Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 968356, 5 pages

http://dx.doi.org/10.1155/2013/968356

Research Article

Oscillatory Criteria for Higher Order Functional Differential Equations with Damping

Peiguang Wang¹ and Hai Cai²

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

Academic Editor: Feliz Minhós

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 [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

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