The Existence and Uniqueness of Solutions and Lyapunov-Type Inequality for CFR Fractional Differential Equations

Abstract

In this paper, we research CFR fractional differential equations with the derivative of order . We prove existence and uniqueness theorems for CFR-type initial value problem. By Green’s function and its corresponding maximum value, we obtain the Lyapunov-type inequality of corresponding equations. As for application, we study the eigenvalue problem in the sense of CFR.

1. Introduction

In the past decades, fractional calculus arose ([1–3]) and received extensive attention of many researchers. In recent years, it becomes more important because the subject of fractional calculus frequently appears in various fields such as science and engineering. Recently, some new fractional differential definitions have been created. With the rising of fractional computation, the research on the quantitative and qualitative properties of fractional differential equations has become a hot topic; see papers [4–15] and the references therein.

In [4–7], the authors presented new fractional derivatives with Mittsg-Leffler kernels and exponential-type kernels. The boundary value problem of fractional equations emerged as new branch in the field of differential equation due to its wide applications. In [8], Abdeljawad defined the higher order fractional derivative in the sense of Abdon and Baleanu and obtained a Lyapunov-type inequality for ABR fractional boundary value problem and if is a nontrivial solution, then where

In [5], Abdeljawad defined the higher order fractional derivative in the sense of Caputo and Fabrizio and obtained a Lyapunov-type inequality for CFR fractional boundary value problem and if is a nontrivial solution, then where

In this paper, we study the existence and uniqueness for initial problems and

We also consider the following boundary value problem:

where is the Riemann-Liouville fractional derivative in the sense of Caputo and Fabrizio and is a continuous function.

Let us introduce the concepts of the Riemann-Liouville fractional integral and the Riemann-Liouville fractional derivative in the sense of Caputo and Fabrizio.

Definition 1 (see [1]). Let and be a real function defined on . The Riemann-Liouville fractional integral of order is defined by where is defined by This is fractionalizing of the integral

Definition 2 (see [5]). Let , , , then the Riemann-Liouville fractional derivative in the sense of Caputo and Fabrizio is defined byThe associated fractional integral is where is a normalization function satisfying B.

Definition 3 (see [5]). Let and be such that , . Set , , then the Riemann-Liouville fractional derivative in the sense of Caputo and Fabrizio has the following form: The associated fractional integral is

We also give a proposition which will be used in this article.

Proposition 4 (see [5]). For and , we have

2. Existence and Uniqueness Theorems

In this section, we establish existence and uniqueness theorems for CFR-type initial value problem and give corresponding proofs. We make some conditions as the mark :

Theorem 5. Consider the initial value problem (7). Suppose that holds; if then system (7) has a unique solution of the form

Proof. First, applying to system (7) and using Proposition 4 with , then we have (20). On the other hand, if we apply to (20) and using Proposition 4, then we obtain (7). It is clear that satisfies the system (7) if and only if it satisfies (20).
We endow the set with the norm . We define the linear operator : Then, for arbitrary , , we have Hence T is a contraction. Form the Banach contraction principle, there exists a unique such that . The proof is complete.

Theorem 6. Consider the initial value problem (8). Suppose that holds; if then system (8) has a unique solution of the form

Proof. Applying to system (8) and using Proposition 4 with , then we have (24). On the other hand, if we apply to (24) and using Proposition 4, then we obtain (8). It is clear that satisfies system (8) if and only if it satisfies (24).
We endow the set with the norm . We define the linear operator T: Then, for arbitrary , , we have Hence T is a contraction. Form the Banach contraction principle, there exists a unique such that . The proof is complete.

3. Lyapunov Inequality for the CFR Boundary Value Problem

In this section, we establish some results for the CFR boundary value problem and give corresponding proofs.

Theorem 7. is a solution of the boundary value problem (9) if and only if satisfies the integral equationwhere is Green’s function defined asand

Proof. From (9), we haveApply integral on the (30); we have Then, by Proposition 4 and Definitions 2 and 3, we obtain Then, we havefor some real constants .
From , we get immediately that . By the boundary condition , we can obtain that . Hence, Using the boundary condition yields Hence, equality (33) becomes By splitting the integral as follows: We have that (27) holds. The proof is completed.

Corollary 8. For , the function in (28) satisfied the following properties: where

Proof. First, we define the function andDifferentiating with respect to , we get Hence, is an increasing function on .
Then, ChooseThen, we have andIt is clear that . Then, Clearly, . Hence, we get the maximum at ,The proof is complete.

From (29), we have the following Corollary 9.

Corollary 9. For , , and , for any , we havewhere

Proof. Form (29), we have By , we obtain (49). The proof is complete.

Theorem 10. If the boundary value problem (9) has a nontrivial continuous solution, then

Proof. Let be a nontrivial solution of the boundary value problem (9) and From Theorem 7, satisfies the integral equation where is defined in (28) and is defined in (29).
By (48), we have Form Corollary 9, we obtain then we get the inequality in (52). This completes the proof.

Theorem 11. If the boundary value problem (9) has a nontrivial continuous solution, then where R(t) is defined in (50).

Proof. Define the function Then, we have Observe that if and only ifIt is easily to see that .
Hence, we getApplying the result in (52), we have The proof is complete.

Example 12. Consider the following fractional differential equation:If is an eigenvalue to the boundary value problem (9), then

Proof. By using Corollary 9, we have From Theorem 11, we have Hence, we have That concludes the proof.

4. Conclusions

In this paper, compared with existing results of fractional differential equations, we extend the order from to . We prove existence and uniqueness theorems for initial value problem in the frame of CFR-type derivative of order and by using Banach Contraction Theorem. Then, we use our extension to obtain new Lyapunov-type inequality for CFR fractional boundary value problem with order by Green’s function and its corresponding maximum value. As for application, we give an example on the eigenvalue problem.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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