A Qualitative Analysis of a Non-Linear Coupled System under Two Types of Fractional Derivatives along with Mixed Boundary Conditions

Abstract

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications.

1. Introduction

Fractional calculus is rapidly expanding, with its appealing applications being extensively utilized in various domains of applied science. For example, in [1], the authors provided many illustrations and theoretical applications of fractional differential equation (FDE) theory, as well as Weyl fractional calculus and its use. Some physical problems with fractional differential or integral equations are included. The authors in [2] examined fractional integral theory and its applications. Comparisons are made among all fractional integro-differentiation types. Fractional calculus is applied to first-order integral equations with power and power logarithmic kernels, particularly functions in kernels, Euler–Poisson–Darboux-type equations, and FDEs. The authors in [3] studied fractional derivatives and fractional-order mathematical models more adequately. In [4], the authors studied Cauchy-type problems involving fractional derivatives and a theory of sequential linear fractional differential equations, including a generalization of the classical Frobenius method and an exciting set of applications. The presence of fractional calculus can also be observed in financial systems [5], optimal control [6], epidemiological models [7], chaotic systems [8], and engineering [9]. The researchers in [10] studied the time-fractional generalized Boussinesq equation, focusing on Rossby solitary waves in a stratified fluid with a dissipation effect. They also explored the conservation laws associated with this equation and examined the existence of exact solutions. The Langevin equation within the framework of the generalized proportional fractional derivative was investigated in [11]. Fractional configurations of boundary value problems also provide a wide range of mathematical models to describe physical, chemical, and biological processes, as evidenced by recent publications. The study conducted by the authors of [12] aimed to evaluate and solve the model of nonlinear FDEs that describe the most common and deadly coronavirus, which is COVID-19, and in [13], the authors extended the thermostat model’s second-order differential equation to the fractional hybrid equation and inclusion versions. In addition to these models that reflect real-world phenomena, numerous researchers have focused on exploring the existence and uniqueness theories of solutions for a general coupled system of fractional differential problems (FDPs) equipped with boundary conditions under local and non-local terms. In [14], linear FDE systems were examined, and an operational matrix of the Haar wavelet method converted it into algebraic equations. A novel structure of the generalized multi-point thermostat control model based on its standard model was developed in [15,16].

To provide a more comprehensive understanding of the current state of research in fractional differential equations, this study delves deeper into the comparative analysis of different methodologies applied to solve fractional differential problems. Traditional integer-order differential equations have been widely studied and utilized across various scientific and engineering fields due to their simplicity and well-established analytical methods. However, these traditional models often fall short when describing memory and hereditary properties inherent in many real-world processes. Fractional differential equations (FDEs), which generalize integer-order equations to non-integer order ones, offer a more accurate representation of such complex systems.

Recently, several methods have been developed and refined to solve FDEs, including the Caputo and Liouville–Riemann (LR) fractional derivatives, each offering unique advantages depending on the problem’s nature. The Caputo derivative is particularly useful for initial value problems, as it allows for the inclusion of standard initial conditions, making it more applicable to physical problems. On the other hand, the LR derivative is more suited for problems where boundary conditions are specified at different points, providing broader applicability in various scientific fields.

In recent work, Sina et al. [17] undertook a study involving the construction of the following coupled sequential Navier fractional differential boundary value problem:

$$\left\{\begin{array}{c}{ }^C{D}^{{\mu }_1}\left({ }^C{D}^{{\mu }_2}{\phi }_1\right)\left(\nu \right)=G\left(\nu ,{\phi }_2\left(\nu \right),{ }^C{D}^{{\mu }_2}{\phi }_2\left(\nu \right)\right),\nu \in [0,1],\hfill \\ { }^C{D}^{{\mu }_1^{\ast }}\left({ }^C{D}^{{\mu }_2^{\ast }}{\phi }_2\right)\left(\nu \right)=H\left(\nu ,{\phi }_1\left(\nu \right),{ }^C{D}^{{\mu }_2^{\ast }}{\phi }_1\left(\nu \right)\right),\nu \in [0,1],\hfill \\ \gamma {\phi }_1\left(0\right)=\delta {\phi }_1\left(1\right)=\alpha { }^C{D}^{{\mu }_2}{\phi }_1\left(0\right)=\beta { }^C{D}^{{\mu }_2}{\phi }_1\left(1\right)=0,\hfill \\ {\gamma }^{\ast }{\phi }_2\left(0\right)={\delta }^{\ast }{\phi }_2\left(1\right)={\alpha }^{\ast }{ }^C{D}^{{\mu }_2^{\ast }}{\phi }_2\left(0\right)={\beta }^{\ast }{ }^C{D}^{{\mu }_2^{\ast }}{\phi }_2\left(1\right)=0,\hfill \end{array}\right.$$

(1)

such that ${\mu }_1,{\mu }_1^{\ast }\in (1,2]$, and $\delta ,\gamma ,\beta ,\alpha ,{\gamma }^{\ast },{\alpha }^{\ast },{\delta }^{\ast },{\beta }^{\ast }\in {\mathbb{R}}_{+}$. In addition, the operator ${ }^C{D}^{(·)}$ is used to compute derivatives of fractional orders according to the Caputo definition. Additionally, they considered continuous single-valued functions G and H, defined on $[0,1]\times {\mathbb{R}}^2$, and used values in $\mathbb{R}$.

Thabet et al. [18] introduced a novel formulation, marking the first instance of a coupled system of pantograph FDPs employing the conformable Caputo operators defined as follows:

$$\left\{\begin{array}{c}{ }^{CC}{D}_{{\nu }_0}^{\mu ,{\xi }_1}\varphi \left(\nu \right)={\omega }_1(\nu ,\psi \left(\nu \right),\psi \left(\lambda \nu \right),\nu \in [{\nu }_0,Z],\hfill \\ { }^{CC}{D}_{{\nu }_0}^{\mu ,{\xi }_2}\psi \left(\nu \right)={\omega }_2(\nu ,\varphi \left(\nu \right),\varphi \left(\lambda \nu \right),\hfill \\ \varphi \left({\nu }_0\right)=0,a_1\varphi \left(Z\right)+a_2{ }^{RC}{I}_{{\nu }_0}^{\mu ,\theta }\varphi \left(c\right)=m_1,\hfill \\ \psi \left({\nu }_0\right)=0,b_1\psi \left(Z\right)+b_2{ }^{RC}{I}_{{\nu }_0}^{\mu ,\theta }\psi \left(d\right)=m_2,\hfill \end{array}\right.$$

(2)

where ${ }^{CC}{D}_{{\nu }_0}^{\mu ,{\xi }_i}$ stands for the conformable Caputo derivatives of order ${\xi }_i\in (1,2)$ with $\mu \in (0,1)$ for $i=1,2$; ${ }^{RC}{I}_{{\nu }_0}^{\mu ,\theta }$ denotes the Liouville–Riemann conformable integral of order $\theta >0$; ${\nu }_0<c,d<Z,$$a_1,a_2,b_1,b_2,m_1,m_2\in \mathbb{R}$; $0<\lambda <1$; and ${\omega }_i:[{\nu }_0,Z]\times {\mathbb{R}}^2⟶\mathbb{R}$ are continuous for $i=1,2.$

Inspired by the above works, this paper focuses on exploring the qualitative analysis of a new non-linear coupled system of FDPs under two symmetrical types in the Caputo and LR fractional derivative sense. Specifically, we delve into the formulation of a non-linear coupled fractional boundary value problem, which can be described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{ }^{LR}{D}_{0^{+}}^{{\varrho }_1}{\phi }_1\left(\nu \right)\hfill & =\widehat{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right)),\nu \in [0,1],\hfill \\ { }^C{D}_{0^{+}}^{{\varrho }_2}{\phi }_2\left(\nu \right)\hfill & =\check{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(\nu \right)),\nu \in [0,1],\hfill \end{array}\right.\end{array}$$

(3)

subjected to the following mixed boundary conditions:

$$\left\{\begin{array}{c}{\displaystyle {\phi }_1\left(0\right)=0,{\phi }_1\left(1\right)={\epsilon }_1{ }^{LR}{D}_{0^{+}}^{{\sigma }_1}{\phi }_2\left({\xi }_1\right)+{\epsilon }_2{ }^{LR}{D}_{0^{+}}^{{\sigma }_2}{\phi }_2\left({\xi }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_1\left(\nu \right)d\left(\nu \right),{\epsilon }_1,{\epsilon }_2\in \mathbb{R},}\hfill \\ {\displaystyle {\phi }_2\left(1\right)={\mathfrak{q}}_1{I}_{0^{+}}^{{\mu }_1}{\phi }_1\left({\eta }_1\right)+{\mathfrak{q}}_2{I}_{0^{+}}^{{\mu }_2}{\phi }_1\left({\eta }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_2\left(\nu \right)d\left(\nu \right),{\mathfrak{q}}_1,{\mathfrak{q}}_2\in \mathbb{R},}\hfill \end{array}\right.$$

(4)

where ${ }^{LR}{D}_{0^{+}}^{\theta }$ and ${ }^C{D}_{0^{+}}^{\theta }$ denote the standard LR and Caputo fractional derivatives of order $\theta$, respectively. Also, $1<{\varrho }_1\le 2$, $0<{\varrho }_2<1$, $0<{\sigma }_i,{\omega }_i<{\varrho }_i$, for $i\in \{1,2\}$, $\psi :(0,1)\to [0,\infty )$ is a bounded function; $\widehat{\ell },\check{\ell }\in C\left([0,1]\times {\mathbb{R}}^3,\mathbb{R}\right)$; $I_{0^{+}}^{{\mu }_i}$ is the LR-fractional integral with order ${\mu }_i>0$; and $0<{\xi }_1,{\xi }_2,{\eta }_1,{\eta }_2<1$.

The study of Ulam–Hyers stability is recent, and it is related to an exchange of questions and answers between Ulam and Hyers. At first, the research was limited to the stability of solutions of functional differential equations. With the advent of fractional calculus, researchers began to investigate more and more problems involving the stability of solutions of fractional functional differential equations in this direction. Here, we declare that our proposed non-linear coupled system of (3) and (4) is new in the framework of symmetrical kernels LR and Caputo fractional operators supplemented by mixed boundary conditions. Additionally, this system covers many new existing problems in the literature. One of the main goals is to study Ulam–Hyers stability.

This paper is organized as follows. In Section 2, we recall the essential preliminaries by presenting basic definitions and an auxiliary proposition. Section 3 is dedicated to the presentation of the existence and uniqueness results for the problem under investigation. Section 4 discusses Ulam–Hyers-type stability. To further illustrate these results, we discuss a series of illustrative examples in Section 5. Finally, we summarize the main steps of our work with a conclusion.

2. Some Useful Definitions and Basic Tools

All fundamental definitions and necessary notations will be presented in this section. Let the Banach space Z be defined as $(Z,\parallel .\parallel )$, where the norm $\parallel w\parallel$ is given by ${max}_{\nu \in [0,1]}\left|w\right|$. Consequently, the space $U=Z\times Z$ is also a Banach space, with the norms denoted by

$$\parallel (w_1,w_2)\parallel =\parallel w_1\parallel +\parallel w_2\parallel .$$

Definition 1

([1,3,4]). Given the function $\chi :(0,\infty )\to \mathbb{R}$, we have the following:

• The LR-fractional integral of order ℓ$(\ell >0)$ is given by

  $$I^{\ell }\chi \left(\nu \right)={\int }_0^{\nu }\frac{{(\nu -\kappa )}^{\ell -1}}{\Gamma (\ell )}\chi \left(\kappa \right)d\kappa .$$
• The LR-fractional derivative of a function χ of order ℓ is expressed as

  $${ }^{LR}{D}^{\ell }\chi \left(\nu \right)=\frac{1}{\Gamma (n-\ell )}{\left(\frac{d}{d\nu }\right)}^n{\int }_0^{\nu }\frac{\chi \left(\kappa \right)}{{(\nu -\kappa )}^{\ell -n+1}}d\kappa ,$$

  with $n-1<\ell \le n.$
• For any given $T>0$ and any continuous function χ on $[0,T]$, the following property holds:

  $$I^{\ell }{ }^{LR}{D}^{\ell }\chi \left(\nu \right)=\chi \left(\nu \right)+{\widehat{p}}_1{\nu }^{\ell -1}+{\widehat{p}}_2{\nu }^{\ell -2}+\cdots +{\widehat{p}}_n{\nu }^{\ell -n},$$

  where ${\widehat{p}}_k\in \mathbb{R}$ and $n-1<\ell <n,k=1,2,\dots ,n.$
• If the function $\chi \left(\nu \right)\in C_{\mathbb{R}}^n[0,+\infty )$, then the Caputo derivative of order ℓ is given by

  $${ }^C{D}^{\ell }\chi \left(\nu \right)=\frac{1}{\Gamma (n-\ell )}{\int }_0^{\nu }\frac{{\chi }^{\left(n\right)}\left(\kappa \right)}{{(\nu -\kappa )}^{\ell +1-n}}d\kappa =I^{n-\ell }{\chi }^{\left(n\right)}\left(\nu \right)$$

  for $\nu >0$ and $n-1<\ell <n.$
• Letting $n-1<\ell <n$, we have

  $${ }^C{D}^{\ell }\chi \left(\nu \right)={ }^{LR}{D}^{\ell }\left(\chi \left(\nu \right)+{\widehat{c}}_0+{\widehat{c}}_1{\nu }^1+\cdots +{\widehat{c}}_{n-1}{\nu }^{n-1}\right),$$

  (5)

   where ${\widehat{c}}_i\in \mathbb{R},i=0,2,\dots ,n-1.$

Definition 2

([19]). For the operators $S_1$, $S_2:U\to Z,$ the system of operator equations, such that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phi }_1\left(\nu \right)\hfill & =S_1({\phi }_1,{\phi }_2)\left(\nu \right),\hfill \\ {\phi }_2\left(\nu \right)\hfill & =S_2({\phi }_1,{\phi }_2)\left(\nu \right),\hfill \end{array}\right.\end{array}$$

(6)

is Ulam–Hyers-type stable if $a_i>0$, $(i=1,2,3,4)$, ${\Delta }_i>0$, $(i=1,2)$ are real numbers, and, for every $(\overline{{\phi }_1},\overline{{\phi }_2})\in U,$ we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\parallel {\phi }_1-S_1(\overline{{\phi }_1},\overline{{\phi }_2})\parallel \le {\Delta }_1,\hfill \\ \parallel {\phi }_2-S_2(\overline{{\phi }_1},\overline{{\phi }_2})\parallel \le {\Delta }_2,\hfill \end{array}\right.\end{array}$$

(7)

$\exists (\overline{{\phi }_1},\overline{{\phi }_2})\in U$ of (6), where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\parallel {\phi }_1-\overline{{\phi }_1}\parallel \le a_1{\Delta }_1+a_2{\Delta }_2,\hfill \\ \parallel {\phi }_2-\overline{{\phi }_2}\parallel \le a_3{\Delta }_1+a_4{\Delta }_2.\hfill \end{array}\right.\end{array}$$

(8)

Definition 3

([19]). Let ${\alpha }_i$$(i=1,2,3,\dots ,n)$ be eigenvalues of an ($n\times n$) matrix $\mathbb{M}$, with spectral radius of ${\rm Y}\left(\mathbb{M}\right)$ defined by

$$\begin{array}{c}\hfill {\rm Y}\left(\mathbb{M}\right)=max\left\{\right|{\alpha }_i|,\mathrm{for}i=1,2,\dots ,n\}.\end{array}$$

(9)

In addition, if ${\rm Y}\left(\mathbb{M}\right)<1$, then $\mathbb{M}$ converges to 0.

Theorem 1

([19]). For the two operators $S_1,S_2:U\to Z$

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\parallel S_1({\phi }_1,{\phi }_2)-S_1(\overline{{\phi }_1},\overline{{\phi }_2})\parallel \le a_1\parallel {\phi }_1-\overline{{\phi }_1}\parallel +a_2\parallel {\phi }_2-\overline{{\phi }_2}\parallel ,\hfill \\ \parallel S_2({\phi }_1,{\phi }_2)-S_2(\overline{{\phi }_1},\overline{{\phi }_2})\parallel \le a_3\parallel {\phi }_1-\overline{{\phi }_1}\parallel +a_4\parallel {\phi }_2-\overline{{\phi }_2}\parallel ,\hfill \\ & \forall ({\phi }_1,{\phi }_2)(\overline{{\phi }_1},\overline{{\phi }_2})\in U,\hfill \end{array}\right.\end{array}$$

(10)

and if the matrix

$$\mathbb{M}=\left[\begin{array}{cc}{a}_1& a_1\\ a_3& a_4\end{array}\right],$$

goes to zero, consequently, the solutions of the system described by Equation (6) are then determined to exhibit Ulam–Hyers-type stability.

Theorem 2

([20]). (Nonlinear alternative of Leary–Schauder type.)

Let E be a closed convex subset of Banach space U. Let A be an open subset of E with $0\in A$, and let $F:\overline{A}\to E$ be a continuous and compact mapping.

In this case, either:

(1) 

F has a fixed point in $\overline{A}$, or

(2) 

$\exists x\in \partial A$ and $\exists \kappa \in (0,1)$ s.t. $x=\kappa Fx$.

Lemma 1.

Let $\mathit{h}:[0,1]\to \mathbb{R}$. Then, $({\phi }_1,{\phi }_2)$ satisfies the linear version of the system (3) and (4), which is given by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{ }^{LR}{D}_{0^{+}}^{{\varrho }_1}{\phi }_1\left(\nu \right)=\widehat{\mathit{h}}\left(\nu \right),\nu \in [0,1],\hfill \\ { }^C{D}_{0^{+}}^{{\varrho }_2}{\phi }_2\left(\nu \right)=\check{\mathit{h}}\left(\nu \right),\nu \in [0,1],\hfill \\ {\displaystyle {\phi }_1\left(0\right)=0,{\phi }_1\left(1\right)={\epsilon }_1{ }^{LR}{D}_{0^{+}}^{{\sigma }_1}{\phi }_2\left({\xi }_1\right)+{\epsilon }_2{ }^{LR}{D}_{0^{+}}^{{\sigma }_2}{\phi }_2\left({\xi }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_1\left(\nu \right)d\left(\nu \right),{\epsilon }_1,{\epsilon }_2\in \mathbb{R},}\hfill \\ {\displaystyle {\phi }_2\left(1\right)={\mathfrak{q}}_1{I}_{0^{+}}^{{\mu }_1}{\phi }_1\left({\eta }_1\right)+{\mathfrak{q}}_2{I}_{0^{+}}^{{\mu }_2}{\phi }_1\left({\eta }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_2\left(\nu \right)d\left(\nu \right),{\mathfrak{q}}_1,{\mathfrak{q}}_2\in \mathbb{R},}\hfill \end{array}\right.\end{array}$$

(11)

if and only if

$$\begin{array}{cc}\hfill {\phi }_1\left(\nu \right)& =I_{0^{+}}^{{\varrho }_1}\widehat{\mathit{h}}\left(\nu \right)+\frac{{\nu }^{{\varrho }_1-1}}{b_3}[{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathit{h}}\left({\xi }_1\right)+{\epsilon }_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathit{h}}\left({\xi }_2\right)+{\mathfrak{q}}_1{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathit{h}}\left({\eta }_1\right)\hfill \\ \hfill +& {\mathfrak{q}}_2{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathit{h}}\left({\eta }_2\right)-I_{0^{+}}^{{\varrho }_1}\widehat{\mathit{h}}\left(1\right)-b_1{I}_{0^{+}}^{\varrho 2}\check{\mathit{h}}\left(1\right)]+\frac{{\nu }^{{\varrho }_1-1}}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_1\left(\nu \right)+b_1{\phi }_2\left(\nu \right))d\left(\nu \right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\phi }_2\left(\nu \right)& =I_{0^{+}}^{{\varrho }_2}\check{\mathit{h}}\left(\nu \right)+\frac{1}{b_3}[{\mathfrak{q}}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathit{h}}\left({\eta }_1\right)+{\mathfrak{q}}_2{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathit{h}}\left({\eta }_2\right)+{\epsilon }_1{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathit{h}}\left({\xi }_1\right)\hfill \\ & +{\epsilon }_2{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathit{h}}\left({\xi }_2\right)-b_2{I}_{0^{+}}^{{\varrho }_1}\widehat{\mathit{h}}\left(1\right)-I_{0^{+}}^{\varrho 2}\check{\mathit{h}}\left(1\right)]+\frac{1}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_2\left(\nu \right)+b_2{\phi }_1)d\left(\nu \right),\hfill \end{array}$$

(13)

where

$$\begin{array}{cc}\hfill b_1& =\frac{{\epsilon }_1{\xi }_1^{-{\sigma }_1}}{1-{\sigma }_1}+\frac{{\epsilon }_2{\xi }_2^{-{\sigma }_2}}{1-{\sigma }_2}.\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill b_2& =\frac{{\mathfrak{q}}_1\Gamma \left({\mu }_1\right)}{\Gamma ({\varrho }_1+{\mu }_1)}{\eta }_1^{{\varrho }_1+{\mu }_1-1}+\frac{{\mathfrak{q}}_2\Gamma \left({\mu }_2\right)}{\Gamma ({\varrho }_1+{\mu }_2)}{\eta }_2^{{\varrho }_1+{\mu }_2-1}.\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill b_3& =1-b_1{b}_2\ne 0.\hfill \end{array}$$

(16)

Proof. 

The first implication is that (11) implies (12) and (13).

Let us consider that $({\phi }_1,{\phi }_2)$ represents a solution to the system defined by Equation (11). Then, we have the following:

$$\begin{array}{cc}\hfill & { }^{LR}{D}_{0^{+}}^{{\varrho }_1}{\phi }_1\left(\nu \right)=\widehat{\mathbf{h}}\left(\nu \right),\hfill \end{array}$$

(17)

and

$$\begin{array}{cc}\hfill & { }^C{D}_{0^{+}}^{{\varrho }_2}{\phi }_2\left(\nu \right)=\check{\mathbf{h}}\left(\nu \right).\hfill \end{array}$$

(18)

By taking the Riemann–Liouville fractional integrals $I_{0^{+}}^{{\varrho }_1}$ and $I_{0^{+}}^{{\varrho }_2}$ from both sides of Equations (17) and (18), respectively, we obtain

$$\begin{array}{cc}\hfill {\phi }_1\left(\nu \right)& =I_{0^{+}}^{\varrho 1}\widehat{\mathbf{h}}\left(\nu \right)+c_1{\nu }^{{\varrho }_1-1}+c_2{\nu }^{{\varrho }_1-2},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\phi }_2\left(\nu \right)& =I_{0^{+}}^{\varrho 2}\check{\mathbf{h}}\left(\nu \right)+c_3,\hfill \end{array}$$

(20)

where $c_1$, $c_2$, and $c_3$ are real constants. Making use of the condition ${\phi }_1\left(0\right)=0$ in (19), we obtain $c_2=0$. Accordingly, Equation (19) becomes the following:

$$\begin{array}{cc}\hfill {\phi }_1\left(\nu \right)& =I_{0^{+}}^{\varrho 1}\widehat{\mathbf{h}}\left(\nu \right)+c_1{\nu }^{{\varrho }_1-1}.\hfill \end{array}$$

(21)

By using the Riemann–Liouville fractional derivative and integral of order $n_1^{\ast }$ and $n_2^{\ast }$, respectively, such that $n_1^{\ast }\in \{{\sigma }_1,{\sigma }_2\}$ and $n_2^{\ast }\in \{{\mu }_1,{\mu }_2\}$ in Equations (20) and (21), we obtain

$$\begin{array}{c}\hfill { }^{LR}{D}_{0^{+}}^{n_1^{\ast }}{\phi }_2\left(\nu \right)=I_{0^{+}}^{{\varrho }_2-n_1^{\ast }}\check{\mathbf{h}}\left(\nu \right)+c_3\frac{{\nu }^{-n_1^{\ast }}}{1-n_1^{\ast }},\mathrm{with}1-n_1^{\ast }\ne 0\end{array}$$

(22)

and

$$\begin{array}{c}\hfill I_{0^{+}}^{n_2^{\ast }}{\phi }_1\left(\nu \right)=I_{0^{+}}^{{\varrho }_1+n_2^{\ast }}\widehat{\mathbf{h}}\left(\nu \right)+c_1\frac{\Gamma \left(n_2^{\ast }\right)}{\Gamma ({\varrho }_1+n_2^{\ast })}{\nu }^{{\varrho }_1+n_2^{\ast }-1}.\end{array}$$

(23)

By replacing the values $n_1^{\ast }={\sigma }_1$, $n_1^{\ast }={\sigma }_2$, $n_2^{\ast }={\mu }_1$, and $n_2^{\ast }={\mu }_2$ and using the condition

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\phi }_1\left(1\right)={\epsilon }_1{ }^C{D}_{0^{+}}^{{\sigma }_1}{\phi }_2\left({\xi }_1\right)+{\epsilon }_2{ }^C{D}_{0^{+}}^{{\sigma }_2}{\phi }_2\left({\xi }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_1\left(\nu \right)d\left(\nu \right),}\hfill \\ {\displaystyle {\phi }_2\left(1\right)={\mathfrak{q}}_1{I}_{0^{+}}^{{\mu }_1}{\phi }_1\left({\eta }_1\right)+{\mathfrak{q}}_2{I}_{0^{+}}^{{\mu }_2}{\phi }_1\left({\eta }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_2\left(\nu \right)d\left(\nu \right),}\hfill \end{array}\right.\end{array}$$

(24)

we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{I}_{0^{+}}^{\varrho 1}\widehat{\mathbf{h}}\left(\nu \right)+c_1\hfill & ={\epsilon }_1\left(I_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathbf{h}}\left({\xi }_1\right)+c_3\frac{{\xi }_1^{-{\sigma }_1}}{1-{\sigma }_1}\right)+{\epsilon }_2\left(I_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathbf{h}}\left({\xi }_2\right)+c_3\frac{{\xi }_2^{-{\sigma }_2}}{1-{\sigma }_2}\right)\hfill \\ & {\displaystyle +{\int }_0^1\psi \left(\nu \right){\phi }_1\left(\nu \right)d\left(\nu \right),}\hfill \\ I_{0^{+}}^{\varrho 2}\check{\mathbf{h}}\left(1\right)+c_3\hfill & ={\mathfrak{q}}_1\left(I_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathbf{h}}\left({\mu }_1\right)+c_1\frac{\Gamma \left({\mu }_1\right){\mu }_1^{{\varrho }_1+{\mu }_1-1}}{\Gamma ({\varrho }_1+{\mu }_1)}\right)+{\mathfrak{q}}_2\left(I_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathbf{h}}\left({\mu }_2\right)+c_1\frac{\Gamma \left({\mu }_2\right){\mu }_2^{{\varrho }_1+{\mu }_2-1}}{\Gamma ({\varrho }_1+{\mu }_2)}\right)\hfill \\ & {\displaystyle +{\int }_0^1\psi \left(\nu \right){\phi }_2\left(\nu \right)d\left(\nu \right).}\hfill \end{array}\right.\end{array}$$

(25)

After calculation and simplification, we find

$$\begin{array}{c}\hfill c_1-c_3{b}_1={\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathbf{h}}\left({\xi }_1\right)+{\epsilon }_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathbf{h}}\left({\xi }_2\right)-I_{0^{+}}^{{\varrho }_1}\widehat{\mathbf{h}}\left(1\right)+{\int }_0^1\psi \left(\nu \right){\phi }_1\left(\nu \right)d\left(\nu \right)\end{array}$$

(26)

and

$$\begin{array}{c}\hfill c_3-c_1{b}_2={\mathfrak{q}}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathbf{h}}\left({\eta }_1\right)+{\mathfrak{q}}_2{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathbf{h}}\left({\eta }_2\right)-I_{0^{+}}^{\varrho 2}\check{\mathbf{h}}\left(1\right)+{\int }_0^1\psi \left(\nu \right){\phi }_2\left(\nu \right)d\left(\nu \right),\end{array}$$

(27)

where $b_1$ and $b_2$ are defined by (14) and (15), respectively. This implies that

$$\begin{array}{cc}\hfill c_1=& \frac{1}{b_3}[b_1{\mathfrak{q}}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathbf{h}}\left({\eta }_1\right)+b_1{\mathfrak{q}}_2{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathbf{h}}\left({\eta }_2\right)-b_1{I}_{0^{+}}^{\varrho 2}\check{\mathbf{h}}\left(1\right)+{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathbf{h}}\left({\xi }_1\right)\hfill \\ & +{\epsilon }_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathbf{h}}\left({\xi }_2\right)-I_{0^{+}}^{{\varrho }_1}\widehat{\mathbf{h}}\left(1\right)]+\frac{1}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_1\left(\nu \right)+b_1{\phi }_2\left(\nu \right))d\left(\nu \right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill c_3=& \frac{1}{b_3}[{\mathfrak{q}}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathbf{h}}\left({\eta }_1\right)+{\mathfrak{q}}_2{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathbf{h}}\left({\eta }_2\right)+{\epsilon }_1{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathbf{h}}\left({\xi }_1\right)+{\epsilon }_2{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathbf{h}}\left({\xi }_2\right)\hfill \\ & -b_2{I}_{0^{+}}^{{\varrho }_1}\widehat{\mathbf{h}}\left(1\right)-I_{0^{+}}^{\varrho 2}\check{\mathbf{h}}\left(1\right)]+\frac{1}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_2\left(\nu \right)+b_2{\phi }_1\left(\nu \right))d\left(\nu \right),\hfill \end{array}$$

(29)

with $b_3\ne 0$ defined by (16). Upon substituting the given values of constants $c_1$ and $c_3$ into Equations (19) and (20), we obtain Equations (12) and (13).

The reverse implication is that (12) and (13) imply (11), and then we have

$$\begin{array}{cc}\hfill {\phi }_1\left(\nu \right)& =I_{0^{+}}^{{\varrho }_1}\widehat{\mathbf{h}}\left(\nu \right)+\frac{{\nu }^{{\varrho }_1-1}}{b_3}[{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathbf{h}}\left({\xi }_1\right)+{\epsilon }_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathbf{h}}\left({\xi }_2\right)+{\mathfrak{q}}_1{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathbf{h}}\left({\eta }_1\right)\hfill \\ \hfill +& {\mathfrak{q}}_2{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathbf{h}}\left({\eta }_2\right)-I_{0^{+}}^{{\varrho }_1}\widehat{\mathbf{h}}\left(1\right)-b_1{I}_{0^{+}}^{\varrho 2}\check{\mathbf{h}}\left(1\right)]+\frac{{\nu }^{{\varrho }_1-1}}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_1\left(\nu \right)+b_1{\phi }_2\left(\nu \right))d\left(\nu \right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\phi }_2\left(\nu \right)& =I_{0^{+}}^{{\varrho }_2}\check{\mathbf{h}}\left(\nu \right)+\frac{1}{b_3}[{\mathfrak{q}}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathbf{h}}\left({\eta }_1\right)+{\mathfrak{q}}_2{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathbf{h}}\left({\eta }_2\right)+{\epsilon }_1{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathbf{h}}\left({\xi }_1\right),\hfill \\ & +{\epsilon }_2{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathbf{h}}\left({\xi }_2\right)-b_2{I}_{0^{+}}^{{\varrho }_1}\widehat{\mathbf{h}}\left(1\right)-I_{0^{+}}^{\varrho 2}\check{\mathbf{h}}\left(1\right)]+\frac{1}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_2\left(\nu \right)+b_2{\phi }_1)d\left(\nu \right).\hfill \end{array}$$

(31)

By using the Riemann–Liouville fractional derivative and Caputo fractional derivative of order ${\varrho }_1$ and ${\varrho }_2$ on Equations (12) and (13), respectively, we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{ }^{LR}{D}_{0^{+}}^{{\varrho }_1}{\phi }_1\left(\nu \right)=\widehat{\mathbf{h}}\left(\nu \right),\hfill \\ { }^C{D}_{0^{+}}^{{\varrho }_2}{\phi }_2\left(\nu \right)=\check{\mathbf{h}}\left(\nu \right).\hfill \end{array}\right.\end{array}$$

(32)

Now, we verify that the integral equations satisfy the three conditions.

• From Equation (12), we find ${\phi }_1\left(0\right)=0$.
• For the second condition

  $$\begin{array}{cc}\hfill {\phi }_1\left(1\right)& {\displaystyle ={\epsilon }_1{ }^C{D}_{0^{+}}^{{\sigma }_1}{\phi }_2\left({\xi }_1\right)+{\epsilon }_2{ }^C{D}_{0^{+}}^{{\sigma }_2}{\phi }_2\left({\xi }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_1\left(\nu \right)d\left(\nu \right),}\hfill \end{array}$$

  (33)

from Equation (12), we have

$$\begin{array}{cc}\hfill {\phi }_1\left(1\right)& =I_{0^{+}}^{{\varrho }_1}\widehat{\mathbf{h}}\left(1\right)+\frac{1}{b_3}[{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathbf{h}}\left({\xi }_1\right)+{\epsilon }_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathbf{h}}\left({\xi }_2\right)+{\mathfrak{q}}_1{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathbf{h}}\left({\eta }_1\right)\hfill \\ \hfill +& {\mathfrak{q}}_2{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathbf{h}}\left({\eta }_2\right)-I_{0^{+}}^{{\varrho }_1}\widehat{\mathbf{h}}\left(1\right)-b_1{I}_{0^{+}}^{\varrho 2}\check{\mathbf{h}}\left(1\right)]+\frac{1}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_1\left(\nu \right)+b_1{\phi }_2\left(\nu \right))d\left(\nu \right).\hfill \end{array}$$

(34)

On the other hand, by using the Riemann–Liouville fractional derivative of order $n_1^{\ast }\in \{{\sigma }_1,{\sigma }_2\}$, we obtain

$$\begin{array}{cc}\hfill { }^{LR}{D}_{0^{+}}^{n_1^{\ast }}{\phi }_2\left(\nu \right)& =I_{0^{+}}^{{\varrho }_2-n_1^{\ast }}\check{\mathbf{h}}\left(\nu \right)+\frac{{\nu }^{-n_1^{\ast }}}{1-n_1^{\ast }}\frac{1}{b_3}[{\mathfrak{q}}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\mathbf{h}}\left({\eta }_1\right)+{\mathfrak{q}}_2{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\mathbf{h}}\left({\eta }_2\right)\hfill \\ & +{\epsilon }_1{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\mathbf{h}}\left({\xi }_1\right)+{\epsilon }_2{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\mathbf{h}}\left({\xi }_2\right)-b_2{I}_{0^{+}}^{{\varrho }_1}\widehat{\mathbf{h}}\left(1\right)-I_{0^{+}}^{\varrho 2}\check{\mathbf{h}}\left(1\right)]\hfill \\ & +\frac{{\nu }^{-n_1^{\ast }}}{1-{\sigma }_1}\frac{1}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_2\left(\nu \right)+b_2{\phi }_1)d\left(\nu \right).\hfill \end{array}$$

(35)

Substituting the value of ${ }^{LR}{D}_{0^{+}}^{{\sigma }_1}{\phi }_2\left({\xi }_1\right)$ and ${ }^{LR}{D}_{0^{+}}^{{\sigma }_2}{\phi }_2\left({\xi }_2\right)$ into the following equation,

$$\begin{array}{c}\hfill {\displaystyle {\epsilon }_1{ }^{LR}{D}_{0^{+}}^{{\sigma }_1}{\phi }_2\left({\xi }_1\right)+{\epsilon }_2{ }^{LR}{D}_{0^{+}}^{{\sigma }_2}{\phi }_2\left({\xi }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_1\left(\nu \right)d\left(\nu \right),}\end{array}$$

(36)

and simplifying Equation (36) and using equality $\frac{b_1{b}_2}{b_3}=\frac{1}{b_3}-1$, we obtain

$$\begin{array}{cc}\hfill {\epsilon }_1{ }^{LR}{D}_{0^{+}}^{{\sigma }_1}{\phi }_2\left({\xi }_1\right)+{\epsilon }_2{ }^{LR}{D}_{0^{+}}^{{\sigma }_2}& {\displaystyle {\phi }_2\left({\xi }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_1\left(\nu \right)d\left(\nu \right)={\phi }_1\left(1\right).}\hfill \end{array}$$

(37)

3.

Similarly, we obtain

$$\begin{array}{cc}\hfill {\displaystyle {\mathfrak{q}}_1{I}_{0^{+}}^{{\mu }_1}{\phi }_1\left({\eta }_1\right)+{\mathfrak{q}}_2{I}_{0^{+}}^{{\mu }_2}{\phi }_1\left({\eta }_2\right)+{\int }_0^1\psi \left(\nu \right){\phi }_2\left(\nu \right)d\left(\nu \right)=}& {\phi }_2\left(1\right).\hfill \end{array}$$

(38)

Hence, the proof is completed. □

The following assumptions are provided as prerequisites for our work:

$\left({\mathbb{V}}_1\right)$ For all ${\phi }_1,\overline{{\phi }_1},{\phi }_2,\overline{{\phi }_2}\in Z$ and for every $\nu \in [0,1]$, $\exists L_{\widehat{\ell }}$ and $L_{\check{\ell }}$, there are two strictly positive constants such that

$$|\widehat{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right))-\widehat{\ell }(\nu ,\overline{{\phi }_1}\left(\nu \right),\overline{{\phi }_2}\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left(\nu \right))|\le L_{\widehat{\ell }}\left(\right|{\phi }_1-\overline{{\phi }_1}|+2|{\phi }_2-\overline{{\phi }_2}\left|\right),$$

$$|\check{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right))-\check{\ell }(\nu ,\overline{{\phi }_1}\left(\nu \right),\overline{{\phi }_2}\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left(\nu \right))|\le L_{\check{\ell }}\left(2\right|{\phi }_1-\overline{{\phi }_1}|+|{\phi }_2-\overline{{\phi }_2}\left|\right).$$

$\left({\mathbb{V}}_2\right)$ For positive real number r and for $\nu \in [0,1]$: $\left|\psi \right|\le r.$

$\left({\mathbb{V}}_3\right)$ For positive real number s and for $\nu \in [0,1]$:

$$max\left\{{\int }_0^1\psi \left(\nu \right){\phi }_1\left(\nu \right)d\nu ,{\int }_0^1\psi \left(\nu \right){\phi }_2\left(\nu \right)d\nu \right\}\le s.$$

$\left({\mathbb{V}}_4\right)$ For some positive real numbers $c_{\widehat{\ell }}$, $d_{\widehat{\ell }}$, $m_{\widehat{\ell }}$, $c_{\check{\ell }}$, $d_{\check{\ell }}$, and $m_{\check{\ell }}$:

$$|\widehat{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right))|\le c_{\widehat{\ell }}|{\phi }_1|+d_{\widehat{\ell }}\left|{\phi }_2\right|+m_{\widehat{\ell }},$$

$$|\check{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right))|\le c_{\check{\ell }}|{\phi }_1|+d_{\check{\ell }}\left|{\phi }_2\right|+m_{\check{\ell }}.$$

Owing to Lemma 1, the solution to the system of (3) and (4) is given as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phi }_1\left(\nu \right)=\hfill & \widehat{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right))\hfill \\ \hfill & +\frac{{\nu }^{{\varrho }_1-1}}{b_3}[{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\ell }({\xi }_1,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_1\right))\hfill \\ \hfill & +{\epsilon }_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\ell }({\xi }_2,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_2\right))\hfill \\ & +{\mathfrak{q}}_1{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\ell }({\eta }_1,{\phi }_1\left({\eta }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_1\right))\hfill \\ \hfill & +{\mathfrak{q}}_2{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\ell }({\eta }_2,{\phi }_1\left({\eta }_2\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_2\right))\hfill \\ & -I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(1\right))\hfill \\ & -b_1{I}_{0^{+}}^{\varrho 2}\check{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(1\right))]\hfill \\ \hfill & +\frac{{\nu }^{{\varrho }_1-1}}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_1\left(\nu \right)+b_1{\phi }_2\left(\nu \right))d\left(\nu \right),\hfill \\ {\phi }_2\left(\nu \right)=\hfill & I_{0^{+}}^{{\varrho }_2}\check{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_1\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(\nu \right))\hfill \\ & +\frac{1}{b_3}[{\mathfrak{q}}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\ell }({\eta }_1,{\phi }_1\left({\eta }_1\right),{\phi }_2\left({\eta }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_1\right))\hfill \\ \hfill & +{\mathfrak{q}}_2{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\ell }({\eta }_2,{\phi }_1\left({\eta }_2\right),{\phi }_2\left({\eta }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_2\right))\hfill \\ & +{\epsilon }_1{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\ell }({\xi }_1,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_1\right))\hfill \\ \hfill & +{\epsilon }_2{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\ell }({\xi }_2,{\phi }_1\left({\xi }_2\right),{\phi }_2\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_2\right))\hfill \\ & -b_2{I}_{0^{+}}^{{\varrho }_1}\widehat{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left(1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(1\right))\hfill \\ & -I_{0^{+}}^{{\varrho }_2}\check{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left(1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(1\right))]\hfill \\ & +\frac{1}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_2\left(\nu \right)+b_2{\phi }_1)d\left(\nu \right).\hfill \end{array}\right.\end{array}$$

Let $K_1,K_2:U\to U$, such that

$$\begin{array}{cc}\hfill K_1({\phi }_1,{\phi }_2)& =I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right))\hfill \\ \hfill & +\frac{{\nu }^{{\varrho }_1-1}}{b_3}[{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\ell }({\xi }_1,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_1\right))\hfill \\ & +{\epsilon }_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\ell }({\xi }_2,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_2\right))\hfill \\ & +{\mathfrak{q}}_1{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\ell }({\eta }_1,{\phi }_1\left({\eta }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_1\right))\hfill \\ & +{\mathfrak{q}}_2{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\ell }({\eta }_2,{\phi }_1\left({\eta }_2\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_2\right))\hfill \\ & -I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(1\right))\hfill \\ & -b_1{I}_{0^{+}}^{\varrho 2}\check{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(1\right))]\hfill \\ & +\frac{{\nu }^{{\varrho }_1-1}}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_1\left(\nu \right)+b_1{\phi }_2\left(\nu \right))d\left(\nu \right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill K_2({\phi }_1,{\phi }_2)& =I_{0^{+}}^{{\varrho }_2}\check{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_1\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(\nu \right))\hfill \\ & +\frac{1}{b_3}[{\mathfrak{q}}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\widehat{\ell }({\eta }_1,{\phi }_1\left({\eta }_1\right),{\phi }_2\left({\eta }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_1\right))\hfill \\ & +{\mathfrak{q}}_2{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\widehat{\ell }({\eta }_2,{\phi }_1\left({\eta }_2\right),{\phi }_2\left({\eta }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_2\right))\hfill \\ \hfill & +{\epsilon }_1{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\ell }({\xi }_1,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_1\right))\hfill \\ & +{\epsilon }_2{b}_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\check{\ell }({\xi }_2,{\phi }_1\left({\xi }_2\right),{\phi }_2\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_2\right))\hfill \\ & -b_2{I}_{0^{+}}^{{\varrho }_1}\widehat{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left(1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(1\right))\hfill \\ & -I_{0^{+}}^{{\varrho }_2}\check{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left(1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(1\right))]\hfill \\ & +\frac{1}{b_3}{\int }_0^1\psi \left(\nu \right)({\phi }_2\left(\nu \right)+b_2{\phi }_1)d\left(\nu \right),\hfill \end{array}$$

and $K({\phi }_1,{\phi }_2)=(K_1({\phi }_1,{\phi }_2),K_2({\phi }_1,{\phi }_2))$. Hence, the solutions to (3) and (4) correspond to the fixed points of the mapping K.

3. Existence and Uniqueness of Solutions

To establish the existence and uniqueness results, we need to apply Lemma 1.

Theorem 3.

Under hypotheses ${\mathbb{V}}_3$ and ${\mathbb{V}}_4$, and if the functions $\widehat{\ell },\check{\ell }:[0,1]\times {\mathbb{R}}^3\to \mathbb{R}$ are continuous, then $K\left(U\right)\subset U$ is completely continuous.

Proof .

Let

$$\begin{array}{cc}\hfill {\mathring{G}}_1=& \frac{1}{b_3}\left(\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}+\frac{1}{\Gamma ({\varrho }_2+1)}\right)(c_{\check{\ell }}+d_{\check{\ell }})\hfill \\ & +\frac{1}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_1+1)}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}\right)(c_{\widehat{\ell }}+d_{\widehat{\ell }}),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\mathring{G}}_2=& \frac{1}{b_3}\left(\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}+\frac{1}{\Gamma ({\varrho }_2+1)}\right)m_{\check{\ell }}\hfill \\ & +\frac{1}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_1+1)}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}\right)m_{\widehat{\ell }}+\frac{s(b_1+1)}{b_3},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\mathring{G}}_3=& \frac{1}{b_3}[\left(\frac{{\mathfrak{q}}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2}{\Gamma ({\varrho }_1+{\mu }_2+1)}+\frac{b_2}{\Gamma ({\varrho }_1+1)}\right)(c_{\widehat{\ell }}+d_{\widehat{\ell }})\hfill \\ & +\left(\frac{b_3+1}{\Gamma ({\varrho }_2+1)}+\frac{{\epsilon }_1{b}_2}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2{b}_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}\right)(c_{\check{\ell }}+d_{\check{\ell }})],\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\mathring{G}}_4=& \frac{1}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_2+1)}+\frac{{\epsilon }_1{b}_2}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2{b}_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}\right)m_{\check{\ell }}\hfill \\ & +\frac{1}{b_3}\left(\frac{{\mathfrak{q}}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2}{\Gamma ({\varrho }_1+{\mu }_2+1)}+\frac{b_2}{\Gamma ({\varrho }_1+1)}\right)m_{\widehat{\ell }}+\frac{s(b_2+1)}{b_3}.\hfill \end{array}$$

(42)

We define B as a closed subset of U: $B=\{({\phi }_1,{\phi }_2)\in U:\parallel ({\phi }_1,{\phi }_2)\parallel \le r\},$ where

$$\begin{array}{c}\hfill r\ge \frac{{\mathring{G}}_2+{\mathring{G}}_4}{1-{\mathring{G}}_1-{\mathring{G}}_3}.\end{array}$$

(43)

For arbitrary $({\phi }_1,{\phi }_2)\in B$, we have

$$\begin{array}{cc}\hfill |K_1({\phi }_1,{\phi }_2)|& \le I_{0^{+}}^{{\varrho }_1} |\widehat{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right))|\hfill \\ & +\frac{{\nu }^{{\varrho }_1-1}}{|b_3|}[|{\epsilon }_1|I_{0^{+}}^{{\varrho }_2-{\sigma }_1}\left|\check{\ell }({\xi }_1,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_1\right))\right|\hfill \\ & +|{\epsilon }_2|I_{0^{+}}^{{\varrho }_2-{\sigma }_2}\left|\check{\ell }({\xi }_2,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_2\right))\right|\hfill \\ & +|{\mathfrak{q}}_1{b}_1|I_{0^{+}}^{{\varrho }_1+{\mu }_1}\left|\widehat{\ell }({\eta }_1,{\phi }_1\left({\eta }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_1\right))\right|\hfill \\ & +|{\mathfrak{q}}_2{b}_1|I_{0^{+}}^{{\varrho }_1+{\mu }_2}\left|\widehat{\ell }({\eta }_2,{\phi }_1\left({\eta }_2\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_2\right))\right|\hfill \\ & +I_{0^{+}}^{{\varrho }_1}\left|\widehat{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(1\right))\right|\hfill \\ & +|b_1|I_{0^{+}}^{\varrho 2}\left|\check{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(1\right))\right|]\hfill \\ & +\frac{{\nu }^{{\varrho }_1-1}}{|b_3|}{\int }_0^1\left|\psi \left(\nu \right)({\phi }_1\left(\nu \right)+b_1{\phi }_2\left(\nu \right))\right|d\left(\nu \right)\hfill \\ & \le \frac{1}{\Gamma ({\varrho }_1+1)}(c_{\widehat{\ell }}+d_{\widehat{\ell }})r+\frac{1}{\Gamma ({\varrho }_1+1)}{m}_{\widehat{\ell }}\hfill \\ & +\frac{r}{b_3}[\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}(c_{\check{\ell }}+d_{\check{\ell }})+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}(c_{\check{\ell }}+d_{\check{\ell }})\hfill \\ & +\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}(c_{\widehat{\ell }}+d_{\widehat{\ell }})+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}(c_{\widehat{\ell }}+d_{\widehat{\ell }})\hfill \\ & +\frac{1}{\Gamma ({\varrho }_1+1)}(c_{\widehat{\ell }}+d_{\widehat{\ell }})+\frac{1}{\Gamma ({\varrho }_2+1)}(c_{\check{\ell }}+d_{\check{\ell }})]\hfill \\ & +\frac{1}{b_3}[\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}{m}_{\check{\ell }}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}{m}_{\check{\ell }}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}{m}_{\widehat{\ell }}\hfill \\ & +\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}{m}_{\widehat{\ell }}+\frac{1}{\Gamma ({\varrho }_1+1)}{m}_{\widehat{\ell }}+\frac{1}{\Gamma ({\varrho }_2+1)}{m}_{\check{\ell }}]+\frac{s(b_1+1)}{b_3},\hfill \end{array}$$

which means that

$$\begin{array}{cc}\hfill |K_1({\phi }_1,{\phi }_2)|& \le {\mathring{G}}_{1}r+{\mathring{G}}_2,\hfill \end{array}$$

(44)

where ${\mathring{G}}_1$ and ${\mathring{G}}_2$ are defined in (39) and (40), respectively.

Similarly,

$$\begin{array}{cc}\hfill |K_2({\phi }_1,{\phi }_2)|& \le {\mathring{G}}_{3}r+{\mathring{G}}_4,\hfill \end{array}$$

(45)

where ${\mathring{G}}_3$ and ${\mathring{G}}_4$ are defined in (41) and (42), respectively.

Therefore, from (44), (45), and (43), one has

$$\begin{array}{cc}\hfill \left|K\right({\phi }_1,{\phi }_2\left)\right|& \le r.\hfill \end{array}$$

(46)

Thus, (46) implies that the matrix K is uniformly bounded over the set B.

In order to establish the complete continuity of $K:U\to U$, we consider ${\nu }_1$, ${\nu }_2\in [0,1]$, $0\le {\nu }_1\le {\nu }_2\le 1$, and, using $\left({\mathbb{V}}_3\right)$ and $\left({\mathbb{V}}_4\right)$, we have

$$\begin{array}{cc}\hfill & |K_1({\phi }_1\left({\nu }_2\right),{\phi }_2\left({\nu }_2\right))-K_1({\phi }_1\left({\nu }_1\right),{\phi }_2\left({\nu }_1\right))|\hfill \\ \hfill & \le \frac{1}{\Gamma \left({\varrho }_1\right)}{\int }_0^{{\nu }_1}{({\nu }_2-s)}^{{\varrho }_1-1}-{({\nu }_1-s)}^{{\varrho }_1-1}\left|\widehat{\ell }(s,{\phi }_1\left(s\right),{\phi }_2\left(s\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(s\right))\right|ds\hfill \\ & +\frac{1}{\Gamma \left({\varrho }_1\right)}{\int }_{{\nu }_1}^{{\nu }_2}{({\nu }_2-s)}^{\varrho -1}\left|\widehat{\ell }(s,{\phi }_1\left(s\right),{\phi }_2\left(s\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(s\right))\right|ds\hfill \\ & +\frac{{\nu }_2^{{\varrho }_1-1}-{\nu }_1^{{\varrho }_1-1}}{b_3}[|{\epsilon }_1|I_{0^{+}}^{{\varrho }_2-{\sigma }_1}\left|\check{\ell }({\xi }_1,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_1\right))\right|\hfill \\ & +|{\epsilon }_2|I_{0^{+}}^{{\varrho }_2-{\sigma }_2}\left|\check{\ell }({\xi }_2,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_2\right))\right|\hfill \\ & +|{\mathfrak{q}}_1{b}_1|I_{0^{+}}^{{\varrho }_1+{\mu }_1}\left|\widehat{\ell }({\eta }_1,{\phi }_1\left({\eta }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_1\right))\right|\hfill \\ & +|{\mathfrak{q}}_2{b}_1|I_{0^{+}}^{{\varrho }_1+{\mu }_2}\left|\widehat{\ell }({\eta }_2,{\phi }_1\left({\eta }_2\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_2\right))\right|\hfill \\ & +I_{0^{+}}^{{\varrho }_1}\left|\widehat{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(1\right))\right|\hfill \\ & +|b_1|I_{0^{+}}^{\varrho 2}\left|\check{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(1\right))\right|]+\frac{{\nu }_2^{{\varrho }_1-1}-{\nu }_1^{{\varrho }_1-1}}{b_3}s(b_1+1)\hfill \\ \hfill & \le \frac{1}{{\varrho }_1+1}\left(c_{\widehat{\ell }}r+d_{\widehat{\ell }}r+m_{\widehat{\ell }}\right)(({\nu }_2^{{\varrho }_1}-{\nu }_1^{{\varrho }_1})-2{({\nu }_2-{\nu }_1)}^{{\varrho }_1-1})\hfill \\ \hfill & +\frac{{\nu }_2^{{\varrho }_1-1}-{\nu }_1^{{\varrho }_1-1}}{b_3}\left\{r\right((\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}\hfill \\ \hfill & +\frac{1}{\Gamma ({\varrho }_2+1)})(c_{\check{\ell }}+d_{\check{\ell }})+(\frac{1}{\Gamma ({\varrho }_1+1)}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}\hfill \\ \hfill & +\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}\left)(c_{\widehat{\ell }}+d_{\widehat{\ell }})\right)(\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}\hfill \\ \hfill & +\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}+\frac{1}{\Gamma ({\varrho }_2+1)})m_{\check{\ell }}+(\frac{1}{\Gamma ({\varrho }_1+1)}\hfill \\ \hfill & +\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}\left)m_{\widehat{\ell }}\right\}+\frac{{\nu }_2^{{\varrho }_1-1}-{\nu }_1^{{\varrho }_1-1}}{b_3}s(b_1+1).\hfill \end{array}$$

(47)

Clearly, it can be observed that the right-hand side of inequality (47) converges to zero as ${\nu }_1$ approaches ${\nu }_2$. Additionally, the matrix $K_1$ is both bounded and continuous. Consequently, it can be deduced that $K_1$ is uniformly bounded.

$$\parallel K_1({\phi }_1\left({\nu }_1\right),{\phi }_2\left({\nu }_1\right))-K_1({\phi }_1\left({\nu }_2\right),{\phi }_2\left({\nu }_2\right))\parallel \to 0\mathrm{as}{\nu }_1\mathrm{tends}\mathrm{to}{\nu }_2.$$

Likewise, it is also possible to demonstrate using a similar approach that

$$\parallel K_2({\phi }_1\left({\nu }_1\right),{\phi }_2\left({\nu }_1\right))-K_2({\phi }_1\left({\nu }_2\right),{\phi }_2\left({\nu }_2\right))\parallel \to 0\mathrm{as}{\nu }_1\mathrm{tends}\mathrm{to}{\nu }_2.$$

As a result, the mapping $K:U\to U$ is determined to be equi-continuous. By applying the well known Arzela–Ascoli theorem, we conclude that K is a completely continuous mapping.   □

Theorem 4.

Under hypotheses $\left({\mathbb{V}}_1\right)-\left({\mathbb{V}}_4\right)$, and if $J<1$, then the system of (3) and (4) has a unique solution, with

$$\begin{array}{c}\hfill J=max\left\{J_1,J_2\right\},\end{array}$$

(48)

where

$$\begin{array}{cc}\hfill J_1=& \frac{2L_{\check{\ell }}}{b_3}\left(\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}+\frac{1}{\Gamma ({\varrho }_2+1)}\right)\hfill \\ & +\frac{2L_{\widehat{\ell }}}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_1+1)}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}\right)+\frac{r(1+b_1)}{|b_3|},\hfill \end{array}$$

(49)

and

$$\begin{array}{cc}\hfill J_2=& \frac{2L_{\check{\ell }}}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_2+1)}+\frac{{\epsilon }_1{b}_2}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2{b}_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}\right)\hfill \\ & +\frac{2L_{\widehat{\ell }}}{b_3}\left(\frac{{\mathfrak{q}}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2}{\Gamma ({\varrho }_1+{\mu }_2+1)}+\frac{b_2}{\Gamma ({\varrho }_1+1)}\right)+\frac{r(1+b_2)}{|b_3|}.\hfill \end{array}$$

(50)

Proof. 

From Theorem 3, we show that $K\left(U\right)\subset U$.

In the next stage, let us assume that ${\phi }_1,\overline{{\phi }_1},{\phi }_2,\overline{{\phi }_2}\in Z$ and, for every $\nu \in [0,1]$, let

$$\begin{array}{cc}\hfill & \parallel K_1({\phi }_1,{\phi }_2)-K_1(\overline{{\phi }_1},\overline{{\phi }_2})\parallel \hfill \\ \hfill & \le \underset{\nu \in [0,1]}{max}|I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right))-I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(\nu ,\overline{{\phi }_1}\left(\nu \right),\overline{{\phi }_2}\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left(\nu \right))|\hfill \\ & +\underset{\nu \in [0,1]}{max}\frac{{\nu }^{{\varrho }_1-1}}{b_3}\left[\right|{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\ell }({\xi }_1,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_1\right))\hfill \\ & -{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\ell }({\xi }_1,\overline{{\phi }_1}\left({\xi }_1\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}\overline{{\phi }_1}\left({\xi }_1\right))|\hfill \\ & +|{\epsilon }_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\left(\check{\ell }({\xi }_2,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_2\right))-\check{\ell }({\xi }_2,\overline{{\phi }_1}\left({\xi }_1\right),\overline{{\phi }_2}\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}\overline{{\phi }_1}\left({\xi }_2\right))\right)|\hfill \\ & +|{\mathfrak{q}}_1{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\left(\widehat{\ell }({\eta }_1,{\phi }_1\left({\eta }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_1\right))-\widehat{\ell }({\eta }_1,\overline{{\phi }_1}\left({\eta }_1\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left({\eta }_1\right))\right)|\hfill \\ & +|{\mathfrak{q}}_2{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\left(\widehat{\ell }({\eta }_2,{\phi }_1\left({\eta }_2\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_2\right))-\widehat{\ell }({\eta }_2,\overline{{\phi }_1}\left({\eta }_2\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left({\eta }_2\right))\right)|\hfill \\ & +|I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(1\right))-I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(1,\overline{{\phi }_1}\left(1\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left(1\right))|\hfill \\ & +|b_1{I}_{0^{+}}^{\varrho 2}\check{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(1\right))-b_1{I}_{0^{+}}^{\varrho 2}\check{\ell }(1,\overline{{\phi }_1}\left(1\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}\overline{{\phi }_1}\left(1\right))\left|\right]\hfill \\ & +\underset{\nu \in [0,1]}{max}\frac{{\nu }^{{\varrho }_1-1}}{|b_3|}\left({\int }_0^1\psi \left(\nu \right)({\phi }_1\left(\nu \right)+b_1{\phi }_2\left(\nu \right))d\left(\nu \right)-{\int }_0^1\psi \left(\nu \right)(\overline{{\phi }_1}\left(\nu \right)+b_1\overline{{\phi }_2}\left(\nu \right))d\left(\nu \right)\right)\hfill \\ & \le \frac{1}{b_3}\left(\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}+\frac{1}{\Gamma ({\varrho }_2+1)}\right)L_{\check{\ell }}(2\parallel {\phi }_1-\overline{{\phi }_1}\parallel +\parallel {\phi }_2-\overline{{\phi }_2}\parallel )\hfill \\ & +\frac{1}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_1+1)}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}\right)L_{\widehat{\ell }}(\parallel {\phi }_1-\overline{{\phi }_1}\parallel +2\parallel {\phi }_2-\overline{{\phi }_2}\parallel )\hfill \\ & +\frac{r}{|b_3|}(\parallel {\phi }_1-\overline{{\phi }_1}\parallel +b_1\parallel {\phi }_2-\overline{{\phi }_2}\parallel )\hfill \\ \hfill & \le J_1(\parallel {\phi }_1-\overline{{\phi }_1}\parallel +\parallel {\phi }_2-\overline{{\phi }_2}\parallel ),\hfill \end{array}$$

(51)

where $J_1$ is defined in Equation (49).

Similarly,

$$\parallel K_2({\phi }_1,{\phi }_2)-K_2(\overline{{\phi }_1},\overline{{\phi }_2})\parallel \le J_2(\parallel {\phi }_1-\overline{{\phi }_1}\parallel +\parallel {\phi }_2-\overline{{\phi }_2}\parallel ),$$

(52)

where $J_2$ is defined in Equation (50).

Hence, from (51) and (52), one has

$$\parallel K({\phi }_1,{\phi }_2)-K(\overline{{\phi }_1},\overline{{\phi }_2})\parallel \le J(\parallel {\phi }_1-\overline{{\phi }_1}\parallel +\parallel {\phi }_2-\overline{{\phi }_2}\parallel ),$$

(53)

where $J=max\left\{J_1,J_2\right\}<1$.

Hence, K is a contraction, and thus, K has a unique fixed point, which implies that the concerned system (3) with condition (4) has a unique solution. □

Theorem 5.

Under hypotheses $\left({\mathbb{V}}_3\right)$, $\left({\mathbb{V}}_4\right)$ along with continuous functions $\widehat{\ell },\check{\ell }:[0,1]\times {\mathbb{R}}^3\to \mathbb{R}$, the system of (3) and (4) possesses at least one solution in $B=\{({\phi }_1,{\phi }_2)\in U:\parallel ({\phi }_1,{\phi }_2)\parallel \le r\}$ where

$$r>\frac{{\mathring{G}}_2+{\mathring{G}}_4}{1-{\mathring{G}}_1-{\mathring{G}}_3}and1>{\mathring{G}}_1+{\mathring{G}}_3.$$

Proof. 

Let $B=\{({\phi }_1,{\phi }_2)\in U:\parallel ({\phi }_1,{\phi }_2)\parallel \le r\}$, $r>\frac{{\mathring{G}}_2+{\mathring{G}}_4}{1-{\mathring{G}}_1-{\mathring{G}}_3}$. By Theorem 3, the mapping $K:U\to U$ is completely continuous.

Consider $({\phi }_1,{\phi }_2)\in B$, $\parallel ({\phi }_1,{\phi }_2)\parallel \le r$. Then, in view of Theorem 3, $\parallel K({\phi }_1,{\phi }_2)\parallel \le r$ implies $K({\phi }_1,{\phi }_2)\in \overline{B}$. Hence, $K:\overline{B}\to \overline{B}$.

Take $({\phi }_1,{\phi }_2)\in \partial B$ and $\lambda \in (0,1)$, $({\phi }_1,{\phi }_2)=\lambda K({\phi }_1,{\phi }_2)$. Subsequently, using assumptions ${\mathbb{V}}_4$, ${\mathbb{V}}_5$, and for $\nu \in [0,1]$, we obtain

$$\begin{array}{cc}\hfill |{\phi }_1|& \le \lambda \left((c_{\widehat{\ell }}+d_{\widehat{\ell }})r+m_{\widehat{\ell }}\right)+\frac{\lambda r}{b_3}[\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}(c_{\check{\ell }}+d_{\check{\ell }})+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}(c_{\check{\ell }}+d_{\check{\ell }})\hfill \end{array}$$

$$\begin{array}{cc}& +\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}(c_{\widehat{\ell }}+d_{\widehat{\ell }})+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}(c_{\widehat{\ell }}+d_{\widehat{\ell }})+\frac{1}{\Gamma ({\varrho }_1+1)}(c_{\widehat{\ell }}+d_{\widehat{\ell }})\hfill \\ \hfill & +\frac{1}{\Gamma ({\varrho }_2+1)}(c_{\check{\ell }}+d_{\check{\ell }})]\hfill \\ & +\frac{\lambda }{b_3}[\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}{m}_{\check{\ell }}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}{m}_{\check{\ell }}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}{m}_{\widehat{\ell }}+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}{m}_{\widehat{\ell }}\hfill \end{array}$$

(54)

$$\begin{array}{cc}& +\frac{1}{\Gamma ({\varrho }_1+1)}{m}_{\widehat{\ell }}+\frac{1}{\Gamma ({\varrho }_2+1)}{m}_{\check{\ell }}+s(1+b_1)]\le \lambda \left({\mathring{G}}_{1}r+{\mathring{G}}_2\right),\hfill \end{array}$$

(55)

which gives $\parallel {\phi }_1\parallel \le \lambda \left({\mathring{G}}_{1}r+{\mathring{G}}_2\right)$. Similarly, we obtain $\parallel {\phi }_2\parallel \le \lambda \left({\mathring{G}}_{3}r+{\mathring{G}}_4\right)$.

Therefore, $\parallel ({\phi }_1,{\phi }_2)\parallel <\lambda r.$ However, $\parallel ({\phi }_1,{\phi }_2)\parallel =r$, which implies $({\phi }_1,{\phi }_2)\notin \partial B$, which is a contradiction. Consequently, based on Theorem 2, K has at least one fixed point $({\phi }_1,{\phi }_2)$ within the closure of the set $\overline{B}$. □

4. Ulam–Hyers-Type Stability

In to this section, we introduce the Ulam–Hyers-type stability by utilizing nonlinear terms.

Theorem 6.

If the matrix $\mathbb{M}$ converges to zero and the assumptions ${\mathbb{V}}_1$ and ${\mathbb{V}}_2$ are satisfied, with the additional condition that $J<1$, then it can be concluded that the solutions to Equations (3) and (4) exhibit Ulam–Hyers-type stability.

Proof. 

Take $({\phi }_1,{\phi }_2)$ and $(\overline{{\phi }_1},\overline{{\phi }_2})\in U$, and, for every $\nu \in [0,1]$, let

$$\begin{array}{cc}\hfill & \parallel K_1({\phi }_1,{\phi }_2)-K_1(\overline{{\phi }_1},\overline{{\phi }_2})\parallel \hfill \\ & \le \underset{\nu \in [0,1]}{max}|I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(\nu ,{\phi }_1\left(\nu \right),{\phi }_2\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(\nu \right))-I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(\nu ,\overline{{\phi }_1}\left(\nu \right),\overline{{\phi }_2}\left(\nu \right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left(\nu \right))|\hfill \\ & +\underset{\nu \in [0,1]}{max}\frac{{\nu }^{{\varrho }_1-1}}{b_3}\left[\right|{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\ell }({\xi }_1,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_1\right))\hfill \\ \hfill & -{\epsilon }_1{I}_{0^{+}}^{{\varrho }_2-{\sigma }_1}\check{\ell }({\xi }_1,\overline{{\phi }_1}\left({\xi }_1\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}\overline{{\phi }_1}\left({\xi }_1\right))|\hfill \\ & +|{\epsilon }_2{I}_{0^{+}}^{{\varrho }_2-{\sigma }_2}\left(\check{\ell }({\xi }_2,{\phi }_1\left({\xi }_1\right),{\phi }_2\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left({\xi }_2\right))-\check{\ell }({\xi }_2,\overline{{\phi }_1}\left({\xi }_1\right),\overline{{\phi }_2}\left({\xi }_2\right),{ }^C{D}_{0^{+}}^{{\omega }_2}\overline{{\phi }_1}\left({\xi }_2\right))\right)|\hfill \\ & +|{\mathfrak{q}}_1{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_1}\left(\widehat{\ell }({\eta }_1,{\phi }_1\left({\eta }_1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_1\right))-\widehat{\ell }({\eta }_1,\overline{{\phi }_1}\left({\eta }_1\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left({\eta }_1\right))\right)|\hfill \\ & +|{\mathfrak{q}}_2{b}_1{I}_{0^{+}}^{{\varrho }_1+{\mu }_2}\left(\widehat{\ell }({\eta }_2,{\phi }_1\left({\eta }_2\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left({\eta }_2\right))-\widehat{\ell }({\eta }_2,\overline{{\phi }_1}\left({\eta }_2\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left({\eta }_2\right))\right)|\hfill \\ & +|I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}{\phi }_2\left(1\right))-I_{0^{+}}^{{\varrho }_1}\widehat{\ell }(1,\overline{{\phi }_1}\left(1\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_1}\overline{{\phi }_2}\left(1\right))|\hfill \\ & +|b_1{I}_{0^{+}}^{\varrho 2}\check{\ell }(1,{\phi }_1\left(1\right),{\phi }_2\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}{\phi }_1\left(1\right))-b_1{I}_{0^{+}}^{\varrho 2}\check{\ell }(1,\overline{{\phi }_1}\left(1\right),\overline{{\phi }_2}\left({\xi }_1\right),{ }^C{D}_{0^{+}}^{{\omega }_2}\overline{{\phi }_1}\left(1\right))\left|\right]\hfill \\ & +\underset{\nu \in [0,1]}{max}\frac{{\nu }^{{\varrho }_1-1}}{|b_3|}\left({\int }_0^1\psi \left(\nu \right)({\phi }_1\left(\nu \right)+b_1{\phi }_2\left(\nu \right))d\left(\nu \right)-{\int }_0^1\psi \left(\nu \right)(\overline{{\phi }_1}\left(\nu \right)+b_1\overline{{\phi }_2}\left(\nu \right))d\left(\nu \right)\right)\hfill \\ & \le \frac{1}{b_3}\left(\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}+\frac{1}{\Gamma ({\varrho }_2+1)}\right)L_{\check{\ell }}\left(2\right|{\phi }_1-\overline{{\phi }_1}|+|{\phi }_2-\overline{{\phi }_2}\left|\right)\hfill \\ & +\frac{1}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_1+1)}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}\right)L_{\widehat{\ell }}(\parallel {\phi }_1-\overline{{\phi }_1}\parallel +2|{\phi }_2-\overline{{\phi }_2}\left|\right)\hfill \\ & +\frac{1}{b_3}r(\parallel {\phi }_1-\overline{{\phi }_1}\parallel -b_1\parallel {\phi }_1-\overline{{\phi }_1}\parallel )\hfill \\ & \le a_1\parallel {\phi }_1-\overline{{\phi }_1}\parallel +a_2\parallel {\phi }_2-\overline{{\phi }_2}\parallel ,\hfill \end{array}$$

(56)

where

$$\begin{array}{cc}\hfill a_1=& \frac{2L_{\check{\ell }}}{b_3}\left(\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}+\frac{1}{\Gamma ({\varrho }_2+1)}\right)\hfill \\ & +\frac{L_{\widehat{\ell }}}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_1+1)}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}\right)+\frac{1}{b_3}r,\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill a_2=& \frac{L_{\check{\ell }}}{b_3}\left(\frac{{\epsilon }_1}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}+\frac{1}{\Gamma ({\varrho }_2+1)}\right)\hfill \\ & +\frac{2L_{\widehat{\ell }}}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_1+1)}+\frac{{\mathfrak{q}}_1{b}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2{b}_1}{\Gamma ({\varrho }_1+{\mu }_2+1)}\right)+\frac{b_1}{b_3}r.\hfill \end{array}$$

(58)

Similarly,

$$\begin{array}{cc}\hfill \parallel K_2& ({\phi }_1,{\phi }_2)-K_2(\overline{{\phi }_1},\overline{{\phi }_2})\parallel \le a_3\parallel {\phi }_1-\overline{{\phi }_1}\parallel +a_4\parallel {\phi }_2-\overline{{\phi }_2}\parallel ,\hfill \end{array}$$

(59)

where

$$\begin{array}{cc}\hfill a_3=& \frac{2L_{\check{\ell }}}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_2+1)}+\frac{{\epsilon }_1{b}_2}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2{b}_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}\right)\hfill \\ & +\frac{L_{\widehat{\ell }}}{b_3}\left(\frac{{\mathfrak{q}}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2}{\Gamma ({\varrho }_1+{\mu }_2+1)}+\frac{b_2}{\Gamma ({\varrho }_1+1)}\right)+\frac{1}{b_3}r,\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill a_4=& \frac{L_{\check{\ell }}}{b_3}\left(\frac{b_3+1}{\Gamma ({\varrho }_2+1)}+\frac{{\epsilon }_1{b}_2}{\Gamma ({\varrho }_2-{\sigma }_1+1)}+\frac{{\epsilon }_2{b}_2}{\Gamma ({\varrho }_2-{\sigma }_2+1)}\right)\hfill \\ & +\frac{2L_{\widehat{\ell }}}{b_3}\left(\frac{{\mathfrak{q}}_1}{\Gamma ({\varrho }_1+{\mu }_1+1)}+\frac{{\mathfrak{q}}_2}{\Gamma ({\varrho }_1+{\mu }_2+1)}+\frac{b_2}{\Gamma ({\varrho }_1+1)}\right)+\frac{b_2}{b_3}r.\hfill \end{array}$$

(61)

So, from (56) and (59), we obtain

$$\begin{array}{cc}\hfill \parallel K_1({\phi }_1,{\phi }_2)-K_1(\overline{{\phi }_1},\overline{{\phi }_2})\parallel & \le a_1\parallel {\phi }_1-\overline{{\phi }_1}\parallel +a_2\parallel {\phi }_2-\overline{{\phi }_2}\parallel ,\hfill \\ & \\ \hfill \parallel K_2({\phi }_1,{\phi }_2)-K_2(\overline{{\phi }_1},\overline{{\phi }_2})\parallel & \le a_3\parallel {\phi }_1-\overline{{\phi }_1}\parallel +a_4\parallel {\phi }_2-\overline{{\phi }_2}\parallel .\hfill \end{array}$$

(62)

By examining Equation (62), we can derive the matrix $\mathbb{M}$ in the following manner:

$$\mathbb{M}=\left[\begin{array}{cc}{a}_1& a_2\\ a_3& a_4\end{array}\right].$$

Given the information provided that the matrix tends towards zero, we can conclude that the solution to Equations (3) and (4) exhibits Ulam–Hyers-type stability. □

5. Illustrative Examples

Example 1.

We take the fractional boundary value problem for $\nu \in [0,1]$:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{ }^{LR}{D}_{0^{+}}^{1.88}{\phi }_1\left(\nu \right)\hfill & {\displaystyle =\frac{exp\left(\nu \right)}{{\nu }^{2024}+2025}+\frac{|{\phi }_1\left(\nu \right)+{\phi }_2\left(\nu \right)|}{2023\left(\right|{\phi }_1\left(\nu \right)+{\phi }_2\left(\nu \right)|+1)}+\frac{\Gamma \left(0.5\right){ }^C{D}_{0^{+}}^{0.5}{\phi }_2\left(\nu \right)}{2023},}\hfill \\ { }^C{D}_{0^{+}}^{0.77}{\phi }_2\left(\nu \right)\hfill & {\displaystyle =\frac{cos\left(\nu \right)}{\nu +5}+\frac{exp(-\nu )}{2000}sin\left(\right|{\phi }_1\left(\nu \right)+{\phi }_2\left(\nu \right)\left|\right)+\frac{\Gamma \left(0.4\right){ }^C{D}_{0^{+}}^{0.6}{\phi }_1\left(\nu \right)}{2000},}\hfill \end{array}\right.\end{array}$$

(63)

with

$$\left\{\begin{array}{c}{\displaystyle {\phi }_1\left(0\right)=0,{\phi }_1\left(1\right)=0.33{ }^C{D}_{0^{+}}^{0.03}{\phi }_2\left(0.22\right)+2.44{ }^C{D}_{0^{+}}^{0.1}{\phi }_2\left(0.1\right)+{\int }_0^1\frac{\nu }{20}{\phi }_1\left(\nu \right)d\left(\nu \right),}\hfill \\ {\displaystyle {\phi }_2\left(1\right)=0.99I_{0^{+}}^{0.44}{\phi }_1\left(0.11\right)+0.55I_{0^{+}}^{0.33}{\phi }_1\left(0.66\right)+{\int }_0^1\frac{\nu }{20}{\phi }_2\left(\nu \right)d\left(\nu \right)+{\int }_0^1\frac{\nu }{20}{\phi }_2\left(\nu \right)d\left(\nu \right).}\hfill \end{array}\right.$$

(64)

After the calculation, we have

$$L_{\widehat{\ell }}=\frac{1}{2023},L_{\check{\ell }}=\frac{1}{2000},$$

$$J_1\approx 0.0307,J_2\approx 0.0265,$$

as $max\{J_1,J_1\}=0.0307<1$. So, system (63) has a non-repeated solution according to Theorem 4. Furthermore, by the values of $a_i$, $i=1,2,3,4$, we have

$$\mathbb{M}=\left[\begin{array}{cc}0.0264& 0.0073\\ 0.0142& 0.0194\end{array}\right].$$

Upon performing the necessary computations, we find that the eigenvalues of the system are determined to be ${\beta }_1=0.0337$ and ${\beta }_2=0.0121$. Consequently, it can be observed that ${\rm Y}\left(\mathbb{M}\right)=0.0337<1$. As a result, according to Theorem 6, we can conclude that the given system, incorporating a delay term and corresponding to the specified fractional order, is Ulam–Hyers-stable.

Example 2.

We take the system for $\nu \in [0,1]$:

$$\left\{\begin{array}{c}{\displaystyle { }^{LR}{D}_{0^{+}}^{1.99}{\phi }_1\left(\nu \right)=\frac{sin\left(\nu \right)}{\nu +2025}+\frac{1}{{\pi }^4}arctan({\phi }_1\left(\nu \right)+{\phi }_2\left(\nu \right))+\frac{\Gamma \left(0.1\right){ }^C{D}_{0^{+}}^{0.9}{\phi }_2\left(\nu \right)}{{\pi }^4},}\hfill \\ {\displaystyle { }^C{D}_{0^{+}}^{0.99}{\phi }_2\left(\nu \right)=exp(-z^2-{\pi }^4)+\frac{|{\phi }_1|}{20\left(\right|{\phi }_1|+1)}+\frac{|{\phi }_2|}{10|{\phi }_2|+10}+\frac{\Gamma \left(0.7\right)}{20}sin\left({ }^C{D}_{0^{+}}^{0.3}{\phi }_2\left(\nu \right)\right),}\hfill \end{array}\right.$$

(65)

subjected to the following mixed boundary conditions:

$$\left\{\begin{array}{c}{\displaystyle {\phi }_1\left(0\right)=0,{\phi }_1\left(1\right)=1.33{ }^C{D}_{0^{+}}^{0.19}{\phi }_2\left(0.12\right)+10.44{ }^C{D}_{0^{+}}^{0.04}{\phi }_2\left(0.33\right)+{\int }_0^1\frac{{\mathfrak{z}}^{\mathfrak{2}}}{44}{\phi }_1\left(\nu \right)d\left(\nu \right),}\hfill \\ {\displaystyle {\phi }_2\left(1\right)=0.49I_{0^{+}}^{0.25}{\phi }_1\left(0.71\right)+0.75I_{0^{+}}^{0.73}{\phi }_1\left(0.36\right)+{\int }_0^1\frac{{\mathfrak{z}}^{\mathfrak{2}}}{44}{\phi }_2\left(\nu \right)d\left(\nu \right).}\hfill \end{array}\right.$$

(66)

After the calculation, we have

$$L_{\widehat{\ell }}=\frac{1}{{\pi }^4},L_{\check{\ell }}=\frac{1}{10},$$

$$J_1\approx 0.0186,J_2\approx 0.2017,$$

as $max\{J_1,J_1\}=0.2017<1$. According to Theorem 4, it can be deduced that the system of (65) and (66) possesses a unique solution that does not repeat or occur more than once. Furthermore, by substituting the specific values of $a_i$ for $i=1,2,3,4$, we obtain

$$\mathbb{M}=\left[\begin{array}{cc}0.0097& 0.1009\\ 0.2017& 0.1009\end{array}\right].$$

After performing the necessary calculations, we obtain the eigenvalues of the system as ${\beta }_1=-0.0945$ and ${\beta }_2=0.2050$. It is noteworthy that ${\rm Y}\left(\mathbb{M}\right)=0.2050$, which is less than 1. This result allows us to conclude, based on Theorem 6, that the given system, which incorporates a delay term and is associated with the specified fractional order, is Ulam–Hyers-stable.

6. Conclusions and Perspectives

In this study, we have conducted a comprehensive qualitative analysis of the solutions to the non-linear coupled fractional boundary value problem with two types of symmetrical fractional derivatives (Caputo and Liouville–Riemann (LR) fractional derivatives) described by Equations (3) and (4).

The Caputo and Liouville–Riemann (LR) fractional derivatives are two prominent types of fractional derivatives. The Caputo derivative is particularly favored for initial value problems because it allows for the use of traditional initial conditions, making it highly applicable to physical problems. On the other hand, the LR derivative is more suited for boundary value problems due to its flexibility in handling different boundary conditions, making it useful across a broader spectrum of scientific and engineering problems.

Our investigation specifically focuses on this problem with mixed integro-differential conditions. By employing Banach and nonlinear alternatives to Leray–Schauder fixed point theorems, we have thoroughly examined the properties of the problem at hand. Additionally, we have discussed several results pertaining to the Ulam–Hyers stability of the solutions. To illustrate the significance and applicability of our findings, we have provided two relevant examples in Section 5 and thoroughly discussed the obtained results. It is worth noting that the considered problem is stochastic in nature and encompasses various practical problems encountered in the fields of fluid mechanics and dynamics, thus highlighting its relevance in real-world applications.

Briefly, this paper aims to advance the understanding of fractional differential equations by introducing a novel approach to solving non-linear coupled systems with mixed boundary conditions. Our work not only highlights the theoretical aspects of fractional calculus but also emphasizes its practical applications, thereby bridging the gap between mathematical theory and real-world problem-solving.

For future research, one can explore higher-order coupled systems with various fractional derivatives to gain deeper insights into the dynamics and stability of these systems, potentially leading to new theoretical advances. Additionally, developing numerical methods to solve these types of fractional differential problems will enhance their applicability to more complex real-world phenomena.