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We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

Fractional calculus is old as the Newtonian calculus [

In 2014 [

Motivated, as mentioned above, with the need of new fractional derivatives with nice properties we study in this article the eigenvalue problems of Sturm-Liouville into conformable (fractional) calculus. Recently, there are several analytical studies devoted to fractional Sturm-Liouville eigenvalue problems; see [

For a function

In this paper we consider the fractional extension of the Sturm-Liouville eigenvalue problem

We say that

Let

Letting

We have

If

Since

Let

Since

We say that

The eigenfunctions of the fractional eigenvalue problem (

Let

The eigenvalues of the fractional eigenvalue problem (

Let

Let

Let

Applying the product rule one can easily verify that

The eigenvalues of the fractional eigenvalue problem (

Let

The eigenvalues

Multiplying (

Letting

Since the boundary conditions are of Dirichlet or Neumann type then it holds that

Now if

(i) Finding the stationary function

To find the stationary of

The problem in (i) is equivalent to the problem of finding the stationary function of (ii)

We have

Using the above results and the fractional Rayleigh Quotient result we have the following.

For the fractional eigenvalue problem (

Assuming that the eigenvalues of (

Consider the fractional eigenvalue problem (

In the following we apply the fractional Rayleigh Quotient to obtain lower estimates of the first eigenvalue. We start with the atrial function

Consider the fractional eigenvalue problem (

It is well-known that the regular Sturm-Liouville eigenvalue problem with integer derivative possesses an infinite number of eigenvalues. This result is not valid for the fractional one as shown in the previous example. However, the fractional Sturm-Liouville equation in (

We have considered a regular conformable fractional Sturm-Liouville eigenvalue problem. We proved that the eigenvalues are real and simple and the eigenfunctions are orthogonal. We also established the fractional Wronskian result for any two linearly independent solutions of the problem. We obtained a fractional Rayleigh Quotient and applied a fractional variational principle to show that the minimum value of the Quotient is obtained at an eigenfunction. This result is used to estimate the first eigenvalue and the presented example illustrates the efficiency of the result. We illustrated by an example that the existence of eigenfunctions is not guaranteed unlike the result for the regular Sturm-Liouville eigenvalue problem. Most of the obtained results are analogous for the ones of regular Sturm-Liouville eigenvalue problems and they open the door for establishing other results such as the countability of eigenfunctions and completeness of eigenfunctions which are essential in solving fractional differential equations by fractional eigenfunction expansion.

The authors declare that they have no conflicts of interest.

The first author gratefully acknowledges the support of the United Arab Emirates University under the Grant 31S239-UPAR(1) 2016.