A Fractional Dirac System with Eigenparameter-Dependent and Transmission Conditions

Abstract

This work investigates the fractional Dirac system that has transmission conditions, and its boundary condition contains an eigenparameter. Defining a convenient inner product space and a new operator that has the same eigenvalues as the considered problem, we demonstrate that the fractional Dirac system is symmetric in this space. Thus, we have reached some remarkable results for the spectral characteristics of the operator. Furthermore, in the next section of the study, the existence of solutions was examined.

1. Introduction

Since Sturm and Liouville’s first joint publication in 1837, the Sturm–Liouville problem has been investigated by many mathematicians; however, the growth of the theory of fractional calculus has opened further perspectives for the theory of new-type problems, so-called fractional Sturm–Liouville problems. Generally, fractional Sturm–Liouville problems are defined as Sturm–Liouville problems when one or both derivatives are changed with fractional derivatives [1]. For a nice overview of the development of fractional calculus, we refer to [2,3,4,5].

In recent years, the fractional Sturm–Liouville problem has taken prominent attention in the literature, (see [6,7]). The numerical solution of fractional Sturm–Liouville problem has been studied in [8,9,10], and the regular fractional Sturm–Liouville problem is considered in [11] by Rivero et al. Recently, Akdogan et al. [12] and Yakar et al. [13] analyzed the singular fractional Sturm–Liouville problem. It has been noted that most of the work on this topic is concerned with Caputo–type or Riemann–Liouville fractional differential equations.

Essentially, in physics, the Dirac equation is one of the most fundamental building blocks of relativistic quantum theory, modeling the evolution of spin $-{\displaystyle \frac{1}{2}}$ charged particles. There is an enormous amount of literature related to the spectral theory of the Dirac operator, and we refer to [14,15,16,17] with their extensive references.

Since the Dirac equation can be thought of as a system form of the Sturm–Liouville equation, it has been unthinkable that there would be no fractional derivative in such equations. But overall, there appears to be little research on the fractional Dirac operator. As an example, we can give the paper [18], in which a one-dimensional conformable fractional Dirac system is studied. In [19], the authors examined the fractional Dirac system with impulsive conditions. In their paper, they demonstrated the operator’s symmetry and the existence of solutions by using the approach described in [12]. However, they did not consider eigenparameter-dependent boundary conditions in their study. In this paper, we will employ Picard’s approximation method to prove the existence of a solution. This paper’s main contribution is to extend the spectral results for a fractional Dirac system with eigenparameter-dependent and transmission conditions. In this work, we consider the following fractional Dirac system

$$\ell (u)=Au(x)-P(x)u(x)=\lambda u(x),x\in \left[a,c\right)\cup \left(c,b\right]$$

(1)

where

$$A=\left(\begin{array}{cc}0& D_{b^{-}}^{\alpha }\\ {-}^C{D}_{a^{+}}^{\alpha }& 0\end{array}\right),P(x)=\left(\begin{array}{cc}{p}_1(x)& 0\\ 0& p_2(x)\end{array}\right),u(x)=\left(\begin{array}{c}{u}_1(x)\\ u_2(x)\end{array}\right)$$

or equivalently

$$\begin{array}{c}\hfill D_{b^{-}}^{\alpha }{u}_2(x)-p_1(x)u_1(x)=\lambda u_1(x)\\ \hfill {-}^C{D}_{a^{+}}^{\alpha }{u}_1(x)-p_2(x)u_2(x)=\lambda u_2(x).\end{array}$$

Here, the functions $p_i(x)$, $(i=1,2)$ are continuous on $x\in \left[a,c\right)\cup \left(c,b\right]$ and $\lambda$ is a spectral parameter. We also consider the eigenparameter-dependent boundary conditions

$$sin{\theta }_1{u}_1(a)-cos{\theta }_1{I}_{b^{-}}^{1-\alpha }{u}_2(a)=0,$$

(2)

$$b_1{u}_1(b)-a_1{I}_{b^{-}}^{1-\alpha }{u}_2(b)+\lambda \left(sin{\theta }_2{u}_1(b)-cos{\theta }_2{I}_{b^{-}}^{1-\alpha }{u}_2(b)\right)=0,$$

(3)

and the following transmission conditions:

$$u_1(c-)=\gamma u_1(c+),$$

(4)

$$I_{b^{-}}^{1-\alpha }{u}_2(c-)={\gamma }^{-1}{I}_{b^{-}}^{1-\alpha }{u}_2(c+),$$

(5)

where $c\in (a,b)$ is an inner point in $(a,b)$. It is assumed that the functions $p_i(x)$, $(i=1,2)$ have the finite limits at c: $p_i(c\pm ):={lim}_{x\to c^{\pm }}{p}_i(x)$, $(i=1,2)$. Also $a_1,b_1,\gamma \in \mathbb{R}$ and ${\theta }_1,{\theta }_2\in \left[0,\pi \right)$. In [16], an eigenparameter-dependent non-fractional Dirac system with transmission conditions is examined, and a numerical approach is given in [20].

Our work is structured as follows: The next part of the paper will continue by introducing some definitions of fractional derivatives and their calculus, which are needed to describe the investigated problem. Section 3 is devoted to the spectral analysis of the studied problem. We give the fundamental theorems regarding the properties of eigenvalues and eigenfunctions in this section. At the end of the work, the existence of solutions to the problem will be investigated.

2. Fractional Calculus

The aim of this section is to introduce the reader to the elementary tools of fractional calculus. Two of the most widely used definitions for applications among the several kinds of fractional derivatives are the Riemann–Liouville and Caputo fractional derivatives [2,3]. Let us now give the definitions of these two fractional derivatives. First, we will need the fractional Riemann integral.

Definition 1.

Let $[a,b]\subset \mathbb{R}$, $\alpha \in {\mathbb{R}}^{+}$ and $f\in L_1[a,b]$. The left-sided and respectively right-sided Riemann–Liouville integrals of order α are given by the formula

$$(I_{a+}^{\alpha }f)(x)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_a^x{(x-t)}^{\alpha -1}f(t)dt,x>a,$$

$$(I_{b-}^{\alpha }f)(x)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_x^b{(t-x)}^{\alpha -1}f(t)dt,x<b,$$

where $\Gamma (.)$ denotes the Euler Gamma function.

By using the Riemann–Liouville fractional integral, we define the Riemann–Liouville and Caputo derivatives as follows:

Definition 2.

Let $[a,b]\subset \mathbb{R}$, $\alpha \in (0,1)$ and $f\in L_1[a,b]$. The left and right sided Riemann–Liouville derivatives of order α are defined as

$$(D_{a+}^{\alpha }f)(x)=D(I_{a+}^{1-\alpha }f)(x),x>a,$$

(6)

$$(D_{b-}^{\alpha }f)(x)=-D(I_{b-}^{1-\alpha }f)(x),x<b.$$

(7)

respectively. Similarly, the left and right sided Caputo derivatives of order α are given by the following formulas:

$${(}^C{D}_{a+}^{\alpha }f)(x)=(I_{a+}^{1-\alpha }Df)(x),x>a$$

(8)

$${(}^C{D}_{b-}^{\alpha }f)(x)=-(I_{b-}^{1-\alpha }Df)(x),x<b.$$

(9)

The following lemma is about integration by parts for fractional derivatives, which will be used in the next section. For further details, see [21].

Lemma 1.

Let $\alpha \in (m-1,m)$, $m\in \mathbb{Z}$, assume that u is absolutely continuous on $[a,b]$ and $v\in L_p[a,b]$. The fractional differential operators defined in (6)–(9) satisfy the following identities

$$\begin{array}{cc}\hfill {\int }_a^{b}u(x)D_{b-}^{\alpha }v(x)dx& ={\int }_a^{b}v(x){}_{ }{}^{C}D_{a^{+}}^{\alpha }u(x)dx\hfill \\ \hfill & +\sum _{k=0}^{m-1}{(-1)}^{m-k}{u}^{(k)}(x)D^{m-k-1}{I}_{b-}^{m-\alpha }{v(x)|}_{x=a}^b\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\int }_a^{b}u(x)D_{a+}^{\alpha }v(x)dx& ={\int }_a^{b}v(x){}_{ }{}^{C}D_{b^{-}}^{\alpha }u(x)dx\hfill \\ \hfill & +\sum _{k=0}^{m-1}{(-1)}^k{u}^{(k)}(x)D^{m-k-1}{I}_{a+}^{m-\alpha }{v(x)|}_{x=a}^b.\hfill \end{array}$$

(11)

In the next section, we will define a new Hilbert space with a special inner product that will have an important role for the symmetry of the operator and then give several theorems about the properties of the spectrum of the considered problem in that space. We shall begin by defining the operator.

3. Hilbert Space Formulation and Spectral Properties of the Operator

Before proving the main theorems, we give certain facts, which we shall use subsequently. Throughout the paper, we assume that $\left|a_1\right|$ + $\left|b_1\right|\ne 0$. By establishing a new equivalent Hilbert space associated with the parameters in the boundary and transmission conditions, we transform the investigated problem into a symmetric one in this space. Let us define the following inner product in the Hilbert space $\mathbb{H}=L_2\left[a,c\right)\cup L_2\left(c,b\right]\oplus \mathbb{C}$ by

$${〈U,V〉}_{\mathbb{H}}={\int }_a^c{u}^T(x)\overline{v}(x)dx+{\int }_c^b{u}^T(x)\overline{v}(x)dx+{\displaystyle \frac{1}{\sigma }}\tilde{u}\overline{\widehat{v}}$$

(12)

where T denotes the transpose and

$$\begin{array}{cc}\hfill U& =\left({\displaystyle \genfrac{}{}{0pt}{}{u(x)}{\tilde{u}}}\right),V=\left({\displaystyle \genfrac{}{}{0pt}{}{v(x)}{\tilde{v}}}\right)\in \mathbb{H},\hfill \\ \hfill u(x)& =\left({\displaystyle \genfrac{}{}{0pt}{}{u_1(x)}{u_2(x)}}\right),v(x)=\left({\displaystyle \genfrac{}{}{0pt}{}{v_1(x)}{v_2(x)}}\right)\in \mathbb{H},\hfill \end{array}$$

$u_i(x),v_i(x)\in L_2\left[a,c\right)\cup L_2\left(c,b\right],(i=1,2),\tilde{u},\tilde{v}\in \mathbb{C}$. The constant $\sigma$ is defined by

$$\sigma :=det\left(\begin{array}{cc}{b}_1& a_1\\ sin{\theta }_2& cos{\theta }_2\end{array}\right)>0.$$

For convenience, we will utilize the following notations:

$$\begin{array}{cc}\hfill \widehat{u}& =sin{\theta }_2{u}_1(b)-cos{\theta }_2{I}_{b^{-}}^{1-\alpha }{u}_2(b)\hfill \\ \hfill \tilde{u}& =b_1{u}_1(b)-a_1{I}_{b^{-}}^{1-\alpha }{u}_2(b).\hfill \end{array}$$

(13)

We now define the operator $\mathcal{L}:D(\mathcal{L})\subset \mathbb{H}\to \mathbb{H}$ such that its domain $D(\mathcal{L})$ is

$$D(\mathcal{L})=\left\{\begin{array}{c}U=\left({\displaystyle \genfrac{}{}{0pt}{}{u(x)}{\widehat{u}}}\right)\in \mathbb{H}\mid u(x)=\left({\displaystyle \genfrac{}{}{0pt}{}{u_1(x)}{u_2(x)}}\right),u_1(x),u_2(x),{}_{ }{}^{C}D_{a^{+}}^{\alpha }{u}_1(x)\mathrm{and}{D}_{b^{-}}^{\alpha }{u}_2(x)\hfill \\ \mathrm{are}\mathrm{absolutely}\mathrm{continuous}\mathrm{on}\left[a,c\right)\cup \left(c,b\right],u_1(c\pm ),I_{b^{-}}^{1-\alpha }{u}_2(c\pm )\mathrm{have}\mathrm{finite}\mathrm{limits}\end{array}\right\}$$

and its rule is

$$\mathcal{L}\left({\displaystyle \genfrac{}{}{0pt}{}{u(x)}{\widehat{u}}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{\mathsf{\ell }(u)}{-\tilde{u}}}\right).$$

At this point, we can give the definition of symmetry in the usual fashion: $\mathcal{L}$ is called a symmetric operator if for all $X,Y\in D(\mathcal{L})$

$${〈\mathcal{L}X,Y〉}_{\mathbb{H}}={〈X,\mathcal{L}Y〉}_{\mathbb{H}}.$$

Therefore, the following eigenvalue problem in the operator form is going to be equivalent to the fractional Dirac system (1)–(5)

$$\mathcal{L}U=\lambda U.$$

(14)

It is also to be noted here that the fractional Dirac system (1)–(5) has the same eigenvalues as the operator $\mathcal{L}$, and the eigenvectors of system (1)–(5) are the same as with the first two components of the eigenelements corresponding to the same eigenvalue of the operator $\mathcal{L}$. We are now prepared to give the following theorem, which is related to the symmetry of the operator $\mathcal{L}$.

Theorem 1.

The linear operator $\mathcal{L}$ is symmetric.

Proof. 

Let U and V be two vector-valued functions in $D(\mathcal{L})$. Using the inner product defined in (12)

$$\begin{array}{cc}\hfill {〈\mathcal{L}U,V〉}_{\mathbb{H}}& =\underset{a}{\overset{c}{\int }}{D}_{b^{-}}^{\alpha }{u}_2{\overline{v}}_{1}dx-\underset{a}{\overset{c}{\int }}{}_{ }{}^{C}D_{a^{+}}^{\alpha }{u}_1{\overline{v}}_{2}dx+\underset{c}{\overset{b}{\int }}{D}_{b^{-}}^{\alpha }{u}_2{\overline{v}}_{1}dx-\underset{c}{\overset{b}{\int }}{}_{ }{}^{C}D_{a^{+}}^{\alpha }{u}_1{\overline{v}}_{2}dx\hfill \\ \hfill & -\underset{a}{\overset{c}{\int }}{p}_1{u}_1{\overline{v}}_{1}dx-\underset{a}{\overset{c}{\int }}{p}_2{u}_2{\overline{v}}_{2}dx-\underset{c}{\overset{b}{\int }}{p}_1{u}_1{\overline{v}}_{1}dx-\underset{c}{\overset{b}{\int }}{p}_2{u}_2{\overline{v}}_{2}dx+{\displaystyle \frac{1}{\sigma }}\tilde{u}\overline{\widehat{v}}.\hfill \end{array}$$

Using the equalities in (10) and (11), we obtain

$$\begin{array}{cc}\hfill {〈\mathcal{L}U,V〉}_{\mathbb{H}}& =\underset{a}{\overset{c}{\int }}{u}_2{}_{ }{}^{C}D_{a^{+}}^{\alpha }{\overline{v}}_{1}dx-{\left[{\overline{v}}_1{I}_{b^{-}}^{1-\alpha }{u}_2\right]}_a^{c-}+\underset{c}{\overset{b}{\int }}{u}_2{}_{ }{}^{C}D_{a^{+}}^{\alpha }{\overline{v}}_{1}dx-{\left[{\overline{v}}_1{I}_{b^{-}}^{1-\alpha }{u}_2\right]}_{c+}^b\hfill \\ \hfill & -\underset{a}{\overset{c}{\int }}{u}_1{D}_{b^{-}}^{\alpha }{\overline{v}}_{2}dx-{\left[u_1{I}_{b^{-}}^{1-\alpha }{\overline{v}}_2\right]}_a^{c-}-\underset{c}{\overset{b}{\int }}{u}_1{D}_{b^{-}}^{\alpha }{\overline{v}}_{2}dx-{\left[u_1{I}_{b^{-}}^{1-\alpha }{\overline{v}}_2\right]}_{c+}^b\hfill \\ \hfill & -\underset{a}{\overset{c}{\int }}{p}_1{u}_1{\overline{v}}_{1}dx-\underset{a}{\overset{c}{\int }}{p}_2{u}_2{\overline{v}}_{2}dx-\underset{c}{\overset{b}{\int }}{p}_1{u}_1{\overline{v}}_{1}dx-\underset{c}{\overset{b}{\int }}{p}_2{u}_2{\overline{v}}_{2}dx+{\displaystyle \frac{1}{\sigma }}\tilde{u}\overline{\widehat{v}}.\hfill \end{array}$$

By using the theorem in [22] and the abbreviations in (13), we obtain

$$\begin{array}{cc}\hfill {〈\mathcal{L}U,V〉}_{\mathbb{H}}& =\underset{a}{\overset{c}{\int }}\left[-\overline{{}_{ }{}^{C}D_{a^{+}}^{\alpha }{v}_1}-p_2{\overline{v}}_2\right]u_{2}dx+\underset{a}{\overset{c}{\int }}\left[\overline{D_{b^{-}}^{\alpha }{v}_2}-p_1{\overline{v}}_1\right]u_{1}dx\hfill \\ \hfill & +\underset{c}{\overset{b}{\int }}\left[-\overline{{}_{ }{}^{C}D_{a^{+}}^{\alpha }{v}_1}-p_2{\overline{v}}_2\right]u_{2}dx+\underset{c}{\overset{b}{\int }}\left[\overline{D_{b^{-}}^{\alpha }{v}_2}-p_1{\overline{v}}_1\right]u_{1}dx\hfill \\ \hfill & -{\left[{\overline{v}}_1{I}_{b^{-}}^{1-\alpha }{u}_2-u_1\overline{I_{b^{-}}^{1-\alpha }{v}_2}\right]}_a^{c-}-{\left[{\overline{v}}_1{I}_{b^{-}}^{1-\alpha }{u}_2-u_1\overline{I_{b^{-}}^{1-\alpha }{v}_2}\right]}_{c+}^b\hfill \\ \hfill & +{\displaystyle \frac{1}{\sigma }}(b_1{u}_1(b)-a_1{I}_{b^{-}}^{1-\alpha }{u}_2(b)).\left(sin{\theta }_2{\overline{v}}_1(b)-cos{\theta }_2\overline{I_{b^{-}}^{1-\alpha }{v}_2(b)}\right).\hfill \end{array}$$

Combining the first and third integrals, second and fourth integrals and inserting limit values, we obtain

$$\begin{array}{cc}\hfill {〈\mathcal{L}U,V〉}_{\mathbb{H}}& =\underset{a}{\overset{b}{\int }}\left[-\overline{{}_{ }{}^{C}D_{a^{+}}^{\alpha }{v}_1}-p_2{\overline{v}}_2\right]u_{2}dx+\underset{a}{\overset{b}{\int }}\left[\overline{D_{b^{-}}^{\alpha }{v}_2}-p_1{\overline{v}}_1\right]u_{1}dx\hfill \\ \hfill & +u_1(b)\overline{I_{b^{-}}^{1-\alpha }{v}_2(b)}-{\overline{v}}_1(b)I_{b^{-}}^{1-\alpha }{u}_2(b)-u_1(c+)\overline{I_{b^{-}}^{1-\alpha }{v}_2(c+)}\hfill \\ & +{\overline{v}}_1(c+)I_{b^{-}}^{1-\alpha }{u}_2(c+)+u_1(c-)\overline{I_{b^{-}}^{1-\alpha }{v}_2(c-)}-{\overline{v}}_1(c-)I_{b^{-}}^{1-\alpha }{u}_2(c-)\hfill \\ \hfill & -u_1(a)\overline{I_{b^{-}}^{1-\alpha }{v}_2(a)}+{\overline{v}}_1(a)I_{b^{-}}^{1-\alpha }{u}_2(a)\hfill \\ \hfill & +{\displaystyle \frac{1}{\sigma }}(b_1{u}_1(b)-a_1{I}_{b^{-}}^{1-\alpha }{u}_2(b)).\left(sin{\theta }_2{\overline{v}}_1(b)-cos{\theta }_2\overline{I_{b^{-}}^{1-\alpha }{v}_2(b)}\right).\hfill \end{array}$$

(15)

On the other hand, firstly, since U and V both satisfy the condition (2) at $x=a,$ we write

$${\overline{v}}_1(a)I_{b^{-}}^{1-\alpha }{u}_2(a)=u_1(a)\overline{I_{b^{-}}^{1-\alpha }{v}_2(a)}.$$

(16)

Secondly, the sum of first and second terms after integrals and the last term in (15) with the abbreviations in (13) gives

$$\begin{array}{c}{u}_1(b)\overline{I_{b^{-}}^{1-\alpha }{v}_2(b)}-\overline{v_1(b)}{I}_{b^{-}}^{1-\alpha }{u}_2(b)\hfill \\ \hfill +{\displaystyle \frac{1}{\sigma }}(b_1{u}_1(b)-a_1{I}_{b^{-}}^{1-\alpha }{u}_2(b)).\left(sin{\theta }_2{\overline{v}}_1(b)-cos{\theta }_2\overline{I_{b^{-}}^{1-\alpha }{v}_2(b)}\right)\end{array}$$

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{1}{\sigma }}(sin{\theta }_2{u}_1(b)-cos{\theta }_2{I}_{b^{-}}^{1-\alpha }{u}_2(b)).\left(b_1{\overline{v}}_1(b)-a_1\overline{I_{b^{-}}^{1-\alpha }{v}_2(b)}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\sigma }}\widehat{u}\overline{\tilde{v}}.\hfill \end{array}$$

(17)

From using (16) and (17) and the fact that u and v satisfy the same transmission conditions (4) and (5), the equality (15) becomes

$${〈\mathcal{L}U,V〉}_{\mathbb{H}}={〈U,\mathcal{L}V〉}_{\mathbb{H}}.$$

As a result, if we consider $\mathcal{L}$ as the operator defined above whose domain is $D(\mathcal{L})$, then $\mathcal{L}$ is symmetric. □

Because the operator $\mathcal{L}$ is symmetric in the space $\mathbb{H}$, we have the following orthogonality relation.

Corollary 1.

All eigenvalues of the problem (1)–(5) are real, and for each eigenvalue ${\lambda }_n$, there exists a vector-valued eigenfunction $u_n^T(x,{\lambda }_n)=(u_{1n}(x,{\lambda }_n),$ $u_{2n}(x,{\lambda }_n)).$ Also, the vector-valued eigenfunctions corresponding to different eigenvalues in the space $\mathbb{H}$ are orthogonal with respect to the inner product

$${〈u_n,u_m〉}_{\mathbb{H}}={\int }_a^c{u}_n^T{\overline{u}}_{m}dx+{\int }_c^b{u}_n^T{\overline{u}}_{m}dx+{\displaystyle \frac{1}{\sigma }}{\tilde{u}}_n{\overline{\widehat{u}}}_m=0.$$

It should be noted here that the above orthogonality relation does not hold the standard Hilbert space $L_2\left[a,b\right]$.

4. The Existence of Solutions to Fractional Dirac System

The main result of this section is a proof of the existence of the solution of a fractional Dirac system (1) with the boundary conditions (2) and (3) and the transmission conditions (4) and (5).

The existence of solutions to the fractional Dirac system (1)–(5) are stated in this section. For the proof of the following existence theorem, the iterative method will be used.

Theorem 2.

The fractional Dirac system (1) has a solution $F(x,\lambda )$ on $[a,b]$ satisfying the boundary conditions (2) and (3) and the transmission conditions (4) and (5).

Proof. 

First, consider the following fractional Dirac system

$$Au(x)-P(x)u(x)=\lambda u(x),x\in \left[a,c\right)$$

with the initial conditions

$$u_1(a)=cos{\theta }_1,u_2(a)=sin{\theta }_1$$

on $\left[a,c\right)$. Due to the existence of standard initial conditions, which allows us to use the arguments of the theory of differential equations (see [23]), and the proof being similar to that of the proof for $\left(c,b\right]$, we omit the case for $\left[a,c\right)$ without destroying the continuity of the proof, and we will only perform the proof only for the more complicated case $\left(c,b\right]$. Let us show the solution to a fractional Dirac system with $F_1(x,\lambda )={\left(F_{11}(x,\lambda ),F_{21}(x,\lambda )\right)}^T$ which is an entire function of $\lambda$ on $\left[a,c\right)$. Let us consider the fractional Dirac system

$$\begin{array}{ccc}\hfill D_{b^{-}}^{\alpha }{u}_2(x)-p_1(x)u_1(x)& =& \lambda u_1(x)\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {-}^C{D}_{c^{+}}^{\alpha }{u}_1(x)-p_2(x)u_2(x)& =& \lambda u_2(x),x\in \left(c,b\right]\hfill \end{array}$$

(19)

with the eigenparameter-dependent initial condition

$$u_1(c+)={\gamma }^{-1}{F}_{11}(c-,\lambda )$$

(20)

$$I_{b^{-}}^{1-\alpha }{u}_2(c+)=\gamma I_{b^{-}}^{1-\alpha }{F}_{21}(c-,\lambda ).$$

(21)

By applying the fractional integral of order $\alpha$ to both sides of (19) and (18), we obtain the following expressions for $u_1(x)$ and $u_2(x)$, respectively;

$$\begin{array}{c}{u}_1(x)=u_{10}(x,\lambda )+{\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{c}{\overset{x}{\int }}{(x-t)}^{\alpha -1}(p_2+\lambda )u_2(t)dt\hfill \\ u_2(x)=u_{20}(x,\lambda )+{\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{x}{\overset{b}{\int }}{(t-x)}^{\alpha -1}(p_1+\lambda )u_1(t)dt\hfill \end{array}$$

(22)

where $u_{10}(x,\lambda )=-u_1(c)$ and $u_{20}(x,\lambda )={\displaystyle \frac{{(b-x)}^{\alpha -1}}{\Gamma (\alpha )}}{I}_{c^{+}}^{\alpha }{u}_1(b)$. In the following, we apply the successive approximation method, which is useful for constructing a solution of the system of integral Equation (22). A function series ${\left\{u_n(x,\lambda )\right\}}_{n=1}^{\infty }$ defined as follows is necessary for this method,

$$\begin{array}{c}{u}_n(x,\lambda )=\left({\displaystyle \genfrac{}{}{0pt}{}{u_{1n}(x,\lambda )}{u_{2n}(x,\lambda )}}\right)\hfill \\ =\left({\displaystyle \genfrac{}{}{0pt}{}{u_{10}(x,\lambda )}{u_{20}(x,\lambda )}}\right)+\left(\begin{array}{c}-{\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{c}{\overset{x}{\int }}{(x-t)}^{\alpha -1}(p_2+\lambda )u_{2n-1}(t,\lambda )dt\\ {\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{x}{\overset{b}{\int }}{(t-x)}^{\alpha -1}(p_1+\lambda )u_{1n-1}(t,\lambda )dt\end{array}\right)\hfill \end{array}$$

(23)

where $u_{10}(x,\lambda )$ and $u_{20}(x,\lambda )$ are defined as above and each $u_n(x,\lambda )$ is an entire function of $\lambda$ for every $x\in \left(c,b\right]$. Now, set

$$\begin{array}{ccc}\hfill z_n(x,\lambda )& =& u_n(x,\lambda )-u_{n-1}(x,\lambda )\hfill \end{array}$$

(24)

where $z_n^T(x,\lambda )=(z_{1n}(x,\lambda ),z_{2n}(x,\lambda ))$. Then,

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill z_1(x,\lambda )& =\hfill & \hfill \left({\displaystyle \genfrac{}{}{0pt}{}{z_{11}(x,\lambda )}{z_{21}(x,\lambda )}}\right)=\left(\begin{array}{c}-{\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{c}{\overset{x}{\int }}{(x-t)}^{\alpha -1}(p_2+\lambda )u_{20}(t,\lambda )dt\\ {\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{x}{\overset{b}{\int }}{(t-x)}^{\alpha -1}(p_1+\lambda )u_{10}(t,\lambda )dt\end{array}\right)\end{array}\end{array}$$

and hence

$$\begin{array}{c}\parallel z_1(x,\lambda )\parallel \le {\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{c}{\overset{x}{\int }}\left|{(x-t)}^{\alpha -1}(p_2+\lambda )u_{20}(t,\lambda )\right|dt\hfill \\ +{\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{x}{\overset{b}{\int }}\left|{(t-x)}^{\alpha -1}(p_1+\lambda )u_{10}(t,\lambda )\right|dt,\hfill \end{array}$$

(25)

where $1-norm$ is used. Furthermore, let us write

$$M_1(\lambda )=ma{x}_{x\in \left(c,b\right]}|p_1(x)+\lambda |,M_2(\lambda )=ma{x}_{x\in \left(c,b\right]}|p_2(x)+\lambda |,$$

$$N_1=ma{x}_{x\in \left(c,b\right]}|u_{10}(x)|,N_2=ma{x}_{x\in \left(c,b\right]}\left|u_{20}(x)\right|,$$

and in closed contour $\{\lambda \in \mathbb{C}:|\lambda |\le R\}$

$$M_1:=ma{x}_{\left|\lambda \right|\le R}(M_1(\lambda )),M_2:=ma{x}_{\left|\lambda \right|\le R}(M_2(\lambda )).$$

Then, we have the following bound for (25)

$$\begin{array}{c}\hfill \parallel z_1(x,\lambda )\parallel \le {\displaystyle \frac{M_{R}N}{\alpha \Gamma (\alpha )}}\left[{(b-x)}^{\alpha }-{(x-c)}^{\alpha }\right]\end{array}$$

where $N:=max(N_1,N_2)$ and $M_R:=max(M_1,M_2)$. Similarly, inserting (23) into (24) for $n=2$ gives

$$z_2(x,\lambda )=\left({\displaystyle \genfrac{}{}{0pt}{}{z_{12}(x,\lambda )}{z_{22}(x,\lambda )}}\right)=\left(\begin{array}{cc}& {\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{c}{\overset{x}{\int }}{(x-t)}^{\alpha -1}(p_2+\lambda )\left[-u_{21}(t,\lambda )+u_{20}(t,\lambda )\right]dt\\ & {\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{x}{\overset{b}{\int }}{(t-x)}^{\alpha -1}(p_1+\lambda )\left[u_{11}(t,\lambda )-u_{10}(t,\lambda )\right]dt\end{array}\right).$$

Therefore, the norm of $z_2(x,\lambda )$ is

$$\begin{array}{c}\hfill \parallel z_2(x,\lambda )\parallel \le {\displaystyle \frac{1}{\Gamma (\alpha )}}[\underset{c}{\overset{x}{\int }}\left|{(x-t)}^{\alpha -1}(p_2+\lambda )(-u_{21}(t,\lambda )+u_{20}(t,\lambda ))\right|dt\\ \hfill +\underset{x}{\overset{b}{\int }}\left|{(t-x)}^{\alpha -1}(p_1+\lambda )(u_{11}(t,\lambda )-u_{10}(t,\lambda ))\right|dt]\end{array}$$

and hence

$$\begin{array}{c}\hfill \parallel z_2(x,\lambda )\parallel \le {\displaystyle \frac{1}{\Gamma (\alpha )}}[\underset{c}{\overset{x}{\int }}|{(x-t)}^{\alpha -1}{M}_2{\displaystyle \frac{M_{R}N}{\alpha \Gamma (\alpha )}}\left[{(b-x)}^{\alpha }-{(x-c)}^{\alpha }\right]|dt\\ \hfill +\underset{x}{\overset{b}{\int }}|{(t-x)}^{\alpha -1}{M}_1{\displaystyle \frac{M_{R}N}{\alpha \Gamma (\alpha )}}\left[{(b-x)}^{\alpha }-{(x-c)}^{\alpha }\right]|dt]\end{array}$$

so, we obtain

$$\begin{array}{c}\hfill \parallel z_2(x,\lambda )\parallel \le {\displaystyle \frac{M_R^{2}N}{{\alpha }^2{\Gamma }^2(\alpha )}}{\left[{(b-x)}^{\alpha }-{(x-c)}^{\alpha }\right]}^2.\end{array}$$

A simple induction argument shows that

$$\begin{array}{c}\hfill \parallel z_n(x,\lambda )\parallel \le {\displaystyle \frac{M_R^{n}N}{{\alpha }^n{\Gamma }^n(\alpha )}}{\left[{(b-x)}^{\alpha }-{(x-c)}^{\alpha }\right]}^n.\end{array}$$

(26)

Now, let us write an infinite series of the form

$$u_0(x,\lambda )+\sum _{k=1}^{\infty }{z}_k(x,\lambda ).$$

(27)

The n-th partial sum of this series is $u_n(x,\lambda )$, that is,

$$u_n(x,\lambda )=u_0(x,\lambda )+\sum _{k=1}^n{z}_k(x,\lambda ).$$

(28)

As a result, we can conclude that the sequence ${\left\{u_n(x,\lambda )\right\}}_{n=1}^{\infty }$ converges if and only if the series (27) converges. It follows from (26) that the series (27) is uniformly convergent with respect to x on $\left(c,b\right]$ and $\lambda$ in the closed contour $\left\{\lambda \in \mathbb{C}:\left|\lambda \right|\le R\right\}$. Let the sum of the series (27) be $F_2(x,\lambda )={(F_{12}(x,\lambda ),F_{22}(x,\lambda ))}^T$, i.e.,

$$F_2(x,\lambda )=u_0(x,\lambda )+\sum _{k=1}^{\infty }{z}_k(x,\lambda )$$

(29)

and so (28) gives

$$\underset{n\to \infty }{lim}{u}_n(x,\lambda )=F_2(x,\lambda ).$$

For the rest of proof, it remains to show that the limit function $F_2(x,\lambda )$ satisfies the fractional Dirac system (18) and (19). From (29) and the definition of the operator A, we have

$$\begin{array}{ccc}\hfill A{F}_2(x,\lambda )& =& \left({\displaystyle \genfrac{}{}{0pt}{}{{}_{ }{}^{C}D_{c^{+}}^{\alpha }{u}_{11}(x,\lambda )}{D_{b^{-}}^{\alpha }{u}_{21}(x,\lambda )}}\right)+\sum _{k=2}^{\infty }\left(\begin{array}{c}{}_{ }{}^{C}D_{c^{+}}^{\alpha }{z}_{1k}(x,\lambda )\\ D_{b^{-}}^{\alpha }{z}_{2k}(x,\lambda )\end{array}\right).\hfill \end{array}$$

(30)

If we put $n=1$ in (23) and take the appropriate derivatives, we write the first term on the right-hand side of (30) as follows,

$$\begin{array}{c}\left({\displaystyle \genfrac{}{}{0pt}{}{{}_{ }{}^{C}D_{c^{+}}^{\alpha }{u}_{11}(x,\lambda )}{D_{b^{-}}^{\alpha }{u}_{21}(x,\lambda )}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{{}_{ }{}^{C}D_{c^{+}}^{\alpha }{u}_{10}(x,\lambda )}{D_{b^{-}}^{\alpha }{u}_{20}(x,\lambda )}}\right)\hfill \\ +\left(\begin{array}{c}{}_{ }{}^{C}D_{c^{+}}^{\alpha }\left(-{\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{c}{\overset{x}{\int }}{(x-t)}^{\alpha -1}(p_2+\lambda )u_{20}(t,\lambda )dt\right)\\ D_{b^{-}}^{\alpha }\left({\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{x}{\overset{b}{\int }}{(t-x)}^{\alpha -1}(p_1+\lambda )u_{10}(t,\lambda )dt\right)\end{array}\right)\end{array}$$

and hence

$$\left({\displaystyle \genfrac{}{}{0pt}{}{{}_{ }{}^{C}D_{c^{+}}^{\alpha }{u}_{11}(x,\lambda )}{D_{b^{-}}^{\alpha }{u}_{21}(x,\lambda )}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{{}_{ }{}^{C}D_{c^{+}}^{\alpha }{u}_{10}(x,\lambda )}{D_{b^{-}}^{\alpha }{u}_{20}(x,\lambda )}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{(p_1+\lambda )u_{10}(x,\lambda )}{-(p_2+\lambda )u_{20}(x,\lambda )}}\right).$$

(31)

Similarly, by using (23), (24) and fractional derivative operations according to the definition of A, the second term on the right-hand side of (30) can be written

$$\begin{array}{c}\hfill \left(\begin{array}{c}{}_{ }{}^{C}D_{c^{+}}^{\alpha }{z}_{1k}(x,\lambda )\\ D_{b^{-}}^{\alpha }{z}_{2k}(x,\lambda )\end{array}\right)=\left(\begin{array}{c}(p_1+\lambda )z_{1k-1}(x,\lambda )\\ -(p_2+\lambda )z_{2k-1}(x,\lambda )\end{array}\right).\end{array}$$

(32)

Writing (32) in series form and then re-indexing, we obtain

$$\sum _{k=2}^{\infty }\left(\begin{array}{c}{}_{ }{}^{C}D_{c^{+}}^{\alpha }{z}_{1k}(x,\lambda )\\ D_{b^{-}}^{\alpha }{z}_{2k}(x,\lambda )\end{array}\right)=\left(\begin{array}{cc}0& -(p_2+\lambda )\\ (p_1+\lambda )& 0\end{array}\right)\sum _{k=1}^{\infty }\left(\begin{array}{c}{z}_{1k}(x,\lambda )\\ z_{2k}(x,\lambda )\end{array}\right).$$

(33)

Substituting (31) and (33) into (30), we have

$$\begin{array}{ccc}\hfill A{F}_2(x,\lambda )& =& \left({\displaystyle \genfrac{}{}{0pt}{}{{}_{ }{}^{C}D_{c^{+}}^{\alpha }{u}_{10}(x,\lambda )}{D_{b^{-}}^{\alpha }{u}_{20}(x,\lambda )}}\right)\hfill \\ & +& \left(\begin{array}{cc}0& -(p_2+\lambda )\\ (p_1+\lambda )& 0\end{array}\right)\left[\left(\begin{array}{c}{u}_{10}(x,\lambda )\\ u_{20}(x,\lambda )\end{array}\right)+\sum _{k=1}^{\infty }\left(\begin{array}{c}{}_{ }{}^{C}D_{c^{+}}^{\alpha }{z}_{1k}(x,\lambda )\\ D_{b^{-}}^{\alpha }{z}_{2k}(x,\lambda )\end{array}\right)\right]\hfill \end{array}$$

and hence

$$A{F}_2(x,\lambda )=\left({\displaystyle \genfrac{}{}{0pt}{}{{}_{ }{}^{C}D_{c^{+}}^{\alpha }{u}_{10}(x,\lambda )}{D_{b^{-}}^{\alpha }{u}_{20}(x,\lambda )}}\right)+\left(\begin{array}{cc}0& -(p_2+\lambda )\\ (p_1+\lambda )& 0\end{array}\right)\left(\begin{array}{c}{F}_{12}(x,\lambda )\\ F_{22}(x,\lambda )\end{array}\right).$$

(34)

Since $u_{10}(x,\lambda )$ is a constant function, ${}_{ }{}^{C}D_{c^{+}}^{\alpha }{u}_{10}(x,\lambda )=0$. On the other hand, from the definition of a Riemann–Liouville derivative, we write

$$\begin{array}{c}\hfill D_{b^{-}}^{\alpha }{u}_{20}(x,\lambda )={\displaystyle \frac{I_{c^{+}}^{\alpha }{u}_1(b)}{\Gamma (n-\alpha )\Gamma (\alpha )}}{\displaystyle \frac{d^n}{d{x}^n}}\underset{x}{\overset{b}{\int }}{(t-x)}^{n-\alpha -1}{(b-t)}^{\alpha -1}dt\end{array}$$

Performing the substitution $t=x-u(x-b)$ in the above integral, we obtain

$$\begin{array}{c}\hfill D_{b^{-}}^{\alpha }{u}_{20}(x,\lambda )=-{\displaystyle \frac{I_{c^{+}}^{\alpha }{u}_1(b)}{\Gamma (n-\alpha )\Gamma (\alpha )}}{\displaystyle \frac{d^n}{d{x}^n}}\left[{(x-b)}^{n-1}\right]\underset{0}{\overset{1}{\int }}{(-u)}^{n-\alpha -1}{(u-1)}^{\alpha -1}du.\end{array}$$

Since the n-th derivative of the polynomial of order $(n-1)$ is zero, we obtain $D_{b^{-}}^{\alpha }{u}_{20}(x,\lambda )=0$. So, we conclude that the first term on the right-hand side of (34) is zero. This means that $F_2(x,\lambda )$ is the solution of the fractional Dirac system (18) and (19) on $\left(c,b\right]$. It is also clear that $F_2(x,\lambda )$ satisfies the initial conditions (20) and (21). Consequently, the function $F(x,\lambda )$ defined by

$$F(x,\lambda )=\left\{\begin{array}{cc}{F}_1^T(x,\lambda )=(F_{11}(x,\lambda ),F_{21}(x,\lambda )),\hfill & x\in \left[a,c\right)\hfill \\ F_2^T(x,\lambda )=(F_{12}(x,\lambda ),F_{22}(x,\lambda )),\hfill & x\in \left(c,b\right]\hfill \end{array}\right.$$

satisfies the fractional Dirac system (1) with the boundary conditions (2) and (3) and the transmission conditions (4) and (5). □

Now, we can give the general form of the solution to the fractional Dirac system in the following corollary.

Corollary 2.

For $\lambda \in \mathbb{C}$, the eigenvalue problem (14) has a solution

$$G(x,\lambda )=\left\{\begin{array}{cc}{G}_1^T(x,\lambda )=(G_{11}(x,\lambda ),G_{21}(x,\lambda )),\hfill & x\in \left[a,c\right)\hfill \\ G_2^T(x,\lambda )=(G_{12}(x,\lambda ),G_{22}(x,\lambda )),\hfill & x\in \left(c,b\right]\hfill \end{array}\right.$$

and therefore the fractional Dirac system

$$\begin{array}{c}\hfill D_{b^{-}}^{\alpha }{u}_2(x)-p_1(x)u_1(x)=\lambda u_1(x)\\ \hfill {-}^C{D}_{a^{+}}^{\alpha }{u}_1(x)-p_2(x)u_2(x)=\lambda u_2(x)\end{array}$$

has a solution $u(x)={(u_1(x),u_2(x))}^T$ on $\left[a,c\right)\cup \left(c,b\right]$ satisfying the boundary conditions (2) and (3) and the transmission conditions (4) and (5).

5. Conclusions

In this article, we have extended some spectral features of the Dirac operators to fractional Dirac systems. The most crucial aspect here is that not only one derivative in the operator is fractional, but rather both derivatives are fractional of different types, and the ordinary derivative is not used at all. In the second part of the article, the existence of a solution of the fractional Dirac system is studied. It is interesting that the method of successive approximations used in the existence theorem also works in the fractional case. According to us, the results obtained above can be of interest to specialists.