Journal of Applied Mathematics

Volume 2013 (2013), Article ID 912072, 11 pages

http://dx.doi.org/10.1155/2013/912072

Research Article

Some New Oscillation Criteria for a Class of Nonlinear Fractional Differential Equations with Damping Term

Bin Zheng and Qinghua Feng

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 19 September 2013; Accepted 11 November 2013

Academic Editor: Kai Diethelm

Copyright © 2013 Bin Zheng and Qinghua Feng. 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 are concerned with oscillation of solutions of a class of nonlinear fractional differential equations with damping term. Based on a generalized Riccati function and inequality technique, we establish some new oscillation criteria for it. Some applications are also presented for the established results.

1. Introduction

Fractional differential equations are generalizations of classical differential equations of integer order, and one can find their applications in many fields of science and engineering. In the last few decades, research for various aspects of fractional differential equations, for example, the existence, uniqueness, and stability of solutions of fractional differential equations, the numerical methods for fractional differential equations, and so on, has been paid much attention by many authors (e.g., we refer the reader to see [1–8] and the references therein). In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations. Recent results in this direction include Chen's work [9], in which some new oscillation criteria are established for the following fractional differential equation:

where and are positive functions and is a quotient of two odd positive numbers.

In this paper, we are concerned with oscillation of solutions of the following nonlinear (2+)-order fractional differential equation with damping term:

where , , , satisfying , for , is a quotient of two odd positive integers, , denotes the Liouville right-sided fractional derivative of order of , and .

A nontrivial solution of (2) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (2) is said to be oscillatory in case all its solutions are oscillatory.

Motivated by the idea in [10], we will establish some new oscillation criteria for (2) by a generalized Riccati function and inequality technique in Section 2, and we will present some applications for our results in Section 3. Throughout this paper, denotes the set of real numbers and . For more details about the theory of fractional differential equations, we refer the reader to [11–13]. For the sake of convenience, in the rest of this paper, we set , ,

2. Main Results

The following lemmas are useful for proving our results.

Lemma 1. Assume that is a solution of (2). Then, .

Lemma 2. Assume that is an eventually positive solution of (2) and

Then, there exists a sufficiently large such that

and either on or .

Proof. Since is an eventually positive solution of (2), there exists such that on . So on and we have

Then, is strictly increasing on , and thus is eventually of one sign. We claim that on , where is sufficiently large. Otherwise, assume there exists a sufficiently large such that on . Then, for , we have

By (4), we have

which implies, for some sufficiently large , , . By Lemma 1, we have

By (5), we obtain , which contradicts on . So on . Thus, is eventually of one sign. Now, we assume that , , for some sufficiently . Then, by Lemma 1, for . Since , furthermore we have . We claim that . Otherwise, assume that . Then, on , and, for , by (8) we have

Substituting with in (12), an integration for (12) with respect to from to yields

which means

Substituting with in (14), an integration for (14) with respect to from to yields

that is,

Substituting with in (16), an integration for (16) with respect to from to yields

By (6), one can see that , which is a contradiction. So the proof is complete.

Lemma 3. Assume that is an eventually positive solution of (2) such that

where is sufficiently large. Then, one has

Proof. By Lemma 2, we obtain that is strictly increasing on . So

Using Lemma 1, we obtain that

Then,

Lemma 4 ([14, Theorem 41]). Assume that and are nonnegative real numbers. Then,

Theorem 5. Assume (4)–(6) hold and there exist two functions and such that

for all sufficiently large . Then, every solution of (2) is oscillatory or satisfies .

Proof. Assume (2) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemma 2, we have , , where is sufficiently large, and either on or . Define the generalized Riccati function as follows:

Then, for , we have

By Lemma 3 and the definition of , we get that

Using the following inequality (see [15, Equation (2.17)])

we obtain

A combination of (27) and (29) yields the following:

Setting

Using Lemma 4 in (30), we get that

Substituting with in (32), an integration for (32) with respect to from to yields

which contradicts (24). So the proof is complete.

Theorem 6. Assume (4)–(6) hold and, for all sufficiently large ,

where and are defined as in Theorem 5. Then, every solution of (2) is oscillatory or satisfies .

Proof. Assume (2) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemma 2, we have , where is sufficiently large, and either on or . Let be defined as in Theorem 5. Proceeding as in Theorem 5, we obtain (26). By Lemma 3, we have the following observation:

Using (35) in (26) we get that

Substituting with in (36), an integration for (36) with respect to from to yields

which contradicts (34). So the proof is complete.

Theorem 7. Define . Assume (4)–(6) hold and there exists a function such that

has a nonpositive continuous partial derivative , and

for all sufficiently large , where and are defined as in Theorem 5. Then, every solution of (2) is oscillatory or satisfies .

Proof. Assume (2) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemma 2, we have either on , for some sufficiently large , or . Now we assume . Let be defined as in Theorem 5. By (32), we have

Substituting with in (40), multiplying both sides by , and then integrating with respect to from to yield

Then

So

which contradicts (39). So the proof is complete.

Theorem 8. Let , , and be defined as in Theorem 7. If (4)–(6) hold and

for all sufficiently large , then every solution of (2) is oscillatory or satisfies .

Proof. Assume (2) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemma 2, we have either on , for some sufficiently large , or . Now, we assume . Let be defined as in Theorem 5. By (36), we have

Substituting with in (45), multiplying both sides by , and then integrating with respect to from to yield

Then, similar to the process of Theorem 7, we get that

which contradicts (44). So the proof is complete.

Remark 9. In Theorems 7 and 8, if we take for some special functions such as or , then we can obtain some corollaries, which are omitted here.

Remark 10. The established oscillation criteria for (2) above are new results so far in the literature to the best of our knowledge.

3. Applications

In this section, we will present some applications for the established results above.

Example 1. Consider the following:

where is a constant.

We have in (2) , , , , , . Then, . Moreover, . Then, we have

Furthermore,

On the other hand, for a sufficiently large , we have

So we can take such that for . Taking and in (24), we get that

provided that . So (4)–(6) and (24) all hold, and by Theorem 5 we deduce that every solution of (48) is oscillatory or satisfies under the condition .

Example 2. Consider the following:

where and is a constant.

We have in (2) , , , , , . Then, . Moreover, . Then, we have

Furthermore,

On the other hand, for a sufficiently large , we have

So we can take such that for . Taking , , in (39), we get that

provided that . So (4)–(6) and (39) all hold, and by Theorem 7 we deduce that every solution of (48) is oscillatory or satisfies under the condition .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the Natural Science Foundation of Shandong Province (in China) (Grant no. ZR2013AQ009) and the Doctoral Initializing Foundation of Shandong University of Technology (in China) (Grant no. 4041-413030).

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