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International Journal of Differential Equations
Volume 2010 (2010), Article ID 308357, 14 pages
http://dx.doi.org/10.1155/2010/308357
Research Article

Oscillation Criteria for Even Order Neutral Equations with Distributed Deviating Argument

Department of Mathematics, Qufu Normal University, Qufu 273165, China

Received 24 September 2009; Accepted 24 November 2009

Academic Editor: Leonid Berezansky

Copyright © 2010 Siyu Zhang and Fanwei Meng. 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 present new oscillation criteria for the even order neutral delay differential equations with distributed deviating argument , where . Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those by Wang et al. (2005).

1. Introduction

In this paper, we are concerned with the oscillation behavior of the even order neutral delay differential equations of the form

where , and   is an even positive integer. We assume that

and , for , , ;, for ;, for and is not eventually zero on any half linear ;, for , has a continuous and positive partial derivative on with respect to and nondecreasing with respect to , respectively, for ; is nondecreasing, and the integral of (1.1) is in the sense of Riemann-Stieltijes.

We restrict our attention to those solutions of (1.1) which exist on some half linear and satisfy for any . As usual, such a solution of (1.1) is called oscillatory if the set of its zeros is unbounded from above; otherwise, it is said to be nonoscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory.

The oscillatory behavior of solutions of higher-order neutral differential equations is of both theoretical and practical interest. There have been some results on the oscillatory and asymptotic behavior of even order neutral equations. We mention here [112]. The oscillation problem for nonlinear delay equation such as

as well as for the the linear ordinary differential equation

and the neutral delay differential equation

has been studied by many authors with different methods. In [13], Rogovchenko established some general oscillation criteria for second-order nonlinear differential equation: In [14], the authors discussed the following neutral equations of the form and obtained the following results.

Theorem 1 A (see [14,Theorem ]). Assume that there exist functions and , such that and , and If there exists a function satisfying then every solution of (1.6) is oscillatory.

We will use the function class to study the oscillation criteria for (1.1). Let , and We say that a continuous function belongs to the class if

  for  has a continuous partial derivativesatisfying, for some the condition

The purpose of this paper is to further improve Theorem A by Wang et al. [14], using a generalized Riccati transformation and developing ideas exploited by the Rogovchenko and Tuncay [13], we establish some new oscillation criteria for (1.1), which remove condition (C2) in Theorem A by Wang et al. [14]; this complements and extends the results in [14].

In addition, we will make use of the following conditions.

There exists a positive real number such that for .

Lemma 1.1. If , then

Lemma 1.2 (Kiguradze [15]). Let u(t ) be a positive and times differentiable function on . If is of constant sign and identically zero on any ray for , then there exists a and an integer , with even for or odd for , and for ,

Lemma 1.3 (Philos [16]). Suppose that the conditions of Lemma 1.2 are satisfied, and then there exists a constant in (0,1) such that for sufficiently large and there exists a constant satisfying where

2. When Is Monotone

In this section, we will deal with the oscillation for (1.1) under the assumptions and the following assumption.

exists, and for .

Theorem 2.1. Let , hold. Equation (1.1) is oscillatory provided that such that where

Proof. Suppose to the contrary that there exists a solution of (1.1) such that From (1.1), we also have and for .
It follows that the function is decreasing and we claim that
Otherwise, if there exist a such that , then for all , which implies that integrating the above inequality from to t, we have Let ; from , we get , which implies that and are negative for all large t; from Lemma 1.2, no two consecutive derivative can be eventually negative, for this would imply that , which is a contradiction. Hence for Using this fact together with , we have that Now from ,, and (2.7), we get and thus, from(1.1), we get Further, observing that is nondecreasing with respect to and for , from Lemma 1.2, we have , , and so So, for and . Thus Define From (1.1), (2.11), and Lemma 1.3 we get Then, by Lemma 1.1 we get Let That is, Integrating by parts for any and using properties and , we obtain We obtain From for which implies that Let , and taking upper limits, we have which contradicts the assumption (2.1). This complete the proof of Theorem 2.1.

From Theorem 2.1, we have the following oscillation result.

Corollary 2.2. If condition (2.1) of Theorem 2.1 is replaced by where is defined by (2.2), then (1.1) is oscillatory.

Remark 2.3. By introducing various from Theorem 2.1 or Corollary 2.2, we can obtain some oscillatory criteria of (1.1). For example, let ,, in which is a integer. By choosing it is clear that the conditions of and hold; then, from Theorem 2.1 and Corollary 2.2, we have the following.

Corollary 2.4. Assume that there exists a function such that where is defined by (2.2), then (1.1) is oscillatory.

Corollary 2.5. Assume that there exists a function such that where is defined by (2.2), then (1.1) is oscillatory.

Theorem 2.6. Assume that the conditions of Theorem 2.1 hold, and If there exists a function satisfying where is defined by (2.2), then (1.1) is oscillatory.

Proof. Assume that there exists a nonoscillatory solution of (1.1) on , such that on . Without loss of generality, assume that , . Then, proceeding as in the proof of Theorem 2.1, for , we have Let , and taking upper limits, we have thus, from (2.27), we have then , and Now we can claim that In fact, assume the contrary, that From (2.26), there exists a constant such that this is and there exists a such that , for all . On the other hand, by virtue of (2.34), for any positive number , there exists a , such that, for all Using integration by parts, we conclude that, for all , Since is an arbitrary positive constant, which contradicts (2.32), consequently, (2.33) holds, and, by virtue of for which contradicts (2.28), and therefore, (1.1) is oscillatory.

Remark 2.7. Choosing as in Remark 2.3, it is not difficult to see that condition (2.26) is satisfied because, for any ,

Consequently, one immediately derives from Theorem 2.6 the following useful corollary for the oscillation of (1.1).

Corollary 2.8. Assume that there exist functions and satisfying where is defined by (2.2), then (1.1) is oscillatory.

Theorem 2.9. Assume that the conditions of Theorem 2.1 and (2.26) hold, and then (1.1) is oscillatory.

Proof. Assume that there exists a nonoscillatory solution of (1.1) on , such that on . Without loss of generality, assume that , . Then, proceeding as in the proof of Theorem 2.1, for , we have Let , and taking lower limits, we have
The following proof is similar to Theorem 2.6, so we omit the details. This completes the proof of Theorem 2.9.

3. When Is Not Monotone

In this section, we will deal with the oscillation for (1.1) under the assumptions and the following assumption:

and for .

Theorem 3.1. Let and hold. Equation (1.1) is oscillatory provided that such that where then (1.1) is oscillatory.

Proof. Let be an eventually positive solution of (1.1). As in the proof of Theorem 2.1, there exists , such that (2.3), (2.4), and (2.7) hold. Thus, from (1.1) and , we get Noting that Thus, (3.3) implies that From (2.10) and (3.5) we get Define Differentiating (3.7) and using (3.6), Lemma 1.1, and 1.3 we get The rest proof is similar to that of Theorem 2.1 and hence is omitted. This completes the proof of Theorem 3.1.

Theorem 3.2. Assume that the conditions of Theorem 2.1 and (2.26) hold; if there exists a function satisfying then (1.1) is oscillatory.

Theorem 3.3. Let all assumptions of Theorem 2.6 be satisfied except that in condition Theorem 3.2 is replaced with then (1.1) is oscillatory.

Acknowledgment

This research was partial supported by the NNSF of China (10771118).

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