1. Introduction
In this paper, we consider a second-order neutral differential equations with deviating arguments of the form
for
, where
is a constant and
,
is a quotient of odd positive integers.
Throughout the paper, it is assumed that the following hypothesis hold true
- (A1)
, and .
- (A2)
, and , for , .
- (A3)
, and there exists a function , such that , where , for , .
- (A4)
, for M is a positive constant.
- (A5)
, and there exists a constant such that for .
We restrict our attention to those solutions
of Equation (
1), which means a function
, and has the property
. We consider only solutions
x of the equation which satisfies
for all
. As usual, a nontrivial solution of Equation (
1) is called oscillatory if it has arbitrarily large zero, otherwise, it called to be nonoscillatory. Equation (
1) is called oscillatory if all of its solutions are oscillatory.
The qualitative properties of ordinary differential equations could be regarded as an age old theme. Such equations are often widely used in physics, population dynamics, economic problems, mechanical control, and other fields. Although it is much more difficult to study differential equations with delay than those without delay, with delay or multiple delay differential equations can better reflect the practical problems. For example, one of the problems is to describe the vibrating masses attached to an elastic bar. In the study of these equations, the solutions of the oscillation are the most popular topic and still receives much attention. The existed researches about delay differential equations can refer to the book [
1] and article [
2]. Different kinds of deviating arguments equations have been studied extensively, since Fite [
3] first studied the oscillatory properties of the deviating arguments equation in 1921. For examples, Zhao and Chatzarakis studied the second-order equations in [
4] and [
5] respectively. Sun considered the third-order nonlinear differential equations [
6]. Li studied the fourth-order differential equations [
7]. Moaaz studied the even-order differential equations [
8].
It is well-known that the study of the equations are based on the canonical case or non-canonical case, and the oscillation results obtained in these two cases are also different [
9,
10,
11,
12]. In this article, we only consider the canonical case, and a large number of articles provided different oscillation criteria for the canonical second-order differential equations (CSODE). For examples, Irena [
9] gave us the Kneser-type oscillation criteria for CSODE with delay. Xu [
13] studies the Kamenev-type oscillation criteria for CSODE with neutral and delay. Gai [
14] studies the Philos-type oscillation results for second-order neutral nonlinear differential equations.
In the following, Xu [
13] considered the second-order neutral equations
Meng [
15] considered the even-order neutral equations
Ye [
16] studied the second-order quasilinear neutral equations
Zhao [
4] studied the second-order neutral equations with distributed deviating argument
In addition, in [
17], Bazighifan used the oscillation of second-order equation to give the criterion of oscillation of higher-order equation. In [
18], Moaaz used the oscillation of the first-order equation to give the criterion of oscillation of second-order equation. In [
19], we get a new method to estimate the fraction function
. Ref. [
2] gives us a new method to set Kamenev-type oscillation criteria. The background outlined above motivates our present research. We also want to study some new methods of oscillation results for second-order delay differential equation, which have more than one neutral delay under the canonical condition. We objective to study the Equation (
1) by applying the known oscillation results of first-order differential equations or the generalized Riccati-type function.
3. Oscillation Results
In this section, we established some new oscillation criteria for Equation (
1). We use the following notations for the simplicity:
Theorem 1. Suppose that hold. If the first-order delay differential equationis oscillatory, then every solutions of Equation (1) are oscillatory. Proof. Suppose to contrary that there exists a nonoscillatory solution of Equation (
1). Without loss of generality, we can assume that
is an eventually positive solution of Equation (
1). From Lemma 2, we have
,
and (
3) hold, where
,
and
. From
in
, we get
Using Equation (
1) and
, note
has a minimum delay
, we have
Next, we give a useful function
We obtain
and
easily. Applying the chain rule, we have
From (
9) and (
10), we have
Integrating the above inequality form
to
t, we obtain
Note
and
, so we get
By simple computation, we have
Then, from
, we get
Combining (
11) and (
12), using
, we have
From the fact that
, the above inequality becomes
Put
, we have
. Then combining (
9) and (
13), we have
Compute
, there exist
, we have
is positive for
, that is to say the inequality
has a positive solution
. Using Philos’s study of nonoscillatory solutions about the first-order delay differential equation [
21], we have Equation (
7) has a positive solution, which is a contradiction. The proof is complete. □
Remark 1. The eventually negative solution in the proof is similar, we don’t give the proof and the corresponding lemma. However, the application of symmetry can unify the proof. If we assume that is an eventually negative solution of Equation (1). Because γ is a quotient of odd positive integers, we get . Multiplying both sides of Equation (1) with , we obtain , where . From and , we get and . Set , then the conclusion will go back to the eventually positive solution case. Corollary 1. If andthen all solutions of Equation (1) are oscillatory. Proof. In view of Ladde et al. [
22] (Theorem 2.1.1), if we have
then we get Equation (
7) is oscillatory. From Theorem 1, we get Equation (
1) is oscillatory. The proof is complete. □
Corollary 2. If does not exist, andhold, then Equation (1) is oscillatory. Proof. The proof is similar to that Ladde et al. [
22] (Theorem 2.1.3). So we ignore. The proof is complete. □
Remark 2. The conclusion of the Theorem 1 is based on the method of Mozza’s paper [18]. The advantage of this method is that the existing oscillation criterion of the first-order delay equation can be directly applied to Equation (7), and then the oscillation results of CSODE can be given. However, from page 440 to 461 in Book [1], we noted most of the conclusions of the first-order delay differential equation must be calculated the function . That is to say, the calculation of this method is more complex. Put in , using the generalized Riccati transformations, we obtain the next Theorem.
Theorem 2. Suppose that and hold. If there exist a function and a constant , such thatthen Equation (1) is oscillatory. Proof. Suppose to contrary that there exists a nonoscillatory solution of Equation (
1). Without loss of generality, we can assume that
is an eventually positive solution of Equation (
1). From Lemma 2 and Theorem 1, we also have (
3) and (
9) hold.
Define the generalized Riccati-type function
by
Clearly
for
. Differentiating
, we obtain
Since
and
, then there exist
such that when
,
Then from
,
,
and (
9), for all
, we have
By Lemma 1 and (
19), we obtain
Multiplying both sides of the above inequality with
for
, then integrating it from
to
t, we get
Thus, we have
which implies that
Since
, we have
Hence,
which contradicts (
17). The proof is complete. □
Put for all hold, where D is a positive constant, we obtain the next Theorem.
Theorem 3. Suppose that hold and . If there exists a function , , such thatwhere , then Equation (1) is oscillatory. Proof. Suppose to contrary that there exists a nonoscillatory solution of Equation (
1). Without loss of generality, we can assume that
is an eventually positive solution of Equation (
1). From Lemma 2 and Theorem 1, we also have (
3) and (
13) hold.
Then, from (
13) and
, we have
Integrating (
22) from
to
t, we obtain
Next, define the generalized Riccati-type function
Obviously,
. Differentiating
, we obtain
Using (
9) and (
23), we have
By Lemma 1 and (
25), we have
Integrating (
26) from
to
t, we have
From (
21), we get
, which contradicts
. The proof is complete. □
The following new Kamenev-type oscillation criteria are obtained by using a similar method in Theorem 2.
Theorem 4. Suppose that hold and . If there exists a function , , and a constant , such thatwhere , then Equation (1) is oscillatory. Proof. From Theorem 3, we also have (
26) hold. Multiplying both sides of the inequality (
26) with
for
, then integrating this inequality from
to
t, we get
That is to say
which contradicts (
27). The proof is complete. □
5. Conclusions
In this paper, by using a well-known first-order delay differential equation and generalized Riccati transformations, we get some new results. By the chain rule, Theorem 1 neglected the condition
which the most articles required. In addition, one of its advantages is that the existing oscillation criterion of the first-order delay equation can be rewritten into the oscillation results of Equation (
1) by Theorem 1. While Theorem 2 is more universal, that is to say, function
is needed to establish new Kamenev-type oscillation criteria. Finally, we will try to get some new oscillation criteria for
, which is no longer just a quotient of odd positive integers of Equation (
1), in the future work.