New Results on the Stability and Existence of Langevin Fractional Differential Equations with Boundary Conditions

Abstract

This manuscript aims to establish the existence, uniqueness, and stability of solutions for Langevin fractional differential equations involving the generalized Liouville-Caputo derivative. Using a novel approach, we derive existence and uniqueness results through fixed-point theorems, extending and generalizing several existing findings in the literature. To demonstrate the applicability of our results, we provide a practical example that validates the theoretical framework.

1. Introduction

In fractional calculus, the concept of integer order has been generalized to include arbitrary orders. Fractional-order models are generally more appropriate and accurate than integer-order models, as they provide a better representation of genetic processes and memory. In recent decades, fractional calculus has attracted considerable interest due to its extensive applications across diverse scientific disciplines. It has been utilized in various fields, such as biology, polymer science, heat conduction, signal processing, nonlinear earthquake oscillations, capacitor theory, image processing, groundwater problems, blood flow dynamics, electrical circuits, biophysics, viscoelasticity, and more. For additional applications of fractional differential equations, please refer to the references cited in [1,2,3,4].

The classical form of the Langevin equation was first presented by Paul Langevin in 1908 [5]. It was the first equation to describe the Brownian motion of particles [5]. However, some areas of fractal disorder were not adequately addressed by the classical Langevin equation. In mathematical physics, this is important for systems such as fractional reaction–diffusion [6,7], harmonic oscillators, noise sources with correlations [8], and the nature of quantum noise [9,10], among others. When there is no macroscopic system and the differential equation for the microscopic time scale is invalid, the fractional Langevin equation (LE) is more useful than the usual LE; see, for instance [11]. The Langevin equation and fractional dynamics are referenced in [12]. The Langevin equation, which has applications in stochastic problems in physics, chemistry, and electrical engineering, is seen in [13] .

Therefore, researchers aim to generalize fractional operators to better capture the hidden aspects of real non-local phenomena. Meanwhile, many researchers focus on fractional integrals and derivatives with non-local and non-singular kernels [14,15,16]. One of the emerging trends in fractional calculus is the study of discrete fractional operators, which have been shown to have valuable applications in various fields [17,18]. The authors of [19] introduced the Caputo version of generalized fractional derivatives. From a mathematical perspective, it is essential to consider fractional derivatives of functions belonging to specific function spaces. Additionally, the authors in [20] generalized the Liouville–Caputo fractional derivatives and their Caputo modification.

The existence and stability of solutions for fractional differential equations of different orders have been the focus of recent research. We now review several notable and contemporary papers on existence and uniqueness results for various types of fractional differential equations, as discussed in [21,22,23,24]. Hyers–Ulam (H-U) stability was initially introduced by Ulam in 1940 [25,26] and can be used to address the difference between approximate and exact solutions. Many researchers have conducted further studies on the stability of fractional equation solutions using this method. For example, see [27,28,29,30].

In [31], we consider a nonlinear fractional Langevin equation involving two fractional orders with given initial conditions.

$$\left\{\begin{array}{cc}{D}^{\beta }(D^{{\alpha }^{★}}+\lambda )\omega \left(\mathfrak{t}\right)=\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right)),\hfill & 0<\mathfrak{r}\le 1,\hfill \\ {\omega }^{\left(i\right)}\left(0\right)={\mu }_i,\hfill & 0\le i<l,\hfill \\ {\omega }^{(i+{\alpha }^{★})}\left(0\right)={\nu }_i,\hfill & 0\le i<n.\hfill \end{array}\right.$$

(1)

where $m-1<{\alpha }^{★}\le m$, $n-1<\beta \le n$, $l=max\{m,n\}$, and $m,n\in \mathbb{N}$. In [32], the study investigates the existence of extremal solutions for the fractional Langevin equation involving nonlinear boundary conditions.

$$\left\{\begin{array}{c}{D}^{\beta }(D^{{\alpha }^{★}}+\lambda )\omega \left(\mathfrak{r}\right)=\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right)),0<{\alpha }^{★},\beta \le 1,\mathfrak{r}\in [0,\mathfrak{L}],\hfill \\ g(\omega \left(0\right),\omega \left(\mathfrak{L}\right))=0,D^{{\alpha }^{★}}\omega \left(0\right)={\omega }^{{\alpha }^{★}}.\hfill \end{array}\right.$$

(2)

where $D^{{\alpha }^{★}}$ is the Caputo fractional derivative of order ${\alpha }^{★}$.

In [33], the study examines the existence and uniqueness of solutions for the anti-periodic boundary value problem of a Langevin equation with two different fractional orders.

$$\left\{\begin{array}{c}{D}^{\sigma }\left(D^{{\alpha }^{★}}+\delta \right)\omega \left(\mathfrak{r}\right)=\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right)),0<\mathfrak{r}<1,0<{\alpha }^{★}\le 1,1<\sigma \le 2,\hfill \\ \omega \left(0\right)+\omega \left(1\right)=0,D^{{\alpha }^{★}}\omega \left(0\right)+D^{{\alpha }^{★}}\omega \left(1\right)=0,D^{2{\alpha }^{★}}\omega \left(0\right)+D^{2{\alpha }^{★}}\omega \left(1\right)=0.\hfill \end{array}\right.$$

(3)

where $\delta$ is a real number, $D^{{\alpha }^{★}}$ denotes the Caputo fractional derivative of order ${\alpha }^{★}$, and $\mathfrak{Y}:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function.

In [34], the existence and uniqueness of solutions with periodic boundary conditions for the fractional Langevin equation using the generalized Caputo derivative are investigated.

$$\left\{\begin{array}{c}{}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }\left({}_{a^{+}}{}^{C}D^{\sigma ,\varsigma }+\delta \right)\omega \left(\mathfrak{r}\right)=\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right)),\mathfrak{r}\in [a,\mathfrak{L}],\hfill \\ \omega \left(a\right)=0,\omega \left(\mathfrak{r}\right)=I_{a^{+}}^{\gamma ,\varsigma }\omega \left(\zeta \right),{}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }\omega \left(\mathfrak{r}\right)={}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }\omega \left(0\right),a<\zeta <\mathfrak{L}.\hfill \end{array}\right.$$

(4)

where ${}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }$ and ${}^{C}D_{a^{+}}^{\sigma ,\varsigma }$ represent the generalized Caputo fractional derivative operators of orders ${\alpha }^{★}\in (1,2]$, $\sigma \in (0,1)$.

We develop and investigate a new version of the Langevin equation that utilizes generalized Liouville–Caputo derivatives, inspired by previous work. We examine the existence, uniqueness, and stability of the solution to the problem:

$$\left\{\begin{array}{cc}{}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }({}^{C}D_{a^{+}}^{\sigma ,\varsigma }+\delta )\omega \left(\mathfrak{r}\right)=\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right)),\hfill & \mathfrak{r}\in \mathbb{J}:=[a,\mathfrak{L}],\delta \in (0,1),\hfill \\ \omega \left(a\right)=0,\omega \left(\theta \right)=0,\omega \left(\mathfrak{r}\right)=\mu I_{a^{+}}^{\gamma ,\varsigma }\omega \left(\zeta \right),\hfill & a<\theta <\zeta <\mathfrak{L}.\hfill \end{array}\right.$$

(5)

where the Liouville-Caputo-type generalized fractional derivatives of orders ${\alpha }^{★}\in (1,2]$, $\sigma \in (0,1)$, and $\varsigma >0$ are represented by the expressions ${}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }$ and ${}^{C}D_{a^{+}}^{\sigma ,\varsigma }$, respectively. The generalized fractional integral operator of order $\gamma >0$ and $\varsigma >0$ is denoted by $I_{a^{+}}^{\gamma ,\varsigma }$, and the given continuous function is $\mathfrak{Y}:[\mathfrak{a},\mathfrak{L}]\times {\mathbb{R}}^3\to \mathbb{R}$.

Preliminaries

The aims of this section, we recall some basic definitions and results that are required for latter.

The Banach space of all continuous functions is denoted by $\mathbf{C}=C(\mathcal{J},\mathbb{R})$. The norm is defined as

$$\begin{array}{c}\hfill \parallel \omega \parallel =\underset{\mathfrak{r}\in \mathcal{J}}{sup}\left|\omega \left(\mathfrak{r}\right)\right|.\end{array}$$

Let $\mathfrak{C}=\{\omega :\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}_{a^{+}}{}^{C}D^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right)\in \mathbf{C}\}$. The Banach space $\mathfrak{C}$ has a norm which is defined as follows:

$$\begin{array}{cc}\hfill {\left|\right|\omega \left|\right|}_{\mathfrak{C}}& =\left|\right|\omega \left(\mathfrak{r}\right)\left|\right|+\left|\right|\omega \left(\delta \mathfrak{r}\right)\left|\right|+\left|\right|{}_{a^{+}}{}^{C}D^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right)\left|\right|\hfill \\ \hfill & =\underset{\mathfrak{r}\in \mathcal{J}}{sup}\left|\omega \left(\mathfrak{r}\right)\right|+\underset{\mathfrak{r}\in \mathcal{J}}{sup}\left|\omega \left(\delta \mathfrak{r}\right)\right|+\underset{\mathfrak{r}\in \mathcal{J}}{sup}\left|{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right)\right|.\hfill \end{array}$$

To begin, let us define the order of fractional derivatives in the following way:

Definition 1

([35]). For $-\infty <a<\mathfrak{r}<b<\infty$, the generalized left-sided fractional integral of order ${\alpha }^{★}>0$ and $\varsigma >0$ for $\mathfrak{Y}\in X_c^p(a,b)$ is defined as follows:

$$\begin{array}{c}\hfill I_{a^{+}}^{{\alpha }^{★},\varsigma }\mathfrak{Y}\left(\mathfrak{r}\right)=\frac{{\varsigma }^{1-{\alpha }^{★}}}{\Gamma \left({\alpha }^{★}\right)}{\int }_a^{\mathfrak{r}}\frac{s^{\varsigma -1}}{{({\mathfrak{r}}^{\varsigma }-s^{\varsigma })}^{1-{\alpha }^{★}}}\mathfrak{Y}\left(s\right)ds.\end{array}$$

(6)

Definition 2

([36]). The generalized fractional derivatives are defined for ${\alpha }^{★}>0$, $n=\lceil \sigma \rceil +1$, $\varsigma >0$, and $0\le a<\mathfrak{r}<b<\infty$ in terms of the generalized fractional integrals, as defined below:

$$\begin{array}{cc}\hfill \left(D_{a^{+}}^{{\alpha }^{★},\varsigma }\mathfrak{Y}\right)\left(\mathfrak{r}\right)& ={\left({\mathfrak{r}}^{1-\varsigma }\frac{d}{d\mathfrak{r}}\right)}^n\left({}_{a^{+}}I^{n-{\alpha }^{★},\varsigma }\mathfrak{Y}\right)\left(\mathfrak{r}\right)\hfill \\ \hfill & =\frac{{\varsigma }^{{\alpha }^{★}-n+1}}{\Gamma (n-{\alpha }^{★})}{\left({\mathfrak{r}}^{1-\varsigma }\frac{d}{d\mathfrak{r}}\right)}^n{\int }_a^{\mathfrak{r}}\frac{s^{\varsigma -1}}{{({\mathfrak{r}}^{\varsigma }-s^{\varsigma })}^{{\alpha }^{★}-n+1}}\mathfrak{Y}\left(s\right)ds.\hfill \end{array}$$

(7)

Definition 3

([20]). Assume ${\alpha }^{★}>0$ and $\mathfrak{Y}\in A{C}_{\delta }^n[a,b]$ for $n=\lfloor \sigma \rfloor +1$. The generalized fractional derivatives of the Liouville–Caputo type, denoted by ${}_{a^{+}}D^{{\alpha }^{★},\varsigma }\mathfrak{Y}$, are defined as follows:

$$\begin{array}{c}\hfill {}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }\mathfrak{Y}\left(\mathfrak{r}\right)=D_{a^{+}}^{{\alpha }^{★},\varsigma }\left(\mathfrak{Y}\left(\mathfrak{r}\right)-\sum _{k=0}^{n-1}{\delta }^k\frac{\mathfrak{Y}\left(\mathfrak{a}\right)}{k!}{\left(\frac{{\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma }}{\varsigma }\right)}^k\right)\delta ={\mathfrak{r}}^{1-\varsigma }\frac{d}{d{\mathfrak{r}}^{\prime }}.\end{array}$$

(8)

Lemma 1

([20]). The definitions of the left generalized Liouville–Caputo derivatives for ${\alpha }^{★}\ge 0$ and $\mathfrak{Y}\in A{C}_{\delta }^n[a,b]$ are as follows:

$$\begin{array}{c}\hfill {}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }\mathfrak{Y}\left(\mathfrak{r}\right)=\frac{1}{\Gamma (n-{\alpha }^{★})}{\int }_a^{\mathfrak{r}}{\left(\frac{({\mathfrak{r}}^{\varsigma }-s^{\varsigma })}{\varsigma }\right)}^{n-{\alpha }^{★}-1}\frac{\left({\delta }^n\mathfrak{Y}\right)\left(s\right)}{s^{1-\varsigma }}ds.\end{array}$$

Lemma 2

([20]). Let $\mathfrak{Y}\in A{C}_{\delta }^n[a,b]$ or $C_{\delta }^n[a,b]$ and ${\alpha }^{★}\in \mathbb{R}$. Then:

$$\begin{array}{c}\hfill I_{a^{+}}^{{\alpha }^{★},\varsigma }{}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }\mathfrak{Y}\left(\mathfrak{r}\right)=\mathfrak{Y}\left(\mathfrak{r}\right)-\sum _{k=0}^{n-1}\frac{\left({\delta }^k\mathfrak{Y}\right)\left(\mathfrak{a}\right)}{k!}{\left(\frac{x^{\varsigma }-{\mathfrak{a}}^{\varsigma }}{\varsigma }\right)}^k.\end{array}$$

Specifically, for $\sigma \in (0,1]$, we have the following:

$$\begin{array}{c}\hfill I_{a^{+}}^{\sigma ,\varsigma }{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\mathfrak{Y}\left(\mathfrak{r}\right)=\mathfrak{Y}\left(\mathfrak{r}\right)-\mathfrak{Y}\left(\mathfrak{a}\right).\end{array}$$

In Figure 1 and Figure 2, the dynamical behavior of the generalized Liouville–Caputo derivatives can be observed in the given functions $\omega \left(\mathfrak{r}\right)=\frac{{\mathfrak{r}}^5}{8}+10$, $\omega \left(\mathfrak{r}\right)=\frac{{\mathfrak{r}}^2}{4}+7$, respectively.

Theorem 1

(Banach FPT [37]). Let $\mathfrak{B}$ be a non-empty subset of a Banach space $\mathfrak{C}$. If $\mathfrak{X}:\mathfrak{B}\to \mathfrak{B}$ is a contraction mapping, then $\mathfrak{X}$ has a unique fixed point.

Theorem 2

(Schauder’s FPT [38]). Let $\mathfrak{B}$ be a non-empty, closed, convex subset of a Banach space $\mathfrak{C}$. The operator $\mathfrak{X}:\mathfrak{B}\to \mathfrak{B}$ has at least one fixed point if $\mathfrak{X}$ is a compact operator.

2. Main Results

The purpose of this section is to discuss the existence and uniqueness of solutions for problem (3).

Lemma 3.

Let $\nu \in \mathbf{C}$, then the boundary value problem

$$\left\{\begin{array}{cc}{}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }({}^{C}D_{a^{+}}^{\sigma ,\varsigma }+\delta )\omega \left(\mathfrak{r}\right)=\nu \left(\mathfrak{r}\right),\hfill & \mathfrak{r}\in \mathbb{J}:=[\mathfrak{a},\mathfrak{L}],\hfill \\ \omega \left(\mathfrak{a}\right)=0,{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right)=0,\omega \left(\mathfrak{r}\right)=\mu I_{a^{+}}^{\gamma ,\varsigma }\omega \left(\zeta \right),\hfill & \mathfrak{a}<\theta <\zeta <\mathfrak{L}\hfill \end{array}\right.$$

(9)

has a solution is given by

$$\begin{array}{cc}\hfill \omega \left(\mathfrak{r}\right)& =I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\mathfrak{r}\right)-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right)+\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)\Pi }\hfill \\ \hfill & \times \left(I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\mathfrak{L}\right)-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right)-\mu I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\nu \left(\zeta \right)+\mu \delta I_{a^{+}}^{\sigma +\gamma ,\varsigma }\omega \left(\zeta \right)\right)\hfill \\ \hfill & -\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{\Pi {({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}\left(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })-\gamma ({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}\right)\hfill \\ \hfill & \times \left[I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\theta \right)-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right)\right],\hfill \end{array}$$

(10)

where

$$\begin{array}{c}\hfill \Pi =\left(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\theta }^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\theta }^{\varsigma })-\gamma ({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}\right)\ne 0.\end{array}$$

Proof. 

Applying Lemma 2 and the operator $I_{a^{+}}^{{\alpha }^{★},\varsigma }$ to the fractional differential equation in Equation (9) yields the following:

$$\begin{array}{c}\hfill \left({}^{C}D_{a^{+}}^{\sigma ,\varsigma }+\delta \right)\omega \left(\mathfrak{r}\right)=I_{a^{+}}^{{\alpha }^{★},\varsigma }\nu \left(\mathfrak{r}\right)+{\mathfrak{b}}_0+{\mathfrak{b}}_1\left(\frac{{\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma }}{\varsigma }\right)\end{array}$$

(11)

for some ${\mathfrak{b}}_0,{\mathfrak{b}}_1\in \mathbb{R}$.

The general solution of the Equation (9) is determined by applying $I_{a^{+}}^{\sigma ,\varsigma }$ to both sides of Equation (11).

$$\begin{array}{c}\hfill \omega \left(\mathfrak{r}\right)=I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\mathfrak{r}\right)-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right)+{\mathfrak{b}}_0\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}+{\mathfrak{b}}_1\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +1}}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}+{\mathfrak{b}}_2\end{array}$$

(12)

where ${\mathfrak{b}}_3\in \mathbb{R}$.

We deduce that ${\mathfrak{b}}_2=0$ by applying the constraint $\omega \left(\mathfrak{a}\right)=0$ in Equation (12). After substituting ${\mathfrak{b}}_2=0$ into Equation (12) and modifying the resulting equation using the operator $I_{a^{+}}^{\gamma ,\varsigma }$, we obtain the following:

$$\begin{array}{c}\hfill I_{a^{+}}^{\gamma ,\varsigma }\omega \left(\mathfrak{r}\right)=I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\nu \left(\mathfrak{r}\right)-\delta I_{a^{+}}^{\sigma +\gamma ,\varsigma }\omega \left(\mathfrak{r}\right)+{\mathfrak{b}}_0\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }}{{\varsigma }^{\sigma +\gamma }\Gamma (\sigma +\gamma +1)}+{\mathfrak{b}}_2\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma +1}}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)}\end{array}$$

(13)

with the help of Equations (12) and (13) and the boundary conditions $\omega \left(\theta \right)=0$ and $\omega \left(\mathfrak{r}\right)=\mu I_{a^{+}}^{\gamma ,\varsigma }\omega \left(\zeta \right)$, one can solve the algebraic system of equations for ${\mathfrak{b}}_0$ and ${\mathfrak{b}}_2$.

$$\begin{array}{cc}\hfill {\mathfrak{b}}_0& =\frac{{\varsigma }^{\sigma }\Gamma (\sigma +1)}{\Pi {({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}[\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +1}}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}\times \left(I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\mathfrak{L}\right)-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right)-\mu I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\nu \left(\zeta \right)\right.\hfill \\ \hfill & +\mu \delta I_{a^{+}}^{\sigma +\gamma ,\varsigma }\omega \left(\zeta \right)-\left(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +1}}{{\varsigma }^{(\sigma +1)}\Gamma (\sigma +2)}-\mu \frac{{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma +1}}{{\varsigma }^{(\sigma +\gamma +1)}\Gamma (\sigma +\gamma +2)}\right)\hfill \\ \hfill & \times I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\theta \right)-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right)]\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathfrak{b}}_1& =-\frac{{\varsigma }^{\sigma }\Gamma (\sigma +1)}{\Pi {({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}[\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +1}}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\times \left(I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\mathfrak{L}\right)-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right)-\mu I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\nu \left(\zeta \right)\right.\hfill \\ \hfill & \left.+\mu \delta I_{a^{+}}^{\sigma +\gamma ,\varsigma }\omega \left(\zeta \right)\right)-\left(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}-\mu \frac{{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }}{{\varsigma }^{\sigma +\gamma }\Gamma (\sigma +\gamma +1)}\right)\times I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\theta \right)-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right)].\hfill \end{array}$$

The result in Equation (10) can be obtained by substituting the values of ${\mathfrak{b}}_0$, ${\mathfrak{b}}_1$, and ${\mathfrak{b}}_2$ into Equation (12). The proof is now complete. □

We demonstrate that the operator $\mathfrak{X}$ possesses a fixed point, which serves as the solution to problem (3). We define operator $\mathfrak{X}:\mathfrak{C}\to \mathfrak{C}$ by the following:

$$\begin{array}{cc}\hfill \mathfrak{X}\omega \left(\mathfrak{r}\right)& =I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right))-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right)+\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)\Pi }\hfill \\ \hfill & \times (I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\mathfrak{Y}(\mathfrak{L},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right))-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right)\hfill \\ \hfill & -\mu I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\mathfrak{Y}(\zeta ,\omega \left(\zeta \right),\omega \left(\delta \zeta \right),{}_{a^{+}}{}^{C}D^{\sigma ,\Psi }\omega \left(\zeta \right))+\mu \delta I_{a^{+}}^{\sigma +\gamma ,\varsigma }\omega \left(\zeta \right))\hfill \\ \hfill & -\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{\Pi {({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}\left(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })-\gamma ({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{\left({\varsigma }^{\sigma +\gamma +1}\right)\Gamma (\sigma +\gamma +2)(\sigma +1)}\right)\hfill \\ & \times \left[I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\mathfrak{Y}(\theta ,\omega \left(\theta \right),\omega \left(\delta \theta \right),{}_{a^{+}}{}^{C}D^{\sigma ,\Psi }\omega \left(\theta \right))-\delta I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right)\right].\hfill \end{array}$$

(14)

We introduce the following hypothesis, which is required for the subsequent results.

$\left({\mathfrak{A}}_1\right)$

Let $\mathfrak{Y}:\mathbb{J}\times {\mathbb{R}}^3\to \mathbb{R}$ be a continuous function.

$\left({\mathfrak{A}}_2\right)$

For every ${\omega }_i,{ð}_i\in \mathbb{R}$ and $\mathfrak{r}\in \mathbb{J}$, there is a constant $\mathcal{A}>0$.

$$\begin{array}{c}\hfill |\mathfrak{Y}(\mathfrak{r},{\omega }_1,{\omega }_2,{\omega }_3)-\mathfrak{Y}(\mathfrak{r},{ð}_1,{ð}_2,{ð}_3)|\le \mathcal{A}\left(|{\omega }_1-{ð}_1|+|{\omega }_2-{ð}_2|+|{\omega }_3-{ð}_3|\right).\end{array}$$

For simplicity, we denote

$$\begin{array}{cc}\hfill \Theta & =\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)+\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}\hfill \\ \hfill & +\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}+\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |}.\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\kappa }_1:=\underset{\mathfrak{r}\in [\mathfrak{a},\mathfrak{r}]}{max}\left|{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })\right|,\end{array}$$

$$\begin{array}{c}\hfill {\kappa }_2=\underset{\mathfrak{r}\in [\mathfrak{a},\mathfrak{r}]}{max}\left|{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }\left[\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })-\gamma ({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{{\varsigma }^{\sigma +\gamma +1)}\Gamma (\sigma +\gamma +2)(\sigma +1)}\right]\right|.\end{array}$$

We discuss the uniqueness results of problem (5) using the Banach fixed point theorem.

Theorem 3.

Assume that conditions $\left(\mathfrak{A}1\right)$–$\left(\mathfrak{A}2\right)$ are satisfied. Then, problem (5) has a unique solution if

$$\begin{array}{c}\hfill \mathcal{A}\Theta <1.\end{array}$$

(15)

Proof. 

Let us define the operator $\mathfrak{X}$ as shown in Equation (14). We show that $\mathfrak{X}$ is a contraction. Define the set

$$\begin{array}{c}\hfill {\mathfrak{B}}_{ϵ}=\{\omega \in \mathfrak{C}:\parallel \omega \parallel \le ϵ\}.\end{array}$$

Then, clearly, ${\mathfrak{B}}_{ϵ}$ is bounded. For $\omega ,ð\in {\mathfrak{B}}_{ϵ}$, we have the following:

$$\begin{array}{cc}\hfill & |\mathfrak{X}\left(\omega \right)\left(\mathfrak{r}\right)-\mathfrak{X}\left(ð\right)\left(\mathfrak{r}\right)|\le I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right))-\mathfrak{Y}(\mathfrak{r},ð\left(\mathfrak{r}\right),ð\left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\mathfrak{r}\right))|\hfill \\ \hfill & +\left|\delta \right|I_{a^{+}}^{\sigma ,\varsigma }|\omega \left(\mathfrak{r}\right)-ð\left(\mathfrak{r}\right)|+\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }\left|({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })\right|}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}(I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\mathfrak{L},\omega \left(\mathfrak{L}\right),\omega \left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right))\hfill \\ \hfill & -\mathfrak{Y}(\mathfrak{L},ð\left(\mathfrak{L}\right),ð\left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\mathfrak{L}\right))|+|\delta |I_{a^{+}}^{\sigma ,\varsigma }|\omega \left(\mathfrak{L}\right)-ð\left(\mathfrak{L}\right)|+\left|\mu \right|I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\hfill \\ \hfill & |\mathfrak{Y}(\zeta ,\omega \left(\zeta \right),\omega \left(\delta \zeta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\zeta \right))-\mathfrak{Y}(\zeta ,ð\left(\zeta \right),ð\left(\delta \zeta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\zeta \right))|+|\mu \left|\right|\delta |I_{a^{+}}^{\sigma +\gamma ,\varsigma }|\omega \left(\zeta \right)-ð\left(\zeta \right)|)\hfill \\ \hfill & +\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{|\Pi |{({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })}^{\sigma }}|\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })-\gamma ({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}|\times \hfill \\ \hfill & \left(I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\theta ,\omega \left(\theta \right),\omega \left(\delta \theta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right))-\mathfrak{Y}(\theta ,ð\left(\theta \right),ð\left(\delta \theta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\theta \right))|+|\delta |I_{a^{+}}^{\sigma ,\varsigma }|\omega \left(\theta \right)-ð\left(\theta \right)|\right)\hfill \\ \hfill & \le {\mathcal{A}\parallel \omega -ð\parallel }_{\mathfrak{C}}(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}\hfill \\ \hfill & {+\mathcal{A}\parallel \omega -ð\parallel }_{\mathfrak{C}}\left(\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\right)\hfill \\ \hfill & +\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |})\hfill \\ \hfill & =\mathcal{A}(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)+\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}\hfill \\ \hfill & +\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}+\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |}){\parallel \omega -ð\parallel }_{\mathfrak{C}}\hfill \end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill \parallel \mathfrak{X}\left(\omega \right)-\mathfrak{X}\left(ð\right)\parallel \le \mathcal{A}\Theta \parallel \omega -ð\parallel \end{array}$$

The operator $\mathfrak{X}$ is a contraction since $\mathcal{A}\Theta <1$, as shown in (15). Therefore, the operator $\mathfrak{X}$ has a unique fixed point according to Theorem 1. Consequently, problem (5) has a unique solution. The proof is complete. □

The solution of problem (5) exists if the operator $\mathfrak{X}$ satisfies the Schauder fixed point theorem.

We present the following hypothesis.

$\left({\mathfrak{A}}_3\right)$

For any $\phi \in C(J,{\mathbb{R}}^{+})$, there exists a continuous function such that

$$\begin{array}{c}\hfill |\mathfrak{Y}(\mathfrak{r},{\omega }_1,{\omega }_2,{\omega }_3)|\le \phi \left(\mathfrak{r}\right){\left|\right|\omega \left|\right|}_{\mathfrak{C}},\forall {\omega }_i\in \mathbb{R}.\end{array}$$

$$\begin{array}{c}\hfill {\phi }^{*}=\underset{\mathfrak{r}\in J}{sup}\left|\phi \left(\mathfrak{r}\right)\right|.\end{array}$$

For simplicity

$$\begin{array}{cc}\hfill \Delta & ={\phi }^{*}(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}\hfill \\ \hfill & +\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}+\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |})<1,\hfill \end{array}$$

Theorem 4.

Let ${\mathfrak{A}}_1$–${\mathfrak{A}}_3$ be satisfied. If $\Delta <1$, then problem (5) has at least one solution.

Proof. 

Consider the set

$$\begin{array}{c}\hfill {\mathfrak{B}}_{ϵ}={\{\omega \in \mathfrak{C}:\parallel \omega \parallel }_{\mathfrak{C}}\le ϵ\}.\end{array}$$

It is clear that the subset $\mathfrak{B}ϵ$ is evidently convex, closed, and bounded. We will show that $\mathfrak{X}$ meets the requirements specified in Theorem 2. The proof will be divided into three steps.

Step 1: We demonstrate that $\mathfrak{X}\mathfrak{B}\subset {\mathfrak{B}}_{ϵ}$. For all $\mathfrak{r}\in [0,1]$ and $\omega \in {\mathcal{B}}_{ϵ}$, we obtain the following:

$$\begin{array}{cc}\hfill \left|\right|\mathfrak{X}\left(\omega \right)\left(\mathfrak{r}\right)\left|\right|& \le \underset{\mathfrak{r}\in J}{sup}\left\{I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right))|+|\delta |I_{a^{+}}^{\sigma ,\varsigma }\left|\omega \left(\mathfrak{r}\right)\right|+\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right.\hfill \\ \hfill & +(I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\mathfrak{L},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right))|+|\delta |I_{a^{+}}^{\sigma ,\varsigma }\left|\omega \left(\mathfrak{L}\right)\right|+\left|\mu \right|I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\hfill \\ \hfill & |\mathfrak{Y}(\zeta ,\omega \left(\zeta \right),\omega \left(\delta \zeta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\zeta \right))+\left|\mu \right|\delta I_{a^{+}}^{\sigma +\gamma ,\varsigma }\omega \left(\zeta \right))+\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{|\Pi |{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}\hfill \\ \hfill & |\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })-\gamma ({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{\left({\varsigma }^{\sigma +\gamma +1}\right)\Gamma (\sigma +\gamma +2)(\sigma +1)}|\hfill \\ & \times \left[I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\theta ,\omega \left(\theta \right),\omega \left(\delta \theta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right))+\left|\delta \right|I_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right)\right]\}\hfill \\ \hfill & \le {\phi }^{*}(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}\hfill \\ \hfill & +\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}+\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |}{\left)\right|\left|\omega \right||}_{\mathfrak{C}}\hfill \\ \hfill & =\Delta ϵ\le ϵ\hfill \end{array}$$

This implies that $\left|\right|\mathfrak{X}\left(\omega \right)\left|\right|\le ϵ$. Thus, the operator $\mathfrak{X}$ maps ${\mathfrak{B}}_{ϵ}$ into itself.

Step 2: Next, we will show that $\mathfrak{X}$ is a continuous function. Let ${\omega }_n$ be a sequence such that ${\omega }_n\to \omega$. We have the following:

$$\begin{array}{cc}\hfill & \left|\right|\mathfrak{X}\left({\omega }_n\right)\left(\mathfrak{r}\right)-\mathfrak{X}\left(\omega \right)\left(\mathfrak{r}\right)\left|\right|\hfill \\ \hfill & \le \underset{\mathfrak{r}\in J}{sup}\{I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\mathfrak{r},{\omega }_n\left(\mathfrak{r}\right),{\omega }_n\left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }{\omega }_n\left(\mathfrak{r}\right))-\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right))|\hfill \\ \hfill & +\left|\delta \right|I_{a^{+}}^{\sigma ,\varsigma }|{\omega }_n\left(\mathfrak{r}\right)-\omega \left(\mathfrak{r}\right)|+\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }\left|({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })\right|}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}(I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\mathfrak{L},{\omega }_n\left(\mathfrak{L}\right),{\omega }_n\left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }{\omega }_n\left(\mathfrak{L}\right))\hfill \\ \hfill & -\mathfrak{Y}(\mathfrak{L},\omega \left(\mathfrak{L}\right),\omega \left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right))|+|\delta |I_{a^{+}}^{\sigma ,\varsigma }|{\omega }_n\left(\mathfrak{L}\right)-\omega \left(\mathfrak{L}\right)|\hfill \\ \hfill & +\left|\mu \right|I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }|\mathfrak{Y}(\zeta ,{\omega }_n\left(\zeta \right),{\omega }_n\left(\delta \zeta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }{\omega }_n\left(\zeta \right))-\mathfrak{Y}(\zeta ,\omega \left(\zeta \right),\omega \left(\delta \zeta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\zeta \right))|\hfill \\ \hfill & +\left|\mu \right|\left|\delta \right|I_{a^{+}}^{\sigma +\gamma ,\varsigma }|\omega \left(\zeta \right)-ð\left(\zeta \right)|)+\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{|\Pi |{({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })}^{\sigma }}|\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}\hfill \\ \hfill & -\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })-\gamma ({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}|\times I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\theta ,{\omega }_n\left(\theta \right),{\omega }_n\left(\delta \theta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }{\omega }_n\left(\theta \right))\hfill \\ \hfill & -\mathfrak{Y}(\theta ,\omega \left(\theta \right),\omega \left(\delta \theta \right),{}^C{D}_{a^{+}}^{\sigma ,\varsigma }_{a^{+}}ð\left(\theta \right))|+|\delta |I_{a^{+}}^{\sigma ,\varsigma }|{\omega }_n\left(\theta \right)-\omega \left(\theta \right)|\}\hfill \\ \hfill & \le \mathcal{A}(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}\hfill \\ \hfill & +\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}+\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |}){\parallel {\omega }_n-\omega \parallel }_{\mathfrak{C}}\hfill \end{array}$$

Thus, by the Lebesgue dominated convergence theorem, $\left|\right|\mathfrak{X}{\omega }_n\left(\mathfrak{r}\right)-\mathfrak{X}\omega \left(\mathfrak{r}\right)\left|\right|\to 0$. Hence, $\mathfrak{X}$ is continuous.

Step 3: This clearly shows the uniform boundedness of the operator $\mathfrak{X}$. Now, we present the equicontinuity of $\mathfrak{X}$. For this, we suppose ${\mathfrak{r}}_1,{\mathfrak{r}}_2\in \mathcal{J}$ such that ${\mathfrak{r}}_1<{\mathfrak{r}}_2$. We have

$$\begin{array}{cc}\hfill & \left|\mathfrak{X}\left(\omega \right)\left({\mathfrak{r}}_2\right)-\mathfrak{X}\left(\omega \right)\left({\mathfrak{r}}_1\right)\right|\hfill \\ \hfill & =\frac{{\varsigma }^{1-({\alpha }^{★}+\sigma )}}{\Gamma ({\alpha }^{★}+\sigma )}\left[{\int }_0^{{\mathfrak{r}}_1}{s}^{\varsigma -1}\left({({\mathfrak{r}}_2^{\varsigma }-s^{\varsigma })}^{{\alpha }^{★}+\sigma -1}-{({\mathfrak{r}}_1^{\varsigma }-s^{\varsigma })}^{{\alpha }^{★}+\sigma -1}\right)\mathfrak{Y}(s,\omega \left(s\right),\omega \left(\delta s\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(s\right))ds\right.\hfill \\ \hfill & +\left.{\int }_{{\mathfrak{r}}_1}^{{\mathfrak{r}}_2}{s}^{\varsigma -1}{({\mathfrak{r}}_2^{\varsigma }-s^{\varsigma })}^{{\alpha }^{★}+\sigma -1}\mathfrak{Y}(s,\omega \left(s\right),\omega \left(\delta s\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(s\right))ds\right]\hfill \\ \hfill & +\left[\frac{{({\mathfrak{r}}_2^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\theta }^{\varsigma }-{\mathfrak{r}}_2^{\varsigma })}{{\varsigma }^{(\sigma +1)}\Gamma (\sigma +2)\Pi }-\frac{{({\mathfrak{r}}_1^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\theta }^{\varsigma }-{\mathfrak{r}}_1^{\varsigma })}{{\varsigma }^{(\sigma +1)}\Gamma (\sigma +2)\Pi }\right]\hfill \\ \hfill & \times \left[I_{a^{+}}^{{\alpha }^{★}+\sigma }\mathfrak{Y}(\mathfrak{L},\omega \left(\mathfrak{L}\right),\omega \left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right))-\mu I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma }\mathfrak{Y}(\zeta ,\omega \left(\zeta \right),\omega \left(\delta \zeta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\zeta \right))\right]\hfill \\ \hfill & -\left[\frac{{({\mathfrak{r}}_2^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{\Pi {({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}_2^{\varsigma })}{{\varsigma }^{(\sigma +1)}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}_2^{\varsigma })-\gamma ({\mathfrak{r}}_2^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}\right]\hfill \\ \hfill & -\left[\frac{{({\mathfrak{r}}_1^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{\Pi {({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}_1^{\varsigma })}{{\varsigma }^{(\sigma +1)}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}_1^{\varsigma })-\gamma ({\mathfrak{r}}_1^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}\right]\hfill \\ \hfill & \times I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\left|\mathfrak{Y}(\theta ,\omega \left(\theta \right),\omega \left(\delta \theta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right))\right|\hfill \\ \hfill & \le \left[|{\mathfrak{r}}_2^{\varsigma ({\alpha }^{★}+\sigma )}-{\mathfrak{r}}_1^{\varsigma ({\alpha }^{★}+\sigma )}|+2{({\mathfrak{r}}_2^{\varsigma }-{\mathfrak{r}}_1^{\varsigma })}^{{\alpha }^{★}+\sigma }\right]+|\frac{{({\mathfrak{r}}_2^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\theta }^{\varsigma }-{\mathfrak{r}}_2^{\varsigma })}{(\sigma +1)\Gamma (\sigma +2)\Pi }-\frac{{({\mathfrak{r}}_1^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\theta }^{\varsigma }-{\mathfrak{r}}_1^{\varsigma })}{(\sigma +1)\Gamma (\sigma +2)\Pi }|\hfill \\ \hfill & \times \left[I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\mathfrak{L},\omega \left(\mathfrak{L}\right),\omega \left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right))|+\mu I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\left|\mathfrak{Y}(\zeta ,\omega \left(\zeta \right),\omega \left(\delta \zeta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\zeta \right))\right|\right]\hfill \\ \hfill & +\left[\frac{{({\mathfrak{r}}_1^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{\Pi {({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}_2^{\varsigma })}{{\varsigma }^{(\sigma +1)}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}_2^{\varsigma })-\gamma ({\mathfrak{r}}_2^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}\right]\hfill \\ \hfill & -\left[\frac{{({\mathfrak{r}}_1^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{\Pi {({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}_1^{\varsigma })}{{\varsigma }^{(\sigma +1)}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}_1^{\varsigma })-\gamma ({\mathfrak{r}}_1^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}\right]\hfill \\ \hfill & \times I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\left|\mathfrak{Y}(\theta ,\omega \left(\theta \right),\omega \left(\delta \theta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right))\right|.\hfill \end{array}$$

This implies that $\left|\mathfrak{X}\left(\omega \right)\left({\mathfrak{r}}_2\right)-\mathfrak{X}\left(\omega \right)\left({\mathfrak{r}}_1\right)\right|\to 0$ as ${\mathfrak{r}}_2\to {\mathfrak{r}}_1$, independently of $\omega \in {\mathfrak{B}}_{ϵ}$. Thus, $\mathfrak{X}$ is equicontinuous, which implies that $\mathfrak{X}$ is relatively compact on ${\mathfrak{B}}_{ϵ}$. Consequently, by the Arzelà–Ascoli theorem, we deduce that $\mathfrak{X}$ is compact on ${\mathfrak{B}}_{ϵ}$. The proof is complete because Theorem 2 guarantees that problem (5) has at least one solution. □

3. Stability

In this section, we study the stability of problem (5).

Assume $\epsilon >0$. Next, we examine the following inequality:

$$\begin{array}{c}\hfill |{}^C{D}_{a^{+}}^{{\alpha }^{★},\varsigma }_{a+}({}^{C}D^{\sigma ,\varsigma }+\delta )\omega \left(\mathfrak{r}\right)-\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right))|<\epsilon ,\mathfrak{r}\in \mathbb{J}.\end{array}$$

(16)

Definition 4.

If there exists $\sigma \in {\mathbb{R}}^{+}$ such that for every $\epsilon >0$ and for each solution $ð\in \mathbf{C}$ of inequality (16), there exists a solution $\omega \in \mathbf{C}$ of problem (5), then problem (5) is said to be H-U stable such that

$$\begin{array}{c}\hfill \left|ð\right(\mathfrak{r})-\omega (\mathfrak{r}\left)\right|\le \epsilon \sigma ,\mathfrak{r}\in \mathbb{J}.\end{array}$$

Definition 5.

If there exists a function $\Phi \in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ such that $\Phi \left(0\right)=0$, and for each $\epsilon >0$ and every solution $ð\in \mathbf{C}$ of the inequality (16), there exists a unique solution $\omega \in \mathbf{C}$ of (5), then problem (5) has generalized H-U stability.

$$\begin{array}{c}\hfill \left|ð\right(\mathfrak{r})-\omega (\mathfrak{r}\left)\right|\le \Phi \left(\epsilon \right),\mathfrak{r}\in \mathbb{J}.\end{array}$$

Remark 1.

A function $ð\in \mathbf{C}$ is a solution of the inequality (16) if and only if there exists a function $\omega \in \mathbf{C}$ (which depends on solution ð) such that

• $\left|\psi \right(\mathfrak{r}\left)\right|\le \epsilon$,
• ${}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }\left({}^{C}D_{a^{+}}^{\sigma ,\varsigma }+\delta \right)ð\left(\mathfrak{r}\right)=\mathfrak{Y}(\mathfrak{r},ð\left(\mathfrak{r}\right),ð\left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\mathfrak{r}\right))+\psi \left(\mathfrak{r}\right),\mathfrak{r}\in \mathbb{J}.$

Lemma 4.

The solution of the given problem

$$\left\{\begin{array}{cc}{}^{C}D_{a^{+}}^{{\alpha }^{★},\varsigma }({}^{C}D_{a^{+}}^{\sigma ,\varsigma }+\delta )ð\left(\mathfrak{r}\right)=\mathfrak{Y}(\mathfrak{r},ð\left(\mathfrak{r}\right),ð\left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\mathfrak{r}\right))+\psi \left(\mathfrak{r}\right),\hfill & \mathfrak{r}\in \mathbb{J}:=[a,\mathfrak{L}],\delta \in (0,1),\hfill \\ ð\left(a\right)=0,{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\theta \right)=0,\omega \left(\mathfrak{r}\right)=\mu I_{a^{+}}^{\gamma ,\varsigma }ð\left(\zeta \right),\hfill & a<\theta <\zeta <\mathfrak{L}\hfill \end{array}\right.$$

(17)

is

$$\begin{array}{cc}\hfill ð\left(\mathfrak{r}\right)& =I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\mathfrak{r}\right)-\delta I_{a^{+}}^{\sigma ,\varsigma }ð\left(\mathfrak{r}\right)+\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)\Pi }\hfill \\ \hfill & \times \left(I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\mathfrak{L}\right)-\delta I_{a^{+}}^{\sigma ,\varsigma }ð\left(\mathfrak{L}\right)-\mu I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\nu \left(\zeta \right)+\mu \delta I_{a^{+}}^{\sigma +\gamma ,\varsigma }ð\left(\zeta \right)\right)\hfill \\ \hfill & -\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{\Pi {({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}\left(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }\left[(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })-\gamma ({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })\right]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}\right)\hfill \\ \hfill & \times \left[I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\nu \left(\theta \right)-\delta I_{a^{+}}^{\sigma ,\varsigma }ð\left(\theta \right)\right]+I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\psi \left(\mathfrak{r}\right).\hfill \end{array}$$

(18)

Theorem 5.

Let $\left({\mathfrak{A}}_1\right)$–$\left({\mathfrak{A}}_3\right)$ hold. Then the problem (5) is H-U stable on $\mathbb{J}$ and consequently generalized H-U stable.

Proof. 

Let $ϵ>0$. Suppose $ð\in \mathbf{C}$ is a solution that satisfies inequality (16), which is defined by (10), and let $\omega \in \mathbf{C}$ be its unique solution. Applying Remark 1, for every $\mathfrak{r}\in \mathbb{J}$, we have the following:

$$\begin{array}{cc}\hfill & \left|\right|ð\left(\mathfrak{r}\right)-\omega \left(\mathfrak{r}\right)\left|\right|\le \underset{\mathfrak{r}\in J}{sup}\{I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\mathfrak{r},ð\left(\mathfrak{r}\right),ð\left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\mathfrak{r}\right))-\mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{r}\right))|\hfill \\ \hfill & +\left|\delta \right|I_{a^{+}}^{\sigma ,\varsigma }|ð\left(\mathfrak{r}\right)-\omega \left(\mathfrak{r}\right)|+\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }\left|({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })\right|}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}(I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\mathfrak{L},ð\left(\mathfrak{L}\right),ð\left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\mathfrak{L}\right))\hfill \\ \hfill & -\mathfrak{Y}(\mathfrak{L},\omega \left(\mathfrak{L}\right),\omega \left(\delta \mathfrak{L}\right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\mathfrak{L}\right))|+|\delta |I_{a^{+}}^{\sigma ,\varsigma }|ð\left(\mathfrak{L}\right)-\omega \left(\mathfrak{L}\right)|+\left|\mu \right|I_{a^{+}}^{{\alpha }^{★}+\sigma +\gamma ,\varsigma }\hfill \\ \hfill & |\mathfrak{Y}(\zeta ,ð\left(\zeta \right),ð\left(\delta \zeta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\zeta \right))-\mathfrak{Y}(\zeta ,\omega \left(\zeta \right),\omega \left(\delta \zeta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\zeta \right))|+|\mu \left|\right|\delta |I_{a^{+}}^{\sigma +\gamma ,\varsigma }|ð\left(\zeta \right)-\omega \left(\zeta \right)|)\hfill \\ \hfill & +\frac{{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{|\Pi |{({\theta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })}^{\sigma }}|\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }({\mathfrak{L}}^{\varsigma }-{\mathfrak{r}}^{\varsigma })}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)}-\frac{\mu {({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }(\sigma +1)({\zeta }^{\varsigma }-{\mathfrak{r}}^{\varsigma })-\gamma ({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })]}{{\varsigma }^{\sigma +\gamma +1}\Gamma (\sigma +\gamma +2)(\sigma +1)}|\hfill \\ \hfill & \times (I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }|\mathfrak{Y}(\theta ,ð\left(\theta \right),ð\left(\delta \theta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }ð\left(\theta \right))-\mathfrak{Y}(\theta ,\omega \left(\theta \right),\omega \left(\delta \theta \right),{}^{C}D_{a^{+}}^{\sigma ,\varsigma }\omega \left(\theta \right))|\hfill \\ \hfill & +\left|\delta \right|I_{a^{+}}^{\sigma ,\varsigma }|ð\left(\theta \right)-\omega \left(\theta \right)|+I_{a^{+}}^{{\alpha }^{★}+\sigma ,\varsigma }\left|\psi \left(\mathfrak{r}\right)\right|\left)\right\}\hfill \\ \hfill & \le \mathcal{A}\left\{\right(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}\hfill \\ \hfill & +\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)+\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}\hfill \\ \hfill & +\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |}\left)\right\}\left|\right|ð-\omega \left|\right|+\epsilon \left(\frac{{\varsigma }^{1-({\alpha }^{★}+\sigma )}{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{({\alpha }^{★}+\sigma )\Gamma ({\alpha }^{★}+\sigma )}\right).\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{cc}\hfill \left|\right|ð\left(\mathfrak{r}\right)-\omega \left(\mathfrak{r}\right)\left|\right|& \le \left\{\mathcal{A}\right(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}\hfill \\ \hfill & +\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)+\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}\hfill \\ \hfill & +\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |}\left)\right\}+\left(\frac{{\varsigma }^{1-({\alpha }^{★}+\sigma )}{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{({\alpha }^{★}+\sigma )\Gamma ({\alpha }^{★}+\sigma )}\right)\}\epsilon \hfill \end{array}$$

If we let

$$\begin{array}{cc}\hfill \sigma =& \left\{\mathcal{A}\right(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +(\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}\hfill \\ \hfill & +\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)+\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}\hfill \\ \hfill & +\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |}\left)\right\}+\left(\frac{{\varsigma }^{1-({\alpha }^{★}+\sigma )}{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{({\alpha }^{★}+\sigma )\Gamma ({\alpha }^{★}+\sigma )}\right)\},\hfill \end{array}$$

then, the H-U stability condition is satisfied.

More generally,

$$\begin{array}{cc}\hfill \Phi =& \left\{\mathcal{A}\right(\frac{{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)\hfill \\ \hfill & +(\frac{\left|\mu \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma +\gamma }{\kappa }_1}{{\varsigma }^{{\alpha }^{★}+2\sigma +\gamma +1}\Gamma ({\alpha }^{★}+\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}+\frac{{({\theta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}}{\kappa }_2}{{\varsigma }^{{\alpha }^{★}+\sigma }\Gamma ({\alpha }^{★}+\sigma +1)|\Pi |}\hfill \\ \hfill & +\frac{\left|\delta \right|{({\mathfrak{L}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma }}{{\varsigma }^{\sigma }\Gamma (\sigma +1)}\left(1+\frac{{\kappa }_1}{{\varsigma }^{\sigma +1}\Gamma (\sigma +2)|\Pi |}\right)+\frac{\left|\mu \right|\left|\delta \right|{({\zeta }^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{\sigma +\gamma }{\kappa }_1}{{\varsigma }^{2\sigma +\gamma +1}\Gamma (\sigma +\gamma +1)\Gamma (\sigma +2)|\Pi |}\hfill \\ \hfill & +\frac{\left|\delta \right|{\kappa }_2}{{\varsigma }^{\sigma }\Gamma (\sigma +1)|\Pi |}\left)\right\}+\left(\frac{{\varsigma }^{1-({\alpha }^{★}+\sigma )}{({\mathfrak{r}}^{\varsigma }-{\mathfrak{a}}^{\varsigma })}^{{\alpha }^{★}+\sigma }}{({\alpha }^{★}+\sigma )\Gamma ({\alpha }^{★}+\sigma )}\right)\}\epsilon ,\hfill \end{array}$$

$\Phi \left(0\right)=0$ the generalized H-U stability condition is also fulfilled. □

4. Illustrative Examples

In this section, we present examples to demonstrate the applicability of the analytical findings.

4.1. Example

Consider the following problem:

$$\left\{\begin{array}{c}{D}_1^{\frac{1}{5},\frac{1}{2}}\left(D_1^{\frac{2}{5},\frac{1}{2}}+\frac{1}{4}\right)\omega \left(\mathfrak{r}\right)=\frac{1}{\sqrt{\mathfrak{r}+625}}\left(\frac{\left|\omega \right(\mathfrak{r}\left)\right|+2}{\left|\omega \right(t\left)\right|+1}+sin\left(\omega \left(\delta \mathfrak{r}\right)\right)+{}^{C}D_1^{\frac{2}{5},\frac{1}{2}}\omega \left(\mathfrak{r}\right)+e^{-2\mathfrak{r}}\right),\hfill \\ \omega \left(1\right)=0,\omega \left(\frac{3}{6}\right)=0,\omega \left(2\right)=\frac{2}{3}{I}^{\frac{3}{2}}\omega \left(\frac{3}{5}\right),\mathfrak{r}\in \mathbb{J}:=[1,2].\hfill \end{array}\right.$$

(19)

Clearly, $\mathfrak{Y}:[1,2]\times {\mathbb{R}}^3\to \mathbb{R}$ is a continuous function. Therefore, we have the following:

$$\begin{array}{c}\hfill \mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}^C\mathcal{D}_{1^{+}}^{\frac{2}{5},\frac{1}{2}}\omega \left(\mathfrak{r}\right))=\frac{1}{\sqrt{\mathfrak{r}+625}}\left(\frac{\left|\omega \right(\mathfrak{r}\left)\right|+2}{\left|\omega \right(\mathfrak{r}\left)\right|+1}+sin\left(\omega \delta \right)+{}^{C}D_{1^{+}}^{\frac{2}{5},\frac{1}{2}}\omega \left(\mathfrak{r}\right)+e^{-2\mathfrak{r}}\right)\end{array}$$

Here, $\varsigma =\frac{1}{2}$, ${\alpha }^{★}=\frac{1}{5}$, $\sigma =\frac{2}{5}$, $\gamma =\frac{3}{2}$, $\delta =\frac{1}{4}$, $\mu =\frac{2}{3}$, a = 1, $\theta =\frac{3}{6}$, $\zeta =\frac{3}{5}$, $\mathfrak{L}$ = 2.

Now

$$\begin{array}{cc}\hfill & |\mathfrak{Y}(\mathfrak{r},{\omega }_1,{\omega }_2,{\omega }_3)-\mathfrak{Y}(\mathfrak{r},{ð}_1,{ð}_2,{ð}_3)|\hfill \\ \hfill & =\frac{1}{\sqrt{\mathfrak{r}+625}}\left(\right|\frac{|{\omega }_1\left(\mathfrak{r}\right)|+2}{|{\omega }_1\left(\mathfrak{r}\right)|+1}+sin\left({\omega }_2\right)+{}^{C}D_{1^{+}}^{\frac{5}{4},\sigma }{\omega }_3\left(\mathfrak{r}\right)+e^{-2\mathfrak{r}}\hfill \\ \hfill & -\frac{|{ð}_1\left(\mathfrak{r}\right)|+2}{|{ð}_1\left(\mathfrak{r}\right)|+1}-sin\left({ð}_2\right)-{}^{C}D_{1^{+}}^{\frac{5}{4},\sigma }{ð}_3\left(\mathfrak{r}\right)-e^{-2\mathfrak{r}}\left|\right)\hfill \\ \hfill & \le \frac{1}{25}\left(|{\omega }_1-{ð}_1|+|{\omega }_2-{ð}_2|+|{\omega }_3-{ð}_3|\right)\hfill \end{array}$$

Therefore, assumption ${\mathfrak{A}}_2$ is met with the value of $\mathcal{A}=\frac{1}{25}$. Using the given date, we find $|\Pi |=1.683$, ${\kappa }_1=0.518$, ${\kappa }_2=0.53$, $\Theta =0.423$.

$$\begin{array}{c}\hfill \mathcal{A}\Theta =0.01692<1.\end{array}$$

Problem (19) has a unique solution because all the conditions of Theorem 3 are satisfied. Additionally, we $\phi \left(\mathfrak{r}\right)=\frac{3+e^{-2\mathfrak{r}}}{\sqrt{\mathfrak{r}+625}}$, where ${\phi }^{*}=0.125$.

$$\begin{array}{c}\hfill \Delta =0.034<1\end{array}$$

As a consequence, problem (19) has at least one solution since each of the conditions of Theorem 4 is satisfied. Additionally, the Theorem 5 and problem (19) are both stable for Hyers–Ulam and generalized Hyers–Ulam. See the Figure 3 for more understanding.

4.2. Example

Consider the following problem:

$$\left\{\begin{array}{c}{}^{C}D_2^{\frac{9}{10},\frac{7}{5}}\left({}^{C}D_2^{\frac{9}{5},\frac{7}{5}}+\frac{1}{7}\right)\omega \left(\mathfrak{r}\right)=\frac{1}{\sqrt{\mathfrak{r}+49}}\left(\frac{\left|\omega \right(\mathfrak{r}\left)\right|+4}{\left|\omega \right(t\left)\right|+3}+sin\left(\omega \left(\delta \mathfrak{r}\right)\right)+{}^{C}D_2^{\frac{9}{5},\frac{7}{5}}\omega \left(\mathfrak{r}\right)+e^{-3\mathfrak{r}}\right),\hfill \\ \omega \left(2\right)=0,\omega \left(\frac{1}{3}\right)=0,\omega \left(3\right)=\frac{3}{2}{I}^{\frac{3}{4}}\omega \left(\frac{21}{10}\right),\mathfrak{r}\in \mathbb{J}:=[2,3].\hfill \end{array}\right.$$

(20)

Clearly, $\mathfrak{Y}:[2,3]\times {\mathbb{R}}^3\to \mathbb{R}$ is a continuous function. Therefore, we have the following:

$$\begin{array}{c}\hfill \mathfrak{Y}(\mathfrak{r},\omega \left(\mathfrak{r}\right),\omega \left(\delta \mathfrak{r}\right),{}^C\mathcal{D}_{2^{+}}^{\frac{9}{5},\frac{7}{5}}\omega \left(\mathfrak{r}\right))=\frac{1}{\sqrt{\mathfrak{r}+49}}\left(\frac{\left|\omega \right(\mathfrak{r}\left)\right|+4}{\left|\omega \right(\mathfrak{r}\left)\right|+3}+sin\left(\omega \delta \right)+{}^{C}D_{2^{+}}^{\frac{9}{5},\frac{7}{5}}\omega \left(\mathfrak{r}\right)+e^{-3\mathfrak{r}}\right)\end{array}$$

Here, $\varsigma =\frac{7}{5}$, ${\alpha }^{★}=\frac{9}{10}$, $\sigma =\frac{9}{5}$, $\gamma =\frac{3}{4}$, $\delta =\frac{1}{7}$, $\mu =\frac{3}{2}$, a = 2, $\theta =\frac{1}{2}$, $\zeta =\frac{21}{10}$, $\mathfrak{L}$ = 3. Now

$$\begin{array}{cc}\hfill & |\mathfrak{Y}(\mathfrak{r},{\omega }_1,{\omega }_2,{\omega }_3)-\mathfrak{Y}(\mathfrak{r},{ð}_1,{ð}_2,{ð}_3)|\hfill \\ \hfill & =\frac{1}{\sqrt{\mathfrak{r}+49}}\left(\right|\frac{|{\omega }_1\left(\mathfrak{r}\right)|+4}{|{\omega }_1\left(\mathfrak{r}\right)|+3}+sin\left({\omega }_2\right)+{}^{C}D_{2^{+}}^{\frac{9}{5},\frac{7}{5}}{\omega }_3\left(\mathfrak{r}\right)+e^{-3\mathfrak{r}}\hfill \\ \hfill & -\frac{|{ð}_1\left(\mathfrak{r}\right)|+4}{|{ð}_1\left(\mathfrak{r}\right)|+3}-sin\left({ð}_2\right)-{}^{C}D_{2^{+}}^{\frac{9}{5},\frac{7}{5}}{ð}_3\left(\mathfrak{r}\right)-e^{-3\mathfrak{r}}\left|\right)\hfill \\ \hfill & \le \frac{1}{7}\left(|{\omega }_1-{ð}_1|+|{\omega }_2-{ð}_2|+|{\omega }_3-{ð}_3|\right)\hfill \end{array}$$

Therefore, assumption ${\mathfrak{A}}_2$ is met with the value of $\mathcal{A}=\frac{1}{7}$. Using the given date, we find $|\Pi |=1.2541$, ${\kappa }_1=15.1142$, ${\kappa }_2=0.3384$, $\Theta =0.0147$.

$$\begin{array}{c}\hfill \mathcal{A}\Theta =0.0021<1.\end{array}$$

Problem (20) has a unique solution because all the conditions of Theorem 3 are satisfied. Additionally, we $\phi \left(\mathfrak{r}\right)=\frac{4+e^{-\mathfrak{r}}}{\sqrt{\mathfrak{r}+49}}$, where ${\phi }^{*}=0.5791$.

$$\begin{array}{c}\hfill \Delta =0.00121611<1\end{array}$$

As a consequence, problem (20) has at least one solution since each of the conditions of Theorem 4 is satisfied. Additionally, the Theorem 5 and problem (20) are both stable for Hyers-Ulam and generalized Hyers-Ulam.

5. Conclusions and Future Goals

We investigated the existence, uniqueness, and stability of solutions to a fractional differential equation involving generalized Liouville–Caputo fractional derivatives. By applying the fixed-point theorem, we established these results. Additionally, we provide an example to illustrate the reliability of our findings. This paper offers a novel and valuable contribution to the literature on fractional differential equations. In future work, the given fractional boundary value problem could be extended to include other fractional derivatives, such as the ABC fractional derivative and the mABC fractional derivative.