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The Scientific World Journal
Volume 2013 (2013), Article ID 685621, 9 pages
http://dx.doi.org/10.1155/2013/685621
Research Article

Oscillation of a Class of Fractional Differential Equations with Damping Term

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

Received 19 May 2013; Accepted 11 July 2013

Academic Editors: J. Banaś and M. M. Cavalcanti

Copyright © 2013 Huizeng Qin and Bin Zheng. 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 the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into another differential equations of integer order with respect to the new variable. Then, using Riccati transformation, inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.

1. Introduction

In the investigations of qualitative properties for differential equations, research of oscillation has gained much attention by many authors in the last few decades (e.g., see [116]). In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations.

In [17], Jumarie proposed a definition for fractional derivative which is known as the modified Riemann-Liouville derivative in the literature. Since then, many authors have investigated various applications of the modified Riemann-Liouville derivative (e.g., see [1821]) including various fractional calculus formulae, the fractional variational iteration method, the Bäcklund transformation method, and the fractional subequation method for soling fractional partial differential equations. In this paper, based on the modified Riemann-Liouville derivative, we are concerned with oscillation of a class of fractional differential equations with damping term as follows: where denotes the modified Riemann-Liouville derivative with respect to the variable , the function , , and denotes continuous derivative of order .

The definition and some important properties for the modified Riemann-Liouville derivative of order are listed as follows (see also in [2024]):

As usual, a solution of (1) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.

We organize the next as follows. In Section 2, using Riccati transformation, inequality, and integration average technique, we establish some new oscillatory criteria for (1), while we present some examples for them in Section 3.

2. Oscillatory Criteria for (1)

In the following, we denote , , , , , , , , , , , , . Let satisfy has continuous partial derivatives and on such that

Lemma 1. Assume is an eventually positive solution of (1), and Then, there exists a sufficiently large such that on and either on or .

Proof. Let , where . Then, by use of (3), we obtain , and furthermore, by use of the first equality in (5), we have Similarly, we have , . So, (1) can be transformed into the following form: Since is an eventually positive solution of (1), then is an eventually positive solution of (12), and there exists such that on . Furthermore, we have Then, is strictly decreasing on , and thus is eventually of one sign. We claim on , where is sufficiently large. Otherwise, assume that there exists a sufficiently large such that on . Then, is strictly decreasing on , and we have By (8), we have . So there exists a sufficiently large with such that , . Furthermore, By (9), we deduce that , which contradicts the fact that is an eventually positive solution of (9). So, on , and on . Thus, is eventually of one sign. Now we assume , for some sufficiently large . Since , furthermore we have . We claim . Otherwise, assume . Then on , and, for , by (12) we have Substituting with in the previous inequality, an integration with respect to from to yields which means Substituting with in (18), an integration for (18) with respect to from to yields that is, Substituting with in (20), an integration for (20) with respect to from to yields By (10), one can see , which causes a contradiction. So, the proof is complete.

Lemma 2. Assume that is an eventually positive solution of (1) such that on , where is sufficiently large. Then, for , we have

Proof. By (13), we obtain that is strictly decreasing on . So, that is which admits (23). On the other hand, we have which can be rewritten as (24). So the proof is complete.

Lemma 3 (see [25, Theorem 41]). Assume that and are nonnegative real numbers. Then, for all .

Theorem 4. Assume that (8)–(10) hold. If there exists such that for any sufficiently large , there exist , , with satisfying where , ,  ; then, (1) is oscillatory or satisfies .

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that on , where is sufficiently large. By Lemma 1, we have , where is sufficiently large, and either on or . Now we assume on . Define the generalized Riccati function: Then, for , we have Using and (23), we obtain Let . Then , and . So (32) is transformed into the following form:
Choose , , arbitrarily in with . Substituting with , multiplying both sides of (33) by , and integrating it with respect to from to for , we get that Dividing both sides of the inequality (34) by and letting , we obtain
On the other hand, substituting with , multiplying both sides of (33) by , and integrating it with respect to from to for , we get that Dividing both sides of the inequality (36) by and letting , we obtain A combination of (35) and (37) yields which contradicts (29). So, the proof is complete.

Theorem 5. Under the conditions of Theorem 4, if for any sufficiently large , then (1) is oscillatory.

Proof. For any , let . In (39), we choose . Then, there exists such that In (40), we choose . Then there exists such that Combining (41) and (42), we obtain (29). The conclusion thus comes from Theorem 4, and the proof is complete.

In Theorems 4 and 5, if we choose , , where is a constant, then we obtain the following two corollaries.

Corollary 6. Under the conditions of Theorem 4, if for any sufficiently large , there exist , , with satisfying then (1) is oscillatory.

Corollary 7. Under the conditions of Theorem 5, if for any sufficiently large , then (1) is oscillatory.

Theorem 8. Assume (8)–(10) hold, and where is defined as in Theorem 4. Then every solution of (1) is oscillatory or satisfies .

Proof. Assume (1) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemma 1, we have , , where is sufficiently large, and either on or . Now we assume that on . Let , be defined as in Theorem 4. Then we obtain (33), and furthermore,
Substituting with in (46) and integrating (46) with respect to from to yield which contradicts (45). So, the proof is complete.

Theorem 9. Assume (8)–(10) hold, and there exists a function such that , for , , for , and has a nonpositive continuous partial derivative . If where is defined as in Theorem 4, then every solution of (1) is oscillatory or satisfies .

Proof. Assume (1) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemma 1, we have , , where is sufficiently large, and either on or . Now we assume on . Let , be defined as in Theorem 4. By (46), we have Substituting with in (49), multiplying both sides by , and then integrating both sides of (49) with respect to from to yield Then, So, which contradicts (48). So the proof is complete.

3. Applications of the Results

Example 10. Consider the following fractional differential equation:
In (1), if we set , , , , , , then we obtain (53). So , , , , . Furthermore, , which implies . On the other hand, , which implies , and then (8) holds. So, there exists a sufficiently large such that on . In (9), In (10), In (48), letting , we obtain Therefore, (53) is oscillatory by Theorem 8.

Example 11. Consider the following fractional differential equation:
In (1), if we set , , , , , , then we obtain (57). So , , , , . Furthermore, , which implies . On the other hand, , which implies . So, there exists a sufficiently large such that on .
From the analysis above, one can see the (8) holds. We now test (9) and (10). In (9), In (10), So, (9) and (10) hold. On the other hand, in (44), after putting , , for any sufficiently large , we have So (44) holds, and then by Corollary 7 we deduce that (57) is oscillatory.

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