Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 390282, 6 pages

http://dx.doi.org/10.1155/2013/390282

Research Article

Oscillation for a Class of Fractional Differential Equation

Zhenlai Han, Yige Zhao, Ying Sun, and Chao Zhang

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China

Received 3 May 2013; Accepted 21 June 2013

Academic Editor: Shurong Sun

Copyright © 2013 Zhenlai Han et al. 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 consider the oscillation for a class of fractional differential equation for where is a real number and is the Liouville right-sided fractional derivative of order of . By generalized Riccati transformation technique, oscillation criteria for a class of nonlinear fractional differential equation are obtained.

1. Introduction

Fractional differential equations have been of great interest recently. Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in self-similar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science. There have appeared lots of works in which fractional derivatives are used for a better description of considered material properties; mathematical modelling based on enhanced rheological models naturally leads to differential equations of fractional order and to the necessity of the formulation of initial conditions to such equations. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications; see [1–6].

It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of linear fractional differential equations. Recently, there are many papers dealing with the qualitative theory, especially the existence of solutions (or positive solutions) of nonlinear initial (or boundary) value problems for fractional differential equation (or system) by the use of techniques of nonlinear analysis (fixed-point theorems, Leray-Schauder theory, Adomian decomposition method, etc.); see [7–11].

The oscillation theory as a part of the qualitative theory of differential equations has been developed rapidly in the last decades, and there has been a great deal of work on the oscillatory behavior of integer order differential equations. However, there are only very few papers dealing with the oscillation of fractional differential equation; see [12–15].

Grace et al. [12] initiated the oscillatory theory of fractional differential equations

where denotes the Riemann-Liouville differential operator of order with and the functions ,  , and are continuous. By the expression of solution and some inequalities, oscillation criteria are obtained for a class of nonlinear fractional differential equations. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo’s differential operator.

Chen [13] considered the oscillation of the fractional differential equation

where is the Liouville right-sided fractional derivative of order of ,   is a quotient of odd positive integers, and are positive continuous functions on for a certain , and is a continuous function such that for a certain constant and for all . They established some oscillation criteria for the equation by using a generalized Riccati transformation technique and an inequality.

In 2013, Chen [15] studied oscillatory behavior of the fractional differential equation with the form

where is the Liouville right-sided fractional derivative of order of .

To the best of our knowledge, nothing is known regarding the oscillatory behavior for the following fractional differential equation:

where is a real number, is the Liouville right-sided fractional derivative of order of defined by for , here is the gamma function defined by for , and the following conditions are assumed to hold:

(H₁) and are two positive continuous functions on for a certain ;(H₂) are continuous functions with for , and there exists some positive constant such that for ;(H₃) are continuous function with for , and there exist positive constants such that for all .

By a solution of (4), we mean a nontrivial function with , and satisfies (4) for . Our attention is restricted to those solutions of (4) which exist on and satisfy for any . A solution of (4) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is nonoscillatory. Equation (4) is said to be oscillatory if all its solutions are oscillatory.

2. Preliminaries

For the convenience of the reader, we give some background materials from fractional calculus theory. These materials can be found in the recent literature; see [12, 13, 16, 17].

Definition 1 (see [16]). The Liouville right-sided fractional integral of order of a function on the half-axis is given by

provided that the right side is pointwise defined on , where is the gamma function.

Definition 2 (see [16]). The Liouville right-sided fractional derivative of order of a function on the half-axis is given by

where , provided that the right side is pointwise defined on .

The following lemma is fundamental in the proofs of our main results.

Lemma 3 (see [13]). Let be a solution of (4) and

Then

Lemma 4 (see [17]). If and are nonnegative, then

3. Main Results

Theorem 5. Suppose that (H₁)–(H₃) and

hold. Furthermore, assume that there exists a positive function such that

where are defined as in . Then every solution of (4) is oscillatory.

Proof. Suppose that is a nonoscillatory solution of (4). Without loss of generality, we may assume that is an eventually positive solution of (4). Then there exists such that

where is defined as in (7). Therefore, it follows from (4) that

Thus, is strictly increasing on and is eventually of one sign. Since for and (H₂), we see that is eventually of one sign. We now claim that

If not, then is eventually positive, and there exists such that . Since is strictly increasing on , it is clear that for . Therefore, from (8), we have

Then, we get

Integrating the above inequality from to , we have

Letting , we see

This contradicts (10). Hence, (14) holds.

Define the function by the generalized Riccati substitution

Then we have for . From (19), (4), (8), and (H₁)–(H₃), it follows that

Taking

from Lemma 4 and (20) we get

Integrating both sides of the inequality (22) from to , we obtain

Taking the limit supremum of both sides of the above inequality as , we get

which contradicts (11). The proof is complete.

Theorem 6. Suppose that (H₁)–(H₃) and (10) hold. Furthermore, suppose that there exist a positive function and a function , where , such that

where , and has a nonpositive continuous partial derivative on with respect to the second variable and satisfies

where , , and are defined as in Theorem 5. Then all solutions of (4) are oscillatory.

Proof. Suppose that is a nonoscillatory solution of (4). Without loss of generality, we may assume that is an eventually positive solution of (4). We proceed as in the proof of Theorem 5 to get (22), that is,

Multiplying the previous inequality by and integrating from to , for , we obtain

Therefore,

which is a contradiction to (26). The proof is complete.

Next, we consider the case

which yields that (10) does not hold. In this case, we have the following results.

Theorem 7. Suppose that (H₁)–(H₃) and (30) hold, is an increasing function, and that there exists a positive function such that (11) holds. Furthermore, assume that for every constant ,

Then every solution of (4) is oscillatory or satisfies

Proof. Assume that is a nonoscillatory solution of (4). Without loss of generality, assume that is an eventually positive solution of (4). Proceeding as in the proof of Theorem 5, there are two cases for the sign of . The proof when is eventually negative is similar to that of Theorem 5 and hence is omitted.

Next, assume that is eventually positive. Then there exists such that for . From (8), we get for . Thus, we get and . We now claim that . Assume not, that is, , then from (H₃) we get

Integrating both sides of the last inequality from to , we have

Hence, from (8), we get

Integrating both sides of the last inequality from to , we obtain

Letting , from (31), we get This contradicts . Therefore, we have , that is, . In view of (7), we see that the proof is complete.

Theorem 8. Suppose that (H₁)–(H₃) and (30) hold and is an increasing function. Let and be defined as in Theorem 6 such that (26) holds. Furthermore, assume that for every constant , (31) holds. Then every solution of (4) is oscillatory or satisfies .

Proof. Assume that is a nonoscillatory solution of (4). Without loss of generality, assume that is an eventually positive solution of (4). Proceeding as in the proof of Theorem 5, there are two cases for the sign of . The proof when is eventually negative is similar to that of Theorem 6 and hence is omitted. The proof when is eventually positive is similar to that of the proof of Theorem 7 and thus is omitted. The proof is complete.

4. Example

Example 1. Consider the fractional differential equation

In (36),  ,  ,  , and  . Take ,  . It is clear that conditions (H₁)–(H₃) and (10) hold. Furthermore, taking , we have

which shows that (11) holds. Therefore, by Theorem 5 every solution of (36) is oscillatory.

Example 2. Consider the fractional differential equation

In (38),  . Take . It is clear that conditions (H₁)–(H₃) and (30) hold. Taking , we have

which shows that (11) holds.

Furthermore, for every constant , we have

which shows that (31) holds. Therefore, by Theorem 7 every solution of (38) is oscillatory or satisfies

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This research is supported by the Natural Science Foundation of China (11071143), the Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119), the Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2011AL007), and the Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

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