Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations
© Zhenlai Han et al. 2010
Received: 10 December 2009
Accepted: 1 July 2010
Published: 19 July 2010
Some sufficient conditions are established for the oscillation of second-order neutral differential equation , , where . The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of second-order neutral delay differential equations, Rat. Mat. 1 (1985), Grace and Lalli (1987), Ruan (1993), H. J. Li (1996), H. J. Li (1997), Xu and Xia (2008).
In recent years, the oscillatory behavior of differential equations has been the subject of intensive study; we refer to the articles [1–13]; Especially, the study of the oscillation of neutral delay differential equations is of great interest in the last three decades; see for example [14–38] and references cited therein. Second-order neutral delay differential equations have applications in problems dealing with vibrating masses attached to an elastic bar and in some variational problems (see ).
In the study of oscillation of differential equations, there are two techniques which are used to reduce the higher-order equations to the first-order Riccati equation (or inequality). One of them is the Riccati transformation technique. The other one is called the generalized Riccati technique. This technique can introduce some new sufficient conditions for oscillation and can be applied to different equations which cannot be covered by the results established by the Riccati technique.
and obtained that if for and then (1.27) is oscillatory. We note that the result given in  fails to apply the cases or for To the best of our knowledge nothing is known regarding the qualitative behavior of (1.1) when
In this paper, we obtain some new oscillation criteria for (1.1). The paper is organized as follows. In the next section, we will use the generalized Riccati transformation technique to give some sufficient conditions for the oscillation of (1.1), and we will give two examples to illustrate the main results. The key idea in the proofs makes use of the idea used in . The method used in this paper is different from that of .
2. Main Results
In this section, we give some new oscillation criteria for (1.1). We start with the following oscillation result.
which contradicts (2.1). This completes the proof.
We note that it suffices to satisfy (2.1) in Theorem 2.1 for any which ensures a certain flexibility in applications. Obviously, if (2.1) is satisfied for some it well also hold for any Parameter introduced in Theorem 2.1 plays an important role in the results that follow, and it is particularly important in the sequel that
As a consequence of Theorem 2.1, we have the following result.
For an application of Corollary 2.3, we give the following example.
It may happen that assumption (2.1) is not satisfied, or it is not easy to verify, consequently, that Theorem 2.1 does not apply or is difficult to apply. The following results provide some essentially new oscillation criteria for (1.1).
which contradicts (2.23). This completes the proof.
Consequently, we have the following result.
From Theorem 2.6, we have the following result.
which contradicts (2.41). This completes the proof.
Thus by Theorem 2.9, we have the following oscillation result.
According to Theorem 2.9, we have the following oscillation result.
The rest of the proof is similar to that of Theorem 2.1, we omit the details. This completes the proof.
For an application of Corollary 2.13, we give the following example.
By (2.61), similar to the proof of Theorem 2.6, we have the following result.
From Theorem 2.15, we have the following result.
Next, by (2.60), similar to the proof of Theorem 2.9, we have the following result.
The statement and the formulation of the results are left to the interested reader.
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), Shandong Postdoctoral Funded Project (200802018) and supported by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), also supported by University of Jinan Research Funds for Doctors (XBS0843).
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