Forced oscillation of certain fractional differential equations
© Chen et al.; licensee Springer 2013
Received: 2 December 2012
Accepted: 19 April 2013
Published: 2 May 2013
The paper deals with the forced oscillation of the fractional differential equation
with the initial conditions () and , where is the Riemann-Liouville fractional derivative of order q of x, , is an integer, is the Riemann-Liouville fractional integral of order of x, and () are/is constants/constant. We obtain some oscillation theorems for the equation by reducing the fractional differential equation to the equivalent Volterra fractional integral equation and by applying Young’s inequality. We also establish some new oscillation criteria for the equation when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. The results obtained here improve and extend some existing results. An example is given to illustrate our theoretical results.
where is the Riemann-Liouville fractional derivative of order q of x, , is an integer, is the Riemann-Liouville fractional integral of order of x, () are/is constants/constant, () are continuous functions, and is a continuous function.
By a solution of (1.1), we mean a nontrivial function which has the property and satisfies (1.1) for . Our attention is restricted to those solutions of (1.1) which exist on and satisfy for any . A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc.; see, for example, [1–6]. There has been a significant development in ordinary and partial differential equations involving both Riemann-Liouville and Caputo fractional derivatives in recent years. The books on the subject of fractional integrals and fractional derivatives by Diethelm , Miller and Ross , Podlubny  and Kilbas et al.  summarize and organize much of fractional calculus and many of theories and applications of fractional differential equations. Many papers have studied some aspects of fractional differential equations such as the existence and uniqueness of solutions to Cauchy type problems, the methods for explicit and numerical solutions, and the stability of solutions, and we refer to [11–18] and the references quoted therein.
For details of the Liouville fractional integrals and fractional derivatives, one can refer to [, Sections 2.2 and 2.3].
when , and , respectively. The results are also stated when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator.
where and are constants. We also get some new oscillatory properties of (1.1) when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. Our results improve and extend some of those in .
2 Preliminaries and a lemma
In this section, we recall several definitions of fractional integrals and fractional derivatives and the well-known Young’s inequality, which will be used throughout this paper. More details on fractional calculus can be found in [7–10].
Definition 2.1 
provided the right-hand side is pointwise defined on , where Γ is the gamma function. Furthermore, we set .
Definition 2.2 
provided the right-hand side is pointwise defined on , where and is an integer. Furthermore, we set .
Definition 2.3 
provided the right-hand side is pointwise defined on , where , is an integer and denotes the usual derivative of integer order m of x. Furthermore, we set .
Let , and , then , where the equality holds if and only if .
Let , , and , then , where the equality holds if and only if .
3 Main results
for every sufficiently large T, where , then every solution of (1.1) is oscillatory.
Take . Next, we consider the cases and , respectively.
It follows from (3.7)-(3.9) that for . Therefore, we find , which contradicts (3.1).
From (3.7), (3.10) and (3.11), we conclude for . Hence, we obtain , which contradicts (3.1).
Finally, we assume that x is an eventually negative solution of (1.1). Then a similar argument leads to a contradiction with (3.2). The proof is complete. □
Remark 3.1 In , the plus sign ‘+’ in (2.9) in Theorem 2.2, (2.13) in Theorem 2.3, (2.17) in Theorem 2.4, (3.6) in Theorem 3.2, (3.8) in Theorem 3.3 and (3.10) in Theorem 3.4 should be the minus sign ‘−’.
Remark 3.2 Theorems 2.2 and 2.3 in  are the special cases of our Theorem 3.1 with and , respectively. Our Theorem 3.1 improves and extends the results of Theorems 2.2-2.4 in  since these theorems did not include the cases and for (1.1).
The following example shows that the condition (3.1) cannot be dropped.
which implies that satisfies the first equality in (3.12). From (3.13) we get , which yields that satisfies the second equality in (3.12). Hence, is a nonoscillatory solution of (3.12).
Next, we consider the case when (1.5) holds, which was not considered in .
for every sufficiently large T, where H is defined as in Theorem 3.1, then every bounded solution of (1.1) is oscillatory.
Take . Next, we consider the cases and , respectively.
Case (i). Let . Then (3.8) and (3.9) are still true. From (3.16), (3.8), (3.9) and (3.18), we conclude for . Thus, we see , which contradicts (3.14).
Case (ii). Let . Then (3.10) and (3.11) are still valid. From (3.16), (3.10), (3.11) and (3.18), we conclude for . Since , we obtain , which contradicts (3.14).
Finally, we suppose that x is a bounded eventually negative solution of (1.1). Then a similar argument leads to a contradiction with (3.15). The proof is complete. □
4 Results with the Caputo fractional derivative
The Riemann-Liouville fractional derivatives played an important role in the development of the theory of fractional derivatives and integrals and for their applications in pure mathematics. But it turns out that the Riemann-Liouville derivatives have certain disadvantages when trying to model real-world phenomena with fractional differential equations. When comparing the Riemann-Liouville definition and the Caputo definition of fractional derivatives, we will see this second one seems to be better suited to such tasks. The main advantages of the Caputo fractional derivatives is that the initial conditions for fractional differential equations with Caputo fractional derivatives take on the same form as for integer-order differential equations, i.e., they contain the limit values of integer-order derivatives of unknown functions at the lower terminal .
Similar to the proof of Theorems 3.1 and 3.2, we can prove the following theorems.
for every sufficiently large T, where H is defined as in Theorem 3.1, then every solution of (4.1) is oscillatory.
for every sufficiently large T, where H is defined as in Theorem 3.1, then every bounded solution of (4.1) is oscillatory.
Remark 4.1 Theorems 3.2 and 3.3 in  are the special cases of our Theorem 4.1 with and , respectively. Our Theorem 4.1 improves and extends the results of Theorems 3.2-3.4 in  since these theorems did not include the cases and for (4.1). The case considered in our Theorem 4.2 was not discussed in  and hence our Theorem 4.2 is a new result.
Dedicated to Professor Hari M Srivastava.
The authors are very grateful to the anonymous referees for their valuable suggestions and comments, which helped the authors to improve the previous manuscript of the article. This work was supported by the National Natural Science Foundation of P.R. China (Grants No. 11271311 and No. 61104072) and the Natural Science Foundation of Hunan of P.R. China (Grant No. 11JJ3010).
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