Research Article | Open Access

Peiguang Wang, Hai Cai, "Oscillatory Criteria for Higher Order Functional Differential Equations with Damping", *Journal of Function Spaces*, vol. 2013, Article ID 968356, 5 pages, 2013. https://doi.org/10.1155/2013/968356

# Oscillatory Criteria for Higher Order Functional Differential Equations with Damping

**Academic Editor:**Feliz Minhós

#### Abstract

We investigate a class of higher order functional differential equations with damping. By using a generalized Riccati transformation and integral averaging technique, some oscillation criteria for the differential equations are established.

#### 1. Introduction

In this paper, we consider the following higher order functional differential equations with distributed deviating arguments of the form as follows: where is an even number, , , , for , and has the same sign as ; when they have the same sign, is nondecreasing, and the integral of (1) is a Stieltjes one.

We restrict our attention to those solutions of (1) which exist on same half liner with for any and satisfy (1). As usual, a solution of (1) is called oscillatory if the set of its zeros is unbounded from above, otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all solutions are oscillatory.

In recent years, there has been an increasing interest in studying the oscillation behavior of solutions for the differential equations with distributed deviating arguments, and a number of results have been obtained (refer to [1–3] and their references). However, to the best of our knowledge, very little is known for the case of higher order differential equations with damping. The purpose of this paper is to establish some new oscillation criteria for (1) by introducing a class of functions defined in [2] and a generalized Riccati technique.

Firstly, we define the following two class functions.

We say that a function belongs to the function class , denoted by , if , where , which satisfies , , and has the partial derivative on that is locally integrable with respect to in .

Let , . We say that a function belongs to the function class , denoted by , if for , in , has continuous partial derivative in with respect to and .

In order to prove the main theorems, we need the following lemmas.

Lemma 1 (see [4]). *Let , if is of constant sign and not identically zero on any ray for , then there exists a , an integer , with even for or odd for ; and for , , , and , .*

Lemma 2 (see [5]). *If the function is as in Lemma 1 and for , then there exists a constant such that for sufficiently large , there exists a constant , satisfying
*

Lemma 3 (see [3]). *Suppose that is a nonoscillatory solution of (1). If
**
then for any large .*

#### 2. Main Results

Theorem 4. *Assume that (3) holds, and*?*there exists a function such that , . , , , where , , and are constants.*?* is nondecreasing with , , and there exist constants and such that
**
If there exists a function , such that for any , and ,
**
where
**
Then (1) is oscillatory.*

*Proof. *Suppose to the contrary that (1) has a nonoscillatory solution . Without loss of generality, we may suppose that is an eventually positive solution. From the conditions of and , there exists a , such that

By Lemma 3, there exists a such that , . Thus, we have

By Lemma 1, there exists a such that , . Further, by Lemma 2, there exist constant and a , such that

Set
then

In view of , and the definition of , , we have
where .

Multiplying (12) by and integrating from to , we have

Integrating by parts and using integral averaging technique, we have
thus
which contradicts (5). This completes the proof of Theorem 4.

If we choose , where . By Theorem 4, we have the following results.

Corollary 5. *Assume that (3), , and hold. If there exist such that for each ,
**
where and are defined by , , and
**
Then (1) is oscillatory.*

If we choose , , and let , by Theorem 4, we have the following corollary.

Corollary 6. *Assume that (3), , and hold. If there exists a constant such that for each ,
**
where is defined as in Corollary 5. Then (1) is oscillatory.*

Theorem 7. *Assume that (3) holds, and*?*there exist functions , such that , , , where are constants, ;*?*there exist constants and , such that
**, , where and are constants, .*

If there exists a function , such that for any , , and , and (5) holds, where is defined as in Theorem 4:

Then (1) is oscillatory.

* Proof. *Suppose to the contrary that (1) has a nonoscillatory solution . Without loss of generality, we may suppose that is an eventually positive solution. Similar to the proof of Theorem 4, there exists a , such that , , , , , and , for , . Set
then

In view of , and the definition of , , we have

The following proof is similar to Theorem 4, and we omit the details. This completes the proof of Theorem 7.

Similar to Corollaries 5 and 6, we have the following corollaries.

Corollary 8. *Assume that (3), , and hold. If there exist such that for each , and (16) holds, where are defined as in Corollary 5:
**
Then (1) is oscillatory.*

Corollary 9. *Assume that (3), , and hold. If there exists a constant such that for each , and (18) holds. where is defined as in Corollary 8, then (1) is oscillatory.*

For the case of the function with monotonicity, we have the following theorem.

Theorem 10. *Assume that (3), hold, and*?*there exist and , such that , where is constants. , , in which and are constants, .**
If there exists a function , such that for any , , and (5) holds, where is defined as in Theorem 4:
**
Then (1) is oscillatory.*

*Proof. *Suppose to the contrary that (1) has a nonoscillatory solution . Without loss of generality, we may suppose that is an eventually positive solution. Similar to the proof of Theorem 4, there exists a , when , and we have , , , , , and , . Set
then

In view of , and the definition of , , we have

The following proof is similar to Theorem 4, we omit the details. This completes the proof of Theorem 10.

Similar to Corollaries 5 and 6, we have the following corollaries.

Corollary 11. *Assume that (3), , and hold. If there exist such that for each , and (16) holds, where are defined as in Corollary 5:
**
Then (1) is oscillatory.*

Corollary 12. *Assume that (3), , and hold. If there exists a constant such that for each , and (18) holds. where is defined as in Corollary 11. Then (1) is oscillatory.*

#### 3. Examples

*Example 13. *Consider the following equation
where , , obviously . Choosing , , then , and

Thus, there exists a constant , such that , that is,

By Corollary 6, then (30) is oscillatory.

*Example 14. *Consider the following equation
where , obviously , . Choosing , , , and , then . By Corollary 12, then (33) is oscillatory.

#### Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. The research was supported by the Natural Science Foundation of China (11271106).

#### References

- P. Wang and M. Wang, “Oscillation of a class of higher order neutral differential equations,”
*Archivum Mathematicum*, vol. 40, no. 2, pp. 201–208, 2004. View at: Google Scholar | Zentralblatt MATH - Y. G. Sun, “Oscillation of second order functional differential equations with damping,”
*Applied Mathematics and Computation*, vol. 178, no. 2, pp. 519–526, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. G. Wang and X. Liu, “Interval oscillation for some kinds of even-order nonlinear differential equations,”
*Chinese Journal of Engineering Mathematics*, vol. 22, no. 6, pp. 1031–1038, 2005. View at: Google Scholar - I. T. Kiguradze, “On the oscillatory character of solutions of the equation ${d}^{m}u/d{t}^{m}+a(t){|u|}^{n}sgnu=0[J]$,”
*Matematicheskil Sbornik*, vol. 65, pp. 172–187, 1964 (Russian). View at: Google Scholar - C. Philos, “A new criterion for the oscillatory and asymptotic behavior of delay differential equations,”
*L'Académie Polonaise des Sciences. Bulletin. Série des Sciences Mathématiques*, vol. 29, no. 7-8, pp. 367–370, 1981. View at: Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2013 Peiguang Wang and Hai Cai. 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.