Abstract and Applied Analysis

Abstract and Applied Analysis / 2013 / Article
Special Issue

Functional Differential and Difference Equations with Applications 2013

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Research Article | Open Access

Volume 2013 |Article ID 393892 | 10 pages | https://doi.org/10.1155/2013/393892

Oscillation Theorems for Even Order Damped Equations with Distributed Deviating Arguments

Academic Editor: Miroslava Růžičková
Received24 Aug 2013
Accepted08 Nov 2013
Published04 Dec 2013

Abstract

A class of even order damped differential equations with distributed deviating arguments are investigated. Several new criteria that ensure the oscillation of solutions are obtained. To demonstrate the validity of the results obtained, two examples are given.

1. Introduction and Lemmas

Oscillatory behavior of solutions for different types of second-order differential equations with damping has been widely discussed by using different techniques. Here, we particularly refer the reader to the papers [19] and the references quoted therein. However, very little is known for the case of higher order damped functional differential equations with deviating arguments, especially the case with distributed deviating arguments. In this paper, we deal with the following class of even order functional differential equations with damping: Our aim is to get the criteria for the oscillatory solutions of (1).

Throughout this paper, we assume that the following conditions hold: is an even positive integer; , is not identically zero on any for , and has the same sign as when have the same sign, , is nondecreasing, and the integral of (1) is a Stieltjes one.

In the sequel, it will be always assumed that solutions of (1) exist for any . A solution of (1) is called eventually positive solution (or negative solution) if there exists a sufficiently large positive number , such that (or ) for all . A nontrivial solution of (1) is called oscillatory if it has arbitrary large zeros; otherwise it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.

Remark 1. Since the integral of (1) is a Stieltjes one, it includes the following equations:

The following lemmas will be useful to the proof of the main results to be presented in this paper.

Lemma 2 (see [10]). Let be a positive and times differentiable function on . If is of constant sign and not identically zero on any ray for , then there exists a and an integer , with even for or odd for ; and for ,

Lemma 3 (see [11]). Suppose that the conditions of Lemma 2 are satisfied, and then there exists a constant such that for sufficiently large , there exists a constant satisfying

We say that a function belongs to a function class , denoted by , if , where , satisfies(i) , for and , for ;(ii)partial derivatives and exist, and where .

2. Oscillation Results for with Monotonicity

Throughout this section, we assume that the following conditions hold. There exist functions , such that , , , and . exists, and for , where are some constants, and .

Lemma 4. Let be an eventually positive solution of (1). Then, there exists a sufficiently large , such that for all

Proof. From the assumption, there exists a sufficiently large , such that for . Further from , there exists such that for all Hence, for all and from , we have for all and Let then it is easy to know that which implies that is nonincreasing on .
Now, we claim that , . Otherwise, there exists such that . Therefore, Using , we see that . Ulteriorly, we can prove , which contradicts , .
Furthermore, from (1), for all , we have Thus, from Lemma 2, there exist and an odd number , such that for , we have By choosing and , we have and for . The proof is completed.

Lemma 5. Let be an eventually positive solution of (1). Then, there exists a sufficiently large , such that for any interval , if let where , then for any ,

Proof. From (1) and (16), we have that for , From Lemma 4, there exists a sufficiently large such that and for . Further from , for all Hence, for all , we have In view of (20) and , for all Thus, for all Therefore, from (18)–(22) and Lemma 3, we obtain for all .
Multiplying (23) by , then integrating it with respect to from to for and using and , we get that Letting in the above, we obtain (17). The proof is completed.

Lemma 6. Let be an eventually positive solution of (1). Then, there exists a sufficiently large such that for any interval , if let be defined by (16) on , then for any ,

Proof. Similar to the proof of Lemma 5, by multiplying (23) by , then integrating it with respect to from to for , and then using and , we get that Letting in the above, we obtain (25). The proof is completed.

The following theorem is an immediate result from Lemmas 5 and 6.

Theorem 7. Assume that for each there exist , and , such that and Then (1) is oscillatory.

Proof. Suppose that (1) has a nonoscillatory solution . Without loss of generality, we assume that is an eventually positive solution of (1). Then from Lemmas 5 and 6, there exists a sufficiently large , such that for any , and for any , and , (17) and (25) hold. By dividing (17) and (25) by and , respectively, and then adding them, we have which contradicts the assumption (27) and completes the proof.

Theorem 8. Assume that for some , and for each , Then (1) is oscillatory.

Proof. For any , let . In (29), we choose . Then there exists such that In (30), we choose , then there exists such that By dividing (31) and (32) by and , respectively, and then adding them, we obtain (27). The conclusion thus comes from Theorem 7. The proof is completed.

For the case of , we have that and thus denote them by . The subclass of containing such is denoted by . Applying Theorem 7 to , and choosing , we obtain the following.

Theorem 9. Assume that for each there exist and such that and Then (1) is oscillatory.

Proof. Let . Then , and for any , we have Hence Thus (33) holds and implies that (27) holds for , and therefore (1) is oscillatory by Theorem 7. The proof is completed.

From the above oscillation criteria, we can obtain different sufficient conditions for oscillation of (1) by different choices of and . For example, let where is a constant. Then, and . From Theorem 8, we have the following result.

Corollary 10. If there exists a function and a constant such that for each , Then (1) is oscillatory.

3. Oscillation Results for without Monotonicity

Throughout this section we assume that the following conditions hold: there exists a function such that , , . there exists a constant and such that for sufficiently large   

Lemma 11. Let be an eventually positive solution of (1). Then, there exists a sufficiently large such that for , we have

The proof is similar to that of Lemma 4, thus we omit the details here.

Lemma 12. Let be an eventually positive solution of (1). Then, there exists a sufficiently large such that for any interval , if let where , then for any ,

Proof. From (1) and (40) we have that for From Lemma 11, there exists a sufficiently large such that for all (39) hold and further from Hence, we have for all , From (44) and , for all Thus, for all Therefore, from (42)–(46) and Lemma 3, we obtain The rest of the proof is similar to that of Lemma 5 and thus we omit the details here.

Similar to the proof in Section 2, we have the following results.

Lemma 13. Let be an eventually positive solution of (1). Then, there exists a sufficiently large such that, for any interval , if let be defined by (40) on , then for any ,

The following theorem is an immediate result from Lemmas 12 and 13.

Theorem 14. Assume that for each there exist , and , such that and Then (1) is oscillatory.

Theorem 15. Assume that for some and , and for each , Then (1) is oscillatory.

Theorem 16. Assume that for each , there exist and such that and Then (1) is oscillatory.

Corollary 17. If there exists a function and a constant such that for each , the following two inequalities hold Then (1) is oscillatory.

4. Examples

In this section we demonstrate the applications of our oscillation criteria through two examples. We will see that the equations in the examples are oscillatory based on the results in Sections 2 and 3.

Example 1. Consider the following nonlinear damped differential equation: where , , , , , , . It is clear that for Applying Corollary 10 with and , we have through a straightforward computation that Therefore (37) hold and we conclude by Corollary 10 that (55) is oscillatory.

Example 2. Consider the following nonlinear damped differential equation: where ,   , . In this example, Clearly, Corollary 10 does not apply to (58). However, with and , we can prove the oscillatory character of (58) by Corollary 17. Noting that