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Mathematical Problems in Engineering
Volume 2013, Article ID 301085, 8 pages
http://dx.doi.org/10.1155/2013/301085
Research Article

Interval Oscillation Criteria for a Class of Fractional Differential Equations with Damping Term

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

Received 20 January 2013; Accepted 10 March 2013

Academic Editor: Sotiris Ntouyas

Copyright © 2013 Chunxia Qi and Junmo Cheng. 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

Some new interval oscillation criteria are established based on the certain Riccati transformation and inequality technique for a class of fractional differential equations with damping term. For illustrating the validity of the established results, we also present some applications for them.

1. Introduction

Fractional differential equations are generalizations of classical differential equations of integer order and can find their applications in many fields of science and engineering. In the last few decades, research on 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 [18] 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 only include Chen’s work [9], in which some new oscillation criteria are established for the following fractional differential equation: where are positive-valued functions and is the quotient of two odd positive numbers.

In this paper, we are concerned with oscillation of solutions of fractional differential equations of the following form: where denotes the Liouville right-sided fractional derivative of order of , and .

A 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.

The organization of the rest of this paper is as follows. In Section 2, we establish some new interval oscillation criteria for (2) by a generalized Riccati transformation and inequality technique and 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 [1012].

2. Main Results

For the sake of convenience, in the rest of this paper, we set , and  .

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

Lemma 2. Assume 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 on , where is sufficiently large. Otherwise, assume there exists a sufficiently large such that on . Then for , we have By (3), we have which implies for some sufficiently large , and  . By Lemma 1, we have By (4), we obtain , which contradicts on . So on . Thus is eventually of one sign. Now we assume for some sufficiently . Then by Lemma 1, for . Since , furthermore we have . We claim . Otherwise, assume . Then on , and, for , by (7) we have Substituting with in (11), an integration for (11) with respect to from to yields which means Substituting with in (13), an integration for (13) with respect to from to yields That is, Substituting with in (15), an integration for (15) with respect to from to yields By (5), one can see , 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 we have

Proof. By Lemma 2 we have   is strictly increasing on . So Using Lemma 1 we obtain that Then

Theorem 4. Assume (3)–(5) 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: Then for , we have By Lemma 3 and the definition of we get that Substituting with in (26), an integration for (26) with respect to from to yields which contradicts (23). So the proof is complete.

Theorem 5. Define . Assume (3)–(5) hold, and there exists a function such that and has a nonpositive continuous partial derivative , and for all sufficiently large , where are defined as in Theorem 4. 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 on for some sufficiently large . Let be defined as in Theorem 4. By (26) we have Substituting with in (30), multiplying both sides by , and then integrating it with respect to from to yield Then So which contradicts (29). So the proof is complete.

In Theorems 5, if we take for some special functions such as or , then we can obtain some corollaries as follows.

Corollary 6. Assume (3)–(5) hold, and for all sufficiently large . Then every solution of (2) is oscillatory or satisfies .

Corollary 7. Assume (3)–(5) hold, and for all sufficiently large . Then every solution of (2) is oscillatory or satisfies .

3. Applications

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

Example 8. Consider where is a quotient of two odd positive integers.

We have in (2) . Then Moreover, we have Furthermore, On the other hand, for a sufficiently large , we have So we can take such that for . Taking in (23), we get that provided that . So (3)–(5) and (23) all hold, and by Theorem 4 we deduce that every solution of (36) is oscillatory or satisfies under condition .

Example 9. Consider where , and is a constant.

We have in (2) . Then So we have Furthermore, On the other hand, Taking in (29), we get that So (3)–(5) and (29) all hold, and by Corollary 6 with we deduce that every solution of (42) is oscillatory or satisfies .

Acknowledgments

This work is partially supported by Planning Fund project of the Ministry of Education of China (10YJA630019). The authors would thank the reviewers very much for their valuable suggestions on this paper.

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