`Journal of Function Spaces and ApplicationsVolume 2013 (2013), Article ID 968356, 5 pageshttp://dx.doi.org/10.1155/2013/968356`
Research Article

Oscillatory Criteria for Higher Order Functional Differential Equations with Damping

1College of Electronic and Information Engineering, Hebei University, Baoding 071002, China
2College of Mathematics and Computer Science, Hebei University, Baoding 071002, China

Received 19 September 2012; Accepted 25 November 2012

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.

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 [13] 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, andthere 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, andthere 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, andthere 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

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