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Journal of Applied Mathematics
Volume 2013, Article ID 971357, 7 pages
http://dx.doi.org/10.1155/2013/971357
Research Article

Oscillation Criteria for Nonlinear Fractional Differential Equations

Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China

Received 5 October 2013; Accepted 3 December 2013

Academic Editor: Nazim Idrisoglu Mahmudov

Copyright © 2013 Run Xu. 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

Several oscillation criteria are established for nonlinear fractional differential equations of the form where is the Liouville right-side fractional derivative of order of and is a quotient of two odd positive integers. We also give some examples to illustrate the main results. To the best of our knowledge, the results are initiation for the oscillatory behavior of the equations.

1. Introduction

In this paper, we are dealing with the oscillation problem of nonlinear fractional differential equations of the form where is a constant, is a ratio of two odd positive integers. is the Liouville right-side fractional derivative of order of defined by

Throughout this paper, we will suppose that the following conditions hold:, , , ;, there exists a function such that for and , .

By a solution of (1), we mean a function such that , and satisfies (1) on . A nontrivial solution of (1) is called oscillatory if it has arbitrary large zero. Otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all of its solutions are oscillatory.

The theory of fractional derivatives goes back to Leibniz’s note in [1], and this led to the appearance of the theory of derivatives and integrals of arbitrary order. The theory had taken a more or less finished form due primarily to Liouville, Grünwald, Letnikov, and Riemann by the end of nineteenth century. We can see some of the books such as [2, 3] on the subject of fractional derivatives and fractional integrals.

Fractional differential equations are generalizations of classical differential equations of integer order. In the last few decades, many researchers found that fractional derivatives and fractional integrals were applied in widespread fields of science and engineering, especially in mathematical modeling and simulation of systems and processes, instead of simply being applied in pure theoretical fields of mathematics. Nowadays, many articles have investigated some aspects of fractional differential equations, such as the existence, the uniqueness and stability of solutions, the methods for explicit and numerical solutions, and the stability of solutions (we refer the reader to see [49] and the references cited in there). In very recent days, the research on oscillation of various fractional differential equations is being a hot topic; see [1016].

In [10], Grace et al. discussed the oscillation of the following question: where denotes the Riemann-Liouville differential operator of order with , and the functions , , and are continuous functions.

In [11], Chen et al. established several oscillation criteria for (3) with some additional initial conditions and , is an integer. They improved and extended some results of [10].

In [12], Chen considered the oscillation of the fractional differential equation with .

In [13], Han et al. have established some oscillation criteria for the equation

In [14], Qi and Cheng studied the following equation: with and established some new interval oscillation criteria by using a generalized Riccati transformation and inequality technique.

In [15], Feng and Meng paid attention to the oscillation of the fractional differential equation

In [16], Chen considered the oscillation of the fractional differential equation

The purpose of this paper is to establish some oscillation criteria for (1) by generalized Riccati function and present some applications for our results.

In order to prove our theories, we use the general weighted functions from the class . We say that a function belongs to the function class , if , where , which satisfies , for , and has nonpositive continuous partial derivative on .

2. Main Results

First, we set then it follows that

We give the following lemmas for our results.

Lemma 1 (see [17]). Let and be nonnegative; then

Lemma 2. Assume that is an eventually positive solution of (1), and then there exists a sufficiently large such that on , and one of the following two conditions hold:(i) on ,(ii) on and .

Proof. From the hypothesis, there exists a such that on , so that on , and we have Then is strictly increasing on , and we can conclude that is eventually of one sign. We claim that on , where is sufficiently large. Otherwise, if there exists a such that , then we can get on ; from those conditions we get that is, Integrating two sides of the previous inequality from to leads to Then from we have , which implies that for a certain constant , , , then by we obtain , which contradicts to on .
So we have on , and is eventually of one sign. There are two possibilities: (i) on , (ii) on , where is sufficiently large.
Assume that , for certain sufficiently large constant ; then , and we have . We claim that . Otherwise, let ; then on , and by (13), we have
Integrating two sides of above inequality from to leads to which means
Integrating two sides of (20) from to yields
That is, Integrating two sides of (22) from to , we have By (12), we have , which contradicts to the fact that . Then we get that , which is . The proof is complete.

Lemma 3. Assume that is an eventually positive solution of (1) such that , on , where is sufficiently large and . Then one has where .

Proof. From (13), we get that is strictly increasing on , so we get Then we get

Theorem 4. Assume that , , and (12) hold; if there exists a function such that for all sufficiently large constants and , where , is defined in Lemma 3, , then every solution of (1) is oscillatory or satisfies .

Proof. Suppose to the contrary that (1) has a nonoscillatory solution on ; without loss of generality, we assume that on , where is sufficiently large. By Lemma 2 we have on , where is sufficiently large and , and either on or . (In the case is eventually negative, it can be proved similarly.)
If on , then the conclusion of Lemma 3 holds. We define then on , and from (1), , and Lemma 3, we have
Let , , ; apply Lemma 1 to (29), we get Integrating two sides of (30) from to , we have
which contradicts to (27).
If on , then from Lemma 2, we get that . This completes the proof.

Corollary 5. Assume that , , and (12) hold; if there exists a function such that for a sufficiently large constant , where , is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies .

Theorem 6. Assume that , , and (12) hold; if there exist functions and such that where is sufficiently large, is defined in Lemma 3, , then every solution of (1) is oscillatory or satisfies .

Proof. Suppose that is a nonoscillatory solution of (1); without loss of generality, we let be an eventually positive solution of (1). According to the proof of Theorem 4, if on , then (29) holds. Multiplying two sides of (29) by and integrating it from to , we get
Taking applying Lemma 1 to (34), we obtain that for , Then we have From above inequality we have Letting we have which contradicts to (33). If on , from Lemma 2, we have . So the proof is complete.

Corollary 7. Assume that , , and (12) hold; if there exist functions and such that where is sufficiently large, , is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies .

With an appropriate choice of the functions and , one can derive from Theorem 6 a number of oscillation criteria for (1). For example, let , , and ; then , . Based on the above results we obtain the following corollary.

Corollary 8. Assume that , , and (12) hold; if where is sufficiently large, is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies .

Corollary 9. Assume that , , and (12) hold; if where is sufficiently large, , is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies .

3. Examples

Example 1. Consider the fractional differential equation where and is a quotient of two odd positive integers.

Here, , , for and , , .

We set ; if is sufficiently large, then , and . Consider so , and (12) hold.

Furthermore, for a sufficiently large constant and , we have

So the conditions of Corollary 5 hold, and we deduce that every solution of (43) is oscillatory or satisfies .

Example 2. Consider the fractional differential equation where and is a quotient of two odd positive integers.

Here, , , for and , , .

We set , ; then , and .

May as well we assume that ; then we have . By calculation we have so , and (12) hold.

Furthermore, and for , the conditions of Corollary 9 hold, and we get that every solution of (46) is oscillatory or satisfies .

Acknowledgment

This research is supported by National Science Foundation of China (11171178 and 11271225). The authors sincerely thank the reviewers for their valuable suggestions and useful comments.

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