One-dimensional conformable fractional Dirac system

Abstract

In this article, we consider a conformable fractional Dirac system. We prove an existence and uniqueness theorem for this system and formulate a self-adjoint boundary-value problem. We also construct the associated Green matrix of the conformable fractional Dirac system, and we give the eigenfunction expansions. Finally, we give some examples.

Introduction

The one-dimensional Dirac system is one of the important equations in quantum mechanics since it formulates the fundamental physics of realistic quantum mechanics. Furthermore, it gives a description of the electron spin and predicts the existence of antimatter. Some properties of the Dirac and q-Dirac systems have been considered in the literature (see [8, 24, 28, 30]).

Conformable fractional calculus is one of developing areas of fractional calculus. The study of conformable fractional calculus is initiated by Khalil et al. [21]. In [21], the authors defined a new simple fractional derivative called the conformable fractional derivative. Later in [1], Abdeljawad defined the right and left conformable fractional derivatives, the fractional chain rule and fractional integrals of higher orders. In recent years, this topic has attracted the attention of several scholars, and a variety of new results can be found [1,2,3,4,5,6,7, 9,10,11,12,13,14,15,16,17,18,19,20,21,22, 26, 27, 29]. However, to the best of our knowledge, the study of conformable fractional Dirac system has received considerably less attention. In [18], Gulsen et al. studied the conformable fractional Dirac system with separated boundary conditions on an arbitrary timescale \({\mathbb {T}}\). They extended some basic spectral properties of the classical Dirac system to the conformable fractional case.

The aim of the present paper is to study a conformable fractional Dirac system of the form

$$\begin{aligned} \left( \begin{array} [c]{cc} 0 &{} -T_{\alpha }\\ T_{\alpha } &{} 0 \end{array} \right) \left( \begin{array} [c]{c} y_{1}\\ y_{2} \end{array} \right) +\left( \begin{array} [c]{cc} p\left( x\right) &{} 0\\ 0 &{} r\left( x\right) \end{array} \right) \left( \begin{array} [c]{c} y_{1}\\ y_{2} \end{array} \right) =\lambda \left( \begin{array} [c]{c} y_{1}\\ y_{2} \end{array} \right) , \end{aligned}$$
(1)

where \(\lambda \) is a complex spectral parameter and \(p\left( .\right) \) and \(r\left( .\right) \) are real-valued continuous and conformable fractional integrable functions on \(\left[ a,b\right] ,\ 0\le x\le b<\infty \). We will study the existence and uniqueness of the solution of this system. Later, we will investigate some spectral properties of the problem, such as formal self-adjointness, orthogonality of eigenvectors, Green’s matrix, eigenvectors forming an orthonormal basis.

The rest of this paper is organized as follows.

In Sect. 2, we give definitions that are used throughout the paper. In Sect. 3, we consider the conformable fractional Dirac system, where we prove an existence and uniqueness theorem for these systems. In Sect. 4, we formulate a self-adjoint conformable fractional Dirac system. In Sect. 5, we construct the associated Green matrix of the conformable fractional Dirac system. In Sect. 6, we give the eigenfunction expansions. In the last section, we give some illustrative examples.

Preliminaries

In this section, we want to recall some necessary definitions and properties of conformable fractional calculus (see [1, 21]).

Definition 1

[1, 21] Let \(\alpha \ \)be a positive number with \(0<\alpha <1.\ \)A function \(f:[0,\infty )\rightarrow {\mathbb {R}} \) the conformable fractional derivative of order \(\alpha \) of f at \(x>0\) was defined by

$$\begin{aligned} T_{\alpha }f(x)=\underset{\varepsilon \rightarrow 0}{\lim }\frac{f(x+\varepsilon x^{1-\alpha })-f(x)}{\varepsilon }, \end{aligned}$$
(2)

and the fractional derivative at 0 is defined

$$\begin{aligned} (T_{\alpha }f)(0)=\lim _{x\rightarrow 0^{+}}(T_{\alpha }f(x)). \end{aligned}$$

Definition 2

[1, 21] The left conformable fractional derivative starting from a of a function \(f:[a,\infty )\rightarrow {\mathbb {R}}\) of order \(\alpha \) is defined by

$$\begin{aligned} (T_{\alpha }^{a}f)(x)=\underset{\varepsilon \rightarrow 0}{\lim }\frac{f(x+\varepsilon \left( x-a\right) ^{1-\alpha })-f\left( x\right) }{\varepsilon },\quad ~0<\alpha \le 1. \end{aligned}$$
(3)

When \(a=0\), we write \(T_{\alpha }\). If \((T_{\alpha }f)(x)\) exists on (ab) then

$$\begin{aligned} (T_{\alpha }^{a}f)(a)=\underset{x\rightarrow a^{+}}{\lim }(T_{\alpha }^{a}f)(x). \end{aligned}$$

Definition 3

[1, 21] The right conformable fractional derivative of order \(0<\alpha \le 1\) of f is defined by

$$\begin{aligned} (_{\alpha }^{b}Tf)(x)=-\underset{\varepsilon \rightarrow 0}{\lim }\frac{f(x+\varepsilon (b-x)^{1-\alpha })-f(x)}{\varepsilon }, \end{aligned}$$
(4)

where f is terminating at b and \((^{b}T_{\alpha }f)(x)=\lim _{x\rightarrow b^{-}}(^{b}T_{\alpha }f)(x).\)

Lemma 4

[1, 21] Let fg be conformable differentiable of order \(\alpha \ \left( 0<\alpha \le 1\right) \) at a point \(x.\ \)Then,

(i):

\(T_{\alpha }\left( \lambda f+\beta g\right) =\lambda T_{\alpha }\left( f\right) +\beta T_{\alpha }\left( g\right) ,\ \lambda ,\beta \in {\mathbb {R}},\)

(ii):

 \(T_{\alpha }\left( fg\right) =fT_{\alpha }\left( g\right) +gT_{\alpha }\left( f\right) ,\)

( iii):

\(T_{\alpha }\left( \frac{f}{g}\right) =\frac{gT_{\alpha }\left( f\right) -fT_{\alpha }\left( g\right) }{g^{2}},\)

( iv):

fis differentiable then\(T_{\alpha }^{a}\left( f\right) \left( x\right) =\left( x-a\right) ^{1-\alpha }f^{\prime }\left( x\right) \),

( v):

\(T_{\alpha }\left( x^{n}\right) =nx^{n-\alpha }\), for all\(n\in {\mathbb {R}}.\)

Definition 5

[1, 21] The conformable fractional integral starting from a of a function f of order \(0<\alpha \le 1\) is defined by

$$\begin{aligned} (I_{\alpha }^{a}f)\left( x\right) = {\displaystyle \int \limits _{a}^{x}} f(t)\mathrm{d}\alpha \left( t,a\right) = {\displaystyle \int \limits _{a}^{x}} (t-a)^{\alpha -1}f(t)\mathrm{d}t. \end{aligned}$$

Similarly, in the right case we have

$$\begin{aligned} (^{b}I_{\alpha }f)\left( x\right) = {\displaystyle \int \limits _{x}^{b}} f(t)\mathrm{d}\alpha \left( b,t\right) = {\displaystyle \int \limits _{x}^{b}} (b-t)^{\alpha -1}f(t)\mathrm{d}t. \end{aligned}$$

Lemma 6

[1, 21] Assume that f is a continuous function on \(\left( a,\infty \right) \) and \(0<\alpha <1.\) Then, we have

$$\begin{aligned} T_{\alpha }^{a}I_{\alpha }^{a}f\left( x\right) =f\left( x\right) , \end{aligned}$$

for all \(x>a.\)

Theorem 7

[1, 21] Let \(f,g:[a,b]\rightarrow {\mathbb {R}}\) be two functions such that f and g are conformable fractional differentiable. Then, we have

$$\begin{aligned} \int _{a}^{b}f(x)T_{\alpha }^{a}\left( g\right) \left( x\right) \mathrm{d}\alpha \left( x,a\right) +\int _{a}^{b}g\left( x\right) T_{\alpha } ^{a}\left( f\right) \left( x\right) \mathrm{d}\alpha \left( x,a\right) =f\left( b\right) g\left( b\right) -f\left( a\right) g\left( a\right) . \end{aligned}$$

Let \(L_{\alpha }^{2}(a,b)\ \)be the space of all complex-valued functions defined on [ab] such that

$$\begin{aligned} \left\| f\right\| :=\left( \int _{a}^{b}\left| f\left( x\right) \right| ^{2}\mathrm{d}^{\alpha }x\right) ^{1/2}=\left( \int _{a}^{b}\left| f\left( x\right) \right| ^{2}x^{\alpha -1}\mathrm{d}x\right) ^{1/2}<\infty . \end{aligned}$$

The space \(L_{\alpha }^{2}(a,b)\ \)is a Hilbert space (see [29]) with the inner product

$$\begin{aligned} \left( f,g\right) _{1}:=\int _{a}^{b}f\left( x\right) \overline{g\left( x\right) }\mathrm{d}^{\alpha }x,\ f,g\in L_{\alpha }^{2}(a,b). \end{aligned}$$

Conformable fractional Dirac systems

In this section, we prove an existence theorem for the one- dimensional conformable fractional Dirac system. Consider this system:

$$\begin{aligned} \tau y:= & {} \left\{ \begin{array} [c]{c} -T_{\alpha }y_{2}+p\left( x\right) y_{1}\\ T_{\alpha }y_{1}+r\left( x\right) y_{2}, \end{array} \right. \nonumber \\ \tau y= & {} \lambda y,\ y=\left( \begin{array} [c]{c} y_{1}\\ y_{2} \end{array} \right) ,\quad 0\le x\le b<\infty , \end{aligned}$$
(5)

where \(\lambda \) is a complex parameter, \(p\left( .\right) \) and \(r\left( .\right) \) are real-valued continuous and conformable fractional integrable functions on \(\left[ 0,b\right] \).

Now, we introduce convenient Hilbert space \({\mathcal {H}}=L_{\alpha } ^{2}((0,b);{\mathbb {C}}^{2})\ \)of vector-valued functions using the inner product

$$\begin{aligned} \left( f,g\right) :=\int _{0}^{b}(f(x),g(x))_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }x, \end{aligned}$$

where \(\left( .,.\right) _{{\mathbb {C}}^{2}}\) denotes the standard inner product in \({\mathbb {C}}^{2}\):

$$\begin{aligned} \left( \xi ,\gamma \right) _{{\mathbb {C}}^{2}}=\sum _{j=1}^{2}\xi _{j} {\overline{\gamma }}_{j}. \end{aligned}$$

Theorem 8

For \(c_{1},c_{2}\in {\mathbb {C}},\) Eq. (5) has a unique solution

$$\begin{aligned} \Phi \left( x,\lambda \right) =\left( \begin{array} [c]{c} \Phi _{1}\left( x,\lambda \right) \\ \Phi _{2}\left( x,\lambda \right) \end{array} \right) \end{aligned}$$

in \({\mathcal {H}}\) which satisfies

$$\begin{aligned} \Phi _{1}\left( 0,\lambda \right) =c_{1},\ \Phi _{2}\left( 0,\lambda \right) =c_{2},\ \lambda \in {\mathbb {C}}. \end{aligned}$$
(6)

For each \(x\in \left[ 0,b\right] ,\)\(\Phi \left( x,\lambda \right) \) is an entire function of \(\lambda .\)

Proof

Let \(p\left( x\right) =r\left( x\right) =0.\) Then, the solution of Eq. (5) is

$$\begin{aligned} U_{z}\left( x,\lambda \right) =\left( \begin{array} [c]{c} U_{z1}\left( x,\lambda \right) \\ U_{z2}\left( x,\lambda \right) \end{array} \right) =c_{1}\varphi _{1}\left( x,\lambda \right) +c_{2}\varphi _{2}\left( x,\lambda \right) \ \left( x\in \left[ 0,b\right] ,\ \lambda \in {\mathbb {C}}\right) , \end{aligned}$$

where

$$\begin{aligned} \varphi _{1}\left( x,\lambda \right) =\left( \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ -\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) \quad \text { and }\quad \varphi _{2}\left( x,\lambda \right) =\left( \begin{array} [c]{c} \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) , \quad \ \lambda \in {\mathbb {C}}. \end{aligned}$$

By using the method of variation of the constants, we find the following system of integral equations:

$$\begin{aligned} U\left( x,\lambda \right)&=c_{1}\varphi _{1}\left( x,\lambda \right) +c_{2}\varphi _{2}\left( x,\lambda \right) \\&\quad +\int _{0}^{x}\left[ \varphi _{2}\left( x,\lambda \right) \varphi _{1} ^\mathrm{T}\left( t,\lambda \right) -\varphi _{1}\left( x,\lambda \right) \varphi _{2}^\mathrm{T}\left( t,\lambda \right) \right] \Omega \left( t\right) U\left( t,\lambda \right) \mathrm{d}^{\alpha }t \end{aligned}$$

where \(x\in \left[ 0,b\right] ,\ \lambda \in {\mathbb {C}},\ \mathrm{T}\) denotes the matrix transpose and

$$\begin{aligned} \Omega \left( x\right) =\left( \begin{array} [c]{cc} p\left( x\right) &{} 0\\ 0 &{} r\left( x\right) \end{array} \right) . \end{aligned}$$

\(\square \)

For \(x\in \left[ 0,b\right] \ \)and\(\ \lambda \in {\mathbb {C}},\) we have

$$\begin{aligned} U\left( x,\lambda \right)&=\left( \begin{array} [c]{c} U_{1}\left( x,\lambda \right) \\ U_{2}\left( x,\lambda \right) \end{array} \right) =U_{z}\left( x,\lambda \right) \nonumber \\&\quad +\left( \begin{array} [c]{c} \int _{0}^{x}\left[ \begin{array} [c]{c} -\cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] p\left( t\right) U_{1}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\ \int _{0}^{x}\left[ \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] p\left( t\right) U_{1}\left( t,\lambda \right) \mathrm{d}^{\alpha }t \end{array} \right) \nonumber \\&\quad -\left( \begin{array} [c]{c} +\int _{0}^{x}\left[ \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] r\left( t\right) U_{2}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\ +\int _{0}^{x}\left[ \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ -\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] r\left( t\right) U_{2}\left( t,\lambda \right) \mathrm{d}^{\alpha }t \end{array} \right) . \end{aligned}$$
(7)

By using the method of successive approximations, we will construct a solution of Eq. (7). Let us define a sequence \(\left\{ U_{n}\left( x,\lambda \right) \right\} _{n=1}^{\infty }\) as follows:

$$\begin{aligned} U_{n+1}\left( x,\lambda \right)= & {} U_{z}\left( x,\lambda \right) \\&+\left( \begin{array} [c]{c} \int _{0}^{x}\left[ \begin{array} [c]{c} -\cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] p\left( t\right) U_{1n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\ \int _{0}^{x}\left[ \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] p\left( t\right) U_{1n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t \end{array} \right) \\&-\left( \begin{array} [c]{c} \int _{0}^{x}\left[ \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] r\left( t\right) U_{2n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\ -\int _{0}^{x}\left[ \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ -\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] r\left( t\right) U_{2n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t \end{array} \right) , \end{aligned}$$

where \(\ x\in \left[ 0,b\right] \ \)and\(\ \lambda \in {\mathbb {C}}.\ \)

Now, we will show that for each fixed \(\lambda \in {\mathbb {C}}\) the uniform limit of the sequence\(\ \left\{ U_{n}\left( x,\lambda \right) \right\} _{n=1}^{\infty }\) (as \(n\rightarrow \infty )\) exists and defines the solution of (5) and (6).

Let \(\lambda \in {\mathbb {C}}\) be fixed. There exist positive numbers \(N\left( \lambda \right) \) and A such that

$$\begin{aligned} \left| \varphi _{ij}\left( x,\lambda \right) \right|&\le \frac{\sqrt{N\left( \lambda \right) }}{2},\ i,j=1,2,\ \\ \max _{x\in \left[ 0,b\right] }\left| p\left( x\right) \right|&=A_{1},\ \max _{x\in \left[ 0,b\right] }\left| r\left( x\right) \right| =A_{2},\ A=\max \left\{ A_{1},A_{2}\right\} ,\\ \left| U_{1j}\left( x,\lambda \right) \right|&\le \left( \left| c_{1}\right| +\left| c_{2}\right| \right) \frac{\sqrt{N\left( \lambda \right) }}{2}=\frac{\widetilde{N\left( \lambda \right) }}{2},\quad j=1,2,\ x\in \left[ 0,b\right] . \end{aligned}$$

Then,

$$\begin{aligned}&\left\| U_{2}\left( x,\lambda \right) -U_{1}\left( x,\lambda \right) \right\| \nonumber \\&\quad \le \left| \begin{array} [c]{c} \int _{0}^{x}\left[ \begin{array} [c]{c} -\cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] p\left( t\right) U_{11}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\ +\int _{0}^{x}\left[ \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] r\left( t\right) U_{12}\left( t,\lambda \right) \mathrm{d}^{\alpha }t \end{array} \right| \nonumber \\&\qquad +\left| \begin{array} [c]{c} \int _{0}^{x}\left[ \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] p\left( t\right) U_{11}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\ -\int _{0}^{x}\left[ \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ -\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] r\left( t\right) U_{12}\left( t,\lambda \right) \mathrm{d}^{\alpha }t \end{array} \right| \nonumber \\&\quad \le \left( N\left( \lambda \right) A_{1}\frac{\widetilde{N\left( \lambda \right) }}{2}+N\left( \lambda \right) A_{2}\frac{\widetilde{N\left( \lambda \right) }}{2}\right) \left| \int _{0}^{x}\mathrm{d}^{\alpha }t\right| \nonumber \\&\qquad +\left( N\left( \lambda \right) A_{1}\frac{\widetilde{N\left( \lambda \right) }}{2}+N\left( \lambda \right) A_{2}\frac{\widetilde{N\left( \lambda \right) }}{2}\right) \left| \int _{0}^{x}\mathrm{d}^{\alpha }t\right| \nonumber \\&\quad \le 2N\left( \lambda \right) A\widetilde{N\left( \lambda \right) } \frac{x^{\alpha }}{\alpha }, \end{aligned}$$
(8)

where the norm \(\left\| .\right\| \) is the convenient norm in \({\mathbb {C}}^{2}\). Similarly, we have

$$\begin{aligned} \left\| U_{3}\left( x,\lambda \right) -U_{2}\left( x,\lambda \right) \right\| \le 2^{2}\widetilde{N\left( \lambda \right) }\frac{A^{2} N^{2}\left( \lambda \right) x^{2\alpha }}{2\alpha ^{2}}, \end{aligned}$$
(9)

and by using mathematical induction, we obtain

$$\begin{aligned} \left\| U_{n+1}\left( x,\lambda \right) -U_{n}\left( x,\lambda \right) \right\| \le 2^{n}(\widetilde{N\left( \lambda \right) })^{n} \frac{(AN\left( \lambda \right) x^{\alpha })^{n}}{\alpha ^{n}n!}\ (n\in {\mathbb {N}}:=\{1,2,...\}). \end{aligned}$$

Since the series

$$\begin{aligned} \sum _{n=1}^{\infty }2^{n}(\widetilde{N\left( \lambda \right) })^{n} \frac{(AN\left( \lambda \right) x^{\alpha })^{n}}{\alpha ^{n}n!} \end{aligned}$$

is uniformly convergent, by Weierstrass’s test, the series

$$\begin{aligned} U_{1}\left( x,\lambda \right) +\sum _{n=1}^{\infty }\left\{ U_{n+1}\left( x,\lambda \right) -U_{n}\left( x,\lambda \right) \right\} \end{aligned}$$
(10)

converges uniformly on \(\left[ 0,b\right] .\) Then, the nth partial sum

$$\begin{aligned} U_{n}\left( x,\lambda \right) =U_{1}\left( x,\lambda \right) +\sum _{k=1}^{n-1}\left\{ U_{k+1}\left( x,\lambda \right) -U_{k}\left( x,\lambda \right) \right\} \end{aligned}$$

of this series approaches a function \(\Phi \left( x,\lambda \right) \) uniformly on \(\left[ 0,b\right] \) as \(n\rightarrow \infty ,\) i.e.,

$$\begin{aligned} \lim _{n\rightarrow \infty }U_{n}\left( x,\lambda \right) =\Phi \left( x,\lambda \right) . \end{aligned}$$

By using induction on n, we can prove that \(U_{n}\left( x,\lambda \right) \) and \(T_{\alpha }U_{n}\left( x,\lambda \right) \) are continuous, where

$$\begin{aligned} T_{\alpha }U_{n}\left( x,\lambda \right)&=c_{1}T_{\alpha }\varphi _{1}\left( x,\lambda \right) +c_{2}T_{\alpha }\varphi _{2}\left( x,\lambda \right) \\&\quad +\int _{0}^{x}\left[ T_{\alpha }\varphi _{2}\left( x,\lambda \right) \varphi _{1}^\mathrm{T}\left( t,\lambda \right) -T_{\alpha }\varphi _{1}\left( x,\lambda \right) \varphi _{2}^\mathrm{T}\left( t,\lambda \right) \right] \Omega \left( t\right) U_{n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t \end{aligned}$$

and\(\ n=1,2,....\ \)We check at once that \(U_{n}\left( x,\lambda \right) \) and \(T_{\alpha }U_{n}\left( x,\lambda \right) \) are continuous, i.e., \(\Phi \left( x,\lambda \right) \in {\mathcal {H}}.\)

It is clear that \(\Phi \left( x,\lambda \right) \) satisfies (6), i.e.,

$$\begin{aligned} \Phi \left( 0,\lambda \right) =\left( \begin{array} [c]{c} \Phi _{1}\left( 0,\lambda \right) \\ \Phi _{2}\left( 0,\lambda \right) \end{array} \right) =\left( \begin{array} [c]{c} c_{1}\\ c_{2} \end{array} \right) . \end{aligned}$$

Now we prove that the function \(\Phi \left( x,\lambda \right) \) satisfies Eq. (5). If \(x\ne 0,\) then

$$\begin{aligned} T_{\alpha }\Phi \left( x,\lambda \right)&=\left( \begin{array} [c]{c} T_{\alpha }\Phi _{1}\left( x,\lambda \right) \\ T_{\alpha }\Phi _{2}\left( x,\lambda \right) \end{array} \right) =c_{1}T_{\alpha }\varphi _{1}\left( x,\lambda \right) +c_{2}T_{\alpha }\varphi _{2}\left( x,\lambda \right) \\&\quad +\int _{0}^{x}\left[ T_{\alpha }\varphi _{2}\left( x,\lambda \right) \varphi _{1}^\mathrm{T}\left( t,\lambda \right) -T_{\alpha }\varphi _{1}\left( x,\lambda \right) \varphi _{2}^\mathrm{T}\left( t,\lambda \right) \right] \Omega \left( t\right) \Phi \left( t,\lambda \right) \mathrm{d}^{\alpha }t. \end{aligned}$$

By Eq. (5), we get

$$\begin{aligned} T_{\alpha }\cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right)&=-\lambda \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) ,\\ T_{\alpha }\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right)&=\lambda \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) . \end{aligned}$$

Therefore, we get

$$\begin{aligned} T_{\alpha }\Phi _{1}\left( x,\lambda \right)= & {} c_{1}T_{\alpha }\cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) +c_{2}T_{\alpha }\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\&+\int _{0}^{x}\left[ \begin{array} [c]{c} -T_{\alpha }\cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0}^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +T_{\alpha }\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0}^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] p\left( t\right) \Phi _{1}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\&-\int _{0}^{x}\left[ \begin{array} [c]{c} T_{\alpha }\cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0}^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +T_{\alpha }\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0}^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] r\left( t\right) \Phi _{2}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\&-\,r(x)\Phi _{2}\left( x,\lambda \right) =-c_{1}\lambda \sin \left( \int _{0} ^{x}\lambda \mathrm{d}^{\alpha }t\right) +c_{2}\lambda \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\&+\int _{0}^{x}\left[ \begin{array} [c]{c} \lambda \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0}^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ +\lambda \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0}^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] p\left( t\right) \Phi _{1}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\&-\int _{0}^{x}\left[ \begin{array} [c]{c} -\lambda \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0}^{t}\lambda \mathrm{d}^{\alpha }s\right) \\ \lambda \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0}^{t}\lambda \mathrm{d}^{\alpha }s\right) \end{array} \right] r\left( t\right) \Phi _{2}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\&\quad -\,r(x)\Phi _{2}\left( x,\lambda \right) =\left( -r\left( x\right) +\lambda \right) \Phi _{2}\left( x,\lambda \right) . \end{aligned}$$

The other equation in (5) is proved similarly.

Now we will show that the problem defined by (5) and (6) has a unique solution. Assume that \(\psi _{1}\left( x,\lambda \right) \ \) and\(\ \psi _{2}\left( x,\lambda \right) \) are two solutions of this problem. Let us define \(\chi \left( x,\lambda \right) =\psi _{1}\left( x,\lambda \right) -\psi _{2}\left( x,\lambda \right) \ \)where\(\ x\in \left[ 0,b\right] .\) Then, \(\chi \left( x,\lambda \right) \) is a solution of this problem. It is evident that \(\chi _{1}\left( 0,\lambda \right) =\chi _{2}\left( 0,\lambda \right) =0.\) An easy computation shows that

$$\begin{aligned} \chi \left( x,\lambda \right) =\left( \begin{array} [c]{c} \chi _{1}\left( x,\lambda \right) \\ \chi _{2}\left( x,\lambda \right) \end{array} \right) =\left( \begin{array} [c]{c} -\int _{0}^{x}\left( -r\left( t\right) +\lambda \right) \chi _{2}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\\ \\ \int _{0}^{x}\left( -p\left( t\right) +\lambda \right) \chi _{1}\left( t,\lambda \right) \mathrm{d}^{\alpha }t \end{array} \right) . \end{aligned}$$
(11)

Since \(\chi \left( x,\lambda \right) ,\ r\left( x\right) ,\ p\left( x\right) \) are continuous, there exist positive numbers \(N_{x,\lambda }\) and \(M_{x,\lambda }\) such that

$$\begin{aligned} \sup _{x\in [0,b]}\left| \chi _{1}\left( x,\lambda \right) \right|&=N_{\lambda }^{\left( 1\right) },\ \sup _{x\in [0,b]}\left| \chi _{2}\left( x,\lambda \right) \right| =N_{\lambda }^{\left( 2\right) },\ \\ N_{\lambda }&=\max \left\{ N_{\lambda }^{\left( 1\right) },N_{\lambda }^{\left( 2\right) }\right\} ,\\ \sup _{x\in [0,b]}\left| \lambda -r\left( x\right) \right|&=M_{\lambda }^{\left( 1\right) },\ \sup _{x\in [0,b]}\left| \lambda -p\left( x\right) \right| =M_{\lambda }^{\left( 2\right) },\ \\ M_{\lambda }&=\max \left\{ M_{\lambda }^{\left( 1\right) },M_{\lambda }^{\left( 2\right) }\right\} . \end{aligned}$$

Then, we have

$$\begin{aligned} \left\| \chi \left( x,\lambda \right) \right\|&=\left| -\int _{0}^{x}\left( -r\left( t\right) +\lambda \right) \chi _{2}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\right| \\&+\left| \int _{0}^{x}\left( -p\left( t\right) +\lambda \right) \chi _{1}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\right| \\&\le 2N_{\lambda }M_{\lambda }\left| \int _{0}^{x}\mathrm{d}^{\alpha }t\right| =2N_{\lambda }M_{\lambda }\frac{x^{\alpha }}{\alpha }. \end{aligned}$$

Applying mathematical induction on k,  we obtain

$$\begin{aligned} \left\| \chi \left( x,\lambda \right) \right\| \le 2^{k}M_{\lambda } ^{k}N_{\lambda }^{k}\frac{x^{k\alpha }}{\alpha ^{k}k!}, \end{aligned}$$
(12)

where \(k\in \mathbb {N\ }\)and\(\ x\in \left[ 0,b\right] .\)

Assuming (11) to hold for k,  we will prove it for \(k+1.\) According to (11), we have

$$\begin{aligned} \left\| \chi \left( x,\lambda \right) \right\|&=\left| -\int _{0}^{x}\left( -r\left( t\right) +\lambda \right) \chi _{2}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\right| \\&\quad +\left| \int _{0}^{x}\left( -p\left( t\right) +\lambda \right) \chi _{1}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\right| \\&\le 2^{k+1}M_{\lambda }^{k+1}N_{\lambda }^{k+1}\left| \int _{0}^{x} \frac{t^{k\alpha }}{\alpha ^{k}k!}\mathrm{d}^{\alpha }t\right| \\&=2^{k+1}M_{\lambda }^{k+1}N_{\lambda }^{k+1}\frac{x^{\left( k+1\right) \alpha }}{\alpha ^{k+1}\left( k+1\right) !}. \end{aligned}$$

Hence, we show that (12) holds for all \(k\in {\mathbb {N}}.\) Since \(\lim _{k\rightarrow \infty }2^{k}M_{\lambda }^{k}N_{\lambda }^{k+1}\frac{x^{k\alpha }}{\alpha ^{k}k!}=0,\) we have \(\chi \left( x,\lambda \right) =0\ \)for all \(x\in \left[ 0,b\right] .\) This proves the uniqueness.

We next show that \(\Phi \left( x,\lambda \right) ,\ \)where\(\ x\in \left[ 0,b\right] ,\) is entire in \(\lambda .\) It is sufficient to prove that \(\Phi \left( x,\lambda \right) \) is analytic in each disk \(\Gamma _{M}\), where \(\Gamma _{M}:=\left\{ \lambda \in {\mathbb {C}}:\left| \lambda \right| \le M\right\} .\) Now, by induction on\(\ n,\ \)we will prove that

$$\begin{aligned}&U_{n}\left( x,\lambda \right) \ \text {is analytic on }\Gamma _{M}\ \text {for all}\ x\in \left[ 0,b\right] , \end{aligned}$$
(13)
$$\begin{aligned}&\frac{\partial }{\partial \lambda }U_{n}\left( x,\lambda \right) \ \text {is continuous at }\left( 0,\lambda \right) \ \text {for all }\lambda \in \Gamma _{M}. \end{aligned}$$
(14)

It is evident that \(\varphi _{1}\left( x,\lambda \right) \ \)and\(\ \varphi _{2}\left( x,\lambda \right) \) are entire functions of \(\lambda \) for \(x\in \left[ 0,b\right] .\) Furthermore, \(\frac{\partial }{\partial \lambda }U_{n}\left( x,\lambda \right) \ \)is continuous at \(\left( 0,\lambda \right) \ \)for all \(\lambda \in {\mathbb {C}}.\) Thus, (13) and (14) hold for \(n=1.\) Assume that (13) and (14) hold for \(n\ge 1.\) Then, we get

$$\begin{aligned}&\frac{\partial }{\partial \lambda }U_{n+1}\left( x_{0},\lambda \right) \big |_{\lambda =\lambda _{0}}=\frac{\partial }{\partial \lambda }\varphi _{2}\left( x_{0},\lambda \right) \big |_{\lambda =\lambda _{0}}\int _{0}^{x_{0}}\varphi _{1}^\mathrm{T}\left( t,\lambda \right) \Omega \left( t\right) U_{n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t \nonumber \\&\quad +\,\frac{\partial }{\partial \lambda }U_{1}\left( x_{0},\lambda \right) \big |_{\lambda =\lambda _{0}} \nonumber \\&\quad -\,\frac{\partial }{\partial \lambda }\varphi _{1}\left( x_{0},\lambda \right) \big |_{\lambda =\lambda _{0}}\int _{0}^{x_{0}}\varphi _{2}^\mathrm{T}\left( t,\lambda \right) \Omega \left( t\right) U_{n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\nonumber \\&\quad +\,\varphi _{2}\left( x_{0},\lambda \right) \frac{\partial }{\partial \lambda }\left( \int _{0}^{x_{0}}\varphi _{1}^\mathrm{T}\left( t,\lambda \right) \Omega \left( t\right) U_{n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\right) \big |_{\lambda =\lambda _{0}}\nonumber \\&\quad -\,\varphi _{1}\left( x_{0},\lambda \right) \frac{\partial }{\partial \lambda }\left( \int _{0}^{x_{0}}\varphi _{2}^\mathrm{T}\left( t,\lambda \right) \Omega \left( t\right) U_{n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\right) \big |_{\lambda =\lambda _{0}}. \end{aligned}$$
(15)

From (14), we conclude that \(\frac{\partial }{\partial \lambda }\varphi _{i}^\mathrm{T}\left( t,\lambda \right) \Omega \left( t\right) U_{n-1}\left( t,\lambda \right) \ \)is continuous at \(\left( 0,\lambda _{0}\right) \ \)for each\(\ i=1,2\). Consequently, there exist constants \(C,\delta >0\) such that

$$\begin{aligned} \left\| \frac{\partial }{\partial \lambda }\left( \varphi _{i}^\mathrm{T}\left( x_{0},\lambda \right) \Omega \left( x_{0}\right) U_{n}\left( x_{0} ,\lambda \right) \right) \right\| \le C,\quad ~n\in {\mathbb {N}}, \end{aligned}$$

for all \(\lambda \ \)in the disk \(\left| \lambda -\lambda _{0}\right| \le \delta .\) From this, we see that the series corresponding to the \(\alpha \)-integrals

$$\begin{aligned} \int _{0}^{x_{0}}\varphi _{i}^\mathrm{T}\left( t,\lambda \right) \Omega \left( t\right) U_{n}\left( t,\lambda \right) \mathrm{d}^{\alpha }t\ (i=1,2,...) \end{aligned}$$

are uniformly convergent in a neighborhood of \(\lambda =\lambda _{0}.\) So, we can interchange the order of differentiation and integration in (15). Since \(x_{0}\ \)and \(\lambda _{0}\) are arbitrary, we have

$$\begin{aligned}&\frac{\partial }{\partial \lambda }U_{n+1}\left( x,\lambda \right) \nonumber \\&\quad =\frac{\partial }{\partial \lambda }U_{z}\left( x,\lambda \right) -\int _{0}^{x_{0}}\frac{\partial }{\partial \lambda }\left( \varphi _{1}\left( x,\lambda \right) \varphi _{2}^\mathrm{T}\left( t,\lambda \right) U_{n}\left( t,\lambda \right) \right) \Omega \left( t\right) \mathrm{d}^{\alpha }t\nonumber \\&\qquad -\int _{0}^{x_{0}}\frac{\partial }{\partial \lambda }\left( \varphi _{2}\left( x,\lambda \right) \varphi _{1}^\mathrm{T}\left( t,\lambda \right) U_{n}\left( t,\lambda \right) \right) \Omega \left( t\right) \mathrm{d}^{\alpha }t \end{aligned}$$
(16)

for all \(x\in \left[ 0,b\right] \ \)and\(\ \lambda \in \Gamma _{M}.\ \)From (14), the integrals in (16) are continuous at \(\left( 0,\lambda \right) .\) Thus, \(\frac{\partial }{\partial \lambda }U_{n+1}\left( x,\lambda \right) \ \)is continuous at \(\left( 0,\lambda \right) .\) Let \(x_{0}\in \left[ 0,b\right] \) be arbitrary. Then, there exist \(\Upsilon \left( x_{0}\right) ,\ \widetilde{\Upsilon \left( x_{0}\right) }>0\) such that

$$\begin{aligned} \left| \varphi _{ij}\left( x_{0},\lambda \right) \right| \le \frac{\sqrt{\Upsilon \left( x_{0}\right) }}{2}\quad \text {and}\quad \left| U_{1j}\left( x_{0},\lambda \right) \right| \le \widetilde{\Upsilon \left( x_{0}\right) }, \end{aligned}$$
(17)

where \(i,j=1,2\ \)and\(\ \lambda \in \Gamma _{M}.\ \)Finally, the mathematical induction yields

$$\begin{aligned} \left\| U_{n+1}\left( x_{0},\lambda \right) -U_{n}\left( x_{0} ,\lambda \right) \right\| \le 2^{n}\frac{(\widetilde{\Upsilon \left( x_{0}\right) }\Upsilon \left( x_{0}\right) x_{0}^{\alpha })^{n}}{n!\alpha ^{n}}. \end{aligned}$$

Consequently, series (10) converges uniformly to \(\Phi \left( x,\lambda \right) \) in \(\Gamma _{M}\ \)with \(x=x_{0}.\) So, \(\Phi \left( x,\lambda \right) \) is analytic in \(\Gamma _{M}.\)

Self-adjoint conformable fractional Dirac system

In this section, we study a self-adjoint conformable fractional Dirac system in \({\mathcal {H}}\) and give some spectral properties of this system.

Now, we consider the conformable fractional Dirac system

$$\begin{aligned} \tau y:= & {} \left\{ \begin{array} [c]{c} -T_{\alpha }y_{2}+p\left( x\right) y_{1}\\ T_{\alpha }y_{1}+r\left( x\right) y_{2}, \end{array} \right. \nonumber \\ \tau y= & {} \lambda y,\ y=\left( \begin{array} [c]{c} y_{1}\\ y_{2} \end{array} \right) ,\quad 0\le x\le b<\infty , \end{aligned}$$
(18)

with boundary conditions

$$\begin{aligned} L_{1}\left( y\right)&:=k_{11}y_{1}\left( 0\right) +k_{12}y_{2}\left( 0\right) =0, \end{aligned}$$
(19)
$$\begin{aligned} L_{2}\left( y\right)&:=k_{21}y_{1}\left( b\right) +k_{22}y_{2}\left( b\right) =0, \end{aligned}$$
(20)

where \(\lambda \) is a complex parameter, \(p\left( .\right) \) and \(r\left( .\right) \) are real-valued continuous and conformable fractional integrable functions on \(\left[ 0,b\right] ,\ k_{ij}\in \mathbb {R\ }\left( i,j=1,2\right) \ \)and the rank of the matrix \(( k_{ij}) \) is 2.

Theorem 9

The boundary-value problem defined by (18)–(20) is formally self-adjoint on \({\mathcal {H}}\).

Proof

Firstly, we will prove the Green’s formula. Let \(y\left( .\right) ,z\left( .\right) \in {\mathcal {H}}.\) Then, we have

$$\begin{aligned} \left( \tau y,z\right) -\left( y,\tau z\right)&=\int _{0}^{b}\left( -T_{\alpha }y_{2}+p\left( x\right) y_{1}\right) \overline{z_{1}}d^{\alpha }x\\&\quad +\int _{0}^{b}\left( T_{\alpha }y_{1}+r\left( x\right) y_{2}\right) \overline{z_{2}}\mathrm{d}^{\alpha }x\\&\quad -\int _{0}^{b}y_{1}\overline{\left( -T_{\alpha }z_{2}+p\left( x\right) z_{1}\right) }\mathrm{d}^{\alpha }x\\&\quad -\int _{0}^{b}y_{2}\overline{\left( T_{\alpha }z_{1}+r\left( x\right) z_{2}\right) }\mathrm{d}^{\alpha }x\\&=-\int _{0}^{b}\left[ \left( T_{\alpha }y_{2}\right) \overline{z_{1} }+y_{2}\overline{\left( T_{\alpha }z_{1}\right) }\right] \mathrm{d}^{\alpha }x\\&\quad +\int _{0}^{b}\left[ \left( T_{\alpha }y_{1}\right) \overline{z_{2}} +y_{1}\overline{\left( T_{\alpha }z_{2}\right) }\right] \mathrm{d}^{\alpha }x. \end{aligned}$$

Hence, we obtain

$$\begin{aligned}&\left( \tau y,z\right) -\left( y,\tau z\right) \\&\quad =-\int _{0}^{b}T_{\alpha }\left( \overline{z_{1}\left( x\right) } y_{2}\left( x\right) \right) \mathrm{d}^{\alpha }x+\int _{0}^{b}T_{\alpha }\left( y_{1}\left( x\right) \overline{z_{2}\left( x\right) }\right) \mathrm{d}^{\alpha }x\\&\quad =\int _{0}^{b}T_{\alpha }\left[ y_{1}\left( x\right) \overline{z_{2}\left( x\right) }-\overline{z_{1}\left( x\right) }y_{2}\left( x\right) \right] \mathrm{d}^{\alpha }x. \end{aligned}$$

Let us define \(\left[ y,z\right] _{x}:=y_{1}\left( x\right) \overline{z_{2}\left( x\right) }-\overline{z_{1}\left( x\right) }y_{2}\left( x\right) .\ \)Thus, we get

$$\begin{aligned} \left( \tau y,z\right) -\left( y,\tau z\right) =\left[ y,z\right] _{b}-\left[ y,z\right] _{0}. \end{aligned}$$
(21)

Now, we prove that the operator L is formally self-adjoint.

Let \(y\left( .\right) ,z\left( .\right) \in {\mathcal {H}}.\ \)Then, we have

$$\begin{aligned} \left( \tau y,z\right) -\left( y,\tau z\right) =\left[ y,z\right] _{b}-\left[ y,z\right] _{0}. \end{aligned}$$

By conditions (19) and (20), we get \(\left[ y,z\right] _{b}=0 \) and \(\left[ y,z\right] _{0}=0.\) So, we obtain

$$\begin{aligned} \left( \tau y,z\right) =\left( y,\tau z\right) , \end{aligned}$$
(22)

which proves the theorem \(\square \)

Lemma 10

All eigenvalues of the boundary-value problem (18)–(20) are real.

Proof

Let \(\mu \) be an eigenvalue with an eigenvector \(z\left( x\right) .\) From (22), we have

$$\begin{aligned} \left( \tau z,z\right) =\left( z,\tau z\right) =\left( z,\mu z\right) ={\overline{\mu }}\left( z,z\right) . \end{aligned}$$
(23)

On the other hand,

$$\begin{aligned} \left( \tau z,z\right) =\left( \mu z,z\right) =\mu \left( z,z\right) . \end{aligned}$$
(24)

From (23) and (24), we get

$$\begin{aligned} \mu \left( z,z\right) ={\overline{\mu }}\left( z,z\right) ,~\left( \mu -{\overline{\mu }}\right) \left( z,z\right) =0. \end{aligned}$$

Since \(z\left( x\right) \ne 0,\) we have \(\mu ={\overline{\mu }}.\)\(\square \)

Lemma 11

If \(\mu _{1}\) and \(\mu _{2}\) are two different eigenvalues of the problem defined by (18)–(20), then the corresponding eigenvectors \(v_{1}\) and \(v_{2}\ \) are orthogonal.

Proof

Let \(\mu _{1}\) and \(\mu _{2}\) be two different real eigenvalues with corresponding eigenvectors \(v_{1}\) and \(v_{2},\) respectively. From (22), we obtain

$$\begin{aligned} \left( \mu _{1}-\mu _{2}\right) \int _{0}^{b}(v_{1}\left( x\right) ,v_{2}\left( x\right) )_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }x=0. \end{aligned}$$

Since \(\mu _{1}\ne \mu _{2},\) we obtain that \(v_{1}\left( x\right) \) and \(v_{2}\left( x\right) \) are orthogonal. \(\square \)

Now let \(y\left( x\right) =\left( \begin{array} [c]{c} y_{1}\left( x\right) \\ y_{2}\left( x\right) \end{array} \right) ,\ z\left( x\right) =\left( \begin{array} [c]{c} z_{1}\left( x\right) \\ z_{2}\left( x\right) \end{array} \right) .\) Then, we define the Wronskian of \(y\left( x\right) \) and \(z\left( x\right) \) by

$$\begin{aligned} W\left( y,z\right) (x)=y_{1}\left( x\right) z_{2}\left( x\right) -z_{1}\left( x\right) y_{2}\left( x\right) . \end{aligned}$$

Theorem 12

The Wronskian of any solution of Eq. (18) is independent of x.

Proof

Let \(y\left( x\right) \ \)and\(\ z\left( x\right) \) be two solutions of Eq. (18). By Green’s formula (21), we have

$$\begin{aligned} \int _{0}^{x}(\tau y,{\overline{z}})_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }t-\int _{0} ^{x}(y,\tau {\overline{z}})_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }t=\left[ y,{\overline{z}}\right] _{x}-\left[ y,{\overline{z}}\right] _{0}. \end{aligned}$$

Since \(\tau y=\lambda y\) and \(\tau {\overline{z}}={\overline{\lambda }}{\overline{z}} \), we have

$$\begin{aligned}&\int _{0}^{x}(\lambda y,{\overline{z}})_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }t-\int _{0} ^{x}(y,{\overline{\lambda }}{\overline{z}})_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }t=\left[ y,{\overline{z}}\right] _{x}-\left[ y,{\overline{z}}\right] _{0}, \\&\left( \lambda -\lambda \right) \int _{0}^{x}(y,{\overline{z}})_{{\mathbb {C}}^{2} }\mathrm{d}^{\alpha }t=\left[ y,{\overline{z}}\right] _{x}-\left[ y,{\overline{z}}\right] _{0}=0. \end{aligned}$$

Then, we have \(W(y,z)(x)=\left[ y,{\overline{z}}\right] _{x}=\left[ y,{\overline{z}}\right] _{0}=W\left( y,z\right) \left( 0\right) ,\ \)i.e., the Wronskian is independent of x. \(\square \)

Corollary 13

If \(y\left( x\right) \) and \(\ z\left( x\right) \) are both solutions of Eq. (18), then either \(W\left( y,z\right) (x)=0\) or \(W\left( y,z\right) (x)\ne 0\) for all \(x\in \left[ 0,b\right] .\)

Theorem 14

Any two solutions of Eq. (18) are linearly dependent if and only if their Wronskian is zero.

Proof

Let \(y\left( x\right) \ \)and\(\ z\left( x\right) \) be two linearly dependent solutions of Eq. (18). Then, there exists a constant \(c\ne 0\) such that \(y\left( x\right) =cz\left( x\right) .\) Hence

$$\begin{aligned} W\left( y,z\right) =\left| \begin{array} [c]{cc} y_{1}\left( x\right) &{} y_{2}\left( x\right) \\ z_{1}\left( x\right) &{} z_{2}\left( x\right) \end{array} \right| =\left| \begin{array} [c]{cc} cz_{1}\left( x\right) &{} cz_{2}\left( x\right) \\ z_{1}\left( x\right) &{} z_{2}\left( x\right) \end{array} \right| =0. \end{aligned}$$

Conversely, the Wronskian \(W\left( y,z\right) =0\) and therefore, \(y\left( x\right) =cz\left( x\right) ,\) i.e., \(y\left( x\right) \) and \(\ z\left( x\right) \) are linearly dependent. \(\square \)

Lemma 15

All eigenvalues of the problem defined by (18)–(20) are simple from the geometric point of view.

Proof

Let \(\mu \) be an eigenvalue with eigenvectors \(z_{1}\left( x\right) \) and \(z_{2}\left( x\right) .\ \)By condition (19), we have

$$\begin{aligned} W\left( z_{1},z_{2}\right) \left( 0\right) =z_{11}\left( 0\right) z_{22}\left( 0\right) -z_{12}\left( 0\right) z_{21}\left( 0\right) =0. \end{aligned}$$

Then, the set \(\left\{ z_{1}\left( x\right) ,\ z_{2}\left( x\right) \right\} \) is linearly dependent. \(\square \)

Now, our next objective is to determine the eigenvalues and the corresponding eigenvectors. Let \(\phi _{1}\left( x,\lambda \right) =\left( \begin{array} [c]{c} \phi _{11}\left( x,\lambda \right) \\ \phi _{12}\left( x,\lambda \right) \end{array} \right) \) and \(\phi _{2}\left( x,\lambda \right) =\left( \begin{array} [c]{c} \phi _{21}\left( x,\lambda \right) \\ \phi _{22}\left( x,\lambda \right) \end{array} \right) \) be linearly independent solutions of (18) which satisfies the initial conditions

$$\begin{aligned} \phi _{ij}\left( 0,\lambda \right) =\delta _{ij},\quad i,j=1,2,\ \lambda \in {\mathbb {C}}, \end{aligned}$$

where \(\delta _{ij}\ (i,j=1,2)\) is the Kronecker delta. Then, every solution of Eq. (18) has the form

$$\begin{aligned} y\left( x,\lambda \right) =K_{1}\phi _{1}\left( x,\lambda \right) +K_{2} \phi _{2}\left( x,\lambda \right) , \end{aligned}$$

where \(K_{1}\) and \(K_{2}\) do not depend on x. If we can find a nontrivial solution of the linear system

$$\begin{aligned} K_{1}L_{1}\left( \phi _{1}\right) +K_{2}L_{1}\left( \phi _{2}\right)&=0,\\ K_{1}L_{2}\left( \phi _{1}\right) +K_{2}L_{2}\left( \phi _{2}\right)&=0, \end{aligned}$$

then the solution \(y\left( x,\lambda \right) \) is called an eigenvector of (18). Hence, \(\lambda \in {\mathbb {R}}\) is an eigenvalue if and only if

$$\begin{aligned} \Delta \left( \lambda \right) =\left| \begin{array} [c]{cc} L_{1}\left( \phi _{1}\right) &{} L_{1}\left( \phi _{2}\right) \\ L_{2}\left( \phi _{1}\right) &{} L_{2}\left( \phi _{2}\right) \end{array} \right| =0. \end{aligned}$$

The function \(\Delta \left( \lambda \right) \) is called the characteristic determinant associated with the conformable fractional Dirac system defined by (18)–(20). The eigenvalues of the problem defined by (18)–(20) are the zeros of the function \(\Delta \left( \lambda \right) .\) Since\(\ \phi _{1}\left( x,\lambda \right) \) and \(\phi _{2}\left( x,\lambda \right) \) are entire in \(\lambda \) for each fixed \(x\in \left[ 0,b\right] \), \(\Delta \left( \lambda \right) \) is an entire function in \(\lambda .\) Hence, the eigenvalues of the system defined by (18)–(20) are at most countable with no finite limit points.

Theorem 16

All eigenvalues \(\mu _{n}\ \)of problem (18)–(20) are simple zeros of the function \(\Delta \left( \lambda \right) .\)

Proof

Let us define \(\theta _{1}\left( x,\lambda \right) =\left( \begin{array} [c]{c} \theta _{11}\left( x,\lambda \right) \\ \theta _{12}\left( x,\lambda \right) \end{array} \right) \) and \(\theta _{2}\left( x,\lambda \right) =\left( \begin{array} [c]{c} \theta _{21}\left( x,\lambda \right) \\ \theta _{22}\left( x,\lambda \right) \end{array} \right) \) by

$$\begin{aligned} \theta _{1}\left( x,\lambda \right)&=L_{1}\left( \phi _{2}\right) \phi _{1}\left( x,\lambda \right) -L_{1}\left( \phi _{1}\right) \phi _{2}\left( x,\lambda \right) ,\nonumber \\ \theta _{2}\left( x,\lambda \right)&=L_{2}\left( \phi _{2}\right) \phi _{1}\left( x,\lambda \right) -L_{2}\left( \phi _{1}\right) \phi _{2}\left( x,\lambda \right) . \end{aligned}$$
(25)

Then, \(\theta _{1}\left( x,\lambda \right) \) and \(\theta _{2}\left( x,\lambda \right) \) are solutions of (18) such that

$$\begin{aligned} \theta _{1}\left( 0,\lambda \right) =\left( \begin{array} [c]{c} a_{12}\\ -a_{11} \end{array} \right) \quad \text {and } \quad \theta _{2}\left( a,\lambda \right) =\left( \begin{array} [c]{c} a_{22}\\ -a_{21} \end{array} \right) . \end{aligned}$$
(26)

On the other hand, we have

$$\begin{aligned}&W\left( \theta _{1}\left( x,\lambda \right) ,\theta _{2}\left( x,\lambda \right) \right) \nonumber \\&\quad =\left| \begin{array} [c]{cc} \theta _{11}\left( x,\lambda \right) &{} \theta _{12}\left( x,\lambda \right) \\ \theta _{21}\left( x,\lambda \right) &{} \theta _{22}\left( x,\lambda \right) \end{array} \right| \nonumber \\&\quad =\theta _{11}\left( x,\lambda \right) \theta _{22}\left( x,\lambda \right) -\theta _{12}\left( x,\lambda \right) \theta _{21}\left( x,\lambda \right) \nonumber \\&\quad =\left( L_{1}\left( \phi _{2}\right) \phi _{11}\left( x,\lambda \right) -L_{1}\left( \phi _{1}\right) \phi _{21}\left( x,\lambda \right) \right) \nonumber \\&\qquad \times \left( L_{2}\left( \phi _{2}\right) \phi _{12}\left( x,\lambda \right) -L_{2}\left( \phi _{1}\right) \phi _{22}\left( x,\lambda \right) \right) \nonumber \\&\qquad -\left( L_{1}\left( \phi _{2}\right) \phi _{12}\left( x,\lambda \right) -L_{1}\left( \phi _{1}\right) \phi _{22}\left( x,\lambda \right) \right) \nonumber \\&\qquad \times \left( L_{2}\left( \phi _{2}\right) \phi _{11}\left( x,\lambda \right) -L_{2}\left( \phi _{1}\right) \phi _{21}\left( x,\lambda \right) \right) \nonumber \\&\quad =L_{1}\left( \phi _{2}\right) L_{2}\left( \phi _{1}\right) \left( -\phi _{11}\left( x,\lambda \right) \phi _{22}\left( x,\lambda \right) +\phi _{12}\left( x,\lambda \right) \phi _{21}\left( x,\lambda \right) \right) \nonumber \\&\qquad +L_{1}\left( \phi _{1}\right) L_{2}\left( \phi _{2}\right) \left( \phi _{11}\left( x,\lambda \right) \phi _{22}\left( x,\lambda \right) -\phi _{12}\left( x,\lambda \right) \phi _{21}\left( x,\lambda \right) \right) \nonumber \\&\quad =\left( \phi _{11}\left( x,\lambda \right) \phi _{22}\left( x,\lambda \right) -\phi _{12}\left( x,\lambda \right) \phi _{21}\left( x,\lambda \right) \right) \nonumber \\&\qquad \times \left( L_{1}\left( \phi _{1}\right) L_{2}\left( \phi _{2}\right) -L_{1}\left( \phi _{2}\right) L_{2}\left( \phi _{1}\right) \right) \nonumber \\&\quad =W\left( \phi _{1}\left( x,\lambda \right) ,\phi _{2}\left( x,\lambda \right) \right) \Delta \left( \lambda \right) =\Delta \left( \lambda \right) . \end{aligned}$$
(27)

Let \(\lambda _{0}\ \)be an eigenvalue of the problem defined by (18 )–(20). Since \(\lambda _{0}\in {\mathbb {R}}\), \(\theta _{i}\left( x,\lambda _{0}\right) \)\(\left( i=1,2\right) \ \)can be taken to be real valued. Then, by (27), \(\theta _{1}\left( x,\lambda _{0}\right) \) and \(\theta _{2}\left( x,\lambda _{0}\right) \) are linearly dependent eigenfunctions. Hence, there exists a nonzero constant \(\eta _{0}\) such that

$$\begin{aligned} \theta _{2}\left( x,\lambda _{0}\right) =\eta _{0}\theta _{1}\left( x,\lambda _{0}\right) . \end{aligned}$$

From (25) and (26), we have

$$\begin{aligned} \theta _{21}\left( 0,\lambda _{0}\right) =\eta _{0}a_{12}=\eta _{0}\theta _{11}\left( 0,\lambda \right) ,\ \theta _{22}\left( 0,\lambda _{0}\right) =-\eta _{0}a_{11}=-\eta _{0}\theta _{12}\left( 0,\lambda \right) . \end{aligned}$$
(28)

If we take \(y\left( x\right) =\theta _{2}\left( x,\lambda \right) \) and \(z\left( x\right) =\theta _{2}\left( x,\lambda _{0}\right) \ \)in (21), then we get

$$\begin{aligned}&\left( \lambda -\lambda _{0}\right) \int _{0}^{b}(\theta _{2}\left( x,\lambda \right) ,\theta _{2}\left( x,\lambda _{0}\right) )_{{\mathbb {C}}^{2} }\mathrm{d}^{\alpha }x\\&\quad =-\left( \theta _{21}\left( b,\lambda \right) \theta _{22}\left( b,\lambda _{0}\right) -\theta _{21}\left( b,\lambda _{0}\right) \theta _{22}\left( b,\lambda \right) \right) \\&\quad =-\left( \theta _{21}\left( b,\lambda \right) \left( -\eta _{0}\theta _{12}\left( b,\lambda \right) \right) -\eta _{0}\theta _{11}\left( b,\lambda \right) \theta _{22}\left( b,\lambda \right) \right) \\&\quad =\eta _{0}\left( \theta _{11}\left( b,\lambda \right) \theta _{22}\left( b,\lambda \right) -\theta _{12}\left( b,\lambda \right) \theta _{21}\left( b,\lambda \right) \right) \\&\quad =\eta _{0}W\left( \theta _{1}\left( x,\lambda \right) ,\theta _{2}\left( x,\lambda \right) \right) =\eta _{0}\Delta \left( \lambda \right) . \end{aligned}$$

Since \(\Delta \left( \lambda \right) \) is an entire function in \(\lambda \), we have

$$\begin{aligned} \frac{d}{d\lambda }\Delta \left( \lambda \right) :=\lim _{\lambda \rightarrow \lambda _{0}}\frac{\Delta \left( \lambda \right) }{\lambda -\lambda _{0}} =\frac{1}{\eta _{0}}\int _{0}^{b}\theta _{2}^{2}\left( x,\lambda _{0}\right) \mathrm{d}^{\alpha }x\ne 0. \end{aligned}$$

Consequently, \(\lambda _{0}\) is a simple zero of \(\Delta \left( \lambda \right) .\)\(\square \)

Green’s function

In this section, we will study the solution of the system

$$\begin{aligned} -T_{\alpha }y_{2}+\left\{ -\lambda +p\left( x\right) \right\} y_{1}= & {} f_{1}\left( x\right) , \end{aligned}$$
(29)
$$\begin{aligned} T_{\alpha }y_{1}+\left\{ -\lambda +r\left( x\right) \right\} y_{2}= & {} f_{2}\left( x\right) ,\ \end{aligned}$$
(30)

(where \(0\le x\le b<\infty \ \)and \(p\left( .\right) \) and \(r\left( .\right) \) are real-valued continuous and conformable fractional integrable functions on \(\left[ 0,b\right] \ \)and \(f\left( .\right) :=\left( \begin{array} [c]{c} f_{1}\left( .\right) \\ f_{2}\left( .\right) \end{array} \right) \in {\mathcal {H}}\)) which fulfills the boundary conditions

$$\begin{aligned} L_{1}\left( y\right)&:=k_{11}y_{1}\left( 0\right) +k_{12}y_{2}\left( 0\right) =0, \end{aligned}$$
(31)
$$\begin{aligned} L_{2}\left( y\right)&:=k_{21}y_{1}\left( b\right) +k_{22}y_{2}\left( b\right) =0, \end{aligned}$$
(32)

where \(k_{ij}\in \mathbb {R\ }\left( i,j=1,2\right) \ \)and the rank of the matrix \(\left( k_{ij}\right) \) is 2. So, we will construct the Green’s function of system (29)–(32).

Theorem 17

If \(\lambda \) is not an eigenvalue of the problem defined by (18 )–(20), then the nonhomogeneous system (29)–(32) is solvable for any vector-valued function \(f\left( x\right) .\) Conversely, if \(\lambda \) is an eigenvalue of the problem defined by (18)–(20), then the nonhomogeneous system (29)–(32) is generally unsolvable.

Proof

Let us define

$$\begin{aligned} G\left( x,t,\lambda \right) =\left\{ \begin{array} [c]{cc} -\frac{\theta _{2}\left( x,\lambda \right) \theta _{1}^\mathrm{T}\left( t,\lambda \right) }{\Delta \left( \lambda \right) }, &{}\quad 0\le t\le x\\ -\frac{\theta _{1}\left( x,\lambda \right) \theta _{2}^\mathrm{T}\left( t,\lambda \right) }{\Delta \left( \lambda \right) }, &{}\quad x<t\le b \end{array} \right. \end{aligned}$$
(33)

which is called Green’s matrix. Now, we will prove that the function

$$\begin{aligned} y\left( x,\lambda \right) =\int _{0}^{b}G\left( x,t,\lambda \right) f\left( t\right) \mathrm{d}^{\alpha }t \end{aligned}$$
(34)

is the solution of system (29)–(32).

By definition of the Green’s matrix, we get

$$\begin{aligned}&G\left( x,t,\lambda \right) \\&\quad =\left\{ \begin{array} [c]{cc} \begin{array} [c]{c} -\frac{1}{\Delta \left( \lambda \right) }\left( \begin{array} [c]{cc} \theta _{21}\left( x,\lambda \right) \theta _{11}\left( t,\lambda \right) &{} \theta _{21}\left( x,\lambda \right) \theta _{12}\left( t,\lambda \right) \\ \theta _{22}\left( x,\lambda \right) \theta _{11}\left( t,\lambda \right) &{} \theta _{22}\left( x,\lambda \right) \theta _{12}\left( t,\lambda \right) \end{array} \right) ,\\ 0\le t\le x, \end{array} &{} \\ \begin{array} [c]{c} -\frac{1}{\Delta \left( \lambda \right) }\left( \begin{array} [c]{cc} \theta _{11}\left( x,\lambda \right) \theta _{21}\left( t,\lambda \right) &{} \theta _{11}\left( x,\lambda \right) \theta _{22}\left( t,\lambda \right) \\ \theta _{12}\left( x,\lambda \right) \theta _{21}\left( t,\lambda \right) &{} \theta _{12}\left( x,\lambda \right) \theta _{22}\left( t,\lambda \right) \end{array} \right) ,~\\ x<t\le b. \end{array}&\end{array} \right. \end{aligned}$$

Hence

$$\begin{aligned}&G\left( x,t,\lambda \right) f\left( t\right) \\&\quad =\left\{ \begin{array} [c]{cc} \begin{array} [c]{c} -\frac{1}{\Delta \left( \lambda \right) }\left( \begin{array} [c]{c} \theta _{21}\left( x,\lambda \right) \theta _{11}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{21}\left( x,\lambda \right) \theta _{12}\left( t,\lambda \right) f_{2}\left( t\right) \\ \theta _{22}\left( x,\lambda \right) \theta _{11}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{22}\left( x,\lambda \right) \theta _{12}\left( t,\lambda \right) f_{2}\left( t\right) \end{array} \right) ,\\ ~0\le t\le x, \end{array} &{} \\ \begin{array} [c]{c} -\frac{1}{\Delta \left( \lambda \right) }\left( \begin{array} [c]{c} \theta _{11}\left( x,\lambda \right) \theta _{21}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{11}\left( x,\lambda \right) \theta _{22}\left( t,\lambda \right) f_{2}\left( t\right) \\ \theta _{12}\left( x,\lambda \right) \theta _{21}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{12}\left( x,\lambda \right) \theta _{22}\left( t,\lambda \right) f_{2}\left( t\right) \end{array} \right) ,\\ ~x<t\le b. \end{array}&\end{array} \right. \end{aligned}$$

From (34), we have

$$\begin{aligned}&y_{1}\left( x,\lambda \right) \nonumber \\&\quad =-\frac{\theta _{21}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\int _{0}^{x}\left( \theta _{11}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{12}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t\nonumber \\&\qquad -\frac{\theta _{11}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\int _{x}^{b}\left( \theta _{21}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{22}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t, \end{aligned}$$
(35)
$$\begin{aligned}&y_{2}\left( x,\lambda \right) \nonumber \\&\quad =-\frac{\theta _{22}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\int _{0}^{x}\left( \theta _{11}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{12}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t\nonumber \\&\qquad -\frac{\theta _{12}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\int _{x}^{b}\left( \theta _{21}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{22}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t. \end{aligned}$$
(36)

From (35), it follows that

$$\begin{aligned}&T_{\alpha }y_{1}\left( x,\lambda \right) \\&\quad =-\frac{T_{\alpha }\theta _{21}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\int _{0}^{x}\left( \theta _{11}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{12}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t\\&\qquad -\frac{T_{\alpha }\theta _{11}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\int _{x}^{b}\left( \theta _{21}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{22}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t\\&\qquad +\frac{1}{\Delta \left( \lambda \right) }W\left( \theta _{1},\theta _{2}\right) f_{2}\left( x\right) \\&\quad =\frac{\theta _{22}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\left\{ -\lambda +r\left( x\right) \right\} \int _{0}^{x}\left( \theta _{11}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{12}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t\\&\qquad +\frac{\theta _{12}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\left\{ -\lambda +r\left( x\right) \right\} \int _{x}^{b}\left( \theta _{21}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{22}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t+f_{2}\left( x\right) \\&\quad =\left\{ -\lambda +r\left( x\right) \right\} \frac{\theta _{22}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\int _{0}^{x}\left( \theta _{11}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{12}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t\\&\qquad + \left\{ -\lambda +r\left( x\right) \right\} \frac{\theta _{12}\left( x,\lambda \right) }{\Delta \left( \lambda \right) }\int _{x}^{b}\left( \theta _{21}\left( t,\lambda \right) f_{1}\left( t\right) +\theta _{22}\left( t,\lambda \right) f_{2}\left( t\right) \right) \mathrm{d}^{\alpha }t\\&\qquad +f_{2}\left( x\right) =-\left\{ -\lambda +r\left( x\right) \right\} y_{2}\left( x\right) +f_{2}\left( x\right) . \end{aligned}$$

The validity of (29) is proved similarly. Hence, the function \(y\left( x,\lambda \right) \) in (34) is the solution of system (29) and (30). We check at once that (34) satisfies conditions (31) and (32). \(\square \)

Theorem 18

The Green’s matrix defined by formula (33) has the following properties:

(i) The Green’s matrix \(G\left( x,t,\lambda \right) \) is unique.

(ii) \(G\left( x,t,\lambda \right) \) is continuous at the point \(\left( 0,0\right) .\)

(iii) \(G\left( x,t,\lambda \right) =G^\mathrm{T}\left( t,x,\lambda \right) .\)

(iv) Let \(\lambda _{0}\) be a zero of \(\Delta \left( \lambda \right) .\) Then, \(\lambda _{0}\) can be a simple pole of the matrix \(G\left( x,t,\lambda \right) .\ \)Therefore, we have

$$\begin{aligned} G\left( x,t,\lambda \right) =\frac{-\varrho \left( x\right) \varrho \left( t\right) }{\lambda -\lambda _{0}}+{\widetilde{G}}\left( x,t,\lambda \right) , \end{aligned}$$

where \({\widetilde{G}}\left( x,t,\lambda \right) \) is an analytic function of \(\lambda \) in the neighborhood of \(\lambda _{0} \) and \(\varrho \left( x\right) \) is a normalized eigenfunction corresponding to \(\lambda _{0}.\)

Proof

(i) Conversely, suppose that there exists another Green’s matrix \({\widetilde{G}}\left( x,t,\lambda \right) \) for system (29)–(32). By formula (34), we have

$$\begin{aligned} \int _{0}^{b}\left\{ G\left( x,t,\lambda \right) -{\widetilde{G}}\left( x,t,\lambda \right) \right\} f\left( t\right) \mathrm{d}^{\alpha }t=0. \end{aligned}$$

Let us take \(f\left( t\right) =\overline{(G\left( x,t,\lambda \right) -{\widetilde{G}}\left( x,t,\lambda \right) })^\mathrm{T}.\) Then,

$$\begin{aligned} \int _{0}^{b}\left\| G\left( x,t,\lambda \right) -{\widetilde{G}}\left( x,t,\lambda \right) \right\| _{{\mathbb {C}}^{2}}^{2}\mathrm{d}^{\alpha }t=\left\| G\left( x,t,\lambda \right) -{\widetilde{G}}\left( x,t,\lambda \right) \right\| ^{2}=0. \end{aligned}$$

Therefore, we conclude that \(G\left( x,t,\lambda \right) ={\widetilde{G}}\left( x,t,\lambda \right) .\)

(ii), (iv) and (iii) are easily checked. \(\square \)

Eigenfunction expansion formula

In this section, we prove the existence of a countable sequence of eigenvalues of the boundary-value problem (29)–(32) with no finite limit points and the corresponding eigenvectors form an orthonormal basis of \({\mathcal {H}}.\ \)

Now, we will give the following definition and theorems.

Definition 19

A matrix-valued function \(M\left( x,t\right) \) in \({\mathbb {C}}^{2}\) of two variables with \(0\le x\ \)and\(\ t\le b\ \)is called the \(\alpha \)-Hilbert–Schmidt kernel if

$$\begin{aligned} \int _{0}^{b}\int _{0}^{b}\left\| M\left( x,t\right) \right\| ^{2}\mathrm{d}^{\alpha }x\mathrm{d}^{\alpha }t<+\infty . \end{aligned}$$

Theorem 20

[25] If

$$\begin{aligned} \sum _{i,k=1}^{\infty }\left| a_{ik}\right| ^{2}<+\infty , \end{aligned}$$
(37)

then the operator A defined by the formula

$$\begin{aligned} A\left\{ x_{i}\right\} =\left\{ y_{i}\right\} , \end{aligned}$$

where

$$\begin{aligned} y_{i}=\sum _{k=1}^{\infty }a_{ik}x_{k},\quad ~i=1,2,... \end{aligned}$$
(38)

is compact in the sequence space \(l^{2}.\)

Theorem 21

(Hilbert–Schmidt) Let A be a compact self-adjoint operator mapping a Hilbert space H into itself. Then, there is an orthonormal system \(\varphi _{1},\varphi _{2},...\) of eigenvectors of A,  with corresponding nonzero eigenvalues \(\lambda _{1},\lambda _{2},...,\) such that every element \(x\in H\) has a unique representation of the form

$$\begin{aligned} x=\sum _{n}c_{n}\varphi _{n}+x^{\prime }, \end{aligned}$$

where \(x^{\prime }\) satisfies the condition \(Ax^{\prime }=0.\) Moreover,

$$\begin{aligned} Ax=\sum _{n}\lambda _{n}c_{n}\varphi _{n}, \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\lambda _{n}=0 \end{aligned}$$

in the case where there are infinitely many nonzero eigenvalues [23].

Let us denote by \(D_{L}\) the linear set of all vectors \(y\in {\mathcal {H}}\) such that \(L_{1}\left( y\right) =0,\ L_{2}\left( y\right) =0.\) We define the operator L on \(D_{L}\) by the equality \(Ly=\tau y.\) It is immediate that the operator L has the same eigenvalues of the conformable fractional Dirac problem defined by (18)–(20). There is no loss of generality in assuming that \(\lambda =0\) is not an eigenvalue. Then, \(\ker L=\left\{ 0\right\} .\) Thus, the solution of the problem \(\left( Ly\right) \left( x\right) =f\left( x\right) ,\ f\left( .\right) \in {\mathcal {H}}\ \)is given by

$$\begin{aligned} y\left( x\right) =\int _{0}^{b}G\left( x,t\right) f\left( t\right) \mathrm{d}^{\alpha }t, \end{aligned}$$

where

$$\begin{aligned} G\left( x,t\right) =G\left( x,t,0\right) =\left\{ \begin{array} [c]{cc} -\frac{\theta _{2}\left( x\right) \theta _{1}^\mathrm{T}\left( t\right) }{W\left( \theta _{1},\theta _{2}\right) }, &{}\quad 0\le t\le x\\ -\frac{\theta _{1}\left( x\right) \theta _{2}^\mathrm{T}\left( t\right) }{W\left( \theta _{1},\theta _{2}\right) }, &{}\quad x<t\le b. \end{array} \right. \end{aligned}$$
(39)

Theorem 22

\(G\left( x,t\right) \) defined by (39) is a \(\alpha \)-Hilbert–Schmidt kernel.

Proof

By the upper half of formula (39), we have

$$\begin{aligned} \int _{0}^{b}\mathrm{d}^{\alpha }x\int _{0}^{x}\left\| G\left( x,t\right) \right\| ^{2}\mathrm{d}^{\alpha }t<+\infty ; \end{aligned}$$

and by the lower half of (39), we have

$$\begin{aligned} \int _{0}^{b}\mathrm{d}^{\alpha }x\int _{x}^{b}\left\| G\left( x,t\right) \right\| ^{2}\mathrm{d}^{\alpha }t<+\infty \end{aligned}$$

since the inner integral exists and is a linear combination of the products \(\theta _{ij}\left( x\right) \theta _{kl}(t)\ (i,j,k,l=1,2),\) and these products belong to \(L_{\alpha }^{2}\left( 0,b\right) \ \)because each of the factors belongs to \(L_{\alpha }^{2}\left( 0,b\right) \times L_{\alpha } ^{2}\left( 0,b\right) .\) Then, we obtain

$$\begin{aligned} \int _{0}^{b}\int _{0}^{b}\left\| G\left( x,t\right) \right\| ^{2}\mathrm{d}^{\alpha }x\mathrm{d}^{\alpha }t<+\infty . \end{aligned}$$
(40)

\(\square \)

Theorem 23

The operator \({\mathbf {S}}\) defined by the formula

$$\begin{aligned} ({\mathbf {S}}f)\left( x\right) =\int _{0}^{b}G\left( x,t\right) f\left( t\right) \mathrm{d}^{\alpha }t \end{aligned}$$

is compact and self-adjoint.

Proof

Let \(\phi _{i}=\phi _{i}\left( t\right) ,\)\(i\in {\mathbb {N}},\) be a complete, orthonormal basis of \({\mathcal {H}}.\) Since \(G\left( x,t\right) \) is a \(\alpha \)-Hilbert–Schmidt kernel, we can define

$$\begin{aligned} x_{i}&=\left( f,\phi _{i}\right) =\int _{0}^{b}(f\left( t\right) ,\phi _{i}\left( t\right) )_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }t,\ \\ y_{i}&=\left( g,\phi _{i}\right) =\int _{0}^{b}((g\left( t\right) ,\phi _{i}(t))_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }t,\\ a_{ik}&=\int _{0}^{b}\int _{0}^{b}(G\left( x,t\right) \phi _{i}\left( x\right) ,\phi _{k}\left( t\right) )_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }x\mathrm{d}^{\alpha }t,\quad ~i,k\in {\mathbb {N}}. \end{aligned}$$

Then, \({\mathcal {H}}\) is mapped isometrically \(l^{2}.\) Consequently, our integral operator transforms into the operator defined by formula (38) in the space \(l^{2}\ \)by this mapping and condition (40) is translated into condition (37). By Theorem 20, this operator is compact. Therefore, the original operator is compact.

Let \(f,g\in {\mathcal {H}}.\) As \(G\left( x,t\right) =G^\mathrm{T}\left( t,x\right) \ \)and \(G\left( x,t\right) \) is a matrix-valued function in \({\mathbb {C}}^{2}\) defined on \(\left[ 0,b\right] \times \left[ 0,b\right] ,\) we have

$$\begin{aligned} \left( {\mathbf {S}}f,g\right)&=\int _{0}^{b}(({\mathbf {S}}f)\left( x\right) ,g\left( x\right) )_{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }x\\&=\int _{0}^{b}\left( \int _{0}^{b}G\left( x,t\right) f(t)\mathrm{d}^{\alpha }t,g\left( x\right) \right) _{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }x\\&=\int _{0}^{b}\left( f\left( t\right) ,\int _{0}^{b}G\left( x,t\right) g\left( x\right) \mathrm{d}^{\alpha }x\right) _{{\mathbb {C}}^{2}}\mathrm{d}^{\alpha }t=\left( f,{\mathbf {S}}g\right) . \end{aligned}$$

Thus, we have proved that \({\mathbf {S}}\) is self-adjoint. \(\square \)

Theorem 24

The eigenvalues of the operator L form an infinite sequence \(\left\{ \lambda _{n}\right\} _{n=1}^{\infty }\) of real numbers which can be ordered so that

$$\begin{aligned} \left| \lambda _{1}\right|<\left| \lambda _{2}\right|< \cdots<\left| \lambda _{n}\right| < \cdots \rightarrow \infty \text { as }n\rightarrow \infty . \end{aligned}$$

The set of all normalized eigenfunctions of L forms an orthonormal basis for \({\mathcal {H}}.\)

Proof

By Theorems 21 and 23, \({\mathbf {S}}\) has an infinite sequence of nonzero real eigenvalues \(\left\{ \xi _{n}\right\} _{n=1}^{\infty }\ \)with\(\ \lim _{n\rightarrow \infty }\xi _{n}=0.\) Then,

$$\begin{aligned} \left| \lambda _{n}\right| =\frac{1}{\left| \xi _{n}\right| }\rightarrow \infty \text { as }n\rightarrow \infty . \end{aligned}$$

Furthermore, let \(\left\{ \chi _{n}\right\} _{n=1}^{\infty }\) denote an orthonormal set of eigenfuntions corresponding to \(\left\{ \xi _{n}\right\} _{n=1}^{\infty }.\) So

$$\begin{aligned} y&={\mathbf {S}}f=\sum _{n=1}^{\infty }\xi _{n}\left( f,\chi _{n}\right) \chi _{n}=\sum _{n=1}^{\infty }\xi _{n}\left( \tau y,\chi _{n}\right) \chi _{n}\\&=\sum _{n=1}^{\infty }\xi _{n}\left( y,\tau \chi _{n}\right) \chi _{n} =\sum _{n=1}^{\infty }\left( y,\chi _{n}\right) \chi _{n}. \end{aligned}$$

\(\square \)

Applications

Now we will solve conformable fractional Dirac systems.

Example 25

Let us consider the conformable fractional Dirac system

$$\begin{aligned} \left\{ \begin{array} [c]{c} -T_{\alpha }y_{2}=\lambda y_{1},\\ T_{\alpha }y_{1}=\lambda y_{2}, \end{array} \right. \end{aligned}$$
(41)

with the Dirichlet conditions

$$\begin{aligned} L_{1}\left( y\right) =y_{1}\left( 0\right) =0,\ L_{2}\left( y\right) =y_{1}\left( 1\right) =0. \end{aligned}$$
(42)

The solutions of (41) are

$$\begin{aligned} \phi _{1}\left( x,\lambda \right)&=\left( \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \\ -\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) \ \text {and}\\ \ \phi _{2}\left( x,\lambda \right)&=\left( \begin{array} [c]{c} \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \\ \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) . \end{aligned}$$

Then, the eigenvalues of problem (41) are the zeros of the determinant

$$\begin{aligned} \Delta \left( \lambda \right) =\left| \begin{array} [c]{cc} L_{1}\left( \phi _{1}\right) &{} L_{1}\left( \phi _{2}\right) \\ L_{2}\left( \phi _{1}\right) &{} L_{2}\left( \phi _{2}\right) \end{array} \right| =-\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) . \end{aligned}$$

Hence, the eigenvalues \(\left\{ \lambda _{n}\right\} _{n=1}^{\infty }\) are the zeros of \(\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \) and the corresponding set of eigenfunctions

$$\begin{aligned} \left\{ \left( \begin{array} [c]{c} \sin \left( \int _{0}^{x}\lambda _{n}\mathrm{d}^{\alpha }t\right) \\ \\ \cos \left( \int _{0}^{x}\lambda _{n}\mathrm{d}^{\alpha }t\right) \end{array} \right) \right\} _{n=1}^{\infty } \end{aligned}$$

is an orthogonal basis of \(L_{\alpha }^{2}\left( \left( 0,1\right) ;{\mathbb {C}}^{2}\right) .\) It is clear that

$$\begin{aligned} \theta _{1}\left( x,\lambda \right) =\left( \begin{array} [c]{c} \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \\ \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) \end{aligned}$$

and

$$\begin{aligned}&\theta _{2}\left( x,\lambda \right) \\&\quad =\left( \begin{array} [c]{c} \sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{x}\lambda \mathrm{d}^{\alpha }t\right) -\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \\ -\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{x}\lambda \mathrm{d}^{\alpha }t\right) -\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) . \end{aligned}$$

Hence, if \(\lambda \) is not an eigenvalue, then Green’s matrix is given by

$$\begin{aligned} G\left( x,t,\lambda \right) =\left\{ \begin{array} [c]{cc} -\frac{\theta _{2}\left( x,\lambda \right) \theta _{1}^\mathrm{T}\left( t,\lambda \right) }{\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) }, &{} \quad 0\le t\le x\\ -\frac{\theta _{1}\left( x,\lambda \right) \theta _{2}^\mathrm{T}\left( t,\lambda \right) }{\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) }, &{}\quad x<t\le 1. \end{array} \right. \end{aligned}$$

Example 26

Now we consider Eq. (41) with the Neumann boundary conditions

$$\begin{aligned} L_{1}\left( y\right) =y_{2}\left( 0\right) =0,\ L_{2}\left( y\right) =y_{2}\left( 1\right) =0. \end{aligned}$$

Then, we have

$$\begin{aligned} \Delta \left( \lambda \right) =\left| \begin{array} [c]{cc} L_{1}\left( \phi _{1}\right) &{} L_{1}\left( \phi _{2}\right) \\ L_{2}\left( \phi _{1}\right) &{} L_{2}\left( \phi _{2}\right) \end{array} \right| =\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) . \end{aligned}$$

So the eigenvalues \(\left\{ \lambda _{n}\right\} _{n=1}^{\infty }\) are the zeros of \(\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) .\) Furthermore,

$$\begin{aligned} \theta _{1}\left( x,\lambda \right) =\left( \begin{array} [c]{c} \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \\ -\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) \end{aligned}$$

and

$$\begin{aligned}&\theta _{2}\left( x,\lambda \right) \\&\quad =\left( \begin{array} [c]{c} \cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{x}\lambda \mathrm{d}^{\alpha }t\right) +\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \\ -\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{x}\lambda \mathrm{d}^{\alpha }t\right) +\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) . \end{aligned}$$

If \(\lambda \) is not an eigenvalue, then Green’s matrix is given by

$$\begin{aligned} G\left( x,t,\lambda \right) =\left\{ \begin{array} [c]{cc} \frac{\theta _{2}\left( x,\lambda \right) \theta _{1}^\mathrm{T}\left( t,\lambda \right) }{\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) }, &{}\quad 0\le t\le x\\ \frac{\theta _{1}\left( x,\lambda \right) \theta _{2}^\mathrm{T}\left( t,\lambda \right) }{\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) }, &{}\quad x<t\le 1. \end{array} \right. \end{aligned}$$

Example 27

Here, we will consider Eq. (41) with the conditions

$$\begin{aligned} L_{1}\left( y\right) =y_{1}\left( 0\right) =0,\ L_{2}\left( y\right) =y_{1}\left( 1\right) +y_{2}\left( 1\right) =0. \end{aligned}$$

Then, we have

$$\begin{aligned} \Delta \left( \lambda \right) =\left| \begin{array} [c]{cc} L_{1}\left( \phi _{1}\right) &{} L_{1}\left( \phi _{2}\right) \\ L_{2}\left( \phi _{1}\right) &{} L_{2}\left( \phi _{2}\right) \end{array} \right| =-\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) -\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) . \end{aligned}$$

The eigenvalues of this problem are the solutions of the equation

$$\begin{aligned} \cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) =-\sin \left( \int _{0} ^{1}\lambda \mathrm{d}^{\alpha }t\right) . \end{aligned}$$

The functions \(\theta _{1}\left( x,\lambda \right) \) and \(\theta _{2}\left( x,\lambda \right) \ \)are given by

$$\begin{aligned} \theta _{1}\left( x,\lambda \right) =\left( \begin{array} [c]{c} -\sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \\ -\cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) , \end{aligned}$$
$$\begin{aligned}&\theta _{2}\left( x,\lambda \right) \\&\quad =\left( \begin{array} [c]{c} \sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{x}\lambda \mathrm{d}^{\alpha }t\right) +\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \\ +\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{x}\lambda \mathrm{d}^{\alpha }t\right) -\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) \\&\qquad +\left( \begin{array} [c]{c} -\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0} ^{x}\lambda \mathrm{d}^{\alpha }t\right) -\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \sin \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \\ \\ -\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0} ^{x}\lambda \mathrm{d}^{\alpha }t\right) +\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) \cos \left( \int _{0}^{x}\lambda \mathrm{d}^{\alpha }t\right) \end{array} \right) . \end{aligned}$$

If \(\lambda \) is not an eigenvalue, then Green’s matrix is given by

$$\begin{aligned}&G\left( x,t,\lambda \right) \\&\quad =\left\{ \begin{array} [c]{cc} -\frac{1}{\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) +\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) }\theta _{2}\left( x,\lambda \right) \theta _{1}^\mathrm{T}\left( t,\lambda \right) , &{}\quad 0\le t\le x\\ -\frac{1}{\sin \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) +\cos \left( \int _{0}^{1}\lambda \mathrm{d}^{\alpha }t\right) }\theta _{1}\left( x,\lambda \right) \theta _{2}^\mathrm{T}\left( t,\lambda \right) , &{}\quad x<t\le 1. \end{array} \right. \end{aligned}$$

Remark 28

In [22], the authors discuss systems of conformable linear differential equations with constant coefficients. They give full solution for homogeneous and nonhomogeneous systems.

Conclusion 29

In this study, we have proved an existence and uniqueness theorem for a conformable fractional Dirac system. Later, we have formulated a self-adjoint boundary-value problem. Then, we also constructed the associated Green matrix of the conformable fractional Dirac system. Finally, we have given the eigenfunction expansions and some examples.

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Allahverdiev, B.P., Tuna, H. One-dimensional conformable fractional Dirac system. Bol. Soc. Mat. Mex. 26, 121–146 (2020). https://doi.org/10.1007/s40590-019-00235-5

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Keywords

  • Conformable fractional Dirac system
  • Self-adjoint operator
  • Eigenvalues and eigenfunctions
  • Green’s matrix
  • Eigenfunction expansions

Mathematics Subject Classification

  • 34A08
  • 26A33
  • 34L10
  • 34L40
  • 47A10
  • 47B25