Existence of Solutions for Coupled System of Sequential Liouville–Caputo-Type Fractional Integrodifferential Equations

Abstract

The present investigation aims to establish the existence and uniqueness of solutions for a system containing sequential fractional differential equations. Furthermore, boundary conditions that include the Riemann–Liouville fractional integral are taken into consideration. The existence of unknown functions, fractional derivatives, and fractional integrals at lower orders are necessary for the nonlinearity to exist. In order to provide proofs for the results presented in this study, the Leray–Schauder alternative and the Banach fixed-point theorem are utilised. Finally, examples are used to support the main results.

1. Introduction

In this article, we examine the system of sequential derivative-containing nonlinear Liouville–Caputo fractional integrodifferential equations (LCFIE):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}({}^c{\mathcal{D}}^{{\varsigma }_1}+{\lambda }_1{}^c{\mathcal{D}}^{{\varsigma }_1-1})\mathcal{G}\left(\tau \right)={\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right)),\tau \in (0,1),\hfill \\ ({}^c{\mathcal{D}}^{{\varsigma }_2}+{\chi }_1{}^c{\mathcal{D}}^{{\varsigma }_2-1})\mathcal{J}\left(\tau \right)={\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right)),\tau \in (0,1),\hfill \end{array}\right.\end{array}$$

(1)

and it is enhanced by coupled Riemann–Liouville integral (RLI) boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathcal{G}\left(0\right)=0,{\mathcal{G}}^{\prime }\left(0\right)=0,{\mathcal{G}}^{\prime }\left(1\right)=0,}\hfill \\ \mathcal{G}\left(1\right)={\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\mathcal{G}\left(\upsilon \right)d\upsilon +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\mathcal{J}\left(\upsilon \right)d\upsilon ,\hfill \\ {\displaystyle \mathcal{J}\left(0\right)=0,{\mathcal{J}}^{\prime }\left(0\right)=0,{\mathcal{J}}^{\prime }\left(1\right)=0,}\hfill \\ \mathcal{J}\left(1\right)={\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\mathcal{G}\left(\upsilon \right)d\upsilon +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\mathcal{J}\left(\upsilon \right)d\upsilon .\hfill \end{array}\right.\end{array}$$

(2)

Here, ${\varsigma }_1$, ${\varsigma }_2\in (3,4],{\lambda }_1,{\chi }_1>0,{\mathfrak{p}}_1,{\mathfrak{p}}_2,{\mathfrak{q}}_1,{\mathfrak{q}}_2>0$, ${}^c{\mathcal{D}}^{\kappa }$ represents the Liouville–Caputo fractional derivative (LCFD) of order $\kappa$ (for $\kappa ={\varsigma }_1,{\varsigma }_2,{\varsigma }_2-1,{\varsigma }_1-1,{\mathfrak{p}}_1,{\mathfrak{p}}_2$), ${\phi }_1,{\varphi }_1:[0,1]\times {\mathcal{R}}_e^4$→${\mathcal{R}}_e$ are continuous functions, and ${\mathcal{I}}_{0^{+}}^{\nu }$ denotes the fractional RLI of order $\nu$ (for $\nu ={\mathfrak{q}}_1,{\mathfrak{q}}_2,\mathfrak{v},\mathfrak{z}$). Given certain assumptions on the functions ${\phi }_1$ and ${\varphi }_1$, we aim to establish the existence of at least one solution for Problems (1) and (2).

In the subsequent section, we will provide an overview of some scholarly articles that are relevant to the issue at hand. The concept of SFD ${\mathcal{D}}_{a+}^{\kappa \varsigma }$, where $\kappa$ is a positive integer, is introduced in the monograph authored by Miller and Ross [1] on page 209. The publications [2,3] provide an explanation of the connection between sequential fractional derivatives (SFDs) and non-Riemann–Liouville SFDs. The authors in Reference [2] studied the existence of minimal, maximal, and unique solutions to an initial value problem (IVP) involving the RLSFDs ${\mathcal{D}}_{0+}^{2\varsigma }$, $\varsigma \in (0,1]$, where an is a value within the interval (0, 1]. The researchers employed the upper and lower solutions approach in conjunction with the accompanying monotone iterative method. In their study, the authors in Reference [3] investigated the existence of a solution for a periodic boundary value problem (BVP). Specifically, they examined a problem involving an RLSFD ${\mathcal{D}}_{0+}^{2\varsigma }$, $\varsigma \in (0,1]$. To analyse this problem, the writers employed the upper and lower solutions approach along with the fixed-point theorem of Schauder’s.

In Reference [4], the author demonstrated the existence of solutions for a nonlinear impulsive fractional differential equation (FDE) with RLSFD. These solutions were subject to periodic boundary conditions and were obtained using the monotone iterative method. The nonexistence of solutions in the function space ${\mathcal{L}}^p((1,\infty ),{\mathcal{R}}_e)$ for an IVP that includes linear sequential fractional differential equations (SFDEs) with a Riemann–Liouville derivative and a classical first-order derivative is examined in Reference [5]. In Reference [6], Klimek demonstrated the existence and uniqueness of the solution for a certain category of nonlinear Hadamard SFDEs. This was achieved through the application of the contraction principle, with the inclusion of a set of beginning conditions that incorporate fractional derivatives. In our study, the term “sequential” refers to the property of the operator $({\mathcal{D}}^{\varsigma }+\lambda {\mathcal{D}}^{\varsigma -1})$, being expressible as a composition of the operators ${\mathcal{D}}^{\varsigma -1}(\mathcal{D}+\lambda )$, where $\mathcal{D}$ represents the ordinary derivative.

Ahmad and Nieto [7] introduced this particular operator in their study on the existence and uniqueness of solutions for the Caputo SFDE. In Reference [8], the authors demonstrated the existence of solutions for the sequential integrodifferential problem through the application of methods derived from fixed-point theory. The authors in Reference [9] conducted a study on the presence of solutions for the sequential Caputo FDE, incorporating boundary conditions that encompass a fractional integral of Riemann–Liouville nature. In their study, the authors of Reference [10] demonstrated the existence of solutions for a sequential fractional differential inclusion of the Caputo type. The boundary conditions of this inclusion encompass a fractional RLI. The authors of Reference [11] derived multiple results pertaining to the existence and uniqueness of the SFDE of the Caputo type. The authors of Reference [12] demonstrated the existence of solutions for SFDEs with nonlocal boundary conditions. The authors of Reference [13] derived existence results for solutions of the Caputo fractional sequential integrodifferential equation and inclusion. The existence of coupled systems of FDEs is prevalent in various practical applications, particularly in the field of biosciences (see [14] and its associated literature). In the following discussion, we will outline several fractional systems that are relevant to the topic at hand, denoted as (1) and (2). The authors of [14] employed the Banach contraction mapping concept and the Leray–Schauder alternative to establish the existence and uniqueness of solutions for the nonlinear system of Caputo-type SFDEs. The authors of Reference [15] presented a study where they established the existence of solutions for a system of nonlinear coupled differential equations and inclusions. These equations involve Caputo-type sequential derivatives of fractional order. The authors employed techniques developed from fixed-point theory to achieve this result. The work done by the authors of Reference [16] focused on investigating the presence and stability of a tripled system of SFDEs while considering multipoint boundary conditions. The authors of Reference [17] presented a study whereby they established the presence and durability of solutions for three nonlinear SFDEs with nonlocal boundary conditions. The authors of the cited Reference [18] obtained conclusions regarding the existence of solutions for a coupled system of nonlinear differential equations and inclusions that incorporate SFD.

The authors of the aforementioned study [14] successfully derived existence results for the solutions of a system consisting of Caputo FDEs,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}({{}^c\mathcal{D}}^{{\varsigma }_1}+{\lambda }_1{{}^c\mathcal{D}}^{{\varsigma }_1-1})\mathcal{G}\left(\tau \right)=\mathfrak{f}(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right)),\tau \in [0,1],\hfill \\ ({{}^c\mathcal{D}}^{{\varsigma }_2}+{\chi }_1{{}^c\mathcal{D}}^{{\varsigma }_2-1})\mathcal{J}\left(\tau \right)=\mathfrak{g}(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right)),\tau \in [0,1],\hfill \\ \mathcal{G}\left(0\right)={\mathcal{G}}^{{}^{\prime }}\left(0\right)=0,\mathcal{G}\left(\zeta \right)=a{\mathcal{I}}^{\beta }\mathcal{G}\left(\eta \right),\hfill \\ \mathcal{J}\left(0\right)={\mathcal{J}}^{{}^{\prime }}\left(0\right)=0,\mathcal{J}\left(z\right)=b{\mathcal{I}}^{\gamma }\mathcal{J}\left(\theta \right),\hfill \end{array}\right.\end{array}$$

(3)

where ${\varsigma }_1,{\varsigma }_2\in (2,3]$, ${\lambda }_1,{\chi }_1>0$, $\beta ,\gamma >0$, $0<\theta <z<1$, and $0<\eta <\zeta <1$. It can be observed that Boundary Conditions (3) indicate that the value of $\mathcal{G}$ at the point $\zeta$ is alone determined by the function $\mathcal{J}$, while the value of $\mathcal{J}$ at the point z is only determined by the function $\mathcal{G}$. The dependence of $\mathcal{G}\left(1\right)$ and $\mathcal{J}\left(1\right)$ on the functions $\mathcal{G}$ and $\mathcal{J}$ is evident in our Boundary Conditions (2). Furthermore, inside our System (1), the nonlinearities ${\phi }_1$ and ${\phi }_2$ exhibit a dependency on certain differential and integral terms, but in (3), no such dependency is found.

The authors of Reference [19] established the existence and uniqueness of solutions for a system of Hadamard-type SFDEs, including nonlocal coupled strip conditions

$$\begin{array}{c}\hfill \left\{\begin{array}{c}({{}^H\mathcal{D}}^{{\varsigma }_1}+{\lambda }_1{{}^H\mathcal{D}}^{{\varsigma }_1-1})\mathcal{G}\left(\tau \right)=\mathfrak{f}(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{{}^H\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right)),\tau \in [0,1],\hfill \\ ({{}^H\mathcal{D}}^{{\varsigma }_2}+{\chi }_1{{}^H\mathcal{D}}^{{\varsigma }_2-1})\mathcal{J}\left(\tau \right)=\mathfrak{g}(\tau ,\mathcal{G}\left(\tau \right),{{}^H\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right)),\tau \in [0,1],\hfill \\ \mathcal{G}\left(1\right)=0,\mathcal{G}\left(e\right){=}^H{\mathcal{I}}^{\gamma }\mathcal{G}\left(\eta \right),\hfill \\ \mathcal{J}\left(1\right)=0,\mathcal{J}\left(e\right){=}^H{\mathcal{I}}^{\beta }\mathcal{J}\left(\zeta \right),\hfill \end{array}\right.\end{array}$$

(4)

where ${\lambda }_1,{\chi }_1>0$, ${\varsigma }_1,{\varsigma }_2\in (1,2]$, ${\mathfrak{p}}_1,{\mathfrak{p}}_2\in (0,1)$, $\gamma >0$, $\eta ,\zeta \in (1,e)$, and $\beta >0$. It is clear from the Boundary Conditions (4) that the function $\mathcal{J}$ alone determines the value of $\mathcal{G}$ at the point e, whereas the function $\mathcal{G}$ solely determines the value of $\mathcal{J}$ at the point e. The dependence of $\mathcal{G}\left(1\right)$ and $\mathcal{J}\left(1\right)$ on the functions $\mathcal{G}$ and $\mathcal{J}$ is evident in our Boundary Conditions (2). Furthermore, inside our System (1), the nonlinearities ${\phi }_1$ and ${\phi }_2$ exhibit a dependency on certain differential and integral terms, but in (4), only differential terms are found. Moreover, within our computational Systems (1) and (2), we employ LCFDs and RLIs. However, in the context of Problem (4), Hadamard fractional derivatives and integrals are employed.

Subramanian et al. [20] conducted an analysis on the existence results for a system of coupled higher-order fractional integro-differential equations. These equations were subject to nonlocal integral and multi-point boundary conditions, which were dependent on lower-order fractional derivatives and integrals:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{{}^c\mathcal{D}}^{{\varsigma }_1}\mathcal{G}\left(\tau \right)=\mathfrak{f}(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{{}^c\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right)),\tau \in [0,T],\hfill \\ {{}^c\mathcal{D}}^{{\varsigma }_2}\mathcal{J}\left(\tau \right)=\mathfrak{g}(\tau ,\mathcal{G}\left(\tau \right),{{}^c\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right)),\tau \in [0,T],\hfill \\ \mathcal{G}\left(0\right)={\varphi }_1\left(\mathcal{J}\right),{\mathcal{G}}^{{}^{\prime }}\left(0\right)={ϵ}_1{\int }_0^{{\nu }_1}{\mathcal{J}}^{{}^{\prime }}\left(\theta \right)d\theta ,{\mathcal{G}}^{{}^{″}}\left(0\right)=0,···,{\mathcal{G}}^{n-2}\left(0\right)=0,\hfill \\ \mathcal{G}\left(T\right)={\lambda }_1{\int }_0^{{\delta }_1}\mathcal{J}\left(\theta \right)d\theta +{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j\mathcal{J}\left({\vartheta }_j\right),\hfill \\ \mathcal{J}\left(0\right)={\varphi }_2\left(\mathcal{G}\right),{\mathcal{J}}^{{}^{\prime }}\left(0\right)={ϵ}_2{\int }_0^{{\nu }_2}{\mathcal{G}}^{{}^{\prime }}\left(\theta \right)d\theta ,{\mathcal{J}}^{{}^{″}}\left(0\right)=0,···,{\mathcal{J}}^{n-2}\left(0\right)=0,\hfill \\ \mathcal{J}\left(T\right)={\lambda }_2{\int }_0^{{\delta }_2}\mathcal{G}\left(\theta \right)d\theta +{\mu }_2\sum _{j=1}^{k-2}{\omega }_j\mathcal{G}\left({\phi }_j\right).\hfill \end{array}\right.\end{array}$$

(5)

The nonlinearity is dependent on both the unknown functions and their fractional derivatives and integrals at a lower level. The consequence of existence is derived by the use of the Leray–Schauder alternative, whilst the result of uniqueness is established by employing the concept of a Banach contraction mapping.

The authors conducted a study [21] to examine the system of sequential Caputo fractional integrodifferential equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}({{}^c\mathcal{D}}^{{\varsigma }_1}+{\lambda }_1{{}^c\mathcal{D}}^{{\varsigma }_1-1})\mathcal{G}\left(\tau \right)=\mathfrak{f}(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{p}}_1}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{p}}_2}\mathcal{J}\left(\tau \right)),\tau \in (0,1),\hfill \\ ({{}^c\mathcal{D}}^{{\varsigma }_2}+{\chi }_1{{}^c\mathcal{D}}^{{\varsigma }_2-1})\mathcal{J}\left(\tau \right)=\mathfrak{g}(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{J}\left(\tau \right)),\tau \in (0,1),\hfill \\ \mathcal{G}\left(0\right)={\mathcal{G}}^{{}^{\prime }}\left(0\right)=0,{\mathcal{G}}^{{}^{\prime }}\left(1\right)=0,\mathcal{G}\left(1\right)={\int }_0^1\mathcal{G}\left(s\right)d{\mathcal{H}}_1\left(s\right)+{\int }_0^1\mathcal{J}\left(s\right)d{\mathcal{H}}_2\left(s\right),\hfill \\ \mathcal{J}\left(0\right)={\mathcal{J}}^{{}^{\prime }}\left(0\right)=0,{\mathcal{J}}^{{}^{\prime }}\left(1\right)=0,\mathcal{J}\left(1\right)={\int }_0^1\mathcal{G}\left(s\right)d{\mathcal{K}}_1\left(s\right)+{\int }_0^1\mathcal{J}\left(s\right)d{\mathcal{K}}_2\left(s\right),\hfill \end{array}\right.\end{array}$$

(6)

where ${\varsigma }_1,{\varsigma }_2\in (3,4]$, ${\lambda }_1,{\chi }_1>0$, and ${\mathfrak{p}}_1,{\mathfrak{p}}_2,{\mathfrak{q}}_1,{\mathfrak{q}}_2>0$. In the last conditions of (6), they have the Riemann–Stieltjes integrals (RSIs) with bounded variation functions ${\mathcal{H}}_1,{\mathcal{K}}_1,{\mathcal{H}}_2,{\mathcal{K}}_2$. Additionally, the nonlinearities ${\phi }_1$ and ${\phi }_2$ in our System (1) depend on certain differential and integral terms, but in (6), there are only integral terms. Moreover, within our computational Systems (1) and (2), we employ LCFDs and RLIs. However, in the context of Problem (6), CFDs and RSIs are employed. Another important difference between these two problems is given by the following conditions: the value of unknown functions with the right end point 1 is proportional to the sum of RLIs of unknown functions with different strip lengths $(0,\eta )$, $(0,\zeta )$ in (2), but in (6), the value of unknown functions with the right end point 1 is proportional to the sum of RSIs of unknown functions with the same strip lengths $(0,1)$.

The structure of the paper is organised in the following manner. In the second section, we analyse a linear fractional BVP that is connected to our Problems (1) and (2). Section 3 focuses on our primary results regarding Equations (1) and (2), while Section 4 provides two illustrative instances that demonstrate the results of our research. At last, the results of our research are presented in Section 5 of this study.

2. Preliminaries

First, we outline some basic concepts of fractional calculus.

Definition 1 

([22]). The fractional order of RLI $\varsigma \in {\mathcal{R}}_e(\varsigma >0)$ for a locally integrable, real-valued function $\mathcal{X}$ on $\infty \le \mathfrak{a}\le \mathfrak{b}+\infty$, denoted by ${\mathcal{I}}_{\mathfrak{a}}^{\varsigma }\left(\mathfrak{s}\right)$, is defined by

$$\begin{array}{c}\hfill {\mathcal{I}}_{\mathfrak{a}}^{\varsigma }\mathcal{X}\left(\mathfrak{z}\right)={\int }_{\mathfrak{a}}^{\mathfrak{z}}\frac{{(\mathfrak{z}-\mathfrak{s})}^{\varsigma -1}}{\Gamma \left(\varsigma \right)}\mathcal{X}\left(\mathfrak{s}\right)d\mathfrak{s},\end{array}$$

and the Gamma function is represented here by $\Gamma (·)$.

Definition 2 

([23]). The fractional order of the Caputo-type ς for an $(r-1)$- times absolutely continuous function $\mathcal{X}:[\mathfrak{a},\infty )\to {\mathcal{R}}_e$ is defined as

$$\begin{array}{c}\hfill {}^c{\mathcal{D}}^{\varsigma }\mathcal{X}\left(\mathfrak{z}\right)={\int }_{\mathfrak{a}}^{\mathfrak{z}}\frac{{(\mathfrak{z}-\mathfrak{s})}^{r-\varsigma -1}}{\Gamma (r-\varsigma )}{\mathcal{X}}^{\left(r\right)}\left(\mathfrak{s}\right)d\mathfrak{s},r-1<\varsigma <r,r=\left[\varsigma \right]+1,\end{array}$$

where the integral portion of the real number ς is denoted by [ς].

3. Auxiliary Results

This section focuses on the investigation of linear FDEs.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}({}^c{\mathcal{D}}^{{\varsigma }_1}+{\lambda }_1{}^c{\mathcal{D}}^{{\varsigma }_1-1})\mathcal{G}\left(\tau \right)={\mathfrak{h}}_1\left(\tau \right),\tau \in (0,1),\hfill \\ ({}^c{\mathcal{D}}^{{\varsigma }_2}+{\chi }_1{}^c{\mathcal{D}}^{{\varsigma }_2-1})\mathcal{J}\left(\tau \right)={\mathfrak{h}}_2\left(\tau \right),\tau \in (0,1),\hfill \end{array}\right.\end{array}$$

(7)

which are supplemented with boundary condition (2), with ${\mathfrak{h}}_1,{\mathfrak{h}}_2\in [0,1].$ We denote these by

$$\begin{array}{cc}\hfill & {\Omega }_1=\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1}),{\Omega }_2=\frac{1}{{\lambda }_1^2}(2{\lambda }_1-2+e^{-{\lambda }_1}),\hfill \\ \hfill & {\Omega }_3=\frac{1}{{\chi }_1}(1-e^{-{\chi }_1}),{\Omega }_4=\frac{1}{{{\chi }_1}^2}(2{\chi }_1-2+e^{-{\chi }_1}),\hfill \\ \hfill & {\Omega }_5=\frac{1}{{\lambda }_1^2}[({\lambda }_1-1+e^{-{\lambda }_1})-{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\upsilon -1+e^{-{\lambda }_1\upsilon })d\upsilon ],\hfill \\ \hfill & {\Omega }_6=\frac{1}{{\lambda }_1^3}[({\lambda }_1^2-2{\lambda }_1+2-2e^{-{\lambda }_1})-{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}({\lambda }_1^2{\upsilon }^2-2{\lambda }_1\upsilon +2-2e^{{\lambda }_1\upsilon })d\upsilon ],\hfill \\ \hfill & {\Omega }_7=\frac{1}{{{\chi }_1}^2}{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}({\chi }_1\upsilon -1+e^{-{\chi }_1\upsilon })d\upsilon ,\hfill \\ \hfill & {\Omega }_8=\frac{1}{{{\chi }_1}^3}{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}({{\chi }_1}^2{\upsilon }^2-2{\chi }_1\upsilon +2-2e^{-{\chi }_1\upsilon })d\upsilon ,\hfill \\ \hfill & {\Omega }_9=\frac{1}{{\lambda }_1^2}{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\upsilon -1+e^{-{\lambda }_1\upsilon })d\upsilon ,\hfill \\ \hfill & {\Omega }_{10}=\frac{1}{{\lambda }_1^3}{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}({\lambda }_1^2{\upsilon }^2-2{\lambda }_1\upsilon +2-2e^{{\lambda }_1\upsilon })d\upsilon ,\hfill \\ \hfill & {\Omega }_{11}=\frac{1}{{{\chi }_1}^2}(\mu -1+e^{-\mu })-\frac{1}{{{\chi }_1}^2}{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}({\chi }_1\upsilon -1+e^{-{\chi }_1\upsilon })d\upsilon ,\hfill \\ \hfill & {\Omega }_{12}=\frac{1}{{{\chi }_1}^3}({{\chi }_1}^2-2{\chi }_1+2-2e^{-\mu })-\frac{1}{{{\chi }_1}^3}{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}({{\chi }_1}^2{\upsilon }^2-2{\chi }_1\upsilon +2-2e^{-{\chi }_1\upsilon })d\upsilon ,\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill {\mathcal{I}}_1=& {\lambda }_1{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\epsilon \right)d\epsilon \right)d\upsilon -{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\upsilon \right)d\upsilon ,\hfill \\ \hfill {\mathcal{I}}_2=& {\chi }_1{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\epsilon \right)d\epsilon \right)d\upsilon -{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\upsilon \right)d\upsilon ,\hfill \\ \hfill {\mathcal{I}}_3=& {\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & -{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\epsilon \right)d\epsilon \right)d\upsilon ,\hfill \\ \hfill {\mathcal{I}}_4=& {\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & -{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\epsilon \right)d\epsilon \right)d\upsilon ,\hfill \\ \hfill {\Delta }_1=& ({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})-({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8),\hfill \\ \hfill \Delta =& {\Omega }_2{\Omega }_4{\Delta }_1.\hfill \end{array}$$

(9)

Lemma 1. 

If ${\Delta }_1\ne 0$, then the solution $(\mathcal{G},\mathcal{J})\in {\left({\mathcal{C}}^4[0,1]\right)}^2$ of the BVP

$$\begin{array}{c}\hfill \left\{\begin{array}{c}({}^c{\mathcal{D}}^{{\varsigma }_1}+{\lambda }_1{}^c{\mathcal{D}}^{{\varsigma }_1-1})\mathcal{G}\left(\tau \right)={\mathfrak{h}}_1\left(\tau \right),\tau \in (0,1),\hfill \\ ({}^c{\mathcal{D}}^{{\varsigma }_2}+{\chi }_1{}^c{\mathcal{D}}^{{\varsigma }_2-1})\mathcal{J}\left(\tau \right)={\mathfrak{h}}_2\left(\tau \right),\tau \in (0,1),\hfill \end{array}\right.\end{array}$$

(10)

supplemented with the boundary conditions

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathcal{G}\left(0\right)=0,{\mathcal{G}}^{\prime }\left(0\right)=0,{\mathcal{G}}^{\prime }\left(1\right)=0,}\hfill \\ \mathcal{G}\left(1\right)={\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\mathcal{G}\left(\upsilon \right)d\upsilon +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\mathcal{J}\left(\upsilon \right)d\upsilon ,\hfill \\ {\displaystyle \mathcal{J}\left(0\right)=0,{\mathcal{J}}^{\prime }\left(0\right)=0,{\mathcal{J}}^{\prime }\left(1\right)=0,}\hfill \\ \mathcal{J}\left(1\right)={\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\mathcal{G}\left(\upsilon \right)d\upsilon +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\mathcal{J}\left(\upsilon \right)d\upsilon ,\hfill \end{array}\right.\end{array}$$

(11)

is given by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathcal{G}\left(\tau \right)=\sum _{i=1}^4{\mathsf{\Phi }}_i\left(\tau \right){\mathcal{I}}_i+{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\epsilon \right)d\epsilon \right)d\upsilon ,}\hfill \\ {\displaystyle \mathcal{J}\left(\tau \right)={\sum }_{i=1}^4{\mathsf{\Psi }}_i\left(\tau \right){\mathcal{I}}_i+{\int }_0^{\tau }{e}^{-{\chi }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\epsilon \right)d\epsilon \right)d\upsilon ,}\hfill \end{array}\right.\end{array}$$

(12)

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\mathsf{\Phi }}_i\left(\tau \right)=\frac{1}{{\lambda }_1^2}\left({\Lambda }_i\right)({\lambda }_1\tau -1+e^{-{\lambda }_1\tau })+\frac{1}{{\lambda }_1^3}\left({\mathsf{\Theta }}_i\right)({\lambda }_1^2{\tau }^2-2{\lambda }_1\tau +2-2e^{-{\lambda }_1\tau }),i=1,2,3,4.}\hfill \\ {\displaystyle {\mathsf{\Psi }}_i\left(\tau \right)=\frac{1}{{{\chi }_1}^2}\left({\Xi }_i\right)({\chi }_1\tau -1+e^{-{\chi }_1\tau })+\frac{1}{{{\chi }_1}^3}\left({\mathsf{{\rm Y}}}_i\right)({{\chi }_1}^2{\tau }^2-2{\chi }_1\tau +2-2e^{-{\chi }_1\tau }),i=1,2,3,4.}\hfill \end{array}\right.\end{array}$$

(13)

Proof. 

The system denoted as (10) can be expressed in the following manner:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}({}^c{\mathcal{D}}^{{\varsigma }_1}+{\lambda }_1{}^c{\mathcal{D}}^{{\varsigma }_1-1})\mathcal{G}\left(\tau \right)={\mathfrak{h}}_1\left(\tau \right),\tau \in (0,1),\hfill \\ ({}^c{\mathcal{D}}^{{\varsigma }_2}+{\chi }_1{}^c{\mathcal{D}}^{{\varsigma }_2-1})\mathcal{J}\left(\tau \right)={\mathfrak{h}}_2\left(\tau \right),\tau \in (0,1).\hfill \end{array}\right.\end{array}$$

(14)

The general solutions of system (14) are

$$\begin{array}{cc}\hfill \mathcal{G}\left(\tau \right)=& {\mathfrak{c}}_0{e}^{-{\lambda }_1\tau }+\frac{{\mathfrak{c}}_1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })+\frac{{\mathfrak{c}}_2}{{\lambda }_1^2}({\lambda }_1\tau -1+e^{-{\lambda }_1\tau })+\frac{{\mathfrak{c}}_3}{{\lambda }_1^3}({\lambda }_1^2{\tau }^2-2{\lambda }_1\tau +2-2e^{-{\lambda }_1\tau })\hfill \\ & +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\epsilon \right)d\epsilon \right)d\upsilon ,\hfill \\ \hfill \mathcal{J}\left(\tau \right)=& {\mathfrak{d}}_0{e}^{-{\chi }_1\tau }+\frac{{\mathfrak{d}}_1}{{\chi }_1}(1-e^{-{\chi }_1\tau })+\frac{{\mathfrak{d}}_2}{{{\chi }_1}^2}({\chi }_1\tau -1+e^{-{\chi }_1\tau })+\frac{{\mathfrak{d}}_3}{{{\chi }_1}^3}({{\chi }_1}^2{\tau }^2-2{\chi }_1\tau +2-2e^{-{\chi }_1\tau })\hfill \\ & +{\int }_0^{\tau }{e}^{-{\chi }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\epsilon \right)d\epsilon \right)d\upsilon .\hfill \end{array}$$

Using the BCs $\mathcal{G}\left(0\right)={\mathcal{G}}^{\prime }\left(0\right)=0,\mathcal{J}\left(0\right)={\mathcal{J}}^{\prime }\left(0\right)=0$ from (11), we derive that ${\mathfrak{c}}_0={\mathfrak{c}}_1=0$ and ${\mathfrak{d}}_0={\mathfrak{d}}_1=0$. Hence, we infer

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \mathcal{G}\left(\tau \right)=}\hfill & {\displaystyle \frac{{\mathfrak{c}}_2}{{\lambda }_1^2}({\lambda }_1\tau -1+e^{-{\lambda }_1\tau })+\frac{{\mathfrak{c}}_3}{{\lambda }_1^3}({\lambda }_1^2{\tau }^2-2{\lambda }_1\tau +2-2e^{-{\lambda }_1\tau })}\hfill \\ & {\displaystyle +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\epsilon \right)d\epsilon \right)d\upsilon ,}\hfill \\ \mathcal{J}\left(\tau \right)=\hfill & \frac{{\mathfrak{d}}_2}{{{\chi }_1}^2}({\chi }_1\tau -1+e^{-{\chi }_1\tau })+\frac{{\mathfrak{d}}_3}{{{\chi }_1}^3}({{\chi }_1}^2{\tau }^2-2{\chi }_1\tau +2-2e^{-{\chi }_1\tau })\hfill \\ & {\displaystyle +{\int }_0^{\tau }{e}^{-{\chi }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\epsilon \right)d\epsilon \right)d\upsilon .}\hfill \end{array}\right.\end{array}$$

(15)

The following is determined by differentiating system (15):

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle {\mathcal{G}}^{\prime }\left(\tau \right)=}\hfill & {\displaystyle \frac{{\mathfrak{c}}_2}{{\lambda }_1^2}({\lambda }_1-{\lambda }_1{e}^{-{\lambda }_1\tau })+\frac{{\mathfrak{c}}_3}{{\lambda }_1^3}(2{\lambda }_1^2\tau -2{\lambda }_1+2e^{-{\lambda }_1\tau })}\hfill \\ & {\displaystyle -{\lambda }_1{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\epsilon \right)d\epsilon \right)d\upsilon +{\int }_0^{\tau }\frac{{(\tau -\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\upsilon \right)d\upsilon ,}\hfill \\ {\mathcal{J}}^{\prime }\left(\tau \right)=\hfill & \frac{{\mathfrak{d}}_2}{{{\chi }_1}^2}({\chi }_1-{\chi }_1{e}^{-{\chi }_1\tau })+\frac{{\mathfrak{d}}_3}{{{\chi }_1}^3}(2{{\chi }_1}^2\tau -2{\chi }_1+2{\chi }_1{e}^{-{\chi }_1\tau })\hfill \\ & {\displaystyle -{\chi }_1{\int }_0^{\tau }{e}^{-{\chi }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\epsilon \right)d\epsilon \right)d\upsilon +{\int }_0^{\tau }\frac{{(\tau -\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\upsilon \right)d\upsilon .}\hfill \end{array}\right.\end{array}$$

(16)

We acquire the following expressions by demanding the constraints, namely ${\mathcal{G}}^{\prime }\left(1\right)={\mathcal{J}}^{\prime }\left(1\right)=0$, as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{{\mathfrak{c}}_2}{{\lambda }_1}(1-e^{-{\lambda }_1})+\frac{{\mathfrak{c}}_3}{{\lambda }_1^2}(2{\lambda }_1^2-2+2e^{-{\lambda }_1})=}\hfill & {\displaystyle {\lambda }_1{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\epsilon \right)d\epsilon \right)d\upsilon }\hfill \\ & {\displaystyle -{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\upsilon \right)d\upsilon ,}\hfill \\ \frac{{\mathfrak{d}}_2}{{\chi }_1}(1-e^{-{\chi }_1})+\frac{{\mathfrak{d}}_3}{{{\chi }_1}^2}(2{\chi }_1-2+2{\chi }_1{e}^{-{\chi }_1})=\hfill & {\displaystyle {\chi }_1{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\epsilon \right)d\epsilon \right)d\upsilon }\hfill \\ & {\displaystyle -{\int }_0^1\frac{{(\tau -\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\upsilon \right)d\upsilon .}\hfill \end{array}\right.\end{array}$$

(17)

Now, using the last boundary conditions from (2), namely ${\displaystyle \mathcal{G}\left(1\right)={\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\mathcal{G}\left(\upsilon \right)d\upsilon }$$+{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\mathcal{J}\left(\upsilon \right)d\upsilon$ and ${\displaystyle \mathcal{J}\left(1\right)={\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\mathcal{G}\left(\upsilon \right)d\upsilon +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\mathcal{J}\left(\upsilon \right)d\upsilon }$, we deduce

$$\begin{array}{cc}\hfill & {\mathfrak{c}}_2\left[\frac{1}{{\lambda }_1^2}[({\lambda }_1-1+e^{-{\lambda }_1})-{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\upsilon -1+e^{-{\lambda }_1\upsilon })d\upsilon ]\right]\hfill \\ & +{\mathfrak{c}}_3\left[\frac{1}{{\lambda }_1^3}[({\lambda }_1^2-2{\lambda }_1+2-2e^{-{\lambda }_1})-{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}({\lambda }_1^2{\upsilon }^2-2{\lambda }_1\upsilon +2-2e^{{\lambda }_1\upsilon })d\upsilon ]\right]\hfill \\ & -{\mathfrak{d}}_2\left[\frac{1}{{{\chi }_1}^2}{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}({\chi }_1\upsilon -1+e^{-{\chi }_1\upsilon })d\upsilon \right]\hfill \\ & +{\mathfrak{d}}_3\left[\frac{1}{{{\chi }_1}^3}{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}({{\chi }_1}^2{\upsilon }^2-2{\chi }_1\upsilon +2-2e^{-{\chi }_1\upsilon })d\upsilon \right],\hfill \\ \hfill =& {\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & -{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\epsilon \right)d\epsilon \right)d\upsilon ,\hfill \end{array}$$

and

$$\begin{array}{cc}& -{\mathfrak{c}}_2\left[\frac{1}{{\lambda }_1^2}{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\upsilon -1+e^{-{\lambda }_1\upsilon })d\upsilon \right]\hfill \\ & -{\mathfrak{c}}_3\left[\frac{1}{{\lambda }_1^3}{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}({\lambda }_1^2{\upsilon }^2-2{\lambda }_1\upsilon +2-2e^{{\lambda }_1\upsilon })d\upsilon \right]\hfill \\ \hfill & +{\mathfrak{d}}_2\left[\frac{1}{{{\chi }_1}^2}(\mu -1+e^{-\mu })-\frac{1}{{{\chi }_1}^2}{\int }_0^1({\chi }_1\upsilon -1+e^{-{\chi }_1\upsilon })d\upsilon \right]\hfill \\ \hfill & +{\mathfrak{d}}_3\left[\frac{1}{{{\chi }_1}^3}({{\chi }_1}^2-2{\chi }_1+2-2e^{-\mu })-\frac{1}{{{\chi }_1}^3}{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}({{\chi }_1}^2{\upsilon }^2-2{\chi }_1\upsilon +2-2e^{-{\chi }_1\upsilon })d\upsilon \right],\hfill \\ \hfill =& {\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\mathfrak{h}}_1\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \end{array}$$

(18)

$$\begin{array}{cc}& -{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\mathfrak{h}}_2\left(\epsilon \right)d\epsilon \right)d\upsilon .\hfill \end{array}$$

(19)

Therefore, by (8), (9), (17), and (18), we find the system in the unknowns ${\mathfrak{c}}_2$, ${\mathfrak{c}}_3$, ${\mathfrak{d}}_2$ and ${\mathfrak{d}}_3$:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\Omega }_1{\mathfrak{c}}_2+{\Omega }_2{\mathfrak{c}}_3={\mathcal{I}}_1,\hfill \\ & {\Omega }_3{\mathfrak{d}}_2+{\Omega }_4{\mathfrak{d}}_3={\mathcal{I}}_2,\hfill \\ & {\Omega }_5{\mathfrak{c}}_2+{\Omega }_6{\mathfrak{c}}_3-{\Omega }_7{\mathfrak{d}}_2-{\Omega }_8{\mathfrak{c}}_3={\mathcal{I}}_3,\hfill \\ & -{\Omega }_9{\mathfrak{c}}_2-{\Omega }_{10}{\mathfrak{c}}_3+{\Omega }_{11}{\mathfrak{d}}_2+{\Omega }_{12}{\mathfrak{c}}_3={\mathcal{I}}_4.\hfill \end{array}\right.\end{array}$$

(20)

By solving the first two equations of (20), we find ${\mathfrak{c}}_3=\frac{{\mathcal{I}}_1-{\Omega }_1{\mathfrak{c}}_2}{{\Omega }_2}$ and ${\mathfrak{d}}_3=\frac{{\mathcal{I}}_2-{\Omega }_3{\mathfrak{d}}_2}{{\Omega }_2},({\Omega }_2,{\Omega }_4>0).$ Plugging these values of ${\mathfrak{c}}_3$ and ${\mathfrak{d}}_3$ into the last two equations of (20), we arrive at the system that is given below, including the unknown parameters ${\mathfrak{c}}_2$ and ${\mathfrak{d}}_2$:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathfrak{c}}_2{\Omega }_4({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)-{\mathfrak{d}}_2{\Omega }_2({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)={\Omega }_2{\Omega }_4{\mathcal{I}}_3-{\Omega }_4{\Omega }_6{\mathcal{I}}_1+{\Omega }_2{\Omega }_8{\mathcal{I}}_2,\hfill \\ -{\mathfrak{c}}_2{\Omega }_4({\Omega }_4{\Omega }_9-{\Omega }_1{\Omega }_{10})+{\mathfrak{d}}_2{\Omega }_2({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})={\Omega }_2{\Omega }_4{\mathcal{I}}_4+{\Omega }_4{\Omega }_{10}{\mathcal{I}}_1-{\Omega }_2{\Omega }_{12}{\mathcal{I}}_2.\hfill \end{array}\right.\end{array}$$

(21)

The determinant of system (21) is denoted as $\Delta ={\Omega }_2{\Omega }_4{\Delta }_1$, with ${\Delta }_1$ being determined by (9). According to the assumption made in Lemma 1, ${\Delta }_1\ne 0$, thus resulting in $\Delta =0$. Consequently, the solution to system (21) is as follows:

$$\begin{array}{cc}\hfill {\mathfrak{c}}_2=& \frac{{\Omega }_2}{\Delta }\{{\mathcal{I}}_1[-{\Omega }_4{\Omega }_6({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})+{\Omega }_4{\Omega }_{10}({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)]\hfill \\ & +{\mathcal{I}}_2[-{\Omega }_2{\Omega }_8({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})+{\Omega }_2{\Omega }_{12}({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)]\hfill \\ & +{\mathcal{I}}_3{\Omega }_2{\Omega }_4({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})+{\mathcal{I}}_4{\Omega }_2{\Omega }_4({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)\},\hfill \\ & ={\Lambda }_1{\mathcal{I}}_1+{\Lambda }_2{\mathcal{I}}_2+{\Lambda }_3{\mathcal{I}}_3+{\Lambda }_4{\mathcal{I}}_4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathfrak{d}}_2=& \frac{{\Omega }_4}{\Delta }\{{\mathcal{I}}_1[-{\Omega }_4{\Omega }_{10}({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)-{\Omega }_4{\Omega }_6({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})]\hfill \\ & +{\mathcal{I}}_2[-{\Omega }_2{\Omega }_{12}({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)+{\Omega }_2{\Omega }_8({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})]\hfill \\ & +{\mathcal{I}}_3{\Omega }_2{\Omega }_4({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})+{\mathcal{I}}_4{\Omega }_2{\Omega }_4({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)\},\hfill \\ & ={\Xi }_1{\mathcal{I}}_1+{\Xi }_2{\mathcal{I}}_2+{\Xi }_3{\mathcal{I}}_3+{\Xi }_4{\mathcal{I}}_4.\hfill \end{array}$$

Hence, we can determine the values of the constants ${\mathfrak{c}}_3$ and ${\mathfrak{d}}_3$ as

$$\begin{array}{cc}\hfill {\mathfrak{c}}_3=& \frac{1}{{\Omega }_2\Delta }\{{\mathcal{I}}_1{\Omega }_2{\Omega }_4[({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})-({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)]\hfill \\ & -{\mathcal{I}}_1{\Omega }_1{\Omega }_2[{\Omega }_4{\Omega }_6({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})+{\Omega }_4{\Omega }_{10}({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)]\hfill \\ & -{\mathcal{I}}_2{\Omega }_1{\Omega }_2[{\Omega }_2{\Omega }_8({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})+{\Omega }_2{\Omega }_{12}({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)]\hfill \\ & -{\mathcal{I}}_3{\Omega }_1{\Omega }_2^2{\Omega }_4({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})+{\mathcal{I}}_4{\Omega }_1{\Omega }_2^2{\Omega }_{12}({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)\},\hfill \\ \hfill =& {\mathsf{\Theta }}_1{\mathcal{I}}_1+{\mathsf{\Theta }}_2{\mathcal{I}}_2+{\mathsf{\Theta }}_3{\mathcal{I}}_3+{\mathsf{\Theta }}_4{\mathcal{I}}_4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathfrak{d}}_3=& \frac{1}{{\Omega }_4\Delta }\{-{\mathcal{I}}_1{\Omega }_3{\Omega }_4[{\Omega }_4{\Omega }_{10}({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)-{\Omega }_4{\Omega }_6({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})]\hfill \\ & -{\mathcal{I}}_2{\Omega }_2{\Omega }_4[({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})-({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)]\hfill \\ & -{\mathcal{I}}_2{\Omega }_3{\Omega }_4[-{\Omega }_2{\Omega }_{12}({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)+{\Omega }_2{\Omega }_8({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})]\hfill \\ & -{\mathcal{I}}_3{\Omega }_2{\Omega }_4^2{\Omega }_3({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})+{\mathcal{I}}_4{\Omega }_2{\Omega }_4^2{\Omega }_3({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)\},\hfill \\ \hfill =& {\mathsf{{\rm Y}}}_1{\mathcal{I}}_1+{\mathsf{{\rm Y}}}_2{\mathcal{I}}_2+{\mathsf{{\rm Y}}}_3{\mathcal{I}}_3+{\mathsf{{\rm Y}}}_4{\mathcal{I}}_4,\hfill \end{array}$$

$$\begin{array}{cc}& {\Lambda }_1=\frac{1}{{\Delta }_1}[-{\Omega }_6({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})+{\Omega }_{10}({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)],\hfill \\ & {\Lambda }_2=\frac{{\Omega }_2}{{\Delta }_1}({\Omega }_8{\Omega }_{11}-{\Omega }_7{\Omega }_{12}),{\Lambda }_3=\frac{{\Omega }_2}{{\Delta }_1}({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12}),{\Lambda }_4=\frac{{\Omega }_2}{{\Delta }_1}({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8),\hfill \\ & {\Xi }_1=\frac{{\Omega }_4}{{\Delta }_1}({\Omega }_5{\Omega }_{10}-{\Omega }_6{\Omega }_9),{\Xi }_2=\frac{1}{{\Delta }_1}[-{\Omega }_{12}({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)+{\Omega }_8({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})],\hfill \\ & {\Xi }_3=\frac{{\Omega }_4}{{\Delta }_1}({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10}),{\Xi }_4=\frac{{\Omega }_4}{{\Delta }_1}({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6),\hfill \\ & {\mathsf{\Theta }}_1=\frac{1}{{\Delta }_1}[-{\Omega }_5({\Omega }_4{\Omega }_{11}-{\Omega }_3{\Omega }_{12})+{\Omega }_9({\Omega }_4{\Omega }_7-{\Omega }_3{\Omega }_8)],\hfill \\ & {\mathsf{\Theta }}_2=\frac{{\Omega }_1}{{\Delta }_1}(-{\Omega }_8{\Omega }_{11}+{\Omega }_7{\Omega }_{12}),{\mathsf{\Theta }}_3=\frac{{\Omega }_1}{{\Delta }_1}({\Omega }_3{\Omega }_{12}-{\Omega }_4{\Omega }_{11}),{\mathsf{\Theta }}_4=\frac{{\Omega }_1}{{\Delta }_1}({\Omega }_3{\Omega }_8-{\Omega }_4{\Omega }_7),\hfill \\ & {\mathsf{{\rm Y}}}_1=\frac{{\Omega }_3}{{\Delta }_1}({\Omega }_6{\Omega }_9-{\Omega }_5{\Omega }_{10}),{\mathsf{{\rm Y}}}_2=\frac{1}{{\Delta }_1}[{\Omega }_{11}({\Omega }_2{\Omega }_5-{\Omega }_1{\Omega }_6)+{\Omega }_7({\Omega }_2{\Omega }_9-{\Omega }_1{\Omega }_{10})],\hfill \\ & {\mathsf{{\rm Y}}}_3=\frac{{\Omega }_3}{{\Delta }_1}({\Omega }_1{\Omega }_{10}-{\Omega }_2{\Omega }_9),{\mathsf{{\rm Y}}}_4=\frac{{\Omega }_3}{{\Delta }_1}({\Omega }_1{\Omega }_6-{\Omega }_2{\Omega }_5),\hfill \end{array}$$

(22)

where ${\Lambda }_1,{\Xi }_i,{\mathsf{\Theta }}_i,{\mathsf{{\rm Y}}}_i,i=1,\cdots ,4$ are given by (22). Upon substituting the aforementioned constants ${\mathfrak{c}}_2$, ${\mathfrak{d}}_2$, ${\mathfrak{c}}_3$, and ${\mathfrak{d}}_3$ into System (15), we can determine the solution to Problems (2)–(7). Conversely, the reverse of this result can be obtained through straightforward computations. □

Next, we introduce the Leray–Schauder alternative, which will be employed in the proof of the existence of solutions for Problems (1) and (2).

Theorem 1. 

Let $\mathcal{E}$ be a Banach space and $\mathsf{\Psi }:\mathcal{E}\to \mathcal{E}$ be a completely continuous operator. Let ${\phi }_1=\{\mathfrak{l}\in \mathcal{E},\mathfrak{l}=\nu \mathsf{\Psi }\left(\mathfrak{l}\right)$ for some $0<\nu <1\}$. Then, either the set ${\phi }_1$ is unbounded or Ψ has at least one fixed point.

4. Main Results

We consider the space $\mathcal{U}=\{\mathcal{G}\in \mathcal{C}[0,1],{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\in \mathcal{C}[0,1]\}$ and $\mathcal{Q}=\{\mathcal{J}\in \mathcal{C}[0,1],{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\in \mathcal{C}[0,1]\}$ to be equipped, respectively, with the norms ${\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}=\left|\right|\mathcal{G}\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left|\right|$ and ${\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}=\left|\right|\mathcal{J}\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left|\right|$, where $\left|\right|·\left|\right|$ is the sup norm; that is, $\left|\right|\mathfrak{w}\left|\right|={sup}_{\tau \in [0,1]}\left|\mathfrak{w}\right(\tau \left)\right|$ for $\mathfrak{w}\in \mathcal{C}[0,1]$. The spaces $(\mathcal{U},||·|{|}_{\mathcal{U}})$ and $(\mathcal{Q},||·|{|}_{\mathcal{Q}})$ both constitute Banach spaces, and when we consider the product space $\mathcal{U}\times \mathcal{Q}$ equipped with the norm ${\left|\right|(\mathcal{G},\mathcal{J})\left|\right|}_{\mathcal{U}\times \mathcal{Q}}={\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}$, it also forms a Banach space.

Implementing Lemma (1), we introduce the operator $\mathcal{T}:\mathcal{U}\times \mathcal{Q}\to \mathcal{U}\times \mathcal{Q}$ defined by $\mathcal{T}(\mathcal{G},\mathcal{J})=({\mathcal{T}}_1(\mathcal{G},\mathcal{J}),{\mathcal{T}}_2(\mathcal{G},\mathcal{J}))$ for $(\mathcal{G},\mathcal{J})\in \mathcal{U}\times \mathcal{Q}$, where the operators ${\mathcal{T}}_1:\mathcal{U}\times \mathcal{Q}\to \mathcal{U}$ and ${\mathcal{T}}_2:\mathcal{U}\times \mathcal{Q}\to \mathcal{Q}$ are given by

$$\begin{array}{cc}\hfill & {\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left(\tau \right)\hfill \\ \hfill =& {\mathsf{\Phi }}_1\left(\tau \right)\left[{\lambda }_1{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon \right.\hfill \\ & \left.-{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\upsilon \right)d\upsilon \right]\hfill \\ & +{\mathsf{\Phi }}_2\left(\tau \right)\left[{\chi }_1{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon \right.\hfill \\ & \left.-{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\upsilon \right)d\upsilon \right]\hfill \\ & +{\mathsf{\Phi }}_3\left(\tau \right)\left[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\right({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times {\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\mathfrak{m}\right)d\mathfrak{m}\left)d\epsilon \right)d\upsilon \hfill \end{array}$$

$$\begin{array}{c}\hfill +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \end{array}$$

(23)

$$\begin{array}{cc}& -{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon ]\hfill \\ & +{\mathsf{\Phi }}_4\left(\tau \right)\left[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\right({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times {\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\mathfrak{m}\right)d\mathfrak{m}\left)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & -{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon ]\hfill \\ & +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left(\tau \right)\hfill \\ \hfill =& {\mathsf{\Psi }}_1\left(\tau \right)\left[{\lambda }_1{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon \right.\hfill \\ & \left.-{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\upsilon \right)d\upsilon \right]\hfill \\ & +{\mathsf{\Psi }}_2\left(\tau \right)\left[{\chi }_1{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon \right.\hfill \\ & \left.-{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\upsilon \right)d\upsilon \right]\hfill \\ & +{\mathsf{\Psi }}_3\left(\tau \right)\left[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\right({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times {\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\mathfrak{m}\right)d\mathfrak{m}\left)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & -{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon ]\hfill \\ & +{\mathsf{\Psi }}_4\left(\tau \right)\left[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\right({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times {\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))\left(\mathfrak{m}\right)d\mathfrak{m}\left)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\mathfrak{m}\right)d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & -{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon ]\hfill \\ & +{\int }_0^{\tau }{e}^{-{\chi }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))\left(\epsilon \right)d\epsilon \right)d\upsilon .\hfill \end{array}$$

The pair $(\mathcal{G},\mathcal{J})$ constitutes a solution to Problems (1) and (2) when it serves as a fixed point of the operator $\mathcal{T}$. Let us now outline the assumptions that have been employed in this section.

• $\left[{\mathcal{H}}_1\right]$ The functions ${\phi }_1,{\varphi }_1$, defined on the domain $[0,1]\times {\mathcal{R}}_e\times {\mathcal{R}}_e\times {\mathcal{R}}_e\times {\mathcal{R}}_e\to {\mathcal{R}}_e$, are continuous. Additionally, there exist real ${\mathfrak{a}}_i,{\mathfrak{b}}_i\ge 0,i=1,2,3,4$ and ${\mathfrak{a}}_0>0,{\mathfrak{b}}_0>0$ such that

  $$\begin{array}{c}\hfill |{\phi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)|\le {\mathfrak{a}}_0+{\mathfrak{a}}_1|{\mathfrak{l}}_1|+{\mathfrak{a}}_2|{\mathfrak{l}}_2|+{\mathfrak{a}}_3|{\mathfrak{l}}_3|+{\mathfrak{a}}_4\left|{\mathfrak{l}}_4\right|,\\ \hfill |{\varphi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)|\le {\mathfrak{b}}_0+{\mathfrak{b}}_1|{\mathfrak{l}}_1|+{\mathfrak{b}}_2|{\mathfrak{l}}_2|+{\mathfrak{b}}_3|{\mathfrak{l}}_3|+{\mathfrak{b}}_4\left|{\mathfrak{l}}_4\right|,\end{array}$$

  for all $\tau \in [0,1]$ and ${\mathfrak{l}}_i\in {\mathcal{R}}_e,i=1,2,3,4.$
• $\left[{\mathcal{H}}_2\right]$ The functions ${\phi }_1,{\varphi }_1:[0,1]\times {\mathcal{R}}_e\times {\mathcal{R}}_e\times {\mathcal{R}}_e\times {\mathcal{R}}_e\to {\mathcal{R}}_e$ are continuous and non-negative constants ${\mathfrak{W}}_0>0$, ${\mathfrak{K}}_0>0$ such that

  $$\begin{array}{cc}\hfill & |{\phi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)-{\phi }_1(\tau ,{\mathfrak{n}}_1,{\mathfrak{n}}_2,{\mathfrak{n}}_3,{\mathfrak{n}}_4)|\le {\mathfrak{W}}_0\left(\right|{\mathfrak{l}}_1-{\mathfrak{n}}_1|+|{\mathfrak{l}}_2-{\mathfrak{n}}_2|+|{\mathfrak{l}}_3-{\mathfrak{n}}_3|+|{\mathfrak{l}}_4-{\mathfrak{n}}_4\left|\right),\hfill \\ & |{\varphi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)-{\varphi }_1(\tau ,{\mathfrak{n}}_1,{\mathfrak{n}}_2,{\mathfrak{n}}_3,{\mathfrak{n}}_4)|\le {\mathfrak{K}}_0\left(\right|{\mathfrak{l}}_1-{\mathfrak{n}}_1|+|{\mathfrak{l}}_2-{\mathfrak{n}}_2|+|{\mathfrak{l}}_3-{\mathfrak{n}}_3|+|{\mathfrak{l}}_4-{\mathfrak{n}}_4\left|\right),\hfill \end{array}$$

  for all $\tau \in [0,1]$ and ${\mathfrak{l}}_i,{\mathfrak{n}}_i\in {\mathcal{R}}_e,i=1,2,3,4.$ 

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\mathsf{\Phi }}_i^{\prime }\left(\tau \right)=\frac{1}{{\lambda }_1^2}\left({\Lambda }_i\right)(1-{\lambda }_1{e}^{-{\lambda }_1})+\frac{1}{{\lambda }_1^3}\left({\mathsf{\Theta }}_i\right)(2{\lambda }_1-2+2e^{-{\lambda }_1}),i=1,2,3,4.}\hfill \\ {\displaystyle {\mathsf{\Psi }}_i^{\prime }\left(\tau \right)=\frac{1}{{\lambda }_1^2}\left({\Xi }_i\right)(1-{\chi }_1{e}^{-{\chi }_1})+\frac{1}{{\lambda }_1^3}\left({\mathsf{{\rm Y}}}_i\right)(2{\chi }_1-2+2e^{-{\chi }_1}),i=1,2,3,4.}\hfill \end{array}\right.\end{array}$$

  (24)

We ${\widehat{\mathsf{\Phi }}}_i={sup}_{\tau \in [0,1]}|{\mathsf{\Phi }}_i\left(\tau \right)|,{\widehat{\mathsf{\Psi }}}_i={sup}_{\tau \in [0,1]}|{\mathsf{\Psi }}_i\left(\tau \right)|$ for $i=1,2,3,4.$

$$\begin{array}{cc}\hfill {\mathcal{U}}_1=& {\widehat{\mathsf{\Phi }}}_1\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)}+{\widehat{\mathsf{\Phi }}}_3\left[\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}+\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right]\hfill \\ & +{\widehat{\mathsf{\Phi }}}_4\left(\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right)+\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)},\hfill \\ \hfill {\mathcal{U}}_2=& {\widehat{\mathsf{\Phi }}}_2\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)}+{\widehat{\mathsf{\Phi }}}_3\left(\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right)\hfill \\ & +{\widehat{\mathsf{\Phi }}}_4\left[\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}+\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right],\hfill \\ \hfill {\mathcal{A}}_1=& \widehat{{\mathsf{\Psi }}_1}\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)}+{\widehat{\mathsf{\Psi }}}_3\left[\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}+\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right]\hfill \\ & +{\widehat{\mathsf{\Psi }}}_4\left(\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right),\hfill \\ \hfill {\mathcal{A}}_2=& {\widehat{\mathsf{\Psi }}}_2\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)}+{\widehat{\mathsf{\Psi }}}_3\left(\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right)\hfill \\ & +{\widehat{\mathsf{\Psi }}}_4\left[\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}+\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right]+\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)},\hfill \\ \hfill {\mathcal{U}}_1^{*}=& {\widehat{\mathsf{\Phi }}}_1^{\prime }\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)}+{\widehat{\mathsf{\Phi }}}_3^{\prime }\left[\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}+\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right]\hfill \\ & +{\widehat{\mathsf{\Phi }}}_4^{\prime }\left(\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right)+\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{U}}_2^{*}=& {\widehat{\mathsf{\Phi }}}_2^{\prime }\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)}+{\widehat{\mathsf{\Phi }}}_3^{\prime }\left(\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right)\hfill \end{array}$$

(25)

$$\begin{array}{cc}& +{\widehat{\mathsf{\Phi }}}_4^{\prime }\left[\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}+\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right],\hfill \\ \hfill {\mathcal{A}}_1^{*}=& {\widehat{\mathsf{\Psi }}}_1^{\prime }\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)}+{\widehat{\mathsf{\Psi }}}_3^{\prime }\left[\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}+\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right]\hfill \\ & +{\widehat{\mathsf{\Psi }}}_4^{\prime }\left(\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right),\hfill \\ \hfill {\mathcal{A}}_2^{*}=& {\widehat{\mathsf{\Psi }}}_2^{\prime }\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)}+{\widehat{\mathsf{\Psi }}}_3^{\prime }\left(\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right)\hfill \\ & +{\widehat{\mathsf{\Psi }}}_4^{\prime }\left[\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}+\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right]+\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)},\hfill \\ & {\mathcal{N}}_1={\mathcal{U}}_1+{\mathcal{A}}_1+\frac{{\mathcal{U}}_1^{*}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{A}}_1^{*}}{\Gamma (2-{\mathfrak{p}}_1)},\hfill \\ & {\mathcal{N}}_2={\mathcal{U}}_2+{\mathcal{A}}_2+\frac{{\mathcal{U}}_2^{*}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{A}}_2^{*}}{\Gamma (2-{\mathfrak{p}}_1)},\hfill \\ & {\mathcal{N}}_3={\mathfrak{a}}_0{\mathcal{N}}_1+{\mathfrak{b}}_0{\mathcal{N}}_2,\hfill \\ & {\mathcal{N}}_4={\mathfrak{a}}_1{\mathcal{N}}_1+\left({\mathfrak{b}}_1+{\mathfrak{b}}_2+\frac{{\mathfrak{b}}_3}{\Gamma ({\mathfrak{q}}_2+1)}\right){\mathcal{N}}_2,\hfill \\ & {\mathcal{N}}_5=\left({\mathfrak{a}}_2+{\mathfrak{a}}_3+\frac{{\mathfrak{a}}_4}{\Gamma ({\mathfrak{q}}_1+1)}\right){\mathcal{N}}_1+{\mathfrak{b}}_4{\mathcal{N}}_2,\hfill \\ & {\varrho }_1=1+\frac{1}{\Gamma (1+{\mathfrak{q}}_1)},{\varrho }_2=1+\frac{1}{\Gamma (1+{\mathfrak{q}}_2)}.\hfill \end{array}$$

Theorem 2. 

Assume that ${\mathcal{H}}_1$ holds. If

$$\begin{array}{c}\hfill max\{{\mathcal{N}}_4,{\mathcal{N}}_5\}<1,\end{array}$$

(26)

then the BVPs (1) and (2) have at least one solution $\left(\mathcal{G}\right(\tau ),\mathcal{J}(\tau \left)\right)$, $\tau \in [0,1]$.

Proof. 

We begin by establishing that the operator $\mathcal{T}:\mathcal{U}\times \mathcal{Q}\to \mathcal{U}\times \mathcal{Q}$ is a completely continuous (c.c.) operator. The continuity of the functions ${\phi }_1$ and ${\varphi }_1$ allows us to infer that the operators ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ are continuous as well. Consequently, $\mathcal{T}$ emerges as a continuous operator.

We proceed to demonstrate that $\mathcal{T}$ is uniformly bounded. Suppose we have a bounded set $\Pi \subset \mathcal{U}\times \mathcal{Q}$. In this case, there exist positive constants ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ such that

$$\begin{array}{c}\hfill |{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))|\le {\mathcal{L}}_1,\\ \hfill |{\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right))|\le {\mathcal{L}}_2.\end{array}$$

for all $(\mathcal{G},\mathcal{J})\in \mathsf{\Pi }$ and $\tau \in [0,1]$.

Then, for any $(\mathcal{G},\mathcal{J})\in \mathsf{\Pi }$ and $\tau \in [0,1]$, we obtain

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left(\tau \right)|\hfill \\ \hfill =& |{\mathsf{\Phi }}_1\left(\tau \right)\left|\right|\left[{\lambda }_1{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon \right.\hfill \\ & \left.+{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\upsilon ,\mathcal{G}\left(\upsilon \right),\mathcal{J}\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\upsilon \right))\right|d\upsilon \right]\hfill \\ & +|{\mathsf{\Phi }}_2\left(\tau \right)|\left[{\chi }_1{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\epsilon )\right|d\epsilon \right)d\upsilon \right.\hfill \\ & \left.+{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\upsilon ,\mathcal{G}\left(\upsilon \right),\mathcal{J}\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\upsilon \right))\right|d\upsilon \right]\hfill \\ & +|{\mathsf{\Phi }}_3\left(\tau \right)\left|\right[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times |{\phi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon )d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\mathfrak{m}\right))\right|d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon ]\hfill \\ & +|{\mathsf{\Phi }}_4\left(\tau \right)\left|\right[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & |{\phi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon )d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\mathfrak{m}\right))\right|d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon \left]\right|\hfill \\ & +{\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\upsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon ,\hfill \\ \hfill \le & {\mathcal{L}}_1\left\{{\widehat{\mathsf{\Phi }}}_1\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)}+{\widehat{\mathsf{\Phi }}}_3\left[\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}+\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right]\right.\hfill \\ & +\left.{\widehat{\mathsf{\Phi }}}_4\left(\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right)\right\}\hfill \\ & +{\mathcal{L}}_2\left\{{\widehat{\mathsf{\Phi }}}_2\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)}+{\widehat{\mathsf{\Phi }}}_3\left(\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right)\right.\hfill \\ & +\left.{\widehat{\mathsf{\Phi }}}_4\left[\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}+\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right]\right\}\hfill \\ & +{\mathcal{L}}_1\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}={\mathcal{L}}_1{\mathcal{U}}_1+{\mathcal{L}}_2{\mathcal{U}}_2.\hfill \end{array}$$

(27)

Then, $\left|\right|{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left|\right|\le {\mathcal{L}}_1{\mathcal{U}}_1+{\mathcal{L}}_2{\mathcal{U}}_2$, for all $(\mathcal{G},\mathcal{J})\in \mathsf{\Pi }$.

By examining the definition of ${\mathcal{T}}_1(\mathcal{G},\mathcal{J})$, we can observe that

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_1^{\prime }(\mathcal{G},\mathcal{J})\left(\tau \right)|\hfill \\ \hfill =& |{\mathsf{\Phi }}_1^{\prime }\left(\tau \right)\left|\right|\left[{\lambda }_1{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon \right.\hfill \\ & \left.+{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\upsilon ,\mathcal{G}\left(\upsilon \right),\mathcal{J}\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\upsilon \right))\right|d\upsilon \right]\hfill \\ & +|{\mathsf{\Phi }}_2^{\prime }\left(\tau \right)|\left[{\chi }_1{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\epsilon )\right|d\epsilon \right)d\upsilon \right.\hfill \\ & \left.+{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\upsilon ,\mathcal{G}\left(\upsilon \right),\mathcal{J}\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\upsilon \right))\right|d\upsilon \right]\hfill \\ & +|{\mathsf{\Phi }}_3^{\prime }\left(\tau \right)\left|\right[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & |{\phi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon )d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\mathfrak{m}\right))\right|d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \end{array}$$

$$\begin{array}{cc}& {\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon ]\hfill \end{array}$$

(28)

$$\begin{array}{cc}& +|{\Phi }_4^{\prime }\left(\tau \right)\left|\right[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & |{\phi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon )d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\mathfrak{m}\right))\right|d\mathfrak{m}\right)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon \left]\right|\hfill \\ & +{\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\upsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon ,\hfill \\ \hfill \le & {\mathcal{L}}_1\left\{{\widehat{\mathsf{\Phi }}}_1^{\prime }\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)}+{\widehat{\mathsf{\Phi }}}_3^{\prime }\left[\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}+\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right]\right.\hfill \\ & +\left.{\widehat{\mathsf{\Phi }}}_4^{\prime }\left(\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right)\right\}\hfill \\ & +{\mathcal{L}}_2\left\{{\widehat{\mathsf{\Phi }}}_2^{\prime }\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)}+{\widehat{\mathsf{\Phi }}}_3^{\prime }\left(\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right)\right.\hfill \\ & \left.+{\widehat{\mathsf{\Phi }}}_4^{\prime }\left[\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}+\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right]\right\}\hfill \\ & +{\mathcal{L}}_1\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}={\mathcal{L}}_1{\mathcal{U}}_1^{*}+{\mathcal{L}}_2{\mathcal{U}}_2^{*}.\hfill \end{array}$$

By considering the definition of the Caputo fractional derivative of order ${\mathfrak{p}}_2\in (0,1)$, we can infer that

$$\begin{array}{cc}\hfill |{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left(\tau \right)|\le & {\int }_0^{\tau }\frac{{(\tau -\upsilon )}^{-{\mathfrak{p}}_2}}{\Gamma (1-{\mathfrak{p}}_2)}\left|{\mathcal{T}}_1^{\prime }(\mathcal{G},\mathcal{J})\right|d\upsilon \le ({\mathcal{L}}_1{\mathcal{U}}_1^{*}+{\mathcal{L}}_2{\mathcal{U}}_2^{*}){\int }_0^{\tau }\frac{{(\tau -\upsilon )}^{-{\mathfrak{p}}_2}}{\Gamma (1-{\mathfrak{p}}_2)}d\upsilon ,\hfill \\ \hfill \le & \frac{({\mathcal{L}}_1{\mathcal{U}}_1^{*}+{\mathcal{L}}_2{\mathcal{U}}_2^{*})}{\Gamma (2-{\mathfrak{p}}_2)},\forall \tau \in [0,1],\hfill \end{array}$$

from which we obtain

$$\begin{array}{c}\hfill \left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left(\tau \right)\left|\right|\le \frac{({\mathcal{L}}_1{\mathcal{U}}_1^{*}+{\mathcal{L}}_2{\mathcal{U}}_2^{*})}{\Gamma (2-{\mathfrak{p}}_2)},\forall \tau \in [0,1].\end{array}$$

Hence, we conclude

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{T}}_1(\mathcal{G},\mathcal{J}){\left|\right|}_{\mathcal{U}}=& \left|\right|{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left|\right|,\hfill \\ \hfill \le & {\mathcal{L}}_1{\mathcal{U}}_1+{\mathcal{L}}_2{\mathcal{U}}_2+\frac{({\mathcal{L}}_1{\mathcal{U}}_1^{*}+{\mathcal{L}}_2{\mathcal{U}}_2^{*})}{\Gamma (2-{\mathfrak{p}}_2)}.\hfill \end{array}$$

(29)

In the same way, we obtain

$$\begin{array}{cc}\hfill |{\mathcal{Y}}_2(\mathcal{G},\mathcal{J})\left(\tau \right)|\le & {\mathcal{A}}_1\left\{{\widehat{\mathsf{\Psi }}}_1\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)}+{\widehat{\mathsf{\Psi }}}_3\left[\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}+\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right]\right.\hfill \\ & \left.+{\widehat{\mathsf{\Psi }}}_4\left(\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right)\right\}\hfill \\ & +{\mathcal{A}}_2\left\{{\widehat{\mathsf{\Psi }}}_2\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)}+{\widehat{\mathsf{\Psi }}}_3\left(\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right)\right.\hfill \\ & \left.+{\widehat{\mathsf{\Psi }}}_4\left[\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}+\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right]\right\}\hfill \\ & +{\mathcal{A}}_2\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}={\mathcal{L}}_1{\mathcal{A}}_1+{\mathcal{L}}_2{\mathcal{A}}_2,\hfill \end{array}$$

$$\begin{array}{cc}\hfill |{\mathcal{Y}}_2^{\prime }(\mathcal{G},\mathcal{J})\left(\tau \right)|\le & {\mathcal{A}}_1\left\{{\widehat{\mathsf{\Psi }}}_1^{\prime }\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)}+{\widehat{\mathsf{\Psi }}}_3^{\prime }\left[\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}+\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right]\right.\hfill \\ & \left.+{\widehat{\mathsf{\Psi }}}_4^{\prime }\left(\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right)\right\}\hfill \\ & +{\mathcal{A}}_2\left\{{\widehat{\mathsf{\Psi }}}_2^{\prime }\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)}+{\widehat{\mathsf{\Psi }}}_3^{\prime }\left(\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right)\right.\hfill \\ & \left.+{\widehat{\mathsf{\Psi }}}_4^{\prime }\left[\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}+\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right]\right\}\hfill \\ & +{\mathcal{A}}_2\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}={\mathcal{L}}_1{\mathcal{A}}_1^{*}+{\mathcal{L}}_2{\mathcal{A}}_2^{*},\hfill \end{array}$$

$$\begin{array}{c}\hfill |{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left(\tau \right)|\le \frac{({\mathcal{L}}_1{\mathcal{A}}_1^{*}+{\mathcal{L}}_2{\mathcal{A}}_2^{*})}{\Gamma (2-{\mathfrak{p}}_1)},\forall \tau \in [0,1].\end{array}$$

Therefore, we conclude

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{T}}_2(\mathcal{G},\mathcal{J}){\left|\right|}_{\mathcal{Q}}=& \left|\right|{\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left|\right|,\hfill \\ \hfill \le & {\mathcal{L}}_1{\mathcal{A}}_1+{\mathcal{L}}_2{\mathcal{A}}_2+\frac{({\mathcal{L}}_1{\mathcal{A}}_1^{*}+{\mathcal{L}}_2{\mathcal{A}}_2^{*})}{\Gamma (2-{\mathfrak{p}}_1)}.\hfill \end{array}$$

(30)

Based on Inequalities (29) and (30), it can be concluded that both ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ are uniformly bounded. This, in turn, signifies that the operator $\mathcal{T}$ is uniformly bounded. Next, we will demonstrate that $\mathcal{T}$ is equi-continuous. Suppose we have ${\tau }_1,{\tau }_2\in [0,1]$, with ${\tau }_1<{\tau }_2$. In this case, we can establish

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left({\tau }_2\right)-{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left({\tau }_1\right)|\hfill \\ \hfill \le & |{\mathsf{\Phi }}_1\left({\tau }_2\right)-{\mathsf{\Phi }}_1\left({\tau }_1\right)|\left|\left[{\lambda }_1{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon \right.\right.\hfill \\ & \left.\left.+{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\upsilon ,\mathcal{G}\left(\upsilon \right),\mathcal{J}\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\upsilon \right))\right|d\upsilon \right]\right|\hfill \\ & +|{\mathsf{\Phi }}_2\left({\tau }_2\right)-{\mathsf{\Phi }}_2\left({\tau }_1\right)|\left|\left[{\chi }_1{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\epsilon )\right|d\epsilon \right)d\upsilon \right.\right.\hfill \\ & \left.\left.+{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\upsilon ,\mathcal{G}\left(\upsilon \right),\mathcal{J}\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\upsilon \right))\right|d\upsilon \right]\right|\hfill \\ & +|{\mathsf{\Phi }}_3\left({\tau }_2\right)-{\mathsf{\Phi }}_3\left({\tau }_1\right)\left|\right[|{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times |{\phi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon \left)d\upsilon \right|\hfill \\ & +\left|{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\mathfrak{m}\right))\right|d\mathfrak{m}\right)d\epsilon \right)d\upsilon \right|\hfill \\ & +\left|{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon \right|]\hfill \\ & +|{\mathsf{\Phi }}_4\left({\tau }_2\right)-{\mathsf{\Phi }}_4\left({\tau }_1\right)\left|\right[|{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times |{\phi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon \left)d\upsilon \right|\hfill \\ & +\left|{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\left({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\mathfrak{m},\mathcal{G}\left(\mathfrak{m}\right),\mathcal{J}\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\mathfrak{m}\right))\right|d\mathfrak{m}\right)d\epsilon \right)d\upsilon \right|\hfill \\ & +\left|{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left|{\varphi }_1(\epsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon \right|]\hfill \\ & +\left|{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left|{\phi }_1(\upsilon ,\mathcal{G}\left(\epsilon \right),\mathcal{J}\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right))\right|d\epsilon \right)d\upsilon \right|,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \le & {\mathcal{L}}_1\frac{1}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}\left((1-e^{-{\lambda }_1{\tau }_1})(1-e^{-{\lambda }_1({\tau }_2-{\tau }_1)})\right)+{\mathcal{L}}_1\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1({\tau }_2-{\tau }_1)})\hfill \\ & +{\mathcal{L}}_1\left\{{\widehat{\mathsf{\Phi }}}_1\left({\tau }_2\right)-{\widehat{\mathsf{\Phi }}}_1\left({\tau }_1\right)\frac{(2-e^{-{\lambda }_1})}{\Gamma \left({\varsigma }_1\right)}+{\widehat{\mathsf{\Phi }}}_3\left({\tau }_2\right)-{\widehat{\mathsf{\Phi }}}_3\left({\tau }_1\right)\left[\frac{(1-e^{-{\lambda }_1})}{{\lambda }_1\Gamma \left({\varsigma }_1\right)}+\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right]\right.\hfill \\ & \left.+{\widehat{\mathsf{\Phi }}}_4{\left(\tau \right)}_2-{\widehat{\mathsf{\Phi }}}_4{\left(\tau \right)}_1\left(\frac{{\eta }^{{\varsigma }_1+\mathfrak{v}-1}}{{\lambda }_1^2\Gamma \left({\varsigma }_1\right)\Gamma \left(\mathfrak{v}\right)}({\lambda }_1\eta +e^{-{\lambda }_1\eta }-1)\right)\right\}\hfill \\ & +{\mathcal{L}}_2\left\{{\widehat{\mathsf{\Phi }}}_2\left({\tau }_2\right)-{\widehat{\mathsf{\Phi }}}_2\left({\tau }_1\right)\frac{(2-e^{-{\chi }_1})}{\Gamma \left({\varsigma }_2\right)}+{\widehat{\mathsf{\Phi }}}_3\left({\tau }_2\right)-{\widehat{\mathsf{\Phi }}}_3\left({\tau }_1\right)\left(\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right)\right.\hfill \\ & \left.+{\widehat{\mathsf{\Phi }}}_4\left({\tau }_2\right)-{\widehat{\mathsf{\Phi }}}_4\left({\tau }_1\right)\left[\frac{(1-e^{-{\chi }_1})}{{\chi }_1\Gamma \left({\varsigma }_2\right)}+\frac{{\zeta }^{{\varsigma }_2+\mathfrak{z}-1}}{{{\chi }_1}^2\Gamma \left({\varsigma }_2\right)\Gamma \left(\mathfrak{z}\right)}({\chi }_1\zeta +e^{-{\chi }_1\zeta }-1)\right]\right\}.\hfill \end{array}$$

This is because

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_i\left({\tau }_2\right)-{\mathsf{\Phi }}_i\left({\tau }_2\right)=& \left|{\displaystyle \frac{1}{{\lambda }_1^2}\left({\Lambda }_i\right)({\lambda }_1{\tau }_2-1+e^{-{\lambda }_1{\tau }_2})+\frac{1}{{\lambda }_1^3}\left({\mathsf{\Theta }}_i\right)({\lambda }_1^2{\tau }_2^2-2{\lambda }_1{\tau }_2+2-2e^{-{\lambda }_1{\tau }_2})}\right.\hfill \\ & \left.{\displaystyle -\frac{1}{{\lambda }_1^2}\left({\Lambda }_i\right)({\lambda }_1{\tau }_1-1+e^{-{\lambda }_1{\tau }_1})+\frac{1}{{\lambda }_1^3}\left({\mathsf{\Theta }}_i\right)({\lambda }_1^2{\tau }_1^2-2{\lambda }_1{\tau }_1+2-2e^{-{\lambda }_1{\tau }_1})}\right|i=1,2,3,4.\hfill \\ \hfill \le & \frac{1}{{\lambda }_1^2}\left|{\Lambda }_i\right|\left|{\lambda }_1({\tau }_2-{\tau }_1)-e^{{\lambda }_1{\tau }_1}+e^{{\lambda }_1{\tau }_2}\right|\hfill \\ & +\frac{1}{{\lambda }_1^3}\left|{\mathsf{\Theta }}_i\right|\left|{\lambda }_1^2({\tau }_2^2({\tau }_2^2-{\tau }_1^2)-2(e^{-{\lambda }_1{\tau }_2}-e^{-{\lambda }_1{\tau }_1}))-{\lambda }_1({\tau }_2-{\tau }_1)\right|\to 0,\hfill \\ & as{\tau }_2\to {\tau }_1,i=1,2,3,4.\hfill \end{array}$$

Clearly, $|{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left({\tau }_2\right)-{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left({\tau }_1\right)|\to 0$, as ${\tau }_2\to {\tau }_1$. Additionally, we obtain

$$\begin{array}{cc}\hfill |{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left({\tau }_2\right)-{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left({\tau }_1\right)|\le & \frac{1}{\Gamma (1-{\mathfrak{p}}_2)}{\int }_0^{{\tau }_1}\frac{|{({\tau }_1-\upsilon )}^{{\mathfrak{p}}_2}-{({\tau }_2-\upsilon )}^{{\mathfrak{p}}_2}|}{{({\tau }_1-\upsilon )}^{{\mathfrak{p}}_2}{({\tau }_2-\upsilon )}^{{\mathfrak{p}}_2}}\left|{\mathcal{T}}_1^{\prime }(\mathcal{G},\mathcal{J})\left(\upsilon \right)\right|d\upsilon \hfill \\ & +\frac{1}{\Gamma (1-{\mathfrak{p}}_2)}{\int }_{{\tau }_1}^{{\tau }_2}{({\tau }_2-\upsilon )}^{-{\mathfrak{p}}_2}\left|{\mathcal{T}}_1^{\prime }(\mathcal{G},\mathcal{J})\left(\upsilon \right)\right|d\upsilon ,\hfill \\ & =\frac{{\mathcal{L}}_1}{\Gamma (2-{\mathfrak{p}}_1)}\left(2{({\tau }_2-{\tau }_1)}^{1-{\mathfrak{p}}_2}-{\tau }_2^{1-{\mathfrak{p}}_2}+{\tau }_1{)}^{1-{\mathfrak{p}}_2}\right)\to 0,\hfill \\ \hfill as{\tau }_2\to {\tau }_1.\end{array}$$

In a similar manner, we have

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left({\tau }_2\right)-{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left({\tau }_1\right)|\to 0\mathrm{and}\hfill \\ & |{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left({\tau }_2\right)-{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left({\tau }_1\right)|\to 0,as{\tau }_2\to {\tau }_1.\hfill \end{array}$$

Consequently, both the operators $\mathcal{T}1$ and $\mathcal{T}2$ exhibit equi-continuous aspects, implying that $\mathcal{T}$ itself is equi-continuous.

Utilising the Ascoli–Arzela theorem, we can deduce that $\mathcal{T}$ is compact. Therefore, we can conclude that $\mathcal{T}$ is a completely continuous operator.

Now, let us establish that the set $\widehat{{\phi }_1}=\{(\mathcal{G},\mathcal{J})\in \mathcal{U}\times \mathcal{Q},(\mathcal{U},\mathcal{Q})=\nu \mathcal{T}(\mathcal{G},\mathcal{J}),0\le \nu \le 1\}$ is bounded. Consider an arbitrary pair $\{(\mathcal{G},\mathcal{J})\in \widehat{{\phi }_1}$, which implies that $\left\{\right(\mathcal{G},\mathcal{J})=\nu \mathcal{T}(\mathcal{G},\mathcal{J})$ for some $\nu \in [0,1]$. Then, for any $\tau \in [0,1]$, we have $\mathcal{G}\left(\tau \right)=\nu {\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left(\tau \right),\mathcal{J}\left(\tau \right)=\nu {\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left(\tau \right)$.

From these relations, it follows that $\left|\mathcal{G}\left(\tau \right)\right|\le |{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left(\tau \right)|$ and $\left|\mathcal{J}\left(\tau \right)\right|\le |{\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left(\tau \right)|$∀$\tau \in [0,1]$.

Then, by ${\mathcal{H}}_1$, we obtain

$$\begin{array}{cc}\hfill & \left|\mathcal{G}\right(\tau \left)\right|\hfill \\ \hfill \le & |{\mathsf{\Phi }}_1\left(\tau \right)|\left[{\lambda }_1{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}|{\mathfrak{a}}_0|+|{\mathfrak{a}}_1\mathcal{G}\left(\epsilon \right)|+|{\mathfrak{a}}_2\mathcal{J}\left(\epsilon \right)|+|{\mathfrak{a}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right)|+|{\mathfrak{a}}_4{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right)|d\epsilon \right)d\upsilon \right.\hfill \\ & \left.+{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left[|{\mathfrak{a}}_0|+|{\mathfrak{a}}_1\mathcal{G}\left(\upsilon \right)|+|{\mathfrak{a}}_2\mathcal{J}\left(\upsilon \right)|+|{\mathfrak{a}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\upsilon \right)|+|{\mathfrak{a}}_4{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\upsilon \right)|\right]d\upsilon \right]\hfill \\ & +|{\mathsf{\Phi }}_2\left(\tau \right)\left|\right[{\chi }_1{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\hfill \\ & \times \left[|{\mathfrak{b}}_0|+{\mathfrak{b}}_1\mathcal{G}\left(\epsilon \right)|+|{\mathfrak{b}}_2\mathcal{J}\left(\tau \right)\left(\epsilon \right)|+|{\mathfrak{b}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\epsilon \right)|+{\mathfrak{b}}_4{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\epsilon \right)|\right]d\epsilon )d\upsilon \hfill \\ & +{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left[|{\mathfrak{b}}_0|+{\mathfrak{b}}_1\mathcal{G}\left(\upsilon \right)|+|{\mathfrak{b}}_2\mathcal{J}\left(\tau \right)\left(\upsilon \right)|+|{\mathfrak{b}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\upsilon \right)|+{\mathfrak{b}}_4{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\upsilon \right)|\right]d\upsilon ]\hfill \\ & +|{\mathsf{\Phi }}_3\left(\tau \right)\left|\right[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times \left[|{\mathfrak{a}}_0|+|{\mathfrak{a}}_1\mathcal{G}\left(\mathfrak{m}\right)|+|{\mathfrak{a}}_2\mathcal{J}\left(\mathfrak{m}\right)|+|{\mathfrak{a}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\mathfrak{m}\right)|+|{\mathfrak{a}}_4{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\mathfrak{m}\right)|\right]d\mathfrak{m}\left)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\hfill \\ & \times \left[|{\mathfrak{b}}_0|+{\mathfrak{b}}_1\mathcal{G}\left(\mathfrak{m}\right)|+|{\mathfrak{b}}_2\mathcal{J}\left(\tau \right)\left(\mathfrak{m}\right)|+|{\mathfrak{b}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\mathfrak{m}\right)|+{\mathfrak{b}}_4{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\mathfrak{m}\right)|\right]d\mathfrak{m}\left)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left[|{\mathfrak{a}}_0|+|{\mathfrak{a}}_1\mathcal{G}\left(\epsilon \right)|+|{\mathfrak{a}}_2\mathcal{J}\left(\epsilon \right)|+|{\mathfrak{a}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right)|+|{\mathfrak{a}}_4{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right)|\right]d\epsilon \right)d\upsilon ]\hfill \\ & +|{\mathsf{\Phi }}_4\left(\tau \right)\left|\right[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times \left[|{\mathfrak{a}}_0|+|{\mathfrak{a}}_1\mathcal{G}\left(\mathfrak{m}\right)|+|{\mathfrak{a}}_2\mathcal{J}\left(\mathfrak{m}\right)|+|{\mathfrak{a}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\mathfrak{m}\right)|+|{\mathfrak{a}}_4{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\mathfrak{m}\right)|\right]d\mathfrak{m}\left)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\hfill \\ & \times \left[|{\mathfrak{b}}_0|+{\mathfrak{b}}_1\mathcal{G}\left(\mathfrak{m}\right)|+|{\mathfrak{b}}_2\mathcal{J}\left(\tau \right)\left(\mathfrak{m}\right)|+|{\mathfrak{b}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\mathfrak{m}\right)|+{\mathfrak{b}}_4{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\mathfrak{m}\right)|\right]d\mathfrak{m}\left)d\epsilon \right)d\upsilon \hfill \\ & +{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}\left[|{\mathfrak{b}}_0|+{\mathfrak{b}}_1\mathcal{G}\left(\epsilon \right)|+|{\mathfrak{b}}_2\mathcal{J}\left(\tau \right)\left(\epsilon \right)|+|{\mathfrak{b}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\epsilon \right)|+{\mathfrak{b}}_4{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\epsilon \right)|\right]d\epsilon \right)d\upsilon ]\hfill \\ & +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\left[|{\mathfrak{a}}_0|+|{\mathfrak{a}}_1\mathcal{G}\left(\epsilon \right)|+|{\mathfrak{a}}_2\mathcal{J}\left(\epsilon \right)|+|{\mathfrak{a}}_3{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\epsilon \right)|+|{\mathfrak{a}}_4{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\epsilon \right)|\right]d\epsilon \right)d\upsilon ,\hfill \end{array}$$

which, on taking the norm for $\tau \in [0,1]$, yields

$$\begin{array}{cc}\hfill \left|\right|\mathcal{G}\left|\right|\le & \left[{\mathfrak{a}}_0+{\mathfrak{a}}_1{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+\left({\mathfrak{a}}_2+{\mathfrak{a}}_3+\frac{{\mathfrak{a}}_4}{\Gamma ({\mathfrak{q}}_1+1)}\right){\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_1\hfill \\ \hfill +& \left[{\mathfrak{b}}_0+\left({\mathfrak{b}}_1+{\mathfrak{b}}_2+\frac{{\mathfrak{a}}_4}{\Gamma (1+{\mathfrak{p}}_1)}\right){\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\mathfrak{b}}_3{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_2.\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{G}}^{\prime }\left|\right|\le & \left[{\mathfrak{a}}_0+{\mathfrak{a}}_1{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+\left({\mathfrak{a}}_2+{\mathfrak{a}}_3+\frac{{\mathfrak{a}}_4}{\Gamma ({\mathfrak{q}}_1+1)}\right){\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_1^{*}\hfill \\ \hfill +& \left[{\mathfrak{b}}_0+\left({\mathfrak{b}}_1+{\mathfrak{b}}_2+\frac{{\mathfrak{a}}_4}{\Gamma (1+{\mathfrak{p}}_1)}\right){\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\mathfrak{b}}_3{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_2^{*}.\hfill \end{array}$$

This implies that

$$\begin{array}{cc}\hfill \left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left|\right|\le & \frac{1}{\Gamma (2-{\mathfrak{p}}_2)}\left\{\left[{\mathfrak{a}}_0+{\mathfrak{a}}_1{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+\left({\mathfrak{a}}_2+{\mathfrak{a}}_3+\frac{{\mathfrak{a}}_4}{\Gamma ({\mathfrak{q}}_1+1)}\right){\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_1^{*}\right.\hfill \\ & \left.+\left[{\mathfrak{b}}_0+\left({\mathfrak{b}}_1+{\mathfrak{b}}_2+\frac{{\mathfrak{a}}_4}{\Gamma (1+{\mathfrak{p}}_1)}\right){\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\mathfrak{b}}_3{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_2^{*}\right\}.\hfill \end{array}$$

(31)

Thus, we have

$$\begin{array}{cc}\hfill {\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}=& \left|\right|\mathcal{G}\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left|\right|\hfill \\ & \le \left[{\mathfrak{a}}_0+{\mathfrak{a}}_1{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+\left({\mathfrak{a}}_2+{\mathfrak{a}}_3+\frac{{\mathfrak{a}}_4}{\Gamma ({\mathfrak{q}}_1+1)}\right){\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_1\hfill \\ & +\left[{\mathfrak{b}}_0+\left({\mathfrak{b}}_1+{\mathfrak{b}}_2+\frac{{\mathfrak{a}}_4}{\Gamma (1+{\mathfrak{p}}_1)}\right){\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\mathfrak{b}}_3{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_2\hfill \\ & +\frac{1}{\Gamma (2-{\mathfrak{p}}_2)}\left\{\left[{\mathfrak{a}}_0+{\mathfrak{a}}_1{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+\left({\mathfrak{a}}_2+{\mathfrak{a}}_3+\frac{{\mathfrak{a}}_4}{\Gamma ({\mathfrak{q}}_1+1)}\right){\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_1^{*}\right.\hfill \\ & \left.+\left[{\mathfrak{b}}_0+\left({\mathfrak{b}}_1+{\mathfrak{b}}_2+\frac{{\mathfrak{a}}_4}{\Gamma (1+{\mathfrak{p}}_1)}\right){\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\mathfrak{b}}_3{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{U}}_2^{*}\right\}.\hfill \end{array}$$

(32)

Likewise, we can have

$$\begin{array}{cc}\hfill {\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}=& \left|\right|\mathcal{J}\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left|\right|\hfill \\ & \le \left[{\mathfrak{a}}_0+{\mathfrak{a}}_1{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+\left({\mathfrak{a}}_2+{\mathfrak{a}}_3+\frac{{\mathfrak{a}}_4}{\Gamma ({\mathfrak{q}}_1+1)}\right){\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{A}}_1\hfill \\ & +\left[{\mathfrak{b}}_0+\left({\mathfrak{b}}_1+{\mathfrak{b}}_2+\frac{{\mathfrak{a}}_4}{\Gamma (1+{\mathfrak{p}}_1)}\right){\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\mathfrak{b}}_3{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{A}}_2\hfill \\ & +\frac{1}{\Gamma (2-{\mathfrak{p}}_1)}\left\{\left[{\mathfrak{a}}_0+{\mathfrak{a}}_1{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+\left({\mathfrak{a}}_2+{\mathfrak{a}}_3+\frac{{\mathfrak{a}}_4}{\Gamma ({\mathfrak{q}}_1+1)}\right){\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{A}}_1^{*}\right.\hfill \\ & \left.+\left[{\mathfrak{b}}_0+\left({\mathfrak{b}}_1+{\mathfrak{b}}_2+\frac{{\mathfrak{a}}_4}{\Gamma (1+{\mathfrak{p}}_1)}\right){\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\mathfrak{b}}_3{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{Q}}\right]{\mathcal{A}}_2^{*}\right\}.\hfill \end{array}$$

(33)

From (32) and (33), we find

$$\begin{array}{cc}\hfill {\left|\right|(\mathcal{G},\mathcal{J})\left|\right|}_{\mathcal{U}\times \mathcal{Q}}=& {\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\hfill \\ \hfill \le & {\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}\left\{{\mathfrak{a}}_1\left[{\mathcal{U}}_1+{\mathcal{A}}_1+\frac{{\mathcal{U}}_1^{*}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{A}}_1^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]+\left({\mathfrak{b}}_1+{\mathfrak{b}}_2+\frac{{\mathfrak{b}}_3}{\Gamma ({\mathfrak{q}}_1+1)}\right)\right.\hfill \\ & \left.+\left[{\mathcal{U}}_2+{\mathcal{A}}_2+\frac{{\mathcal{U}}_2^{*}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{A}}_2^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]\right\}\hfill \\ & +{\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\left\{{\mathfrak{b}}_4\left[{\mathcal{U}}_2+{\mathcal{A}}_2+\frac{{\mathcal{U}}_2^{*}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{A}}_2^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]+\left({\mathfrak{a}}_2+{\mathfrak{a}}_3+\frac{{\mathfrak{a}}_4}{\Gamma ({\mathfrak{q}}_1+1)}\right)\right.\hfill \\ & \left.+\left[{\mathcal{U}}_1+{\mathcal{A}}_1+\frac{{\mathcal{U}}_1^{*}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{A}}_1^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]\right\}\hfill \\ & +{\mathfrak{a}}_0\left[{\mathcal{U}}_1+{\mathcal{A}}_1+\frac{{\mathcal{U}}_1^{*}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{A}}_1^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]+{\mathfrak{b}}_0\left[{\mathcal{U}}_2+{\mathcal{A}}_2+\frac{{\mathcal{U}}_2^{*}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{A}}_2^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]\hfill \\ & \le {\mathcal{N}}_3+max\{{\mathcal{N}}_4+{\mathcal{N}}_5\}{\left|\right|\mathcal{G},\mathcal{J}\left|\right|}_{\mathcal{U}\times \mathcal{Q}}.\hfill \end{array}$$

(34)

Therefore, by implementing the assumption $max{\mathcal{N}}_4+{\mathcal{N}}_5<1$, we can deduce

$$\begin{array}{c}\hfill {\left|\right|\mathcal{G},\mathcal{J}\left|\right|}_{\mathcal{U}\times \mathcal{Q}}\le \frac{{\mathcal{N}}_3}{1-max\{{\mathcal{N}}_4+{\mathcal{N}}_5\}}.\end{array}$$

Hence, we deduce that the set $\widehat{{\phi }_1}$ is bounded.

Utilising Theorem 2, we can ascertain that the operator $\mathcal{T}$ possesses at least one fixed point, which serves as a solution to our Problems (1) and (2). With this, we conclude the proof. □

Subsequently, we will establish existence and uniqueness results for Problems (1) and (2), employing the Banach contraction mapping principle.

We introduce the concepts

$$\begin{array}{cc}\hfill r_1=& \underset{\tau \in [0,1]}{sup}|{\phi }_1(\tau ,0,0,0,0)|,r_2=\underset{\tau \in [0,1]}{sup}\left|{\varphi }_1(\tau ,0,0,0,0)\right|,\hfill \\ \hfill \Lambda =& {\mathfrak{W}}_0{\rho }_1{\mathcal{U}}_1+{\mathfrak{K}}_0{\rho }_2{\mathcal{A}}_1,\mathcal{M}={\mathfrak{W}}_0{\rho }_1{\mathcal{U}}_1^{*}+{\mathfrak{K}}_0{\rho }_2{r}_2{\mathcal{A}}_1^{*},\hfill \\ \hfill {\Lambda }^{*}=& {\mathfrak{W}}_0{\rho }_1{\mathcal{U}}_2+{\mathfrak{K}}_0{\rho }_2{\mathcal{A}}_2,{\mathcal{M}}^{*}={\mathfrak{W}}_0{\rho }_1{\mathcal{U}}_2^{*}+{\mathfrak{K}}_0{\rho }_2{r}_2{\mathcal{A}}_2^{*},\hfill \\ \hfill {\mathcal{Z}}_1=& r_1{\mathcal{U}}_1+r_2{\mathcal{A}}_1,{\mathcal{Z}}_1^{*}=r_1{\mathcal{U}}_1^{*}+r_2{\mathcal{A}}_1^{*},\widehat{{\mathcal{Z}}_1}=r_1{\mathcal{U}}_2+r_2{\mathcal{A}}_2,\widehat{{\mathcal{Z}}_1^{*}}=r_1{\mathcal{U}}_2^{*}+r_2{\mathcal{A}}_2^{*}.\hfill \end{array}$$

Theorem 3. 

Assume that ${\mathcal{H}}_2$ holds. Furthermore, if

$$\begin{array}{c}\hfill \left[\Lambda +{\Lambda }^{*}+\frac{\mathcal{M}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{M}}^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]<1,\end{array}$$

(35)

then Problems (1) and (2) have a unique solution.

Proof. 

We consider the positive number r given by

$$\begin{array}{c}\hfill {\displaystyle r\ge \frac{\left[{\mathcal{Z}}_1+{\mathcal{Z}}_1^{*}+\frac{\widehat{{\mathcal{Z}}_1}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{\widehat{{\mathcal{Z}}_1^{*}}}{\Gamma (2-{\mathfrak{p}}_1)}\right]}{\left(1-\left[\Lambda +{\Lambda }^{*}+\frac{\mathcal{M}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{M}}^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]\right)}.}\end{array}$$

We show that $\mathcal{T}{\widehat{\mathfrak{B}}}_r\subset {\widehat{\mathfrak{B}}}_r$, where ${\widehat{\mathfrak{B}}}_r={\{(\mathcal{G},\mathcal{J})\in \mathcal{U}\times \mathcal{Q},||\mathcal{G},\mathcal{J}||}_{\mathcal{U}\times \mathcal{Q}}<r\}$. For $\mathcal{G},\mathcal{J}\in {\widehat{\mathfrak{B}}}_r$, we obtain

$$\begin{array}{cc}\hfill & |{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))|\hfill \\ \hfill \le & |{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))-{\phi }_1(\tau ,0,0,0,0)|+|{\phi }_1(\tau ,0,0,0,0)|,\hfill \\ & \le {\mathfrak{W}}_0\left(\right|\mathcal{G}\left(\tau \right)|+|\mathcal{J}\left(\tau \right)|+|{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right)|+|{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right)\left|\right)+r_1,\hfill \\ & \le {\mathfrak{W}}_0\left[{\left|\right|\mathcal{G}\left|\right|}_{\mathcal{U}}+{\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}+\frac{1}{\Gamma ({\mathfrak{q}}_1+1)}{\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}}\right]+r_1,\hfill \\ & \le {\mathfrak{W}}_0{\left(\right|\left|\mathcal{G}\right||}_{\mathcal{U}}+{\rho }_1{\left|\right|\mathcal{J}\left|\right|}_{\mathcal{Q}})+r_1,\hfill \\ & \le {\mathfrak{W}}_0{\rho }_{1}r+r_1.\hfill \end{array}$$

In a similar manner, we have

$$\begin{array}{cc}\hfill & |{\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right))|\le {\mathfrak{K}}_0{\rho }_{2}r+r_2.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill |{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left(\tau \right)|\le & ({\mathfrak{W}}_0{\rho }_{1}r+r_1){\mathcal{U}}_1+({\mathfrak{K}}_0{\rho }_{2}r+r_2){\mathcal{A}}_1,\hfill \\ & =({\mathfrak{W}}_0{\rho }_1{\mathcal{U}}_1+{\mathfrak{K}}_0{\rho }_2{\mathcal{A}}_1)+r_2{\mathcal{A}}_1+{\mathcal{U}}_1{r}_1,\hfill \\ & =\Lambda r+{\mathcal{Z}}_1,forall\tau \in [0,1].\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |{\mathcal{T}}_1^{\prime }(\mathcal{G},\mathcal{J})\left(\tau \right)|\le & ({\mathfrak{W}}_0{\rho }_{1}r+r_1){\mathcal{U}}_1^{*}+({\mathfrak{K}}_0{\rho }_{2}r+r_2){\mathcal{A}}_1^{*},\hfill \\ & =({\mathfrak{W}}_0{\rho }_1{\mathcal{U}}_1^{*}+{\mathfrak{K}}_0{\rho }_2{\mathcal{A}}_1^{*})+r_2{\mathcal{A}}_1^{*}+{\mathcal{U}}_1^{*}{r}_1,\hfill \\ & ={\Lambda }^{*}r+{\mathcal{Z}}_1^{*},forall\tau \in [0,1].\hfill \end{array}$$

which gives us

$$\begin{array}{cc}\hfill |{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left(\tau \right)|\le & {\int }_0^{\tau }\frac{{(\tau -\upsilon )}^{-{\mathfrak{p}}_2}}{\Gamma (1-{\mathfrak{p}}_2)}\left|{\mathcal{T}}_1^{\prime }(\mathcal{G},\mathcal{J})\left(\upsilon \right)\right|d\upsilon ,\hfill \\ & \le \frac{1}{\Gamma (2-{\mathfrak{p}}_2)}(\mathcal{M}r+{\mathcal{Z}}_1^{*}),forall\tau \in [0,1].\hfill \end{array}$$

Therefore, we deduce

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{T}}_1(\mathcal{G},\mathcal{J}){\left|\right|}_{\mathcal{U}}=& \left|\right|{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1(\mathcal{G},\mathcal{J})\left|\right|,\hfill \\ & \le \left(\Lambda +\frac{\mathcal{M}}{\Gamma (2-{\mathfrak{p}}_2)}\right)r+{\mathcal{Z}}_1+\frac{{\mathcal{Z}}_1^{*}}{\Gamma (2-{\mathfrak{p}}_2)}.\hfill \end{array}$$

(36)

In a similar manner, we obtain

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left(\tau \right)|\le {\Lambda }^{*}r+\widehat{{\mathcal{Z}}_1},\hfill \end{array}$$

$$\begin{array}{c}\hfill |{\mathcal{T}}_2^{\prime }(\mathcal{G},\mathcal{J})\left(\tau \right)|\le {\mathcal{M}}^{*}r+{\widehat{{\mathcal{Z}}_1}}^{*},\end{array}$$

$$\begin{array}{cc}\hfill |{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left(\tau \right)|\le & {\int }_0^{\tau }\frac{{(\tau -\upsilon )}^{-{\mathfrak{p}}_1}}{\Gamma (1-{\mathfrak{p}}_1)}\left|{\mathcal{T}}_2^{\prime }(\mathcal{G},\mathcal{J})\left(\upsilon \right)\right|d\upsilon ,\hfill \\ & \le \frac{1}{\Gamma (2-{\mathfrak{p}}_1)}({\mathcal{M}}^{*}r+{\widehat{{\mathcal{Z}}_1}}^{*}),forall\tau \in [0,1].\hfill \end{array}$$

Then, we conclude

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{T}}_2(\mathcal{G},\mathcal{J}){\left|\right|}_{\mathcal{Q}}=& \left|\right|{\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{T}}_2(\mathcal{G},\mathcal{J})\left|\right|,\hfill \\ & \le \left({\Lambda }^{*}+\frac{{\mathcal{M}}^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right)r+\widehat{{\mathcal{Z}}_1}+\frac{\widehat{{\mathcal{Z}}_1^{*}}}{\Gamma (2-{\mathfrak{p}}_1)}.\hfill \end{array}$$

(37)

By Relations (36) and (37), we deduce

$$\begin{array}{cc}\hfill {\left|\right|\mathcal{T}(\mathcal{G},\mathcal{J})\left|\right|}_{\mathcal{U}\times \mathcal{Q}}=& \left|\right|{\mathcal{T}}_1{(\mathcal{G},\mathcal{J})\left|\right|}_{\mathcal{U}}+\left|\right|{\mathcal{T}}_2(\mathcal{G},\mathcal{J}){\left|\right|}_{\mathcal{Q}}\hfill \\ & \le \left(\left[\Lambda +{\Lambda }^{*}+\frac{\mathcal{M}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{M}}^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]\right)r+\left[{\mathcal{Z}}_1+{\mathcal{Z}}_1^{*}+\frac{\widehat{{\mathcal{Z}}_1}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{\widehat{{\mathcal{Z}}_1^{*}}}{\Gamma (2-{\mathfrak{p}}_1)}\right]\hfill \\ & =r.\hfill \end{array}$$

This implies $\mathcal{T}{\widehat{\mathfrak{B}}}_r\subset {\widehat{\mathfrak{B}}}_r$.

Next, we prove that the operator $\mathcal{T}$ is a contraction. For $({\mathcal{G}}_i,{\mathcal{J}}_i)\in {\widehat{\mathfrak{B}}}_r,i=1,2$ and for each $\tau \in [0,1]$, we have

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_1({\mathcal{G}}_1,{\mathcal{J}}_1)\left(\tau \right)-{\mathcal{T}}_2({\mathcal{G}}_2,{\mathcal{J}}_2)\left(\tau \right)|\hfill \\ \hfill =& |{\mathsf{\Phi }}_1\left(\tau \right)|\left[{\lambda }_1{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}|{\phi }_1(\epsilon ,{\mathcal{G}}_1\left(\epsilon \right),{\mathcal{J}}_1\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_1\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_1\left(\epsilon \right))\right.\right.\hfill \\ & \left.-{\phi }_1(\epsilon ,{\mathcal{G}}_2\left(\epsilon \right),{\mathcal{J}}_2\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_2\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_2\left(\epsilon \right))|d\epsilon \right)d\upsilon \hfill \\ & \left.-{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}|{\phi }_1(\upsilon ,{\mathcal{G}}_1\left(\upsilon \right),{\mathcal{J}}_1\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_1\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_1\left(\upsilon \right))-{\phi }_1(\upsilon ,{\mathcal{G}}_2\left(\upsilon \right),{\mathcal{J}}_2\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_2\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_2\left(\upsilon \right))|d\upsilon \right]\hfill \\ & +|{\mathsf{\Phi }}_2\left(\tau \right)|\left[{\chi }_1{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}|{\varphi }_1(\epsilon ,{\mathcal{G}}_1\left(\epsilon \right),{\mathcal{J}}_1\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_1\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_1\left(\epsilon \right))\right.\right.\hfill \\ & \left.-{\varphi }_1(\epsilon ,{\mathcal{G}}_2\left(\epsilon \right),{\mathcal{J}}_2\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_2\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_2\left(\epsilon \right))|d\epsilon \right)d\upsilon \hfill \\ & \left.-{\int }_0^1\frac{{(1-\upsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}|{\varphi }_1(\upsilon ,{\mathcal{G}}_1\left(\upsilon \right),{\mathcal{J}}_1\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_1\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_1\left(\upsilon \right))-{\varphi }_1(\upsilon ,{\mathcal{G}}_2\left(\upsilon \right),{\mathcal{J}}_2\left(\upsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_2\left(\upsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_2\left(\upsilon \right))|d\upsilon \right]\hfill \\ & +|{\mathsf{\Phi }}_3\left(\tau \right)\left|\right[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times |{\phi }_1(\mathfrak{m},{\mathcal{G}}_1\left(\mathfrak{m}\right),{\mathcal{J}}_1\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_1\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_1\left(\mathfrak{m}\right))\hfill \\ & -{\phi }_1(\mathfrak{m},{\mathcal{G}}_2\left(\mathfrak{m}\right),{\mathcal{J}}_2\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_2\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_2\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon )d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}|{\varphi }_1(\mathfrak{m},{\mathcal{G}}_1\left(\mathfrak{m}\right),{\mathcal{J}}_1\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_1\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_1\left(\mathfrak{m}\right))\hfill \\ & -{\varphi }_1(\mathfrak{m},{\mathcal{G}}_2\left(\mathfrak{m}\right),{\mathcal{J}}_2\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_2\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_2\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon )d\upsilon \hfill \\ & -{\int }_0^1{e}^{-{\lambda }_1(1-\upsilon )}({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}|{\phi }_1(\epsilon ,{\mathcal{G}}_1\left(\epsilon \right),{\mathcal{J}}_1\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_1\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_1\left(\epsilon \right))\hfill \\ & -{\phi }_1(\epsilon ,{\mathcal{G}}_2\left(\epsilon \right),{\mathcal{J}}_2\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_2\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_2\left(\epsilon \right))\left|d\epsilon \right)d\upsilon ]\hfill \\ & +|{\mathsf{\Phi }}_4\left(\tau \right)\left|\right[{\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\left({\int }_0^{\upsilon }{e}^{-{\lambda }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}\hfill \\ & \times |{\phi }_1(\mathfrak{m},{\mathcal{G}}_1\left(\mathfrak{m}\right),{\mathcal{J}}_1\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_1\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_1\left(\mathfrak{m}\right))\hfill \\ & -{\phi }_1(\mathfrak{m},{\mathcal{G}}_2\left(\mathfrak{m}\right),{\mathcal{J}}_2\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_2\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_2\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon )d\upsilon \hfill \\ & +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\left({\int }_0^{\upsilon }{e}^{-{\chi }_1(\upsilon -\epsilon )}\right({\int }_0^{\epsilon }\frac{{(\epsilon -\mathfrak{m})}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}|{\varphi }_1(\mathfrak{m},{\mathcal{G}}_1\left(\mathfrak{m}\right),{\mathcal{J}}_1\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_1\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_1\left(\mathfrak{m}\right))\hfill \\ & -{\varphi }_1(\mathfrak{m},{\mathcal{G}}_2\left(\mathfrak{m}\right),{\mathcal{J}}_2\left(\mathfrak{m}\right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_2\left(\mathfrak{m}\right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_2\left(\mathfrak{m}\right))\left|d\mathfrak{m}\right)d\epsilon )d\upsilon \hfill \\ & -{\int }_0^1{e}^{-{\chi }_1(1-\upsilon )}({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_2-2}}{\Gamma ({\varsigma }_2-1)}|{\varphi }_1(\epsilon ,{\mathcal{G}}_1\left(\epsilon \right),{\mathcal{J}}_1\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_1\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_1\left(\epsilon \right))\hfill \\ & -{\varphi }_1(\epsilon ,{\mathcal{G}}_2\left(\epsilon \right),{\mathcal{J}}_2\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_2\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_2\left(\epsilon \right))\left|d\epsilon \right)d\upsilon ]\hfill \\ & +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\upsilon )}\left({\int }_0^{\upsilon }\frac{{(\upsilon -\epsilon )}^{{\varsigma }_1-2}}{\Gamma ({\varsigma }_1-1)}|{\phi }_1(\epsilon ,{\mathcal{G}}_1\left(\epsilon \right),{\mathcal{J}}_1\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_1\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_1\left(\epsilon \right))\right.\hfill \\ & \left.-{\phi }_1(\epsilon ,{\mathcal{G}}_2\left(\epsilon \right),{\mathcal{J}}_2\left(\epsilon \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_2\left(\epsilon \right),{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_2\left(\epsilon \right))|d\epsilon \right)d\upsilon .\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\mathcal{U}}_1{\mathfrak{W}}_0\left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2\left|\right|+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_1-{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_2\left|\right|+\left|\right|{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_1-{\mathcal{I}}^{{\mathfrak{q}}_1}{\mathcal{J}}_2\left|\right|)\hfill \\ & +{\mathcal{A}}_1{\mathfrak{K}}_0\left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_1-{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_2\left|\right|+\left|\right|{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_1-{\mathcal{I}}^{{\mathfrak{q}}_2}{\mathcal{G}}_2\left|\right|+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2\left|\right|)\hfill \\ \hfill \le & {\mathcal{U}}_1{\mathfrak{W}}_0\left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2\left|\right|+{\rho }_1\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_1-{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}{\mathcal{J}}_2\left|\right|)\hfill \\ & +{\mathcal{A}}_1{\mathfrak{K}}_0({\rho }_2\left|\right|{\mathcal{G}}_1-{\mathcal{G}}_2\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_1-{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{G}}_2\left|\right|+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2\left|\right|)\hfill \\ \hfill \le & \Lambda \left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2|{|}_{\mathcal{U}}+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2\left|{|}_{\mathcal{Q}}\right).\hfill \end{array}$$

Then, we obtain

$$\begin{array}{c}\hfill |{\mathcal{T}}_1^{\prime }({\mathcal{G}}_1,{\mathcal{J}}_1)\left(\tau \right)-{\mathcal{T}}_2^{\prime }({\mathcal{G}}_2,{\mathcal{J}}_2)\left(\tau \right)|\le \mathcal{M}(\left|\right|{\mathcal{G}}_1-{\mathcal{G}}_2{\left|\right|}_{\mathcal{U}}+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2{\left|\right|}_{\mathcal{Q}}).\end{array}$$

which gives us

$$\begin{array}{cc}\hfill |{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1({\mathcal{G}}_1,{\mathcal{J}}_1)\left(\tau \right)-{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1({\mathcal{G}}_2,{\mathcal{J}}_2)\left(\tau \right)|\le & {\int }_0^{\tau }\frac{{(\tau -\upsilon )}^{-{\mathfrak{p}}_2}}{\Gamma (1-{\mathfrak{p}}_2)}|{\mathcal{T}}_1^{\prime }({\mathcal{G}}_1,{\mathcal{J}}_1)\left(\upsilon \right)-{\mathcal{T}}_1^{\prime }({\mathcal{G}}_2,{\mathcal{J}}_2)\left(\upsilon \right)|d\upsilon ,\hfill \\ & \le \frac{1}{\Gamma (2-{\mathfrak{p}}_2)}\mathcal{M}\left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2{\left|\right|}_{\mathcal{U}}+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2{\left|\right|}_{\mathcal{Q}}).\hfill \end{array}$$

From the above inequalities, we conclude

$$\begin{array}{c}\left|\right|{\mathcal{T}}_1({\mathcal{G}}_1,{\mathcal{J}}_1)\left(\tau \right)-{\mathcal{T}}_1({\mathcal{G}}_2,{\mathcal{J}}_2)\left(\tau \right){\left|\right|}_{\mathcal{U}}\hfill \\ =\left|\right|{\mathcal{T}}_1({\mathcal{G}}_1,{\mathcal{J}}_1)\left(\tau \right)-{\mathcal{T}}_1({\mathcal{G}}_2,{\mathcal{J}}_2)\left(\tau \right)\left|\right|+\left|\right|{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1({\mathcal{G}}_1,{\mathcal{J}}_1)\left(\tau \right)-{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}{\mathcal{T}}_1({\mathcal{G}}_2,{\mathcal{J}}_2)\left(\tau \right)\left|\right|\hfill \\ \le \left[\Lambda +\frac{1}{\Gamma (2-{\mathfrak{p}}_2)}\mathcal{M}\right]\left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2{\left|\right|}_{\mathcal{U}}+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2{\left|\right|}_{\mathcal{Q}}).\hfill \end{array}$$

(38)

In a similar manner, we deduce

$$\begin{array}{cc}\hfill & \left|\right|{\mathcal{T}}_2({\mathcal{G}}_1,{\mathcal{J}}_1)-{\mathcal{T}}_2({\mathcal{G}}_2,{\mathcal{J}}_2){\left|\right|}_{\mathcal{Q}}\le \left[{\Lambda }^{*}+\frac{1}{\Gamma (2-{\mathfrak{p}}_2)}{\mathcal{M}}^{*}\right]\left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2{\left|\right|}_{\mathcal{U}}+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2{\left|\right|}_{\mathcal{Q}}).\hfill \end{array}$$

(39)

Therefore, by (38) and (39), we obtain

$$\begin{array}{cc}\hfill & \left|\right|\mathcal{T}({\mathcal{G}}_1,{\mathcal{J}}_1)-\mathcal{T}({\mathcal{G}}_2,{\mathcal{J}}_2){\left|\right|}_{\mathcal{Q}}\hfill \\ & =\left|\right|{\mathcal{T}}_1({\mathcal{G}}_1,{\mathcal{J}}_1)-{\mathcal{T}}_1({\mathcal{G}}_2,{\mathcal{J}}_2){\left|\right|}_{\mathcal{U}}+\left|\right|{\mathcal{T}}_2({\mathcal{G}}_1,{\mathcal{J}}_1)-{\mathcal{T}}_2({\mathcal{G}}_2,{\mathcal{J}}_2){\left|\right|}_{\mathcal{Q}}\hfill \\ & \le \left[\Lambda +\frac{1}{\Gamma (2-{\mathfrak{p}}_2)}\mathcal{M}\right]\left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2{\left|\right|}_{\mathcal{U}}+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2{\left|\right|}_{\mathcal{Q}})+\left[{\Lambda }^{*}+\frac{1}{\Gamma (2-{\mathfrak{p}}_2)}{\mathcal{M}}^{*}\right]\left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2{\left|\right|}_{\mathcal{U}}+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2{\left|\right|}_{\mathcal{Q}})\hfill \\ & \le \left(\Lambda +{\Lambda }^{*}+\frac{1}{\Gamma (2-{\mathfrak{p}}_2)}\mathcal{M}+\frac{1}{\Gamma (2-{\mathfrak{p}}_2)}{\mathcal{M}}^{*}\right)\times \left(\right||{\mathcal{G}}_1-{\mathcal{G}}_2{\left|\right|}_{\mathcal{U}}+\left|\right|{\mathcal{J}}_1-{\mathcal{J}}_2{\left|\right|}_{\mathcal{Q}}).\hfill \end{array}$$

Through the utilisation of Condition (35), it can be inferred that $\mathcal{T}$ qualifies as a contraction operator. Consequently, by virtue of Banach’s fixed-point theorem, the operator $\mathcal{T}$ possesses a unique fixed point, corresponding to the unique solution of Problems (1) and (2). With this, we conclude the proof. □

5. Example

Let ${\lambda }_1=2;{\chi }_1=3;\eta =\frac{1}{4};v=1;z=1;\zeta =\frac{1}{5};\tau =1,{\varsigma }_1=\frac{9}{2},{\varsigma }_2=\frac{7}{2},{\mathfrak{p}}_1=\frac{1}{3}{\mathfrak{p}}_2=\frac{1}{2},{\mathfrak{q}}_1=\frac{3}{2},{\mathfrak{q}}_2=\frac{8}{3}.$ We then turn our attention to the following system of fractional differential equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}({}^c{\mathcal{D}}^{{\varsigma }_1}+{\lambda }_1{}^c{\mathcal{D}}^{{\varsigma }_1-1})\mathcal{G}\left(\tau \right)={\phi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_1}\mathcal{J}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_1}\mathcal{J}\left(\tau \right)),\tau \in (0,1),\hfill \\ ({}^c{\mathcal{D}}^{{\varsigma }_2}+{\chi }_1{}^c{\mathcal{D}}^{{\varsigma }_2-1})\mathcal{J}\left(\tau \right)={\varphi }_1(\tau ,\mathcal{G}\left(\tau \right),\mathcal{J}\left(\tau \right),{}^c{\mathcal{D}}^{{\mathfrak{p}}_2}\mathcal{G}\left(\tau \right),{\mathcal{I}}^{{\mathfrak{q}}_2}\mathcal{G}\left(\tau \right)),\tau \in (0,1)\hfill \end{array}\right.\end{array}$$

(40)

complemented by the general coupled integral boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathcal{G}\left(0\right)=0,{\mathcal{G}}^{\prime }\left(0\right)=0,{\mathcal{G}}^{\prime }\left(1\right)=0,\mathcal{G}\left(1\right)={\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\mathcal{G}\left(\upsilon \right)d\upsilon +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\mathcal{J}\left(\upsilon \right)d\upsilon ,}\hfill \\ {\displaystyle \mathcal{J}\left(0\right)=0,{\mathcal{J}}^{\prime }\left(0\right)=0,{\mathcal{J}}^{\prime }\left(1\right)=0,\mathcal{J}\left(1\right)={\int }_0^{\eta }\frac{{(\eta -\upsilon )}^{\mathfrak{v}-1}}{\Gamma \left(\mathfrak{v}\right)}\mathcal{G}\left(\upsilon \right)d\upsilon +{\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\mathfrak{z}-1}}{\Gamma \left(\mathfrak{z}\right)}\mathcal{J}\left(\upsilon \right)d\upsilon ,}\hfill \end{array}\right.\end{array}$$

(41)

We have ${\mathsf{\Omega }}_1=0.432332$, ${\mathsf{\Omega }}_2=0.533834$, ${\mathsf{\Omega }}_3=0.316738$, ${\mathsf{\Omega }}_4=0.449976$, ${\mathsf{\Omega }}_5=0.281525$, ${\mathsf{\Omega }}_6=0.247777$, ${\mathsf{\Omega }}_7=0.00115512$, ${\mathsf{\Omega }}_8=0.000118806$, ${\mathsf{\Omega }}_9=0.00120999$, ${\mathsf{\Omega }}_{10}=0.00029549$, ${\mathsf{\Omega }}_{11}=0.22659$, ${\mathsf{\Omega }}_{12}=0.181378$, $\Delta =0.00192126$, ${\Delta }_1=0.00046151$, ${\Lambda }_1\approx 5.74083$, ${\Lambda }_2\approx 0.0507348$, ${\Lambda }_3\approx 17.6486$, ${\Lambda }_4\approx 0.160883$, ${\Xi }_1\approx 0.0507341$, ${\Xi }_2\approx 4.07506$, ${\Xi }_3\approx 0.121363$, ${\Xi }_4\approx 10.1098$, ${\mathsf{\Theta }}_1\approx -6.52253$, ${\mathsf{\Theta }}_2\approx 0.0410882$, ${\mathsf{\Theta }}_3\approx -10.017$, ${\mathsf{\Theta }}_4\approx 0.0084369$, ${\mathsf{{\rm Y}}}_2\approx 5.09077$, ${\mathsf{{\rm Y}}}_2\approx 0.0357116$, ${\mathsf{{\rm Y}}}_3\approx -0.0854273$, ${\mathsf{{\rm Y}}}_4\approx -7.11626$, ${\mathsf{\Phi }}_1\approx 3.03939$, ${\mathsf{\Phi }}_2\approx 0.00551836$, ${\mathsf{\Phi }}_3\approx 2.84393$, ${\mathsf{\Phi }}_4\approx 0.0474878$, ${\mathsf{\Psi }}_1\approx 0.00507333$, ${\mathsf{\Psi }}_2\approx 0.00415164$, ${\mathsf{\Psi }}_3\approx 0.0121361$, ${\mathsf{\Psi }}_4\approx 1.01096$, ${\mathcal{U}}_1\approx 0.344355$, ${\mathcal{U}}_2\approx 0.00133908$, ${\mathcal{A}}_1\approx 0.000357643$, ${\mathcal{A}}_2=0.0966458$, ${\mathcal{U}}_1^{*}\approx 0.758877$, ${\mathcal{U}}_2^{*}\approx 0.00793721$, ${\mathcal{A}}_1^{*}\approx 0.0000464214$, ${\mathcal{A}}_2^{*}=0.59475844.$

Example 1. 

We consider the functions

$$\begin{array}{cc}\hfill & |{\phi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)|=\frac{\tau }{{\tau }^2+1}\left(3cos\tau +\frac{1}{8}sin({\mathfrak{l}}_1+{\mathfrak{l}}_2)\right)-\frac{1}{8{(\tau +1)}^2}{\mathfrak{l}}_3+\frac{1}{10}arctan{\mathfrak{l}}_4,\hfill \\ & |{\varphi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)|=\frac{\tau }{{(\tau +2)}^3}\left(5e^{-\tau }+\frac{{\mathfrak{l}}_1}{2}+2{\mathfrak{l}}_2\right)-\frac{\tau }{6}sin({\mathfrak{l}}_3+{\mathfrak{l}}_4),\hfill \end{array}$$

for all $\tau \in [0,1]$, ${\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4\in {\mathcal{R}}_e$. We achieve the inequalities

$$\begin{array}{cc}\hfill & |{\phi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)|\le \frac{3}{2}+\frac{1}{16}|{\mathfrak{l}}_1|+\frac{1}{16}|{\mathfrak{l}}_2|+\frac{1}{8}|{\mathfrak{l}}_3|+\frac{1}{10}|{\mathfrak{l}}_4|,\hfill \\ & |{\varphi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)|\le \frac{5}{8}+\frac{1}{16}|{\mathfrak{l}}_1|+\frac{1}{4}|{\mathfrak{l}}_2|+\frac{1}{6}|{\mathfrak{l}}_3|+\frac{1}{6}|{\mathfrak{l}}_4|,\hfill \end{array}$$

for all $\tau \in [0,1],{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4\in {\mathcal{R}}_e$. Hence, we arrive at ${\mathfrak{a}}_0=\frac{3}{2},{\mathfrak{a}}_1=\frac{1}{16},{\mathfrak{a}}_2=\frac{1}{16},{\mathfrak{a}}_3=\frac{1}{10},{\mathfrak{a}}_4=\frac{1}{10},{\mathfrak{b}}_0=\frac{5}{8},{\mathfrak{b}}_1=\frac{1}{16},{\mathfrak{b}}_2=\frac{1}{4},{\mathfrak{b}}_3=\frac{1}{6},{\mathfrak{b}}_4=\frac{1}{6}$.

Given that ${\mathcal{N}}_4\approx 0.34571011$ and ${\mathcal{N}}_5\approx 0.352583822$, we can ascertain that the condition of Theorem 2, $max\{{\mathcal{N}}_4,{\mathcal{N}}_5\}<1$, is indeed met. Therefore, in accordance with Theorem 2, we can conclude that Problems (1) and (2) possess at least one solution, $\tau \in [0,1]$.

Example 2. 

We consider the functions

$$\begin{array}{cc}\hfill & {\phi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)=\frac{\tau }{2}+\frac{1}{8{(\tau +1)}^2}\left(\frac{{\mathfrak{l}}_1}{1+|{\mathfrak{l}}_1|}-{\mathfrak{l}}_2\right)+\frac{1}{32}{sin}^2{\mathfrak{l}}_3-\frac{\tau }{9}arctan{\mathfrak{l}}_4,\hfill \\ & {\varphi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)=\frac{{\tau }^2}{{\tau }^3+1}-\frac{1}{16}sin{\mathfrak{l}}_1+\frac{1}{10}{\mathfrak{l}}_2+\frac{1}{\sqrt{4+{\tau }^2}}cos{\mathfrak{l}}_3-\frac{|{\mathfrak{l}}_4|}{6(1+|{\mathfrak{l}}_4\left)\right|},\hfill \end{array}$$

for all $\tau \in [0,1],{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4\in {\mathcal{R}}_e$.

We obtain the following inequalities

$$\begin{array}{cc}\hfill & |{\phi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)-{\phi }_1(\tau ,{\mathfrak{n}}_1,{\mathfrak{n}}_2,{\mathfrak{n}}_3,{\mathfrak{n}}_4)|\le \frac{1}{8}\left(\right|{\mathfrak{l}}_1-{\mathfrak{n}}_1|+|{\mathfrak{l}}_2-{\mathfrak{n}}_2|+|{\mathfrak{l}}_3-{\mathfrak{n}}_3|+|{\mathfrak{l}}_4-{\mathfrak{n}}_4\left|\right)\hfill \\ & |{\varphi }_1(\tau ,{\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4)-{\varphi }_1(\tau ,{\mathfrak{n}}_1,{\mathfrak{n}}_2,{\mathfrak{n}}_3,{\mathfrak{n}}_4)|\le \frac{1}{6}\left(\right|{\mathfrak{l}}_1-{\mathfrak{n}}_1|+|{\mathfrak{l}}_2-{\mathfrak{n}}_2|+|{\mathfrak{l}}_3-{\mathfrak{n}}_3|+|{\mathfrak{l}}_4-{\mathfrak{n}}_4\left|\right)\hfill \end{array}$$

for all $\tau \in [0,1]$ and ${\mathfrak{l}}_1,{\mathfrak{l}}_2,{\mathfrak{l}}_3,{\mathfrak{l}}_4\in {\mathcal{R}}_e$.

Here, ${\mathfrak{K}}_0=\frac{1}{6}$ and ${\mathfrak{M}}_0=\frac{1}{8}$. Furthermore, we deduce ${\rho }_1\approx 0.752252$${\rho }_2\approx 0.249238$, $\Lambda \approx 0.03243582$, ${\Lambda }^{*}\approx 0.00404732,\mathcal{M}\approx 0.00404732,{\mathcal{M}}^{*}\approx 0.0247046$ and

$$\begin{array}{c}\hfill \left[\Lambda +{\Lambda }^{*}+\frac{\mathcal{M}}{\Gamma (2-{\mathfrak{p}}_2)}+\frac{{\mathcal{M}}^{*}}{\Gamma (2-{\mathfrak{p}}_1)}\right]\approx 0.1447348<1.\end{array}$$

(42)

Hence, all the conditions outlined in Theorem 3 are fulfilled. Consequently, in accordance with Theorem 3, we can establish that Problems (1) and (2) possess a unique solution, denoted by $\tau \in [0,1]$.

6. Conclusions

The primary objective of this research is to investigate the possible existence and uniqueness of a solution to a system of fractional integrodifferential equations. Sequential Liouville–Caputo derivatives and nonlinearities comprising both integral and differential terms are present in Equation (1). Additionally, the system is supplemented with Riemann–Liouville integral boundary conditions (2). It is noted that under these circumstances, the unknown functions $\mathcal{G}$ and $\mathcal{J}$ at point 1 exhibit dependence on both $\mathcal{G}$ and $\mathcal{J}$ across the entire interval $[0,1]$. The Leray–Schauder alternative theorem and the Banach contraction mapping principle were utilised in the demonstration of our primary theorems. The results obtained in our study corresponded to specific problems when the parameters $\mathfrak{v},\mathfrak{z}$ were established. Suppose that Problems (1) and (2) are presented in a manner that the results are obtained by setting $\mathfrak{v}=\mathfrak{z}=1$.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathcal{G}\left(0\right)=0,{\mathcal{G}}^{\prime }\left(0\right)=0,{\mathcal{G}}^{\prime }\left(1\right)=0,\mathcal{G}\left(1\right)={\int }_0^{\eta }\mathcal{G}\left(\upsilon \right)d\upsilon +{\int }_0^{\zeta }\mathcal{J}\left(\upsilon \right)d\upsilon ,}\hfill \\ {\displaystyle \mathcal{J}\left(0\right)=0,{\mathcal{J}}^{\prime }\left(0\right)=0,{\mathcal{J}}^{\prime }\left(1\right)=0,\mathcal{J}\left(1\right)={\int }_0^{\eta }\mathcal{G}\left(\upsilon \right)d\upsilon +{\int }_0^{\zeta }\mathcal{J}\left(\upsilon \right)d\upsilon .}\hfill \end{array}\right.\end{array}$$

The problems mentioned above can be solved using the same approach as Problems (1) and (2), as described in the preceding section.