisfies we denote the function by

In this section, for convenience, we suppose that holds for all on

We start with the following Theorem.

Theorem 2.2. Assume that hold and there exists a mapping which satisfies and that If then every solution of (1.1) is oscillatory.

Proof. Suppose that is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that for Therefore, we may assume existence of such that
Now, we set By we denote differential of functions with respect to the first component. Considering (2.5), we rewrite (1.1) in the form on By Leibnitz’s rule, (2.7) and , we have which implies and is eventually strictly monotonic on Hence there exists such that either or holds for all
We consider the following two possible cases:Case 1 (). Integrating (2.8) from to , we have which implies that Therefore, for for which holds, we have Then we conclude that by Property 3. Hence and Property 4 imply that Since is bounded, is also integrable in So we obtain that there exists a constant such that Let From (2.6), we have Then is bounded and monotonous and exists. We can suppose that since
So there exists such that for arbitrary by we have This implies that which is a contradiction.
Case 2 (). Since is nonincreasing by (2.8), the inequality implies that Hence We claim that On contrary, there exists such that and We get the following contradiction: since Thus Accordingly, by (2.4) and (2.6), it follows that Therefore, holds and we see that which is a contradiction.
Therefore, we completed the proof by considering both possible cases.

Remark 2.3. When and Theorem 2.2 reduces to Theorem in [1]. So Theorem 2.2 extends and improves the corresponding results in [1, 4, 10].

Theorem 2.4. Assume that hold and there exists a mapping which satisfies Furthermore, assume that If then every solution of (1.2) is oscillatory or tends to zero asymptotically.

Proof. Suppose that is a nonoscillatory solution of (1.2). Without loss of generality, we assume that for Therefore, we may assume existence of a constant and such that (2.4) and hold for all Now, for set where is defined on the interval as in (2.5). Then as in (2.8), we have Thus there exists satisfying either or for all
We consider the following two possible cases.Case 1 (). In this case, one can show that as shown in above proofs. Since is bounded, is also integrable in Let By (2.19), we have then is bounded and monotonous and exists. By (2.21), we can obtain that exists. Letting we can obtain that exists. Suppose that where We claim Suppose that By we see that there exists such that Because there exists such that for arbitrary But by we have this implies that which is a contradiction. Therefore, Since for we have that Case 2 (). Then we have that by (2.20). We claim that On contrary, there exists such that and hold and
Taking (2.4), (2.5), (2.18), and (2.19) into account, we get the following contradiction: Hence Accordingly, using (2.18), (2.19), and the fact that on we have that Thus and which is a contradiction.
Therefore, we completed the proof by considering both possible cases.

Remark 2.5. If there exists such that on and exists and is finite, then on satisfies

Corollary 2.6. If all conditions of Theorem 2.4 hold, then every nonoscillatory solution of (1.2) converges to zero at infinity.

Remark 2.7. When Theorem 2.4 reduces to Theorem in [1]. So Theorem 2.4 extends and improves the corresponding results in [1, 4, 10].

3. Examples

In this section, we provide two examples to illustrate our main results.

Example 3.1. Consider the following equation: Here, we have Set the function with then
By simple calculation, we have Therefore, according to Theorem 2.2, every solution of (3.1) is oscillatory. Clearly, the known results in [110] cannot be applied to (3.1).

Example 3.2. Consider the following equation:
Here, we have Set the function with for then
By simple calculation, we have Therefore, according to Theorem 2.4 and Remark 2.5, every solution of (3.4) is oscillatory or tends to zero asymptotically. Clearly, the known results in [110] cannot be applied to (3.4).

Acknowledgment

The authors gratefully acknowledge the referee who pointed out the important reference [1] and gave some valuable suggestions which improved the results in this paper. Project supported by the National Natural Science Foundation of China (10771118), STPF of University in Shandong Province of China (J09LA04) and the Research Foundation of Qufu Normal University (XJ0620).

References

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