The Oscillatory of Linear Conformable Fractional Differential Equations of Kamenev Type

Abstract

In this paper, the oscillatory of the Kamenev-type linear conformable fractional differential equations in the form of is studied, where and . By employing a generalized Riccati transformation technique and integral average method, we obtain some oscillation criteria for the equation. We also give some examples to illustrate the significance of our results.

1. Introduction

The fractional derivative originated in the 17th century and formed a relatively complete form in the 19th century. In the existing literature, we can find many definitions of the fractional integral and fractional differential given by these scholars, such as Riemann–Liouville, Caputo, Weyl, Hadamard, and Chen. In the past decades, fractional differential equations have been widely used in viscoelasticity, electrical networks, signal processing, systems identification, and other natural phenomena and physical problems, fluid flow, rheology, and many other fields. There are many literature studies that have investigated the existence, uniqueness, and stability of fractional differential equations and a series of conclusions about the qualitative of solutions, see [1–9] and their references.

In 2014, a new definition, which is called the conformable derivative, was proposed in the literature [10], and the properties and calculations of conformable derivatives have been studied in the literature [11–14]. It almost satisfied all the properties of the normal derivative: the derivative of constant is zero, the rules for the product and quotient of two functions, and the chain rule. Because it combines the best properties, many research studies have increased their research on it in recent years. Although there is much about the oscillations of fractional differential equations (see [15–28] and the references therein), there is little about the conformable fractional differential equations of Kamenev type.

In 1978, Kamenev [7] studied the oscillation of the following:where and might change signs. And the above equation is oscillation under the condition

In 2019, Shao and Zhaowen [14] established new oscillation criteria of Kamenev type for linear conformable of the above equation, where and might change signs.

Inspired by the above, by extending the order of the equation containing conformable fractional derivative and increasing the number of terms, we make it have new application value.

In this paper, we discuss the oscillatory behavior about linear conformable fractional differential equations of Kamenev type of the following:where .

In this paper, we always assume that the following conditions are in force: (C₁): there exist a constant such that , . (C₂): is a continuous function and satisfies the following:(1) for .(2) has a continuous and nonpositive partial derivative on with respect to the second variable, where .(3)Let be a continuous function and satisfy

2. Main Results

First, we present the definition and properties of conformable fractional derivatives and integral. (D₁): the left conformable fractional derivative starting from of a function : of order is defined by where , we write . If exists on , then If is differentiable, then (D₂): if functions are -differential for and continuous at .(1)For all real constants , we can get (2)(3)For all , (4), for (5)When is a constant, we have  (D₃): let be two functions, and is differentiable. Then,

Theorem 1. Suppose that (C₁) and (C₂) hold, and there exists a function such thatthen equation (3) is oscillatory, where ,

Proof. On the contrary, we suppose that equation (3) has a nonoscillation solution , . Defining V(t) as a generalized Riccati substitution,From (C₁), (3), and (D₂), we obtainthat is,Multiply both sides of (13) by , and integrate from to . In accordance with (D₁), (C₂), (13), and the integration by parts, we getLet and .
Fromwe can havewhere .
For , we haveLetting , we getDividing both sides of the above inequality by and taking the upper limit in both sides as , we have the contradict with (9). The proof is complete.
Letting be an integer with , then . We have , so equation (9) is transformed to the following.

Corollary 1. If there exists a function andthen equation (3) is oscillatory, where

Before we introduce Theorem 2, we present the properties of the conformable fractional integral. (D₄): the left conformable fractional integral of order starting at is defined by We call the conformable fractional integral of a given function is -integrable if it exists. (D₅): assume that is continuous and . For all , we can get (D₆) (Cauchy–Schwarz inequality with conformable fractional derivative). Let be two functions such that and are -integrable. Then,

Theorem 2. Let be defined as in (C₂) and (C₁) also hold. Suppose thathold. For every , if there exists functions , such thatthen equation (3) is oscillatory providedwhere , , and .

Proof. On the contrary, for all , we suppose that equation (3) has a nonoscillation solution . Define V(t) as a generalized Riccati substitution:Proceeding as the proof of Theorem 1, for , from (16), we can getDividing both sides of the above inequality by , we obtainImmediately, it shows the following:So, for all , from (25), we haveIt means that, for every ,Then, for every , we obtainFor , we letFrom (3), we haveNext, we claim thatFrom (23), we can find a constant that will allowIf (36) is not true, then for every , we can find such thatFrom (D₃), we obtain the following:Thus, (4) and (7) give , for , and we have .
However,Noticing that can be arbitrarily close to and is continuous, we choose such that, for ,So, from (39) and (41), we haveand is arbitrary, and we obtainIn the interval , we select a sequence , , which makes , andFrom (35), we can always find a constant which letsThrough (43) and (45), we gainThe above also show thatUsing (46), divide both sides of (45) by , and because is sufficiently large, we receiveSo, we can getBy (D₆), and for every positive integer , we gainThat is,Using (37), we obtainThen, there exists , for such thatSo, we have for sufficiently large ,For any large , (52) is going to be equal toSince , that implieswhich contradicts with (24). This gives (36), that is,and by using , we can gainwhich contradicts with (26). The proof is complete.

Corollary 2. If there exists a function and a function , . So, (24) and (25) are replaced bywhere ; then, equation (3) is oscillatory.

3. Example

Consider the oscillation for the conformable fractional differential equations

Here, is the conformable fractional derivative. Compared with (1.1), here , and let , that is, . Using Corollary 1, we can verify that

Letting , we have

Next, letting , then the above one

That is,where is the classical beta function defined by

So, we have , which shows (19) holds. From Corollary 1, we have equation (3) is oscillatory.

4. Conclusion

In the paper, we concerned with the oscillation of the Kamenev type for linear conformable fractional differential equations by using generalized Riccati transformation, Cauchy–Schwarz inequality for the conformable fractional derivative, and average inequality technique. Because conformable derivative combines the best qualities of fractional derivatives, it is now well developed. We will continue to refine this part of the theory and continue to study the oscillation of fractional differential equations.

Data Availability

The authors declare that the data and material in the paper can be used publicly.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

HL carried out the main results and completed the corresponding proof. RX participated in the proof and helped to complete Section 3-Examples. All authors read and approved the final manuscript.

Acknowledgments

This research was supported by the National Science Foundation of China (11671227 and 11971015) and the Natural Science Foundation of Shandong Province (ZR2019MA034).