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`Abstract and Applied AnalysisVolume 2014 (2014), Article ID 623713, 5 pageshttp://dx.doi.org/10.1155/2014/623713`
Research Article

Necessary and Sufficient Conditions of Oscillation in First Order Neutral Delay Differential Equations

1School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
2School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China
3Department of Mathematics and Information Science, Shandong Agricultural University, Tai’an 271018, China

Received 31 December 2013; Accepted 5 April 2014; Published 27 April 2014

Copyright © 2014 Songbai Guo 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 are concerned with oscillation of the first order neutral delay differential equation with constant coefficients, and we obtain some necessary and sufficient conditions of oscillation for all the solutions in respective cases and .

1. Introduction

Delay differential equations (DDEs) arose widely in many fields, like oscillation theory [19], stability theory [1012], dynamical behavior of delayed network systems [1315], and so on. Theoretical studies on oscillation of solutions for DDEs have fundamental significance (see [16, 17]). For this reason, DDEs have been attracting great interest of many mathematicians during the last few decades.

In this paper, we consider a class of neutral DDEs where is a positive number and , , , and are positive constants. Generally, a solution of (1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is nonoscillatory. It can be seen in the literature that the oscillation theory regarding solutions of (1) has been extensively developed in the recent years.

In [18], Zhang came to the following conclusion.

Theorem I. Assume that and ; then all solutions of (1) are oscillatory.

This result in Theorem I improves the corresponding result in [19]. Afterward, many authors have been devoted to studying this problem and have obtained many better results. For details, Gopalsamy and Zhang [20] obtained the improved result shown in Theorem II.

Theorem II. If and , then all solutions of (1) are oscillatory.

Further, Zhou and Yu [21] proved the following theorem.

Theorem III. Suppose that and ; then all solutions of (1) are oscillatory.
Continuing to improve the research work, Xiao and Li [22] obtained the following.

Theorem IV. Let and ; then all solutions of (1) are oscillatory.
Finally, Lin [23] obtained the result shown in Theorem V.

Theorem V. Assume that and ; then all solutions of (1) are oscillatory.
However, all the conclusions mentioned above are limited to sufficient conditions in the case . The aim of this paper is to establish systematically the necessary and sufficient conditions of oscillation for all solutions of (1) for the cases and .

2. Main Results

It is well known [24] that all solutions of (1) are oscillatory if and only if the characteristic equation of (1) has no real roots.

Theorem 1. Assume that and let Then all solutions of (1) are oscillatory if and only if where is a unique zero of in .

Proof. It is easy to see that, for , we have Thus any real root of (2) must be negative.
Next, let We consider the monotonicity of the function . Differentiation yields where satisfies the following properties:(1) for ;(2) is strictly increasing on since the function is strictly increasing on .
In addition, Thus, we get that function has a unique zero in . Hence for and for , which imply that is decreasing on and increasing on . Therefore, for if and only if (7) has no real roots in . It is easy to see that is the minimum value of in . Consequently, has no real roots in if and only if . Since we obtain the result immediately.

From Theorem 1, we obtain immediately the following.

Corollary 2. If and , then all solutions of (1) are oscillatory if and only if holds, where .

Theorem 3. Suppose that ; then all solutions of (1) are oscillatory if and only if one of the following conditions holds:();(),
where and are the unique zeros of and (see (3) and (4)) in , respectively.

Proof. Let ; then where , which satisfies If , we get obviously that for all . If , we also get for all since . Thus, for all . From this and (11) we get that for all . Consequently, is strictly decreasing on . Further, Therefore, if , we have . Hence . If , we have . Hence, it is easy to find that both functions and have an equal and unique zero . Consequently, is equivalent to .

From Theorem 1, all solutions of (1) are oscillatory if and only if one of (H1) or (H2) holds.

Theorem 4. Assume that ; then all solutions of (1) are oscillatory if one of the following conditions holds:();(),
where is a unique zero of in .

Proof. If , we have that From the proof of Theorem 1, all solutions of (1) are oscillatory.

If , we suppose furthermore that (otherwise, all solutions of (1) are oscillatory by the above conclusion); that is, . Since is a minimum value of the function at , we have that and the result follows.

So far, for we have discussed the necessary and sufficient conditions of oscillation for all solutions of (1). Our results have perfected the results in [23] (see Theorem 4). Next, we will discuss the behavior of oscillation of solutions of (1) in the case .

Lemma 5. Let ; then all solutions of (1) are oscillatory if and only if the equation has no real roots in .

Proof. By (14), we know that for . It is not difficult to see that is strictly decreasing on while is strictly increasing on . Notice that at ; we find that Hence, has no real roots which is equivalent to that has no real roots in .

Theorem 6. Suppose that and ; then all solutions of (1) are oscillatory if and only if where .

Proof. It is similar to the proof of Theorem 1; is the maximum value of for . This and Lemma 5 imply the result.

Theorem 7. Assume that and ; then all solutions of (1) are oscillatory if and only if where is a unique zero of (3) in .

Proof. Firstly, we prove that has a unique zero in . In fact, It is easy to verify that is strictly increasing on . In addition, Therefore, has a unique zero in . Hence, is strictly decreasing on and strictly increasing on , so that has a unique zero in as and .

Now, from (8), it follows that is the maximum value of in . By (10), we know that (19) is equivalent to for .

From Theorem 7, we obtain the following corollary that extends Theorem  1 in [25] for .

Corollary 8. If , , and , then all solutions of (1) are oscillatory.

Proof. The inequality is equivalent to . From the proof of Theorem 7, we get that . This and (17) imply ; therefore, .

Theorem 9. Suppose that and ; then all solutions of (1) are oscillatory if and only if one of the following conditions holds:();();(),
where is a unique zero of in and is the maximum negative zero of .

Proof. By Lemma 5, all solutions of (1) are oscillatory if and only if From (20), we have that and is strictly decreasing on and strictly increasing on . Thus, is the minimum value of in .

If , which is the case of (H1), we have that Combining (23) and (24), we obtain that This means that is strictly decreasing on and, consequently,

If , has a unique zero in and a unique zero in since . Hence is strictly increasing on , strictly decreasing on , and strictly increasing on . Consequently, is the maximum value of in . Now, it is easy to find that (22) holds if .

On the other hand, applying , we can get So is equivalent to . This is the case of (H2).

If , we obtain that has a unique zero in and a unique zero in . Therefore, is strictly decreasing on , strictly increasing on , and strictly decreasing on . Therefore, it is not difficult to find that (22) holds if and only if and it is the case of (H3).

From Theorem 9, we obtain the following corollary immediately.

Corollary 10. If , , and , then all solutions of (1) are oscillatory.

Example 11. Consider the following neutral delay differential equation: It is not difficult to see that , , , and . Consequently, , and so that all the solutions of (28) are oscillatory from Theorem 9.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors wish to thank the editor and anonymous reviewers for their helpful and valuable comments. This work was supported in part by NSF of Hainan Province under Grant 111004.

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