#### Abstract

By using a generalized Riccati transformation technique and an inequality, we establish some oscillation theorems for the fractional differential equation  −  = , for , where is the Liouville right-sided fractional derivative of order of and is a quotient of odd positive integers. The results in this paper extend and improve the results given in the literatures (Chen, 2012).

#### 1. Introduction

Differential equations with fractional-order derivatives have gained importance due to their various applications in science and engineering such as rheology, dynamical processes in self-similar and porous structures, heat conduction, control theory, electroanalytical chemistry, chemical physics, and economics; for example, see . It is well recognized that fractional calculus leads to better results than classical calculus.

Many articles have investigated some aspects of differential equation with fractional-order derivatives, such as the existence and uniqueness for -type fractional neutral differential equations, smoothness and stability of the solutions, and the methods for explicit and numerical solutions; for example, see . However, to the best of the author’s knowledge very little is known regarding the oscillatory behavior of differential equation with fractional-order derivatives up to now except for .

Grace initiated the study of oscillatory theory of FDE, and he considered the equations of the form where denotes the Riemann-Liouville differential operator of order with . In fact, the IVP is equivalent to the Volteria fractional integral equation: He made use of the conditions: where and are real numbers. He talked over the four cases of ; and ; and ; and . Besides that, he replaced with and got some results on the same cases by using an inequality; refer to .

Chen studied the oscillation of the differential equation with fractional-order derivatives: where denotes the Liouville right-sided fractional derivative of order with the form and he obtained four main results under the condition of by using a generalized Riccati transformation technique and an inequality; see .

Using the same method, in 2013, Chen  studied oscillatory behavior of the fractional differential equation with the form where is the Liouville right-sided fractional derivative of order of .

Zheng  considered the oscillation of the nonlinear fractional differential equation with damping term: where denotes the Liouville right-sided fractional derivative of order of . Using a generalized Riccati function and inequality technique, he established some new oscillation criteria.

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

Qi and Cheng  studied the oscillation behavior of the equation with the form where also denotes the Liouville right-sided fractional derivative and some sufficient conditions for the oscillation of the equation were given.

The above works on the oscillation are all on fractional equations with Liouville right-sided fractional derivative by Riccati transformation technique.

We notice that very little attention is paid to oscillation of fractional differential equations with Riemann-Liouville derivative. For the relative works of study for oscillatory behavior of fractional differential equations Riemann-Liouville derivative we refer to [17, 21, 25, 26].

Marian et al.  presented the oscillatory behavior of forced nonlinear fractional difference equation of the form where is a Riemann-Liouville like discrete fractional difference operator of order , and some oscillation criteria are established by the same method with .

In 2013, Chen et al.  improved and extended some work in  by considering the forced oscillation of fractional differential equation: with the conditions where denotes the Riemann-Liouville or Caputo differential operator of order with ,  , and the operator is the Riemann-Liouville fractional integral operator. The authors obtained some new oscillation criteria by the same method with .

In 2014, Wang et al.  extended some oscillation results from integer differential equation to the fractional differential equation: where denotes the standard Riemann-Liouville differential operator of order with , is a positive real-valued function, is a continuous functional defined on satisfying that and denotes Riemann-Liouville integral operator. The authors obtained some new oscillation criteria by the method of Riccati transformation technique.

The main purpose of this paper is giving several oscillation theorems for the fractional differential equation: where is a constant, is a quotient of odd positive integers, and 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.

(A) is a continuous function such that for a certain constant and for all . ,  , and are positive continuous functions on for a certain , and is a nonnegative continuous function on for a certain . There exists , , for . And .

(B) ,   for  .

By a solution of (16) we mean a nontrivial function such that and , satisfying (16) for . We consider only those solutions of (16) that satisfy for any . A solution of (16) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (16) is said to be oscillatory if all its solutions are oscillatory.

Our results obtained here improve and extend the main results of . In , the author studied the oscillation of (16), where and . We are dealing with the oscillation theorems for (16).

For the sake of convenience, we remember

#### 2. Preliminaries and Lemmas

In this section, we present some useful preliminaries and lemmas, which will be used in the proof of our main results.

Definition 1 (see ). The Liouville right-sided fractional integral of order of a function on the half-axis is given by for , provided that the right-hand side is pointwise defined on , where is the gamma function.

Definition 2 (see ). The Liouville right-sided fractional derivative of order of a function on the half-axis is given by provided that the right-hand side is pointwise defined on , where is the ceiling function.

Lemma 3 (see ). If and are nonnegative constants, then where the equality holds if and only if .

Lemma 4 (see ). Let be a solution of (16) and Then

The proof of Lemma 4 is the same as the proof of Lemma  2.1 in .

#### 3. Main Results

In this section, we establish some new oscillation criteria for (16).

Theorem 5. Assume that (A) holds, and Furthermore, assume that there exists a positive function such that where . Then every solution of (16) is oscillatory.

Proof. Suppose that is a nonoscillatory solution of (16). Without loss of generality, we may assume that is an eventually positive solution of (16). Then there exists such that where is defined as in (22). Therefore, it follows from (16) that
Thus, is strictly increasing on and is eventually of one sign. Since for and is a quotient of odd positive integers, we see that is eventually of one sign. We first show
Otherwise, there exists such that , and since is strictly increasing on , it is clear that for . Therefore, we have Due to , from (18), we get Integrating both sides of last inequality from to , from (23), we obtain
So, we get and this contradicts (26). Hence, we have that (28) holds.
From (A), (18), and (23), we have Therefore,
Define the function by a generalized Riccati transformation Then, we have for , and from (16), (34), (35), and (A), it follows that where is defined as in Theorem 5. Let From (21) and (36), we derive Integrating both sides of (38) from to , we have Letting , we get which contradicts (25). The proof is complete.

Theorem 6. Suppose that (A) and (24) hold. Furthermore, suppose that there exists a positive function , and a function , where , such that where , and has a nonpositive continuous partial derivate on , with respect to the second variable, and satisfies where for ; here is defined as in Theorem 5. Then all solutions of (16) are oscillatory.

Proof. Suppose that is a nonoscillatory solution of (16). Without loss of generality, we may assume that is an eventually positive solution of (16). We proceed as in proof of Theorem 5 to get that (36) holds. Multiplying (36) by and integrating from to , for , we derive
From we have where is defined as in Theorem 6. Let From (21) and (45), we get
From , we have and ; then
Therefore, we get Letting , we get which is a contradiction to (42). The proof is complete.

Next, we consider the condition of which yields that (24) does not hold. Under this condition, we have the following results.

Theorem 7. Suppose that (A), (B), and (52) hold, and there exists a positive function such that (25) holds. Furthermore, assume that, for every constant , Then every solution of (16) is oscillatory or satisfies or , where is defined as Lemma 4.

Proof. Assume that is a nonoscillatory solution of (16). Without loss of generality, assume that is an eventually positive solution of (16). Proceeding as in the proof of Theorem 5, we get that (26) and (27) hold. Then there are two cases for the sign of .
When is eventually negative, from the proof of Theorem 5, we get that every solution of (16) is oscillatory.
Next, assume that is eventually positive; then there exists , such that , for . From (18) and (23), we get Therefore,
Since (B) holds and , then we get Letting in (55), we have If , then there exists such that , for . We set . Thus, we get , and .
We now prove . If not, that is, , then from (27), we derive Integrating both sides of (58) from to , we have Therefore, Hence, from (18), (A), and (23), we get Integrating both sides of (61) from to , we have
Then, we obtain This contradicts (26). Therefore, we have ; that is, The proof is complete.

Theorem 8. Suppose that (A), (B), and (52) hold. Let and be defined as in Theorem 6 such that (42) holds. Furthermore, assume that, for every , (53) holds. Then every solution of (16) is oscillatory or satisfies or , where is defined as in Lemma 4.

Proof. Assume that is a nonoscillatory solution of (16). Without loss of generality, assume that is an eventually positive solution of (16). Proceeding as in the proof of Theorem 5, we get that (26) and (27) hold. Then there are two cases for the sign of .
When is eventually negative, the proof is similar to that of Theorem 6. When is eventually positive, the proof is similar to that of Theorem 7. Here we omitted it.

Remark 9. From Theorems 58, we can get many different sufficient conditions for the oscillation of (16) with different choices of the functions and .

#### 4. Examples

Example 10. Consider the differential equation with fractional-order derivatives: where , and is a quotient of odd positive integers.

Here, , , , and . Take , , and . From we see that (A) and (24) hold. Letting , we get which satisfies condition (25). Therefore, by Theorem 5, every solution of (65) is oscillatory.

Example 11. Consider the differential equation with fractional-order derivatives: where , and is a quotient of odd positive integers.

Here, , , , , and . Take , and , . From we find that (A), (B), and (52) hold.

Take , we have which satisfies condition (25). For every constant , , we obtain which implies that (53) holds. Therefore, by Theorem 7, every solution of (68) is oscillatory or satisfies