Research Article | Open Access
Kamenev Type Oscillatory Criteria for Linear Conformable Fractional Differential Equations
Using integral average method and properties of conformable fractional derivative, new Kamenev type oscillation criteria are given firstly for conformable fractional differential equations, which improve known results in oscillation theory. Examples are also given to illustrate the effectiveness of the main results.
Fractional differential equations are used in various fields such as physics, mathematics, biology, biomedical sciences, and finance as well as other disciplines (see [1, 2]). In the last few decades, a lot of attention was paid to finding the more suitable definitions of fractional derivatives. There are many definitions in the existent literature, such as the Riemann-Liouville, Caputo, Riesz, Riesz-Caputo, Weyl, Grunwald-Letnikov, Hadamard, and Chen derivatives, etc. In 2014, R. Khalil et al.  have suggested a new fractional derivative, which is called the conformable derivative. This new definition satisfies almost all the requirements of standard derivative, for example, the conformable derivative of a constant is zero, and the chain rule (see [3–7] and references therein).
In this paper, we consider the linear conformable fractional differential equationwhere , , , and might change signs.
A nontrivial solution of (1) is called oscillatory if it has arbitrarily large zeros; otherwise, it is said to be nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.
When , we have the following second-order differential equation:There are a lot of papers involving the oscillation for (2) and other linear, nonlinear, damped, and forced differential equations or Hamiltonian systems (see [8–11]) since the foundation work of Wintner  (see also for [12–25]). Especially, if , we obtain the second-order linear Hill equation
In 1978, Kamenev  established a new oscillation criterion of differential equation (1), using integral average method, which has the result of Wintner as a particular case. The obtained result in  states that the condition for some integer , is sufficient for the oscillation of (3).
Theorem 1. Let be a continuous function which satisfies for , for ; has a continuous and nonpositive partial derivative on with respect to the second variable. Moreover, let be a continuous function with for all . Then (3) is oscillatory if
As a new research field, the oscillation of fractional differential equations has been widely studied by many authors (e.g., see [27–35] and the references quoted therein). Among these oscillation criteria, most of them are major in the fractional differential equations with forcing terms. The Kamenev type oscillation criteria of fractional differential equations without forcing terms are obtained  under the frame of Caputo derivative. However, to our best knowledge, the study of oscillatory behavior of linear conformable fractional differential equation has not been seen in literature. The purpose of this paper is to obtain new Kamenev type oscillatory criteria for (1) using integral average method based on the properties of conformable fractional derivatives and integral. By investigating some new properties of this derivative, the classical oscillatory problem of (2) can be extended to (1). These oscillation criteria improve the results mentioned above. Examples are also given to illustrate the effectiveness of our main results.
2. Main Results
Firstly, we give some definitions and properties of conformable fractional derivatives and integral, which are important in the proofs of the main results.
Definition 4. The left conformable fractional derivative starting from of a function of order with is defined by when , this derivative of coincides with . If exists on then
Definition 5. Let . Then the left conformable fractional integral of order starting at is defined by If the conformable fractional integral of a given function exists, we call that is integrable.
Lemma 6 (see ). If and , then, for all , we have and
Lemma 7 (see ). (1) for all real constant , .
(3) , for all .
(5) , where is a constant.
Lemma 8 (see ). Let be two functions such that is differentiable. Then
Lemma 9 (Cauchy-Schwarz inequality with conformable fractional derivative). Let be two functions such that and are integrable. Then
Proof. Suppose on the contrary that there exists a nontrivial solution of (1) which is not oscillatory. Without loss of generality, we may suppose that for all . Define for ; by the properties of conformable fractional derivative, we get the Riccati equation Thus, for every , with , we haveUsing Lemma 8, for , we getThus, for every ,This inequality holds for , which meansDividing both sides of (25) by , and taking the upper limit in both sides as , the right hand side is always bounded, which contradicts hypothesis (20). This completes the proof of Theorem 10.
Remark 11. The singularity of integral at has no affection on the oscillation of conformable fractional differential equations since oscillation is a qualitative property at infinite. The results are still true if the lower bound of the integration in (20) is replaced by any other point larger than .
Let us consider the function defined by where is an integer with . Then , for , and by direct calculation, we have
Under a modification of the hypotheses of Theorem 10, we can obtain the following result.
Corollary 12. If there exists an integer such that then (1) is oscillatory.
If in (1), we can get and then we further obtain the following.
Proof. Suppose that (1) possesses a nonoscillatory solution ; without loss of generality, we may also suppose that for . Define ; as in the proof of Theorem 10, we getfor all , with . Then we have Hence,Thus, by condition (33), we obtain for This implies thatfor every , and Then for every , we get i.e., we havewhereandNow we claim thatBy condition (31), there exists a constant withIf (45) is not true, then for each , there exists a such thatThen for all , we have However, using the continuity of and noticing that can be arbitrarily close to , we can choose a such that for every . Hence, we have , for all . Since is arbitrary, we haveNext, let a sequence , in the interval , be selected such that , and By (42), there exists a constant such thatUsing (51) and (53), we haveand henceBy (53) and (54), we obtain provided that is sufficiently large. Therefore, for all large . Then (55) ensures thatUsing Lemma 9 and Cauchy-Schwarz inequality, we have, for any positive integer , and consequently for all large . By (46), we get and then there exists a , such that for every . Hence, for sufficiently large . Therefore, for all large . Using (58), we haveThis implies that which contradicts condition (32). This gives (45). Since can be selected arbitrarily, we have Using (39), we have which contradicts condition (34). This completes the proof of Theorem 14.
Proof. Suppose on the contrary that there exists a prepared solution of (1) which is not oscillatory. Suppose that , for all . Let , on ; as in the proof of Theorem 14, we get (35) for any with . Thus Using condition (72), we get By (39), we have Hence, we have i.e.,where and are defined by (43) and (44). Using condition (72), we obtain and by condition (71), we get Hence there exists a sequence in the interval with and such thatIf (47) holds, following the procedure of the proof of Theorem 14, we conclude that (51) is satisfied. By (77), there exists a constant , such that (53) is fulfilled. Then, as in the proof of Theorem 14, we can arrive at (65), which contradicts (80). This proves that (45) holds. The remainder of the proof proceeds as in the proof of Theorem 14.
Remark 18. Choosing different functions and in Theorems 10, 14, and 16, we can obtain various oscillation criteria for (1). For example, , where is a continuously differentiable function on , , , for ; , ; , , where is an integer with and is a positive continuous function on such that .
Remark 19. If , then Theorem 10, Theorem 14, and Theorem 16 reduce to Theorem 1.1, Theorem 1.2, and Theorem 1.3 in , respectively. Our results improve the results mentioned above, since they can handle the cases not covered by known results.
Example 21. Consider the following conformable fractional differential equation:where , are constants. Using Corollary 12, we can verify that where is the classical Beta function defined by By and we get Hence by Corollary 12, we obtain that the conformable fractional differential equation (83) is oscillatory.
Example 22. Consider the following conformable fractional differential equation:where is constant. Since , using Corollary 13, we can verify that