Diamond-Type Dirac Dynamic System in Mathematical Physics

Abstract

In order to merge continuous and discrete analyses, a number of dynamic derivative equations have been put out in the process of developing a time-scale calculus. The investigations that incorporated combined dynamic derivatives have led to the proposal of improved approximation expressions for computational application. One such expression is the diamond alpha (${\diamond }_{\alpha }$) derivative, which is defined as a linear combination of delta and nabla derivatives. Several dynamic equations and inequalities, as well as hybrid dynamic behavior—which does not occur in the real line or on discrete time scales—are analyzed using this combined concept. In this study, we consider a ${\diamond }_{\alpha }$ Dirac system under boundary conditions on a uniform time scale. We examined some basic spectral properties of the problem we are considering, such as the simplicity, the reality of eigenvalues, orthogonality of eigenfunctions, and self adjointness of the operator. Finally, we construct an expression for the eigenfunction of the ${\diamond }_{\alpha }$ Dirac boundary value problem (BVP) on a uniform time scale.

1. Introduction

A time scale $\mathbb{T}$ is an arbitrary, non-empty, closed subset of $\mathbb{R}$. The time-scale theory was first stated by Hilger [1]. Hilger and Aulbach have obtained important results on this subject [2,3,4]. As a result of the selection of two operators considered on a time scale, $\sigma$ and $\rho$ operators, respectively, we encounter the concepts of delta and nabla calculations. A detailed theory of delta calculus was discussed by Bohner and Peterson [5,6]. The nabla theory can be found in the relevant references [7,8,9,10]. ${\diamond }_{\alpha }$ calculus, which is considered as a combination of delta and nabla calculi, was introduced by Sheng et al. [11]. They studied the basic properties of first and second order ${\diamond }_{\alpha }$ derivatives, which are linear combinations of delta and nabla dynamic calculus on a time scale. Here, diamond calculus corresponds to delta calculus when $\alpha =1$ and nabla calculus when $\alpha =0$. On the other hand, $\alpha =\frac{1}{2}$ represents a central derivative formula on any uniform discrete time scales [11]. There are many studies related to ${\diamond }_{\alpha }$ derivative on a time scale. In 2007, Rogers and Sheng studied the case of ${\diamond }_{\alpha }$ and the ${\diamond }_{\alpha }$ integral as a dynamic derivative on the time scale [12]. In this same year, Sheng studied an approach to the classical second-order derivative via ${\diamond }_{\alpha }$ dynamical differentiation under the necessary conditions on a time scale [13]. In 2008, Ammi and Torres gave the basic properties of ${\diamond }_{\alpha }$ derivatives and proved a generalized version of Jensen’s inequalities on a time scale via ${\diamond }_{\alpha }$ integral [14]. In 2009, Mozyrska and Torres worked on a ${\diamond }_{\alpha }$ exponential function and homogeneous linear dynamic ${\diamond }_{\alpha }$ equations on a time scale [15]. In the same year, Mozyrska and Torres studied ${\diamond }_{\alpha }$ integrals on time scales and proved their mean value theorem [16]. In another study, they discussed the results of generalized polynomials covering the definitions and properties of delta and nabla derivatives on a time scale and generalized these results using the concept of the ${\diamond }_{\alpha }$ derivative [17]. In 2009, Ozkan and Kaymakcalan introduced double integral calculus for two-variable functions through partial ${\diamond }_{\alpha }$ dynamic derivatives and the ${\diamond }_{\alpha }$ dynamic integral [18]. In 2010, Ammi and Torres studied of the Hardy-type dynamic inequalities on a time scale via the ${\diamond }_{\alpha }$ integral [19]. In 2015, Cruz, Martins, and Torres gave an improved version of the ${\diamond }_{\alpha }$ integral and proved new versions of some important inequalities [20].

To obtain the fundamental results for (5), we need to recall a few basic time-scale ideas. The following are the definitions of the forward and backward jump operators of $\sigma ,\rho :\mathbb{T}\to \mathbb{T}$, for $t\in \mathbb{T}:$

$$\sigma \left(t\right)=\inf \left\{s\in \mathbb{T}:s>t\right\},\rho \left(t\right)=\sup \left\{s\in \mathbb{T}:s<t\right\}.$$

If $\sigma \left(t\right)>t,\rho \left(t\right)<t,\rho \left(t\right)<t<\sigma \left(t\right)$, t is a right-scattered point, a left-scattered point, an isolated (discrete) point, respectively. On the other hand, if $t<\sup \mathbb{T}$ and $\sigma \left(t\right)=t$; $t>\inf \mathbb{T}$ and $\rho \left(t\right)=t;$ and $\rho \left(t\right)=t=\sigma \left(t\right)$; t is called the right-dense, left-dense, and dense point, respectively. The graininess function $\mu :\mathbb{T}\to \left[0,\infty \right)$ is defined as $\mu \left(t\right)=\sigma \left(t\right)-t.$ ${\mathbb{T}}^{\kappa }=\mathbb{T}-\left\{m\right\}$ if there is a maximum m point of $\mathbb{T}$; otherwise, ${\mathbb{T}}^{\kappa }=\mathbb{T}$. The $f:\mathbb{T}\to \mathbb{R}$ function is called rd-continuous, provided that $\mathbb{T}$ has a left-sided limit at its right-dense points and at its left-scattered points, and $C_{rd}\left(\mathbb{T}\right)$ will be used to denote the collection of $f:\mathbb{T}\to \mathbb{R}$ rd-continuous functions. For all $t\in {\mathbb{T}}^{\kappa },$ if $1+\mu \left(t\right)p\left(t\right)\ne 0$ holds, we say that function $p:\mathbb{T}\to \mathbb{R}$ is regressive [5].

Let $t\in {\mathbb{T}}^{\kappa }$ and $f:\mathbb{T}\to \mathbb{R}$ be a function. For all $\epsilon >0,$ and for every s in a neighborhood U of point t, if there is a real number $f^{\mathrm{\Delta }}\left(t\right)$, such that

$$\left|\left[f\left(\sigma \right(t\left)\right)-f\left(s\right)\right]-f^{\mathrm{\Delta }}\left(t\right)\left[\sigma \left(t\right)-s\right]\right|\le \epsilon \left|\sigma \left(t\right)-s\right|,\forall s\in U,$$

$f^{\mathrm{\Delta }}\left(t\right)$ is called the $\mathrm{\Delta }$ (delta) derivative of f at point t. Furthermore, if $f^{\mathrm{\Delta }}\left(t\right)$ exists for every $t\in {\mathbb{T}}^{\kappa }$, we state that f is $\mathrm{\Delta }$ differentiable. For all $\epsilon >0,$ and for every s in a neighborhood U of point $t\in {\mathbb{T}}_{\kappa }$, if there is a real number $f^{\nabla }\left(t\right)$, such that

$$\left|\left[f\left(\rho \right(t\left)\right)-f\left(s\right)\right]-f^{\nabla }\left(t\right)\left[\rho \left(t\right)-s\right]\right|\le \epsilon \left|\rho \left(t\right)-s\right|,\forall s\in U,$$

$f^{\nabla }\left(t\right)$ is called the ∇ (delta) derivative of f at point t. Moreover, we declare that f^∇ differentiable on ${\mathbb{T}}_{\kappa }$ if $f^{\nabla }\left(t\right)$ exists for each $t\in {\mathbb{T}}_{\kappa }$ [3,5,6,9].

Now, we will give the important definitions and theorems related to ${\diamond }_{\alpha }$ calculus [11,12,13,14,15,16,17,18,19,20,21,22,23], which we use as the basis for our study on a time scale.

Assume that $f\left(t\right)$ is differentiable in $\mathrm{\Delta }$ and ∇ senses on a time scale $\mathbb{T}$. For $t\in {\mathbb{T}}_{\kappa }^{\kappa }$, where ${\mathbb{T}}_{\kappa }^{\kappa }={\mathbb{T}}_{\kappa }\cap {\mathbb{T}}^{\kappa }$, we define ${\diamond }_{\alpha }$ dynamic derivative $f^{{\diamond }_{\alpha }}\left(t\right)$ by

$$f^{{\diamond }_{\alpha }}\left(t\right)=\alpha f^{\mathrm{\Delta }}\left(t\right)+(1-\alpha )f^{\nabla }\left(t\right),\alpha \in [0,1].$$

(1)

Consequently, if and only if f is both $\mathrm{\Delta }$ and ∇ differentiable, then f is ${\diamond }_{\alpha }$ differentiable. We assume that $t\in \mathbb{T}$ is a dense point. If $f^{\prime }\left(t\right)$ exists, then $f^{{\diamond }_{\alpha }}\left(t\right)=f^{\mathrm{\Delta }}\left(t\right)=f^{\nabla }\left(t\right)=f^{\prime }\left(t\right)$. This shows that ${\diamond }_{\alpha }$ derivative is an important generalization.

If $f,g:\mathbb{T}\to \mathbb{R}$ are ${\diamond }_{\alpha }$ differentiable at $t_0\in \mathbb{T}$, the following properties hold:

${({\gamma }_{1}f+{\gamma }_{2}g)}^{{\diamond }_{\alpha }}\left(t_0\right)=({\gamma }_1{f}^{{\diamond }_{\alpha }}+{\gamma }_2{g}^{{\diamond }_{\alpha }})\left(t_0\right),$ for all ${\gamma }_1,{\gamma }_2\in \mathbb{R},$

${\left(fg\right)}^{{\diamond }_{\alpha }}\left(t_0\right)=(f^{{\diamond }_{\alpha }}g+\alpha f^{\sigma }{g}^{\mathrm{\Delta }}+(1-\alpha )f^{\rho }{g}^{\nabla })\left(t_0\right),$

${\left(\frac{f}{g}\right)}^{{\diamond }_{\alpha }}\left(t_0\right)=\frac{(f^{{\diamond }_{\alpha }}{g}^{\sigma }{g}^{\rho }-\alpha f^{\sigma }{g}^{\rho }{g}^{\mathrm{\Delta }}-(1-\alpha )f^{\rho }{g}^{\sigma }{g}^{\nabla })\left(t_0\right)}{\left(g{g}^{\sigma }{g}^{\rho }\right)\left(t_0\right)},$ where $g{g}^{\sigma }{g}^{\rho }\left(t_0\right)\ne 0.$

Let $a,t\in \mathbb{T}$, and $h:\mathbb{T}\to \mathbb{R}$. The ${\diamond }_{\alpha }$ integral of h is defined as

$${\int }_a^{t}h\left(\zeta \right){\diamond }_{\alpha }\zeta =\alpha {\int }_a^{t}h\left(\zeta \right)\mathrm{\Delta }\zeta +(1-\alpha ){\int }_a^{t}h\left(\zeta \right)\nabla \zeta .$$

(2)

The following property is provided for $a,t\in \mathbb{T}$, $b,c\in \mathbb{R}$.

$${\int }_a^t[bf\left(\zeta \right)+cg\left(\zeta \right)]{\diamond }_{\alpha }\zeta =b{\int }_a^{t}f\left(\zeta \right){\diamond }_{\alpha }\zeta +c{\int }_a^{t}g\left(\zeta \right){\diamond }_{\alpha }\zeta .$$

Let $t,t_i\in \mathbb{T}$, $p:\mathbb{T}\to \mathbb{R}$ is a regressive function, $p\left(t\right)\equiv p,$ and $1+{\nu }^2\left(t\right)p^2\ne 0$ for all $t\in \mathbb{T}$ where $\nu \left(t\right)$ is the the backward graininess function. Then, for $t\in {\mathbb{T}}_{\kappa }^{\kappa }$ [24]

$$si{n}_p^{{\diamond }_{\alpha }}(t,t_i)=\frac{p}{1+{\nu }^2{p}^2}((1+\alpha {\nu }^2{p}^2)co{s}_p(t,t_i)+(1-\alpha )\nu psi{n}_p(t,t_i)),$$

(3)

and

$$co{s}_p^{{\diamond }_{\alpha }}(t,t_i)=\frac{-p}{1+{\nu }^2{p}^2}((1+\alpha {\nu }^2{p}^2)si{n}_p(t,t_i)-(1-\alpha )\nu pco{s}_p(t,t_i)).$$

(4)

The space of all square ${\diamond }_{\alpha }$ integrable functions on $[a,b]$, where $a,b\in \mathbb{T}$ and $\alpha \in [0,1]$, is as follows.

$$L_2^{{\diamond }_{\alpha }}[a,b]=\left\{f:{\int }_a^b{f}^2\left(t\right){\diamond }_{\alpha }t<\infty \right\}.$$

We define the diamond-type inner product of $y(t,\lambda )=\left(\begin{array}{c}{y}_1(t,\lambda )\\ y_2(t,\lambda )\end{array}\right)$ and $z(t,\lambda )=\left(\begin{array}{c}{z}_1(t,\lambda )\\ z_2(t,\lambda )\end{array}\right)$ on $L_2^{{\diamond }_{\alpha }}[a,b]$ by

$${〈y,z〉}_{{\diamond }_{\alpha }}={\int }_a^b[y_1(t,\lambda )z_1(t,\lambda )+y_2(t,\lambda )z_2(t,\lambda )]{\diamond }_{\alpha }t.$$

The space $L_2^{{\diamond }_{\alpha }}[0,N]$ and ${\diamond }_{\alpha }$ inner product will be used in the proof of spectral properties of a ${\diamond }_{\alpha }$ Dirac problem in the next section.

In this study, we consider a ${\diamond }_{\alpha }$ Dirac eigenvalue problem

$$D_{{\diamond }_{\alpha }}y\equiv B{y}^{{\diamond }_{\alpha }}\left(t\right)+Q\left(t\right)y\left(t\right)=\lambda y\left(t\right),t\in [0,N],N\in \mathbb{R},$$

(5)

which is handled with boundary conditions

$$y_1\left(0\right)-h{y}_2\left(0\right)=0,$$

(6)

$$y_1\left(N\right)+H{y}_2\left(N\right)=0,$$

(7)

where

$$Q\left(t\right)=\left(\begin{array}{cc}q\left(t\right)& 0\\ 0& r\left(t\right)\end{array}\right),B=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),$$

$h,H\in \mathbb{R}$; $\lambda$ is a spectral parameter, and $q,r:[0,N]\cap \mathbb{T}\to \mathbb{R}$ are continuous functions. If the necessary adjustments are made in Equation (5), the following system is obtained:

$$\begin{array}{cc}\hfill & y_2^{{\diamond }_{\alpha }}\left(t\right)=(\lambda -q\left(t\right))y_1\left(t\right),\hfill \\ \hfill & y_1^{{\diamond }_{\alpha }}\left(t\right)=(-\lambda +r\left(t\right))y_2\left(t\right).\hfill \end{array}$$

(8)

By setting $\mathbb{T}=\mathbb{R}$ in (8), we obtain following classical Dirac system:

$$\begin{array}{cc}\hfill & y_2^{\prime }\left(t\right)=(\lambda -q\left(t\right))y_1\left(t\right),\hfill \\ \hfill & y_1^{\prime }\left(t\right)=(-\lambda +r\left(t\right))y_2\left(t\right).\hfill \end{array}$$

(9)

Equation (9) is called the first canonical form of the Dirac system. In quantum physics, the Dirac operator is the relativistic Schrödinger operator. Dirac proposed the Dirac operator for some difficulties encountered in quantum physics [25]. There are many studies on the Dirac operator and related problems, which is an important subject of study in this field. Levitan and Sargsjan have discussed the spectral theory of Dirac operators in detail [26]. A study on the eigenfunction expansion for the Dirac operator was made by Joa and Minkin [27]. Bairamov, Aygar, and Olgun found the Jost solutions of self-adjoint Dirac systems and examined their analytical properties and asymptotic behavior. They demonstrated that there is a finite number of simple real eigenvalues in the Dirac system [28]. Keskin and Ozkan studied the properties of eigenvalues and eigenfunctions of the inverse spectral problem for the Dirac operator [29]. The Dirac system has also been studied on the time scale. Gulsen and Yilmaz examined the spectral theory of the Dirac system on the time scale [30]. Hovhannisyan studied the linear Dirac equation on the discrete continuous and quantum time scale [31]. Allahverdiev and Tuna introduced the q analogue of the Dirac equation and examined some spectral properties by investigating the existence and uniqueness of the solution [32]. Gulsen, Yilmaz, and Goktas examine the conformable Dirac system on time scales [33]. Koprubası studied the impulsive discrete Dirac system with hyperbolic eigenparameters [34]. There are many studied theories regarding different versions of the Dirac system [35,36,37,38,39,40,41,42].

This study is organized as follows: After a reminder of some basic definitions and theorems regarding the calculation of ${\diamond }_{\alpha }$ on time scales in the introduction section, in Section 2, we examine some spectral properties of a ${\diamond }_{\alpha }$ Dirac problem on a uniform ($\mu \left(t\right)=\nu \left(t\right)$, for all $t\in {\mathbb{T}}_{\kappa }^{\kappa }$ [12]) time scale, and we obtain an expression for the eigenfunction of the ${\diamond }_{\alpha }$ Dirac system.

2. Main Results

In this section, a ${\diamond }_{\alpha }$ Dirac BVP,

$$y_2^{{\diamond }_{\alpha }}\left(t\right)=(\lambda -q\left(t\right))y_1\left(t\right),y_1^{{\diamond }_{\alpha }}\left(t\right)=(-\lambda +r\left(t\right))y_2\left(t\right),$$

(10)

$$y_1\left(0\right)-h{y}_2\left(0\right)=0,$$

(11)

$$y_1\left(N\right)+H{y}_2\left(N\right)=0,$$

(12)

is discussed on a uniform time scale. The realness of eigenvalues and orthogonality of eigenfunctions on $L_2^{{\diamond }_{\alpha }}[0,N]$ are examined. The symmetry, boundedness, and linearity properties of Dirac operator,

$$D_{{\diamond }_{\alpha }}:X\to Y,D_{{\diamond }_{\alpha }}y=\left(\begin{array}{c}{y}_2^{{\diamond }_{\alpha }}\left(t\right)-[\lambda -q\left(t\right)]y_1\left(t\right)\\ y_1^{{\diamond }_{\alpha }}\left(t\right)-[-\lambda +r\left(t\right)]y_2\left(t\right)\end{array}\right),$$

have been proven on $L_2^{{\diamond }_{\alpha }}[0,N]$, where X and Y are vector spaces. Later, we obtained some eigenfunction expansions for a ${\diamond }_{\alpha }$ Dirac problem. In this section, it is assumed that $\mathbb{T}$ is uniform.

Theorem 1 

([5]). Let us suppose that $f:\mathbb{T}\to \mathbb{R}$ on ${\mathbb{T}}^k$ is Δ differentiable. Then, for any $t\in {\mathbb{T}}^k$ such that $\sigma \left(\rho \right(t\left)\right)=t$, f is ∇ differentiable at t and $f^{\nabla }\left(t\right)=f^{\mathrm{\Delta }}\left(\rho \left(t\right)\right)$. Let us assume that $f:\mathbb{T}\to \mathbb{R}$ on ${\mathbb{T}}_k$ is ∇ differentiable. Then, for any $t\in {\mathbb{T}}_k$ such that $\rho \left(\sigma \right(t\left)\right)=t$, f is Δ differentiable at t and $f^{\mathrm{\Delta }}\left(t\right)=f^{\nabla }\left(\sigma \left(t\right)\right)$.

Lemma 1.

Let $f:\mathbb{T}\to \mathbb{R}$ be ${\diamond }_{\alpha }$ integrable for $\forall t\in {\mathbb{T}}_{\kappa }^{\kappa }$ and $\alpha \in (0,1]$, $d_1,d_2\in \mathbb{T}$. Then, the following equality holds:

$${\int }_{d_1}^{d_2}{f}^{{\diamond }_{\alpha }}\left(t\right){\diamond }_{\alpha }t=(2{\alpha }^2-2\alpha +1)f\left(t\right){|}_{d_1}^{d_2}+\alpha (1-\alpha )(f^{\rho }+f^{\sigma })\left(t\right){|}_{d_1}^{d_2}.$$

Proof. 

Using (1) then (2) and Theorem 1, we obtain

$$\begin{array}{cc}\hfill {\int }_{d_1}^{d_2}{f}^{{\diamond }_{\alpha }}\left(t\right){\diamond }_{\alpha }t& ={\int }_{d_1}^{d_2}(\alpha f^{\mathrm{\Delta }}+(1-\alpha )f^{\nabla })\left(t\right){\diamond }_{\alpha }t\hfill \\ \hfill & ={\alpha }^2{\int }_{d_1}^{d_2}{f}^{\mathrm{\Delta }}\left(t\right)\mathrm{\Delta }t+\alpha (1-\alpha ){\int }_{d_1}^{d_2}{f^{\rho }}^{\mathrm{\Delta }}\left(t\right)\mathrm{\Delta }t\hfill \\ & +\alpha (1-\alpha ){\int }_{d_1}^{d_2}{f^{\sigma }}^{\nabla }\left(t\right)\nabla t+{(1-\alpha )}^2{\int }_{d_1}^{d_2}{f}^{\nabla }\left(t\right)\nabla t\hfill \\ & =(2{\alpha }^2-2\alpha +1)f\left(t\right){|}_{d_1}^{d_2}+\alpha (1-\alpha )(f^{\rho }+f^{\sigma })\left(t\right){|}_{d_1}^{d_2}.\hfill \end{array}$$

□

Lemma 2.

Let $a,b\in {\mathbb{T}}^{\ast }=\mathbb{T}-$ $\{min\mathbb{T}$, $max\mathbb{T}\},$ $h_1,h_2:\mathbb{T}\to \mathbb{R}$ be ${\diamond }_{\alpha }$ integrable for ∀$\zeta \in {\mathbb{T}}_{\kappa }^{\kappa }$ and $\alpha \in (0,1].$ Then, the ${\diamond }_{\alpha }$ integration by parts formula on $\mathbb{T}$ is constructed by

$$\begin{array}{ccc}\hfill \underset{a}{\overset{b}{\int }}\left(h_1^{{\diamond }_{\alpha }}{h}_2\right)\left(\zeta \right){\diamond }_{\alpha }\zeta & =& (1-2\alpha +2{\alpha }^2)\left(h_1{h}_2\right)\left(\zeta \right){\mid }_a^b+\alpha (1-\alpha )\left[{\left(h_1{h}_2\right)}^{\rho }+{\left(h_1{h}_2\right)}^{\sigma }\right]\left(\zeta \right){\mid }_a^b\hfill \\ & & +(1-\alpha )\underset{a}{\overset{b}{\int }}{h}_2^{\nabla }\left(h_1^{\sigma }-h_1^{\rho }\right)\left(\zeta \right){\diamond }_{\alpha }\zeta -\underset{a}{\overset{b}{\int }}\left(h_1^{\sigma }{h}_2^{{\diamond }_{\alpha }}\right)\left(\zeta \right){\diamond }_{\alpha }\zeta .\hfill \end{array}$$

Proof. 

By applying the ${\diamond }_{\alpha }$ integral to the below equality,

$${\left(h_1{h}_2\right)}^{{\diamond }_{\alpha }}\left(\zeta \right)=h_1^{{\diamond }_{\alpha }}\left(\zeta \right)h_2\left(\zeta \right)+\alpha h_1^{\sigma }\left(\zeta \right)h_2^{\mathrm{\Delta }}\left(\zeta \right)+(1-\alpha )h_1^{\rho }\left(\zeta \right)h_2^{\nabla }\left(\zeta \right),$$

on $[a,b],$ we get

$$\underset{a}{\overset{b}{\int }}{\left(h_1{h}_2\right)}^{{\diamond }_{\alpha }}\left(\zeta \right){\diamond }_{\alpha }\zeta =\underset{a}{\overset{b}{\int }}{h}_1^{{\diamond }_{\alpha }}\left(\zeta \right)h_2\left(\zeta \right){\diamond }_{\alpha }\zeta +\alpha \underset{a}{\overset{b}{\int }}{h}_1^{\sigma }\left(\zeta \right)h_2^{\mathrm{\Delta }}\left(\zeta \right){\diamond }_{\alpha }\zeta +(1-\alpha )\underset{a}{\overset{b}{\int }}{h}_1^{\rho }\left(\zeta \right)h_2^{\nabla }\left(\zeta \right){\diamond }_{\alpha }\zeta .$$

In the last equality, isolating the term $\underset{a}{\overset{b}{\int }}{h}_1^{{\diamond }_{\alpha }}\left(\zeta \right)h_2\left(\zeta \right){\diamond }_{\alpha }\zeta$ yields

$$\begin{array}{ccc}\hfill \underset{a}{\overset{b}{\int }}{h}_1^{{\diamond }_{\alpha }}\left(\zeta \right)h_2\left(\zeta \right){\diamond }_{\alpha }\zeta & =& (1-2\alpha +2{\alpha }^2)\left(h_1{h}_2\right)\left(\zeta \right){\mid }_a^b+\alpha (1-\alpha )\left[{\left(h_1{h}_2\right)}^{\rho }+{\left(h_1{h}_2\right)}^{\sigma }\right]\left(\zeta \right){\mid }_a^b\hfill \\ & -& \alpha \underset{a}{\overset{b}{\int }}\left(h_1^{\sigma }{h}_2^{\mathrm{\Delta }}\right)\left(\zeta \right){\diamond }_{\alpha }\zeta -(1-\alpha )\underset{a}{\overset{b}{\int }}\left(h_1^{\rho }{h}_2^{\nabla }\right)\left(\zeta \right){\diamond }_{\alpha }\zeta \hfill \\ & -& (1-\alpha )\underset{a}{\overset{b}{\int }}\left(h_1^{\sigma }{h}_2^{\nabla }\right)\left(\zeta \right){\diamond }_{\alpha }\zeta +(1-\alpha )\underset{a}{\overset{b}{\int }}\left(h_1^{\sigma }{h}_2^{\nabla }\right)\left(\zeta \right){\diamond }_{\alpha }\zeta \hfill \\ & =& (1-2\alpha +2{\alpha }^2)\left(h_1{h}_2\right)\left(\zeta \right){\mid }_a^b+\alpha (1-\alpha )\left[{\left(h_1{h}_2\right)}^{\rho }+{\left(h_1{h}_2\right)}^{\sigma }\right]\left(\zeta \right){\mid }_a^b\hfill \\ & -& \underset{a}{\overset{b}{\int }}{h}_1^{\sigma }\left[\alpha h_2^{\mathrm{\Delta }}+(1-\alpha )h_2^{\nabla }\right]\left(\zeta \right){\diamond }_{\alpha }\zeta +(1-\alpha )\underset{a}{\overset{b}{\int }}{h}_2^{\nabla }\left[h_1^{\sigma }-h_1^{\rho }\right]\left(\zeta \right){\diamond }_{\alpha }\zeta .\hfill \end{array}$$

□

Theorem 2.

For ${\lambda }_1\ne {\lambda }_2$, the distinct eigenfunctions of ${\diamond }_{\alpha }$ Dirac BVP (10)–(12) are orthogonal on $L_2^{{\diamond }_{\alpha }}[0,N]$.

Proof. 

Let $y(t,{\lambda }_1)=\left(\begin{array}{c}{y}_1(t,{\lambda }_1)\\ y_2(t,{\lambda }_1)\end{array}\right)$ and $z(t,{\lambda }_2)=\left(\begin{array}{c}{z}_1(t,{\lambda }_2)\\ z_2(t,{\lambda }_2)\end{array}\right)$ be two eigenfunctions of (10)–(12). Thus, these functions provide (10). Then, we get

$$\begin{array}{cc}\hfill & y_2^{{\diamond }_{\alpha }}-[{\lambda }_1-q\left(t\right)]y_1=0,\hfill \\ \hfill & y_1^{{\diamond }_{\alpha }}-[-{\lambda }_1+r\left(t\right)]y_2=0,\hfill \\ \hfill & z_2^{{\diamond }_{\alpha }}-[{\lambda }_2-q\left(t\right)]z_1=0,\hfill \\ \hfill & z_1^{{\diamond }_{\alpha }}-[-{\lambda }_2+r\left(t\right)]z_2=0.\hfill \end{array}$$

If the above equations are multiplied by $z_1$, $-z_2$, $-y_1$, and $y_2$, respectively, and rearranged, we get

$$z_1{y}_2^{{\diamond }_{\alpha }}-z_2{y}_1^{{\diamond }_{\alpha }}-y_1{z}_2^{{\diamond }_{\alpha }}+y_2{z}_1^{{\diamond }_{\alpha }}+[{\lambda }_1-{\lambda }_2](z_1{y}_1+z_2{y}_2)=0.$$

Since $\mathbb{T}$ is uniform, the following equation can be written as

$$z_1{y}_2^{{\diamond }_{\alpha }}-z_2{y}_1^{{\diamond }_{\alpha }}-y_1{z}_2^{{\diamond }_{\alpha }}+y_2{z}_1^{{\diamond }_{\alpha }}={\diamond }_{\alpha }[-y_1{z}_2+y_2{z}_1].$$

If we use this expression in the previous equation and integrate both sides from 0 to N, we get

$${\int }_0^N{(-y_1{z}_2+y_2{z}_1)}^{{\diamond }_{\alpha }}\left(t\right){\diamond }_{\alpha }t=({\lambda }_2-{\lambda }_1){\int }_0^N(z_1{y}_1+z_2{y}_2)\left(t\right){\diamond }_{\alpha }t.$$

So,

$$\begin{array}{cc}\hfill ({\lambda }_2-{\lambda }_1){\int }_0^N(z_1{y}_1+z_2{y}_2)\left(t\right){\diamond }_{\alpha }t& =(2{\alpha }^2-2\alpha +1)\left(z_1{y}_2\right)\left(t\right){|}_0^N\hfill \\ & +\alpha (1-\alpha ){\left(z_1{y}_2\right)}^{\rho }\left(t\right)+{\left(z_1{y}_2\right)}^{\sigma }\left(t\right){|}_0^N\hfill \\ & -(2{\alpha }^2-2\alpha +1)\left(z_2{y}_1\right)\left(t\right){|}_0^N\hfill \\ & -\alpha (1-\alpha ){\left(z_2{y}_1\right)}^{\rho }\left(t\right)+{\left(z_2{y}_1\right)}^{\sigma }\left(t\right){|}_0^N.\hfill \end{array}$$

Remembering that $\mathbb{T}$ is uniform and $y(x,{\lambda }_1)$ and $z(x,{\lambda }_2)$ fulfill the boundary conditions (11) and (12), we get

$$({\lambda }_2-{\lambda }_1){\int }_0^N(z_1{y}_1+z_2{y}_2)\left(t\right){\diamond }_{\alpha }t=0.$$

As a result, we have

$$({\lambda }_2-{\lambda }_1){\int }_0^N(z_1{y}_1+z_2{y}_2)\left(t\right){\diamond }_{\alpha }t=0,{\lambda }_2\ne {\lambda }_1,$$

or

$${\int }_0^N(z_1{y}_1+z_2{y}_2)\left(t\right){\diamond }_{\alpha }t=0.$$

Since ${〈y,z〉}_{{\diamond }_{\alpha }}=0$, $y(t,{\lambda }_1)$ and $z(t,{\lambda }_2)$ are orthogonal on $L_2^{{\diamond }_{\alpha }}[0,N]$. □

Theorem 3.

The eigenvalues of ${\diamond }_{\alpha }$ Dirac BVP (10)–(12) are all real.

Proof. 

Assume that the complex eigenvalue is ${\lambda }_1$. So, ${\overline{\lambda }}_1=u-iv$ is also an eigenvalue. The following equation is obtained from the orthogonality theorem, where $y\overline{(}t,{\lambda }_1)$ is an eigenfunction.

$$({\lambda }_1-\overline{{\lambda }_1)}{\int }_0^N(y_1\overline{y_1}+y_2\overline{y_2})(t,{\lambda }_1){\diamond }_{\alpha }t=0$$

and, so

$$({\lambda }_1-{\overline{\lambda }}_1){\int }_0^N({\left|y_1\right|}^2+{\left|y_2\right|}^2)(t,{\lambda }_1){\diamond }_{\alpha }t=0.$$

Since ${\lambda }_1\ne {\overline{\lambda }}_1$, $y_1(t,{\lambda }_1)=y_2(t,{\lambda }_1)$. This is a contradiction since eigenfunctions cannot be zero. So, all eigenvalues are real. □

Theorem 4 

([43]). Let X be a Hilbert space and given the bounded linear operator $A:X\to X$. A necessary and sufficient condition for an operator A to be self-adjoint is that A is symmetric.

Lemma 3.

Minkowski’s inequality for ${\diamond }_{\alpha }$ integral [44]. If $p>1$ and $f_1,f_2:\mathbb{T}\to \mathbb{R}$ are ${\diamond }_{\alpha }-$ integrable on ${[a,b]}_{\mathbb{T}}$, then

$${\left({\int }_a^b{\left|f_1\left(\zeta \right)+f_2\left(\zeta \right)\right|}^p{\diamond }_{\alpha }\zeta \right)}^{\frac{1}{p}}\le {\left({\int }_a^b{\left|f_1\left(\zeta \right)\right|}^p{\diamond }_{\alpha }\zeta \right)}^{\frac{1}{p}}+{\left({\int }_a^b{\left|f_2\left(\zeta \right)\right|}^p{\diamond }_{\alpha }\zeta \right)}^{\frac{1}{p}}.$$

Theorem 5.

${\diamond }_{\alpha }$ Dirac operator $D_{{\diamond }_{\alpha }}$ is bounded on $L_2^{{\diamond }_{\alpha }}[0,N]$.

Proof. 

By the definition of $D_{{\diamond }_{\alpha }}$ and norm on $L_2^{{\diamond }_{\alpha }}$, and lemma 3, the following inequality can be obtained:

$$\begin{array}{cc}\hfill {∥D_{{\diamond }_{\alpha }}y∥}_{L_2^{{\diamond }_{\alpha }}}& ={\left({\int }_0^N\left({\left|y_2^{{\diamond }_{\alpha }}-[\lambda -q]y_1\right|}^2+{\left|y_1^{{\diamond }_{\alpha }}-[-\lambda +r]y_1\right|}^2\right){\diamond }_{\alpha }t\right)}^{{ }^1/{ }_2}\hfill \\ \hfill & \le {\left({\int }_0^N{\left|y_2^{{\diamond }_{\alpha }}-[\lambda -q]y_1\right|}^2{\diamond }_{\alpha }t\right)}^{{ }^1/{ }_2}+{\left({\int }_0^N{\left|y_1^{{\diamond }_{\alpha }}-[-\lambda +r]y_2\right|}^2{\diamond }_{\alpha }t\right)}^{{ }^1/{ }_2}\hfill \\ \hfill & \le 2{∥y∥}_{L_2^{{\diamond }_{\alpha }}}.\hfill \end{array}$$

Thus, $D_{{\diamond }_{\alpha }}$ is bounded on $L_2^{{\diamond }_{\alpha }}[0,N]$. □

Theorem 6.

${\diamond }_{\alpha }$ Dirac operator $D_{{\diamond }_{\alpha }}$ is linear.

Proof. 

Let $y^{{\diamond }_{\alpha }}=\left(\begin{array}{c}{y}_1^{{\diamond }_{\alpha }}\\ y_2^{{\diamond }_{\alpha }}\end{array}\right)$, $z^{{\diamond }_{\alpha }}=\left(\begin{array}{c}{z}_1^{{\diamond }_{\alpha }}\\ z_2^{{\diamond }_{\alpha }}\end{array}\right)$ and ${\beta }_1,{\beta }_2\in \mathbb{R}.$ From the definition of $D_{{\diamond }_{\alpha }}$, we get

$$\begin{array}{cc}\hfill D_{{\diamond }_{\alpha }}({\beta }_{1}y+{\beta }_{2}z)& =\left(\begin{array}{c}{\left({\beta }_1{y}_2+{\beta }_2{z}_2\right)}^{{\diamond }_{\alpha }}-(\lambda -q)({\beta }_1{y}_1+{\beta }_2{z}_1)\\ {\left({\beta }_1{y}_1+{\beta }_2{z}_1\right)}^{{\diamond }_{\alpha }}-(-\lambda +r)({\beta }_1{y}_2+{\beta }_2{z}_2)\end{array}\right)\hfill \\ \hfill & ={\beta }_1\left(\begin{array}{c}{y}_2^{{\diamond }_{\alpha }}-(\lambda -q)y_1\\ y_1^{{\diamond }_{\alpha }}-(-\lambda +r)y_2\end{array}\right)+{\beta }_2\left(\begin{array}{c}{z}_2^{{\diamond }_{\alpha }}-(\lambda -q)z_1\\ z_1^{{\diamond }_{\alpha }}-(-\lambda +r)z_2\end{array}\right)\hfill \\ \hfill & ={\beta }_1{D}_{{\diamond }_{\alpha }}\left(y\right)+{\beta }_2{D}_{{\diamond }_{\alpha }}\left(z\right).\hfill \end{array}$$

Therefore, $D_{{\diamond }_{\alpha }}$ is linear. □

Theorem 7.

${\diamond }_{\alpha }$ Dirac operator is symmetric on $L_2^{{\diamond }_{\alpha }}[0,N]$.

Proof. 

Let eigenfunctions $x=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$ and $y=\left(\begin{array}{c}{y}_1\\ y_2\end{array}\right)$ be defined for the system (5). It is necessary to demonstrate that ${〈x,D_{{\diamond }_{\alpha }}y〉}_{{\diamond }_{\alpha }}={〈D_{{\diamond }_{\alpha }}x,y〉}_{{\diamond }_{\alpha }}$. Considering the system (5), we obtain

$$\begin{array}{cc}\hfill x^T{D}_{{\diamond }_{\alpha }}y-y^T{D}_{{\diamond }_{\alpha }}x& =x^{T}B{y}^{{\diamond }_{\alpha }}-y^{T}B{x}^{{\diamond }_{\alpha }}\hfill \\ \hfill & =\left(\begin{array}{cc}{x}_1& x_2\end{array}\right)\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}{y}_1^{{\diamond }_{\alpha }}\\ y_2^{{\diamond }_{\alpha }}\end{array}\right)-\left(\begin{array}{cc}{y}_1& y_2\end{array}\right)\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}{x}_1^{{\diamond }_{\alpha }}\\ x_2^{{\diamond }_{\alpha }}\end{array}\right)\hfill \\ \hfill & =y_2^{{\diamond }_{\alpha }}{x}_1-y_1^{{\diamond }_{\alpha }}{x}_2-x_2^{{\diamond }_{\alpha }}{y}_1+x_1^{{\diamond }_{\alpha }}{y}_2\hfill \\ \hfill & =\alpha (x_1{y}_2^{\mathrm{\Delta }}+y_2{x}_1^{\mathrm{\Delta }})+(1-\alpha )(x_1{y}_2^{\nabla }+y_2{x}_1^{\nabla })\hfill \\ & -\alpha (x_2{y}_1^{\mathrm{\Delta }}+y_1{x}_2^{\mathrm{\Delta }})-(1-\alpha )(y_1{x}_2^{\nabla }+x_2{y}_1^{\nabla }).\hfill \end{array}$$

Since $\mathbb{T}$ is uniform, the following equation is obtained.

$$x^T\left(t\right)D_{{\diamond }_{\alpha }}y\left(t\right)-y^T\left(t\right)D_{{\diamond }_{\alpha }}x\left(t\right)={\left(x_1{y}_2\right)}^{{\diamond }_{\alpha }}\left(t\right)-{\left(y_1{x}_2\right)}^{{\diamond }_{\alpha }}\left(t\right).$$

Let us apply the ${\diamond }_{\alpha }$ integral of both sides from 0 to N in the last equation. Then,

$$\begin{array}{cc}\hfill {〈x,D_{{\diamond }_{\alpha }}y〉}_{{\diamond }_{\alpha }}-{〈D_{{\diamond }_{\alpha }}x,y〉}_{{\diamond }_{\alpha }}& =[(2{\alpha }^2-2\alpha +1)\left(x_1{y}_2\right)\left(t\right)+\alpha (1-\alpha )\left(x_1{y}_2^{\rho }\right)\left(t\right)\hfill \\ & -\left(x_1{y}_2^{\sigma }\right)\left(t\right)-(2{\alpha }^2-2\alpha +1)\left(y_1{x}_2\right)\left(t\right)\hfill \\ & -\alpha (1-\alpha )\left(y_1{x}_2^{\rho }\right)\left(t\right)-\left(x_2{y}_1^{\sigma }\right)\left(t\right)]{|}_0^N.\hfill \end{array}$$

If boundary conditions (11) and (12) are considered, we obtain

$${〈x,D_{{\diamond }_{\alpha }}y〉}_{{\diamond }_{\alpha }}-{〈D_{{\diamond }_{\alpha }}x,y〉}_{{\diamond }_{\alpha }}=0.$$

From here, the ${\diamond }_{\alpha }$ Dirac operator $D_{{\diamond }_{\alpha }}$ is symmetric on $L_2^{{\diamond }_{\alpha }}[0,N]$. □

Conclusion 1.

The Dirac operator $D_{{\diamond }_{\alpha }}$is self-adjoint since it is bounded, linear, and symmetrical on $L_2^{{\diamond }_{\alpha }}[0,N]$.

Theorem 8.

All eigenvalues of the problem (10)–(12) are simple.

Proof. 

Let $y(t,\lambda )$ and $u(t,\lambda )$ be eigenfunctions of the problem (10)–(12). From (10), the following equations can be written:

$$\begin{array}{cc}\hfill & y_2^{{\diamond }_{\alpha }}=(\lambda -q\left(t\right))y_1,\hfill \\ \hfill & y_1^{{\diamond }_{\alpha }}=(-\lambda +r\left(t\right))y_2,\hfill \\ \hfill & u_2^{{\diamond }_{\alpha }}=(\lambda -q\left(t\right))u_1,\hfill \\ \hfill & u_1^{{\diamond }_{\alpha }}=(-\lambda +r\left(t\right))u_2.\hfill \end{array}$$

If these equations are multiplied by $u_2$, $u_1$, $y_2$, and $y_1$, respectively, and the last equation is subtracted from the 2nd and the 3rd equation is subtracted from first, we obtain

$$u_1{y}_1^{{\diamond }_{\alpha }}-y_1{u}_1^{{\diamond }_{\alpha }}=0,$$

(13)

and

$$u_2{y}_2^{{\diamond }_{\alpha }}-y_2{u}_2^{{\diamond }_{\alpha }}=0.$$

(14)

From (13) and (14), respectively, following is obtained:

$$u_1(\alpha y_1^{\mathrm{\Delta }}+(1-\alpha )y_1^{\nabla })-y_1(\alpha u_1^{\mathrm{\Delta }}+(1-\alpha )u_1^{\nabla })=0,$$

and

$$u_2(\alpha y_2^{\mathrm{\Delta }}+(1-\alpha )y_2^{\nabla })-y_2(\alpha u_2^{\mathrm{\Delta }}+(1-\alpha )u_2^{\nabla })=0.$$

Then,

$$\alpha {\left(\frac{y_1}{u_1}\right)}^{\mathrm{\Delta }}+(1-\alpha ){\left(\frac{y_1}{u_1}\right)}^{\nabla }=0,$$

and

$$\alpha {\left(\frac{y_2}{u_2}\right)}^{\mathrm{\Delta }}+(1-\alpha ){\left(\frac{y_2}{u_2}\right)}^{\nabla }=0.$$

Therefore, the following expressions are obtained:

$${\left(\frac{y_1}{u_1}\right)}^{{\diamond }_{\alpha }}=0,$$

and

$${\left(\frac{y_2}{u_2}\right)}^{{\diamond }_{\alpha }}=0.$$

As a result, $y_1=c{u}_1$ and $y_2=c{u}_2$ are obtained. This gives $y=cu,c\in \mathbb{R}$. So, it completes the proof. □

Theorem 9.

The asymptotic estimates for eigenfunction $y(t,t_0)$ corresponding to the ${\diamond }_{\alpha }$ Dirac problem (5)–(7) are

$$\begin{array}{cc}\hfill & y_1(t,t_0)=hco{s}_{\lambda }(t,t_0)-\frac{h(1-\alpha )\nu \lambda }{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,t_0)+\frac{1+{\nu }^2{\lambda }^2}{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,t_0)\hfill \\ \hfill & +\frac{\lambda {\nu }^2{\lambda }^2(1-\alpha )}{1+\alpha {\nu }^2{\lambda }^2}{\int }_0^t[y_2(s,t_0)co{s}_{\lambda }(t,t_0)+y_1(s,t_0)si{n}_{\lambda }(t,t_0)]{\diamond }_{\alpha }s\hfill \\ \hfill & +\frac{(1-\alpha )\nu {\lambda }^2}{1+\alpha {\nu }^2{\lambda }^2}{\int }_0^t[y_2(s,t_0)si{n}_{\lambda }(t,t_0)-y_1(s,t_0)co{s}_{\lambda }(t,t_0)]{\diamond }_{\alpha }s\hfill \\ \hfill & +{\int }_0^t\left(r\left(s\right)y_2(s,t_0)c{s}_{\lambda }^{-}(t,s)-\frac{(1+{\nu }^2{\lambda }^2)}{1+\alpha {\nu }^2{\lambda }^2}q\left(s\right)y_1(s,t_0)si{n}_{\lambda }(t,s)\right){\diamond }_{\alpha }s,\hfill \end{array}$$

(15)

and

$$\begin{array}{cc}\hfill & y_2(t,t_0)=-hsi{n}_{\lambda }(t,t_0)+co{s}_{\lambda }(t,t_0)+\frac{(1-\alpha )\nu \lambda }{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,t_0)\hfill \\ \hfill & +\frac{(1-\alpha )\nu {\lambda }^2}{1+\alpha {\nu }^2{\lambda }^2}{\int }_0^t[y_1(s,t_0)si{n}_{\lambda }(t,t_0)+y_2(s,t_0)co{s}_{\lambda }(t,t_0)]{\diamond }_{\alpha }s\hfill \\ \hfill & +{\int }_0^t[-r\left(s\right)y_2(s,t_0)si{n}_{\lambda }(t,s)-q\left(s\right)y_1(s,t_0)c{s}_{\lambda }^{+}(t,s)]{\diamond }_{\alpha }s,\hfill \end{array}$$

(16)

where

$c{s}_{\lambda }^{+}(t,s)=co{s}_{\lambda }(t,s)+\frac{(1-\alpha )\nu \lambda }{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,s),$ $c{s}_{\lambda }^{-}(t,s)=co{s}_{\lambda }(t,s)-\frac{(1-\alpha )\nu \lambda }{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,s),$ $y(t,t_0)=\left(\begin{array}{c}{y}_1(t,t_0)\\ y_2(t,t_0)\end{array}\right),$ $\nu \left(t\right)=\nu ,$ and $t_0=0.$

Proof. 

First, let us consider homogeneous solution of the system (8) $y_2^{{\diamond }_{\alpha }{\diamond }_{\alpha }}+{\lambda }^2{y}_2=0$, as

$$y_2(t,t_0)=c_{1}co{s}_{\lambda }(t,t_0)+c_{2}si{n}_{\lambda }(t,t_0).$$

(17)

If we use $y_2^{{\diamond }_{\alpha }}=\lambda y_1$, the following occurs:

$$c_{1}co{s}_{\lambda }^{{\diamond }_{\alpha }}(t,t_0)+c_{2}si{n}_{\lambda }^{{\diamond }_{\alpha }}(t,t_0)=\lambda y_1(t,t_0).$$

If (3) and (4) are used, we obtain

$$\begin{array}{c}\hfill y_1(t,t_0)=c_1\left(\frac{-1}{1+{\nu }^2{\lambda }^2}[(1+\alpha {\nu }^2{\lambda }^2)si{n}_{\lambda }(t,t_0)-(1-\alpha )\nu \lambda co{s}_{\lambda }(t,t_0)]\right)\\ \hfill +c_2\left(\frac{1}{1+{\nu }^2{\lambda }^2}[(1+\alpha {\nu }^2{\lambda }^2)co{s}_{\lambda }(t,t_0)+(1-\alpha )\nu \lambda si{n}_{\lambda }(t,t_0)]\right).\end{array}$$

(18)

These two homogeneous solutions can be written as follows by using a variation of parameters method [6].

$$\begin{array}{cc}\hfill & c_1^{{\diamond }_{\alpha }}\left(\frac{-1}{1+{\nu }^2{\lambda }^2}[(1+\alpha {\nu }^2{\lambda }^2)si{n}_{\lambda }(t,t_0)-(1-\alpha )\nu \lambda co{s}_{\lambda }(t,t_0)]\right)\hfill \\ & +c_2^{{\diamond }_{\alpha }}\left(\frac{1}{1+{\nu }^2{\lambda }^2}[(1+\alpha {\nu }^2{\lambda }^2)co{s}_{\lambda }(t,t_0)+(1-\alpha )\nu \lambda si{n}_{\lambda }(t,t_0)]\right)=-q\left(t\right)y_1(t,t_0),\hfill \end{array}$$

$$c_1^{{\diamond }_{\alpha }}co{s}_{\lambda }(t,t_0)+c_2^{{\diamond }_{\alpha }}si{n}_{\lambda }(t,t_0)=r\left(t\right)y_2(t,t_0).$$

If the equations are multiplied by $co{s}_{\lambda }(t,t_0)$ and

$$\frac{1}{1+{\nu }^2{\lambda }^2}[(1+\alpha {\nu }^2{\lambda }^2)si{n}_{\lambda }(t,t_0)-(1-\alpha )\nu \lambda co{s}_{\lambda }(t,t_0)],$$

respectively, and the ${\diamond }_{\alpha }$ integral is applied in the range $[0,t]$, we obtain

$$\begin{array}{cc}\hfill & c_2={\int }_0^t\left(r\left(s\right)y_2(s,t_0)s{c}_{\lambda }(s,t_0)-\frac{(1+{\nu }^2{\lambda }^2)}{1+\alpha {\nu }^2{\lambda }^2}q\left(s\right)y_1(s,t_0)co{s}_{\lambda }(s,t_0)\right){\diamond }_{\alpha }s,\hfill \end{array}$$

and in a similar way, one can obtain

$$\begin{array}{cc}\hfill & c_1={\int }_0^t\left(r\left(s\right)y_2(s,t_0)c{s}_{\lambda }{(s,t_0)}^{+}+\frac{(1+{\nu }^2{\lambda }^2)}{1+\alpha {\nu }^2{\lambda }^2}q\left(s\right)y_1(s,t_0)si{n}_{\lambda }(s,t_0)\right){\diamond }_{\alpha }s,\hfill \end{array}$$

where $s{c}_{\lambda }(t,s)=si{n}_{\lambda }(t,s)-\frac{(1-\alpha )\nu \lambda }{1+\alpha {\nu }^2{\lambda }^2}co{s}_{\lambda }(t,s).$ If these results are substituted into the solutions (17) and (18), the following solutions are obtained.

$$\begin{array}{cc}\hfill y_1(t,t_0)& =\frac{-1}{1+{\nu }^2{\lambda }^2}[(1+\alpha {\nu }^2{\lambda }^2)si{n}_{\lambda }(t,t_0)-(1-\alpha )\nu \lambda co{s}_{\lambda }(t,t_0)]\hfill \\ \hfill & \times {\int }_0^t\left(r\left(s\right)y_2(s,t_0)c{s}_{\lambda }{(s,t_0)}^{+}+\frac{(1+{\nu }^2{\lambda }^2)}{1+\alpha {\nu }^2{\lambda }^2}q\left(s\right)y_1(s,t_0)si{n}_{\lambda }(s,t_0)\right){\diamond }_{\alpha }s\hfill \\ \hfill & +\frac{1}{1+{\nu }^2{\lambda }^2}[(1+\alpha {\nu }^2{\lambda }^2)co{s}_{\lambda }(t,t_0)+(1-\alpha )\nu \lambda si{n}_{\lambda }(t,t_0)]\hfill \\ \hfill & \times {\int }_0^t\left(r\left(s\right)y_2(s,t_0)s{c}_{\lambda }(s,t_0)-\frac{(1+{\nu }^2{\lambda }^2)}{1+\alpha {\nu }^2{\lambda }^2}q\left(s\right)y_1(s,t_0)co{s}_{\lambda }(s,t_0)\right){\diamond }_{\alpha }s,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & y_2(t,t_0)=co{s}_{\lambda }(t,t_0){\int }_0^t\left(r\left(s\right)y_2(s,t_0)c{s}_{\lambda }{(s,t_0)}^{+}+\frac{(1+{\nu }^2{\lambda }^2)}{1+\alpha {\nu }^2{\lambda }^2}q\left(s\right)y_1(s,t_0)si{n}_{\lambda }(s,t_0)\right){\diamond }_{\alpha }s\hfill \\ \hfill & +si{n}_{\lambda }(t,t_0){\int }_0^t\left(r\left(s\right)y_2(s,t_0)s{c}_{\lambda }(s,t_0)-\frac{(1+{\nu }^2{\lambda }^2)}{1+\alpha {\nu }^2{\lambda }^2}q\left(s\right)y_1(s,t_0)co{s}_{\lambda }(s,t_0)\right){\diamond }_{\alpha }s,\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill y_1(t,t_0)& =& -{\int }_0^t[r\left(s\right)y_2(s,t_0)si{n}_{\lambda }(t,s)+c{s}_{\lambda }{(t,s)}^{+}q\left(s\right)y_1(s,t_0)]{\diamond }_{\alpha }s,\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill y_2(t,t_0)& =& {\int }_0^t[c{s}_{\lambda }{(t,s)}^{-}r\left(s\right)y_2(s,t_0)-\frac{1+\alpha {\nu }^2{\lambda }^2}{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,s)q\left(s\right)y_1(s,t_0)]{\diamond }_{\alpha }s.\hfill \end{array}$$

(20)

Taking into consideration the system (5),

$$y_1(t,t_0)=-{\int }_0^t[(\lambda y_2+y_1^{{\diamond }_{\alpha }})si{n}_{\lambda }(t,s)+c{s}_{\lambda }{(t,s)}^{+}(\lambda y_1-y_2^{{\diamond }_{\alpha }})]{\diamond }_{\alpha }s,$$

(21)

$$y_2(t,t_0)={\int }_0^t[c{s}_{\lambda }{(t,s)}^{-}(\lambda y_2+y_1^{{\diamond }_{\alpha }})-\frac{1+\alpha {\nu }^2{\lambda }^2}{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,s)(\lambda y_1-y_2^{{\diamond }_{\alpha }})]{\diamond }_{\alpha }s,$$

(22)

is obtained first, and

$$\begin{array}{ccc}\hfill y_1(t,t_0)& =& hsi{n}_{\lambda }(t,t_0)+y_2(t,t_0)-co{s}_{\lambda }(t,t_0)-\frac{(1-\alpha )\nu \lambda }{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,t_0)\hfill \\ & -& \frac{(1-\alpha )\nu {\lambda }^2}{1+\alpha {\nu }^2{\lambda }^2}{\int }_0^t[y_1(s,t_0)si{n}_{\lambda }(t,s)+y_2(s,t_0)co{s}_{\lambda }(t,s)]{\diamond }_{\alpha }s,\hfill \end{array}$$

(23)

$$\begin{array}{c}{y}_2(t,t_0)=-hco{s}_{\lambda }(t,t_0)+y_1(t,t_0)-\frac{h(1-\alpha )\nu \lambda }{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,t_0)-\frac{1+{\nu }^2{\lambda }^2}{1+\alpha {\nu }^2{\lambda }^2}si{n}_{\lambda }(t,t_0)\\ -\frac{(1-\alpha ){\nu }^2{\lambda }^3}{1+\alpha {\nu }^2{\lambda }^2}{\int }_0^t[y_2(s,t_0)co{s}_{\lambda }(t,t_0)+y_1(s,t_0)si{n}_{\lambda }(t,t_0)]si{n}_{\lambda }(t,s){\diamond }_{\alpha }s\\ -\frac{(1-\alpha )\nu {\lambda }^2}{1+\alpha {\nu }^2{\lambda }^2}{\int }_0^t[y_2(s,t_0)si{n}_{\lambda }(t,t_0)-y_1(s,t_0)co{s}_{\lambda }(t,t_0)]co{s}_{\lambda }(t,s){\diamond }_{\alpha }s\end{array}$$

(24)

is subsequently found by applying the integration by parts approach for (21) and (22) considering Lemma 3. Consequently, by taking into account (22) and (24) and (21) and (22), we have arrived at (15) and (16). □

3. Conclusions

In this study, some basic features of the ${\diamond }_{\alpha }$ calculation are examined and the ${\diamond }_{\alpha }$ Dirac system is discussed on a uniform time scale. Then, basic spectral features are examined. An eigenfunction expansion has been obtained for the ${\diamond }_{\alpha }$ Dirac problem, which is considered with a certain boundary condition. Under special assumptions, these solutions can yield different results.

In mathematics and science, spectral theory is essential, especially when examining linear operators on different function spaces. We want to extend the application of spectral theory to a larger class of mathematical objects beyond the classical continuous and discrete situations by investigating the spectral features of Dirac systems with diamond-alpha derivatives on a uniform time scale. The behavior of quantum systems with non-classical dynamics can be better understood by taking into account the spectrum characteristics of Dirac systems on a uniform time scale. New mathematical approaches and methodologies for the time-scale analysis of differential equations and operators can be developed via the study of the spectral theory of Dirac systems with diamond-alpha derivatives. In the future, the spectral theory could be extended to more intricate systems than the simple Dirac equation with diamond-alpha derivatives. This might entail taking into account extra interactions, boundary constraints, and expansions to higher-dimensional spaces. Theoretical physics and mathematical analysis may be affected in this way.