Oscillation for a Class of Right Fractional Differential Equations on the Right Half Line with Damping

Abstract

In this paper, we discuss a class of fractional differential equations of the form is the Liouville right-sided fractional derivative of order . We obtain some oscillation criteria for the equation by employing a generalized Riccati transformation technique. Some examples are given to illustrate the significance of our results.

1. Introduction

The theory of fractional derivatives was originated from G.W. Leibniz’s conjecture. To this day, the theory about fractional calculus and fractional differential equation have been well developed; see [1–7]. In the beginning, the theory of fractional derivatives developed mainly as a pure theoretical filed of mathematics, which can be used only for mathematicians. However, in the past few decades, fractional differential equations were widely used in many fields, such as fluid flow, rheology, electrical networks, and many other branches of science. Great attention was paid to study the properties of solutions of fractional differential equations.

Because only few differential equations can be solved, many researches focus on the analysis of qualitative theory for fractional differential equations, such as the existence, uniqueness of solutions, numerical solutions, stability, and oscillation of solutions; see [8–34] and the references therein. Among them, there have been many results for the oscillation of solutions for fractional differential equations.

In 2013, Chen [16] studied oscillatory behavior of the fractional differential equation in the form offor , where is the Liouville right-sided fractional derivative of order .

In 2013, Han [17] brought up the oscillation of fractional differential equationsfor , where is a real number, and is the Liouville right-sided fractional derivative of .

In 2013, Xu [18] studied the oscillation of nonlinear fractional differential equations of the formwhere is a constant, and is a ratio of two odd positive integers.

In 2013, based on the modified Riemann-Liouville derivative, Qin and Zheng [19] discussed the oscillation of a class of fractional differential equations with damping term as follows:for , where denotes the modified Riemann-Liouville derivative regarding the variable , the function , and denotes continuous derivative of order.

In 2014, Jehad Alzabut and Thabet Abdeljawad [20] studied the oscillatory theory of fractional difference equations in the formwhere is the Riemann-Liouville difference operator of order and is the Riemann-Liouville sum operator where and is the greatest integer less than or equal to .

In 2017, B. Abdalla, K. Abodayeh, T. Abdeljawad, J. Alzabut [21] studied the oscillation of solutions of nonlinear forced fractional difference equations in the formwhere is the greatest integer less than or equal to , , and and are the Riemann-Liouville sum and difference operators.

In 2018, Bai and Xu [22] discussed the oscillation problem of a class of nonlinear fractional difference equations with the damping term in the fromwhere is a quotient of two odd positive integers, is a constant, denotes the Riemann-Liouville fractional difference operator of order , and .

In 2018, Bahaaeldin Abdalla and Thabet Abdeljawad [23] studied the oscillation of Hadamard fractional differential equation of the formwhere , is the left-fractional Hadamard derivative of order , in the Riemann-Liouville setting.

In 2018, J. Alzabut, T. Abdeljawad, H. Alrabaiah [24] considered the following forced and damped nabla fractional difference equationwhere and are the Riemann-Liouville fractional difference and sum operators of of order , respectively, is a real number, is constant, and are real sequences from is a positive real sequence from and such that for all .

In 2018, B. Abdalla, J. Alzabut, T. Abdeljawad [25] investigated the oscillation of solutions for fractional difference equations with mixed nonlinearities in formsandwhere and are functions defined from to , and are ratios of odd positive integers with .

Inspired by the above results, in this paper, we discuss the oscillatory behavior of the fractional differential equational with dampingwhere is a real number. is the Liouville right-sided fractional derivative of . We always assume that the following conditions are valid.

   and are continuous functions on , .

   are continuous functions with , for , and there exist positive constants such that , for all .

  , are continuous functions with for , and there exists some positive constant such that for

2. Preliminaries

For convenience, some background materials from fractional calculus are given.

From [4], we can get the definition for Liouville right-side fractional integral and Liouville right-side fractional derivative on the whole axis of order for a function as follows,provided the right hand side is pointwise defined on , where is the gamma function defined by , and is the ceiling function.

If , we havefor

The following relations also existed:

Set

and then

3. Main Results

First, we study the oscillation of (12) under the following condition:

Theorem 1. Suppose that and (19) hold; furthermore, assume that there exists a positive function such thatwhere , are defined as in , andThen every solution of (12) is oscillatory.

Proof. Suppose that is a nonoscillation solution of (12); without loss of generality, we may assume that is an eventually positive solution of (12). Then there exists such that and for , where is defined in (16). From , (12), and (16) we haveThus is strictly increasing on . Since for , and from , we see that is eventually of one sign. Now we can claimIf not, then there exists such that . Since is strictly increasing on , it is clear that for . Therefore, from (18), we haveThen, we getIntegrating the above inequality from to , we haveLetting , we seeThis is in contradiction with (19). Hence, (23) holds.
Define the function as generalized Riccati substitution Then we have for . From (18), (22), (28), and , it follows thatThat is,Taking ,   , and , from and (30) we could conclude thatIntegrating both sides of inequality (31) from to , we obtain Taking the limit supremum of both sides of the above inequality as , we get which is in contradiction with (20).
If is an eventually negative solution of (12), the proof is similar; hence we omit it.
The proof is complete.

Theorem 2. Suppose that and (19) hold. Furthermore, assume that there exists a positive function such thatwhere are defined in Theorem 1. Then every solution of (12) is oscillatory.

Proof. Suppose that is a nonoscillation solution of (12); without loss of generality, we may assume that is an eventually positive solution of (12). Proceeding the same as in the proof of Theorem 1, we get (23). Define the function as follows Then we have for . From (16), (18), (22), (35), and , it follows thatThat is,Taking ,   , and , from and (37) we concludeIntegrating both sides of inequality (38) from to , we obtainTaking the limit supremum of both sides of the above inequality as , we get which is in contradiction with (34).
If is an eventually negative solution of (12), the proof is similar; here we omit it.
The proof is complete.

We define a function class ; set , . We say , if satisfyand has a nonpositive continuous partial derivative on with respect to the second variable.

Theorem 3. Suppose that and (19) hold. Furthermore, assume that there exists a positive function and a function satisfies Then every solution of (12) is oscillatory.

Proof. Suppose that is a nonoscillation solution of (12); without loss of generality, we may assume that is an eventually positive solution of (12). Proceeding as in the proof of Theorem 2 we get (37).
Multiplying (37) by and integrating from to , we getUsing the formula integration by parts, we obtainSubstituting (44) with (43), we haveTaking ,   , and , from we getSubstituting (46) in (45), we haveSince for , we have for Therefore, from the previous inequality, we getSince for , we have for . Hence, it follows from (48) that we haveLetting , we obtainwhich yields a contradiction to (42). The proof is complete.

Second, we study the oscillation of (12) under the following condition:

Theorem 4. Suppose that and (51) hold, and there exists a positive function such that (20) holds. Furthermore, assume that, for every constant ,where are defined as in Theorem 1. Then every solution of (12) is oscillatory or satisfies

Proof. Suppose that is a nonoscillation solution of (12); without loss of generality, we may assume that is an eventually positive solution of (12). Proceeding as in the proof of Theorem 1, we know that is eventually one sign; then there are two cases for the sign of
If is eventually negative, similar to Theorem 1, we have the oscillation of (12). Next, if is eventually positive, then there exists such that for From (18), we get for Thus, we get 0 and . We now claim that . Assuming not, that is, , then from (23) and we getIntegrating both sides of the above inequality from to , we haveHence, from (17), we getIntegrating both sides of the last inequality from to , we obtainLetting , from (52), we get ; this is in contradiction with . Therefore, we have , that is, . The proof is complete.

Theorem 5. Suppose that and (51) hold. Let and be defined as in Theorem 3 such that (42) holds. Furthermore, assume that, for every constant , (52) holds. Then every solution of (12) is oscillatory or satisfies

From Theorem 3, proceeding as in the proof of Theorem 4, we get the results of the theorem.

4. Examples

Example 1. Consider the fractional differential equationIn (57), , , , and .
Sincethen (19) holds.
Taking , . It is clear that conditions hold. Furthermore, taking , we havewhich shows that (34) holds. Therefore, by Theorem 2, every solution of (12) is oscillatory.

Example 2. Consider the fractional differential equationIn (60), , , , and .
Proceeding the same process as Example 1, we see that (19) holds. Taking , , . It is clear that conditions hold. Furthermore, taking , it meets for , .
Sincewhich shows that (42) holds, by Theorem 3, every solution of (12) is oscillatory.

5. Conclusion

In the paper, by using the generalized Riccati transformation and inequality technique, we study a class of order fractional differential equations in the form (12), which contains the damping term and has not been studied before. The oscillation criteria of (12) are obtained and some examples are given to reinforce our results.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Authors’ Contributions

Hui Liu carried out the main results and completed the corresponding proof. Run Xu participated in the proof and helped in completing Section 4. All authors read and approved the final manuscript.

Acknowledgments

This research is supported by National Science Foundation of China (11671227).