Lyapunov-Type Inequalities for a Conformable Fractional Boundary Value Problem of Order

Abstract

We establish new Lyapunov-type inequalities for the following conformable fractional boundary value problem (BVP): where is the conformable fractional derivative of order and is a real-valued continuous function Some applications to the corresponding eigenvalue problem are discussed.

1. Introduction

In [1], Lyapunov proved that if the boundary value problem BVPwhere , has a nontrivial continuous solution, thenMoreover, the constant in (2) is sharp (see [2]).

We emphasize that the above inequality has been proved to be very useful in the study of various problems related to differential equations; see, for instance, [2–5] and the references therein.

Many researchers have studied generalizations and extensions of Lyapunov’s inequality.

In [6], Wintner improved inequality (2) and obtained the following version:where

In [2], Hartamn generalized inequality (2) as follows:In the frame of fractional differential equations, Ferreira (see [7]) proved a Lyapunov-type inequality for the Caputo fractional BVPwhere is a real and continuous function.

He showed that if a nontrivial continuous solution to the above problem exists, thenIn [8], the same author investigated a Lyapunov-type inequality for the Riemann-Liouville fractional BVPHe proved that if (7) has a nontrivial continuous solution, thenFor definitions and properties of Caputo fractional derivatives and Riemann-Liouville fractional, we refer the reader to [9, 10].

Observe that inequalities (6) and (8) lead to Lyapunov’s classical inequality (2) when

Recently, Khalil et al. [11] introduced a new definition of a fractional derivative called conformable fractional derivative (see Definition 1). This derivative is much easier to handle, is well-behaved, and obeys the Leibniz rule and chain rule [12].

In short time, this new fractional derivative definition has attracted many researchers. In [13], Chung used the conformable fractional derivative and integral to discuss fractional Newtonian mechanics.

In [14], the authors proved a generalized Lyapunov-type inequality for a conformable BVP of order They have established that if the BVPwhere is the conformable derivative of order , has a nontrivial continuous solution, thenFor other generalizations and extensions of the classical Lyapunov’s inequality, we refer the reader to [2, 5, 15–23] and the references therein.

In this paper, we establish new Hartman-type and Lyapunov-type inequalities for the following conformable fractional BVP:where is the conformable derivative starting at of order and is a real-valued continuous function on Some applications to the corresponding eigenvalue problem are discussed. The obtained results are new in the context of conformable fractional derivatives.

The outline of the paper is as follows. In Section 2, we recall and collect basic properties on conformable derivatives. This allows us to construct Green’s function of the corresponding linear problem. Some properties of this Green’s function are established. In Section 3, we state and prove our main results. Some applications are discussed.

2. Preliminaries on Conformable Derivatives

In this section, we recall some basic definitions and lemmas, which will be very useful to state our results.

Definition 1 (see [11, 12]). For a given function , the conformable fractional derivative of of order is defined by If , we write If exists on , then define The geometric and physical interpretation of the conformable fractional derivatives was given in Zhao [24].

Remark 2. (i) Let and be differentiable function at , and then (ii) For , we have , for all Therefore , but is not differentiable at

Some important properties for the conformable fractional derivative given in [11, 12] are as follows.

Theorem 3. Let and be -differentiable at a point , and then
(i) , for all
(ii) , for all and
(iii)
(iv)
(v) Assume further that the function is defined in the range of , and then for all with and , one has the following Chain Rule:

The following conformable fractional derivatives of certain functions [11] are worth noting:(i)(ii)(iii)(iv)

Definition 4 (see [11, 12]). Let and be a function such that exists. The conformable fractional derivative of of order is defined by

Definition 5 (see [11, 12]). Let The fractional integral of a function of order is defined by

Lemma 6 (see [11, 12]). Let (i)If is continuous on , then (ii) if and only if , where , for (iii)If is continuous on , then, for ,  where , for

Lemma 7. Let and Then the admits a solution if and only ifwhere is Green’s function defined as

Proof. Using Lemma 6 and Definition 5, we deduce that is a solution of problem (19) if and only ifwhere
This, together with the boundary conditions, implies and Hencewhere is given in (21).

Lemma 8. Let The following property is satisfied by Green’s function (21): For any in ,

Proof. Fix in By differentiating with respect to , we obtain Hence , and we have and for , we have Using the fact that and are in , we deduce that So, the function is nondecreasing on This implies that The proof is completed.

3. Main Results

Theorem 9 (Hartman-type inequality). Assume that the (11) has a nontrivial continuous solution; then

Proof. Let , with and
Consider the Banach space , equipped with the uniform norm
Assume that problem (11) has a nontrivial solution
By (20), we have Note that since is nontrivial, then cannot be the zero function on . This, with Lemma 8, implies, for all , ThereforeNow, since , then we deduce thatwhereThe proof is completed.

Corollary 10. Assume that the (11) has a nontrivial continuous solution; then

Proof. The property follows from Theorem 9 and the fact

Theorem 11 (Lyapunov-type inequality). Assume that the (11) has a nontrivial continuous solution; then

Proof. From Corollary 10, we havewhere
By simple computation, one can check that This fact with (40) gives the required result.

Corollary 12. If is an eigenvalue to the fractional then

Proof. By using Theorem 9, we obtainNow, by simple computation, we have This gives inequality (43).

Corollary 13. Let and , such that Then the fractional has no nontrivial solution.

Proof. The assertion follows from Theorem 11.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research group NO (RG-1435-043). The authors would like to thank the anonymous referees for their careful reading of the paper.