Abstract
Some oscillation criteria are established for the second-order superlinear neutral differential equations , , where , , , , and . Our results are based on the cases or . Two examples are also provided to illustrate these results.
1. Introduction
This paper is concerned with the oscillatory behavior of the second-order superlinear differential equation where is a constant, .
Throughout this paper, we will assume the following hypotheses:(), for ;(), where is a constant;(), , ; (), ; (), and there exists a function such that
By a solution of (1.1), we mean a function for some which has the property that and satisfies (1.1) on . We consider only those solutions which satisfy for all . As is customary, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on , otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
We note that neutral differential equations find numerous applications in electric networks. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines which rise in high-speed computers where the lossless transmission lines are used to interconnect switching circuits; see [1].
In the last few years, there are many studies that have been made on the oscillation and asymptotic behavior of solutions of discrete and continuous equations; see, for example, [2–28]. Agarwal et al. [5], Chern et al. [6], Džurina and Stavroulakis [7], Kusano and Yoshida [8], Kusano and Naito [9], Mirzov [10], and Sun and Meng [11] observed some similar properties between and the corresponding linear equations Baculíková [12] established some new oscillation results for (1.3) when . In 1989, Philos [13] obtained some Philos-type oscillation criteria for (1.4).
Recently, many results have been obtained on oscillation and nonoscillation of neutral differential equations, and we refer the reader to [14–35] and the references cited therein. Liu and Bai [16], Xu and Meng [17, 18], Dong [19], Baculíková and Lacková [20], and Jiang and Li [21] established some oscillation criteria for (1.3) with neutral term under the assumptions , ,
Saker and O'Regan [24] studied the oscillatory behavior of (1.1) when , and .
Han et al. [26] examined the oscillation of second-order nonlinear neutral differential equation where , , , , and the authors obtained some oscillation criteria for (1.7).
However, there are few results regarding the oscillatory problem of (1.1) when and . Our aim in this paper is to establish some oscillation criteria for (1.1) under the case when and .
The paper is organized as follows. In Section 2, we will establish an inequality to prove our results. In Section 3, some oscillation criteria are obtained for (1.1). In Section 4, we give two examples to show the importance of the main results.
All functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all large enough.
2. Lemma
In this section, we give the following lemma, which we will use in the proofs of our main results.
Lemma 2.1. Assume that ,??. If ,??, then holds.
Proof. (i) Suppose that or . Obviously, we have (2.1). (ii) Suppose that , . Define the function by ,??. Then for . Thus, is a convex function. By the definition of convex function, for , we have that is, This completes the proof.
3. Main Results
In this section, we will establish some oscillation criteria for (1.1).
First, we establish two comparison theorems which enable us to reduce the problem of the oscillation of (1.1) to the research of the first-order differential inequalities.
Theorem 3.1. Suppose that (1.5) holds. If the first-order neutral differential inequality has no positive solution for all sufficiently large , where , then every solution of (1.1) oscillates.
Proof. Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that ,??, and for all . Then for . In view of (1.1), we obtain
Thus, is decreasing function. Now we have two possible cases for : (i) eventually, (ii) eventually.
Suppose that for . Then, from (3.2), we get
which implies that
Letting , by (1.5), we find , which is a contradiction. Thus
for .
By applying (1.1), for all sufficiently large , we obtain
Using inequality (2.1), (3.2), (3.5), , and the definition of , we conclude that
It follows from (3.2) and (3.5) that is decreasing and then
Thus, from (3.7) and the above inequality, we find
That is, inequality (3.1) has a positive solution ; this is a contradiction. The proof is complete.
Theorem 3.2. Suppose that (1.5) holds. If the first-order differential inequality has no positive solution for all sufficiently large , where is defined as in Theorem 3.1, then every solution of (1.1) oscillates.
Proof. Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that ,??, and for all . Then for . Proceeding as in the proof of Theorem 3.1, we obtain that is decreasing, and it satisfies inequality (3.1). Set . From , we get In view of the above inequality and (3.1), we see that That is, inequality (3.10) has a positive solution ; this is a contradiction. The proof is complete.
Next, using Riccati transformation technique, we obtain the following results.
Theorem 3.3. Suppose that (1.5) holds. Moreover, assume that there exists such that holds, where is defined as in Theorem 3.1, . Then every solution of (1.1) oscillates.
Proof. Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that , , and for all . Then for . Proceeding as in the proof of Theorem 3.1, we obtain (3.2)–(3.7).
Define a Riccati substitution
Thus for . Differentiating (3.14) we find that
Hence, by (3.14) and (3.15), we see that
Similarly, we introduce another Riccati substitution
Then for . From (3.2), (3.5), and , we have
Differentiating (3.17), we find
Therefore, by (3.17), (3.18), and (3.19), we see that
Thus, from (3.16) and (3.20), we have
It follows from (3.5), (3.7), and that
Integrating the above inequality from to , we obtain
Define
Using inequality
we have
Similarly, we obtain
Thus, from (3.23), we get
which contradicts (3.13). This completes the proof.
As an immediate consequence of Theorem 3.3 we get the following.
Corollary 3.4. Let assumption (3.13) in Theorem 3.3 be replaced by Then every solution of (1.1) oscillates.
From Theorem 3.3 by choosing the function , appropriately, we can obtain different sufficient conditions for oscillation of (1.1), and if we define a function by , and , we have the following oscillation results.
Corollary 3.5. Suppose that (1.5) holds. If where is defined as in Theorem 3.1, then every solution of (1.1) oscillates.
Corollary 3.6. Suppose that (1.5) holds. If where is defined as in Theorem 3.1, then every solution of (1.1) oscillates.
In the following theorem, we present a Philos-type oscillation criterion for (1.1).
First, we introduce a class of functions . Let The function is said to belong to the class (defined by , for short) if(i), for ,??, for ;(ii) has a continuous and nonpositive partial derivative on with respect to .
We assume that and for are given continuous functions such that and differentiable and define
Now, we give the following result.
Theorem 3.7. Suppose that (1.5) holds and is a quotient of odd positive integers. Moreover, let be such that holds, where is defined as in Theorem 3.1. Then every solution of (1.1) oscillates.
Proof. Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that ,??, and for all . Then for . Proceeding as in the proof of Theorem 3.1, we obtain (3.2)–(3.7). Define the Riccati substitution by
Then, we have
Using (3.35), we get
Let
By applying the inequality (see [21, 24])
we see that
Substituting (3.40) into (3.37), we have
That is,
Next, define another Riccati transformation by
Then, we have
From (3.2), (3.5), and , we have that (3.18) holds. Hence, we obtain
Using (3.43), we get
Let
By applying the inequality (3.39), we see that
Substituting (3.48) into (3.46), we have
That is,
By (3.42) and (3.50), we find
In view of the above inequality, (3.5), (3.7), and , we get
which follows that
Using the integration by parts formula and , we have
So, by (3.53), we obtain
Using the inequality
where
we have
due to (3.55), which yields that
which contradicts (3.34). The proof is complete.
From Theorem 3.7, we can obtain different oscillation conditions for all solutions of (1.1) with different choices of ; the details are left to the reader.
Theorem 3.8. Assume that (1.6) and (3.30) hold. Furthermore, assume that . If then every solution of (1.1) oscillates or .
Proof. Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that ,??, and for all . Then for . Proceeding as in the proof of Theorem 3.1, we obtain (3.2). Thus is decreasing function, and there exists a such that , or , .
Case 1. Assume that , for . Proceeding as in the proof of Theorem 3.3 and setting , we can obtain a contradiction with (3.31).
Case 2. Assume that , for . Then there exists a finite limit
where . Next, we claim that . If not, then for any , we have , eventually. Take . We calculate
where
From (3.2) and (3.62), we have
Integrating the above inequality from to , we get
which implies
Integrating the above inequality from to , we have
which yields ; this is a contradiction. Hence, . Note that . Then, . The proof is complete.
4. Examples
In this section, we will give two examples to illustrate the main results.
Example 4.1. Consider the following linear neutral equation:
where and are positive integers.
Let
Hence, . Obviously, all the conditions of Corollary 3.5 hold. Thus by Corollary 3.5, every solution of (4.1) is oscillatory. It is easy to verify that is a solution of (4.1).
Example 4.2. Consider the following linear neutral equation:
where and are positive integers.
Let
Clearly, all the conditions of Theorem 3.8 hold. Thus by Theorem 3.8, every solution of (4.3) is either oscillatory or . It is easy to verify that is a solution of (4.3).
Remark 4.3. Recent results cannot be applied to (4.1) and (4.3) since and .
Remark 4.4. Using the method given in this paper, we can get other oscillation criteria for (1.1); the details are left to the reader.
Remark 4.5. It would be interesting to find another method to study (1.1) when .
Acknowledgments
The authors sincerely thank the referees 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 (11071143, 60904024, 11026112), China Postdoctoral Science Foundation funded project (200902564), by Shandong Provincial Natural Science Foundation (ZR2010AL002, ZR2009AL003, Y2008A28), and also by University of Jinan Research Funds for Doctors (XBS0843).