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 [1–12]. 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).