Analysis of Implicit Solutions for a Coupled System of Hybrid Fractional Order Differential Equations with Hybrid Integral Boundary Conditions in Banach Algebras

Abstract

This paper investigates the existence and uniqueness of implicit solutions in a coupled symmetry system of hybrid fractional order differential equations, along with hybrid integral boundary conditions in Banach Algebras. The methodology centers on a hybrid fixed-point theorem that involves mixed Lipschitz and Carathéodory conditions, serving to establish the existence of solutions. Moreover, it derives sufficient conditions for solution uniqueness and establishes the Hyers–Ulam types of solution stability. This study contributes valuable insights into complex hybrid fractional order systems and their practical implications.

1. Introduction

Nonlinear fractional-order differential equations play a crucial role in modeling a wide range of real-world nonlinear phenomena across various fields like Physical Sciences and Technology. These equations are used to describe complex systems such as air movement, dynamic systems, electromagnetism, and nonlinear process control. Some of these equations do not have analytical solutions, which leads to the usefulness of perturbation of such problems. There are different types of perturbed differential equations; one important perturbation type is the hybrid differential equation that involves quadratic perturbations of nonlinear differential equations. This area of study, which focuses on hybrid equations, has recently gained significant attention due to its applicability to numerous dynamic systems. In particular, fractional hybrid differential equations, which introduce quadratic changes to nonlinear equations, have became more prominent among contemporary researchers due to their extensive implications and practical uses. The investigation of hybrid differential equations has greatly contributed to understanding and addressing the behaviors of diverse physical systems.

In this context, pivotal contributions from Dhage [1,2,3,4] and Zhao [5] have deeply explored the intricate interplay of these equations. Importantly, hybrid fractional differential equations stand out due to their capacity to encapsulate various dynamic scenarios as specialized instances, making them indispensable tools for meticulously modeling intricate real-world phenomena. The extensive literature on hybrid differential equations has been thoroughly documented in seminal works by leading scholars. These include significant monographs authored by Ahmad [6], Allaoui et al. [7], Awad et al. [8,9,10,11], Baleanu [12], Darwish [13], Dhage and Carathéodory [2], Herzallah [14], Krasnoselskii [15], Melliani [16], Zhao [5], and Zheng [17]. These influential publications have played a crucial role in shaping our current understanding of hybrid differential equations, offering profound insights into their nature and behavior. To delve deeper into this subject matter, interested individuals can consult the following references: Asaduzzaman et al. [18], ElHady et al. [19], Hattaf et al. [20], and NIkan et al. [21], along with additional sources such as [22,23,24,25,26].

Concurrently, alongside the exploration of coupled systems of FDEs, mathematical researchers have shown keen interest in harnessing the power of symmetry to unravel intricate real-world enigmas. Coupled systems of fractional differential equations endowed with inherent symmetry have emerged as a captivating area of investigation, owing to their applicability across a broad spectrum of natural phenomena. Pioneering works by Ahmad [27,28], Cheng [29], Hu [30], and Su [31] have illuminated the significance of such coupled systems in diverse contexts.

This paper builds upon significant progress by conducting a comprehensive examination of a system consisting of hybrid fractional-order differential equations within the context of Banach algebras. The focus of this contribution is to clarify the existence and uniqueness of mild solutions by incorporating Caputo differential operators and hybrid boundary conditions. It relies on a hybrid fixed point theorem applied under mixed Lipschitz and Carathéodory conditions as the fundamental basis for demonstrating the existence of solutions. Additionally, the paper establishes conditions for solution uniqueness and explores the stability of solutions in the Hyers–Ulam sense. In essence, this research not only advances our understanding of intricate hybrid fractional-order systems, but also highlights their practical relevance across various scientific disciplines. Through rigorous analysis and innovative methodologies, this study extends the frontiers of knowledge in this emerging field, enhancing our grasp of the complex dynamics involved.

Before starting our research, we need to provide an overview of recent research endeavors that address the exact issue we are currently exploring. This serves as a preparatory phase that enables us to comprehend the significance and distinctiveness of our inquiry.

In the work of Dhage and Lakshmikantham [32], the authors accomplished significant milestones in the field by establishing the existence and uniqueness of solutions, along with fundamental differential inequalities, for hybrid differential equations characterized by the following form:

$$\frac{d}{d\zeta }\left(\frac{x\left(\zeta \right)}{f(\zeta ,x(\zeta \left)\right)}\right)=g(\zeta ,x\left(\zeta \right))\mathrm{almost}\mathrm{everywhere}\zeta \in J,\mathrm{with}x\left({\zeta }_0\right)=x_0\in R.$$

In 2014, Ahmad conducted a comprehensive investigation [27] concerning the existence and uniqueness of solutions pertaining to coupled systems of boundary value problems, specifically focusing on hybrid fractional differential equations. These equations are represented in the form

$$\begin{array}{c}{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left(\frac{x\left(\zeta \right)}{f(\zeta ,x\left(\zeta \right),y\left(\zeta \right))}\right)=h_1(\zeta ,x\left(\zeta \right),y\left(\zeta \right)),\alpha \in (1;2];0<\zeta <1,\\ {}^c{\mathfrak{D}}_{0^{+}}^{\beta }\left(\frac{y\left(\zeta \right)}{g(\zeta ,x\left(\zeta \right),y\left(\zeta \right))}\right)=h_2(\zeta ,x\left(\zeta \right),y\left(\zeta \right)),\beta \in (1;2];0<\zeta <1,\\ x\left(0\right)=x\left(1\right)=0;y\left(0\right)=y\left(1\right)=0,\end{array}$$

where ${}^c{\mathfrak{D}}_{0^{+}}^{\alpha }$ denotes the Caputo fractional derivative of order $\alpha .$

In 2015, Ali et al. conducted a comprehensive investigation on the existence and uniqueness of solutions pertaining to the subsequent multi-point hybrid coupled systems [33].

$$\left\{\begin{array}{c}{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left(\frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right),y\left(\zeta \right))}{f_2(\zeta ,x\left(\zeta \right),y\left(\zeta \right))}\right)=\varphi (\zeta ,x\left(\zeta \right),y\left(\zeta \right)),\alpha \in (1;2];0<\zeta <1,\\ {}^c{\mathfrak{D}}_{0^{+}}^{\beta }\left(\frac{y\left(\zeta \right)-g_1(\zeta ,x\left(\zeta \right),y\left(\zeta \right))}{g_2(\zeta ,x\left(\zeta \right),y\left(\zeta \right))}\right)=\psi (\zeta ,x\left(\zeta \right),y\left(\zeta \right)),\beta \in (1;2];0<\zeta <1,\\ x\left(0\right)=a;x\left(1\right)=b;y\left(0\right)=c;y\left(1\right)=d.\end{array}\right.$$

In 2016, Bashiri [34] examined the existence of solutions for a coupled system of fractional hybrid differential equations in Banach algebra. The author employed a fixed point theorem proposed by Dhage [35] to establish the existence of solutions. Specifically, Bashiri used this theorem to analyze the sum of three operators in the following system:

$$\begin{array}{c}{}^{RL}{\mathfrak{D}}_{0^{+}}^{\alpha }\left(\frac{x\left(\zeta \right)-u(\zeta ,x(\zeta \left)\right)}{f(\zeta ,x(\zeta \left)\right)}\right)=g(\zeta ;y\left(\zeta \right)),\\ {}^{RL}{\mathfrak{D}}_{0^{+}}^{\alpha }\left(\frac{y\left(\zeta \right)-u(\zeta ,y(\zeta \left)\right)}{f(\zeta ,y(\zeta \left)\right)}\right)=g(\zeta ;x\left(\zeta \right)),\\ x\left(0\right)=0,y\left(0\right)=0,\alpha \in \left(0,1\right),\mathrm{and}0<\zeta <T,\end{array}$$

where ${}^{RL}{\mathfrak{D}}_{0^{+}}^{\alpha }$ denotes the the Riemann–Liouville fractional derivative of order $\alpha$.

In their investigation conducted in 2018, Karthikeyan and Buvaneswari [36] examined the existence of solutions for the following set of nonlinear coupled fractional hybrid differential equations, which are intricately associated with the field of Banach algebra:

$$\left\{\begin{array}{c}{}^{RL}{\mathfrak{D}}_{0^{+}}^{\alpha }\left(\frac{x\left(\zeta \right)-\omega (\zeta ,x(\zeta \left)\right)}{u(\zeta ,x(\zeta \left)\right)}\right)=v(\zeta ;y\left(\zeta \right)),\\ {}^{RL}{\mathfrak{D}}_{0^{+}}^{\alpha }\left(\frac{y\left(\zeta \right)-\omega (\zeta ,y(\zeta \left)\right)}{u(\zeta ,y(\zeta \left)\right)}\right)=v(\zeta ;x\left(\zeta \right)),\\ x\left(0\right)=a,y\left(0\right)=b,\alpha \in (0,1],\mathrm{and}0\le \zeta \le 1.\end{array}\right.$$

In their work published in 2021, ElSayed et al. [37] presented the necessary conditions that ensure the existence of solutions for the following specific initial value problem involving coupled hybrid fractional-order differential equations:

$$\left\{\begin{array}{c}{}^{RL}{\mathfrak{D}}_{0^{+}}^{\alpha }\left(\frac{x\left(\zeta \right)-k_1(\zeta ,x\left({\phi }_1\left(\zeta \right)\right))}{g_1(\zeta ,x\left({\phi }_2\left(\zeta \right)\right))}\right)=f_1(\zeta ;{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\zeta ;y\left({\phi }_3\left(\zeta \right)\right)));\alpha ,\beta \in (0,1],\\ {}^{RL}{\mathfrak{D}}_{0^{+}}^{\alpha }\left(\frac{y\left(\zeta \right)-k_2(\zeta ,y\left({\phi }_1\left(\zeta \right)\right))}{g_2(\zeta ,y\left({\phi }_2\left(\zeta \right)\right))}\right)=f_2(\zeta ;{\mathcal{I}}_{0^{+}}^{\beta }{u}_2(\zeta ;x\left({\phi }_3\left(\zeta \right)\right)));\alpha ,\beta \in (0,1],\\ x\left(0\right)=k_1(0;x\left(0\right)),y\left(0\right)=k_2(0;y\left(0\right)),\mathrm{and}0<\zeta <T.\end{array}\right.$$

Building upon the aforementioned investigation, we study the following system of coupled hybrid fractional order differential equations with hybrid integral boundary conditions in Banach algebras (CHFDE):

$$\left\{\begin{array}{c}{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right)={\Im }_1(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\zeta ,y\left(\zeta \right)),\zeta \in J=[0,T],\\ {}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}\right)={\Im }_2(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_2(\zeta ,x\left(\zeta \right)),\zeta \in J=[0,T],\\ {\left.{\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right|}_{\zeta =0}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon ,\\ {\left.{\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right|}_{\zeta =T}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_2\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon ,\\ {\left.{\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}\right|}_{\zeta =0}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{k}_1\left(\upsilon ,y\left(\upsilon \right)\right)d\upsilon ,\\ {\left.{\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}\right|}_{\zeta =T}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{k}_2\left(\upsilon ,y\left(\upsilon \right)\right)d\upsilon ,\end{array}\right.$$

(1)

where ${}^c{\mathfrak{D}}_{0^{+}}^{\alpha }$ denotes the Caputo fractional derivative of order $\alpha \in (1,2]$, ${\mathcal{I}}_{0^{+}}^{\beta }$ is the Riemann–Liouville fractional integral of order $\beta \in (0,1)$ with $g_1,g_2\in C(J\times R,R\setminus \left\{0\right\})$, $f_i,{\Im }_i,h_i,k_i,u_i\in C(J\times R,R)$ for $i=1,2$, and $\Gamma \left(.\right)$ is the classical Gamma function.

The proposed problem covers some symmetry conditions. For instance, if $h_1=h_2$ and $k_1=k_2$, then the conditions of Equation (1) become the symmetry conditions

$${\displaystyle \frac{x\left(0\right)-f_1(0,x\left(0\right))}{g_1(0,x\left(0\right))}}={\displaystyle \frac{x\left(T\right)-f_1(T,x\left(T\right))}{g_1(T,x\left(T\right))}},$$

and

$${\displaystyle \frac{y\left(0\right)-f_2(0,y\left(0\right))}{g_2(0,y\left(0\right))}}={\displaystyle \frac{y\left(T\right)-f_2(T,y\left(T\right))}{g_2(T,y\left(T\right))}},$$

with any value of $\gamma$, and for the fractional derivatives of order $\alpha$ of the unknown functions $\left({\displaystyle \frac{x(.)-f_1(.,x(.))}{g_1(.,x(.))}}\right)$ and $\left({\displaystyle \frac{y(.)-f_2(.,y(.))}{g_2(.,y(.))}}\right)$.

In this study, we make significant contributions to the field of hybrid fractional-order differential equations within the context of Banach algebras. Our primary focus is to establish the existence and uniqueness of mild solutions for the coupled system of hybrid fractional-order differential equations (CHFDE) described in Equation (1), incorporating Caputo differential operators and hybrid boundary conditions. These contributions are vital in extending our understanding of intricate hybrid fractional-order systems. Additionally, we delve into the stability analysis of solutions in the Ulam–Hyers sense. Our work not only advances the theoretical foundations of hybrid fractional differential equations, but also highlights their practical relevance in modeling complex phenomena across various scientific disciplines. Through rigorous analysis and innovative methodologies, this research extends the frontiers of knowledge in this emerging field, enhancing our grasp of the complex dynamics involved.

The manuscript is structured as follows: In Section 2, we provide an overview of important background information that is utilized throughout this work. In Section 3, the main results of this study are presented as follows: First, we prove the sufficient conditions for the existence of mild solutions for the coupled system of hybrid fractional-order differential equations (1) under investigation. These results are derived using a fixed point theorem for three operators in Banach algebra, as introduced by Dhage [35], and under mixed Lipschitz and Carathéodory conditions. Second, we establish the uniqueness of these mild solutions. Lastly, we investigate the Ulam–Hyers stability of the CHFDE (1).

2. Preliminaries

In this section, some basic definitions and preliminaries used throughout this work are introduced.

Definition 1

([38]). For any real number $\alpha >0$ and for all $\zeta \in J=\left[0,T\right]$, the Riemann–Liouville fractional integral ${\mathcal{I}}_{0^{+}}^{\alpha }$ of the function $f\in L^1(J,R)$ is defined by

$${\mathcal{I}}_{0^{+}}^{\alpha }f\left(\zeta \right)={\int }_0^{\zeta }\frac{{(\zeta -\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}f\left(\upsilon \right)d\upsilon ,$$

where $\Gamma (.)$ is Euler’s Gamma function.

Definition 2

([38]). For any real number $\alpha >0$ and for all $\zeta \in J=\left[0,T\right]$, the Caputo fractional derivative ${}^c{\mathfrak{D}}_{0^{+}}^{\alpha }$ of the absolutely continuous function $f\left(\zeta \right)\in C^n\left(J,R\right)$ is defined as

$${}^c{\mathfrak{D}}_{0^{+}}^{\alpha }f\left(\zeta \right)={\mathcal{I}}_{0^{+}}^{1-\alpha }\frac{d}{d\zeta }f\left(\zeta \right)=\frac{1}{\Gamma (n-\alpha )}{\int }_0^{\zeta }{(\zeta -\upsilon )}^{n-\alpha -1}{f}^{\left(n\right)}\left(\upsilon \right)d\upsilon ,$$

where $n=\left[\alpha \right]+1$.

To explore additional properties of fractional calculus operators, references to relevant works include [38,39,40,41,42].

We let $X=C\left(J,R\right)$ represent any Banach algebra comprising all continuous real-valued functions from $J=\left[0,T\right]$ to R, where the norm $∥x∥$ is defined as the supremum of the absolute values of $x\left(\zeta \right)$ for all $\zeta \in J$. Furthermore, we let $L^1\left(J,R\right)$ denote the space of Lebesgue integrable real-valued functions defined on J and equipped with the $L^1$-norm ${∥x∥}_{L^1}={\int }_0^T\left|x\left(\upsilon \right)\right|d\upsilon$.

Definition 3

([43]). (Normed algebra) A normed algebra is a mathematical structure represented by a pair $(A,∥.∥)$, where A is an algebra and $∥.∥$ is a norm defined on A. The norm $∥.∥$ is considered an algebra norm if it satisfies the condition $\left|\right|x·y\left|\right|\le \left|\right|x\left|\right|·\left|\right|y\left|\right|$ for all elements x and y in A. In case $(A,∥.∥)$ fulfills this property, it is termed as a normed algebra. Moreover, if the normed algebra $(A,∥.∥)$ is also complete, it is further classified as Banach algebra.

It is evident that $C(J,R)$ constitutes a complete normed algebra under the specified supremum norm. Furthermore, within any Banach algebra X, the operation denoted as “·” is defined as $(x·y)\left(\zeta \right)=x\left(\zeta \right)·y\left(\zeta \right)$. The Cartesian product space is defined as $E=X\times X$, equipped with a norm defined by $\parallel (x,y)\parallel \hspace{0.17em}=\hspace{0.17em}\parallel x\parallel +\parallel y\parallel$. As a result, the normed linear space $(E,\parallel (.,.)\parallel )$ is elevated to a Banach space. Additionally, when endowed with the multiplication operation “·” defined as $\left(\right(x,y)·(u,v\left)\right)\left(\zeta \right)=(x,y)\left(\zeta \right)·(u,v)\left(\zeta \right)=\left(x\right(\zeta )·u(\zeta ),y(\zeta )·v(\zeta \left)\right)$ for all $\zeta \in J$ and $(x,y),(u,v)\in E$, it also becomes Banach algebra.

Definition 4

([43]). We let $E=X\times X$ be a Banach space with norm $∥.∥$. A mapping $T:E\to E$ is called D-Lipschitz if there exists a continuous and nondecreasing function $\varphi :R^{+}\to R^{+}$ such that

$$\parallel T\left(u\right)-T\left(v\right)\parallel \hspace{0.17em}\le \hspace{0.17em}{\varphi }_T(\parallel u-v\parallel ),$$

for all vectors $u=(x_1,y_1)$ and $v=(x_2,y_2)$ in E with ${\varphi }_T\left(0\right)=0$. Sometimes we call the function ϕ a D-function of T on E.

Definition 5

([43]). A mapping $f:J\times R\to R$ is said to satisfy a condition of $L^1$-Carathéodory or simply called $L^1$-Carathéodory if

• $\zeta \to f\left(\zeta ,x\right)$ is measurable for each $x\in R$,
• $x\to f\left(\zeta ,x\right)$ is continuous almost everywhere for each $\zeta \in J$,
• For any real number $r>0$, there exists a function $g\in L^1(J,R)$ such that $\left|f\right(\zeta ,x\left)\right|\le g\left(\zeta \right)$ a.e. $\zeta \in J$, and for every $x\in R$ with $\left|x\right|\le r$.

Lemma 1

([42]). We consider any real number $\alpha >0$, $n=\left[\alpha \right]+1$ and let $f\in C^n\left(J,R\right)$.

• If ${}^c{\mathfrak{D}}_{0^{+}}^{\alpha }f\left(\zeta \right)=0$, then $f\left(\zeta \right)=c_0+c_1\zeta +c_2{\zeta }^2+\cdots +c_{n-1}{\zeta }^{n-1}$, where $c_i\in R$ for all $i=1,2,...,n$.
• ${\mathcal{I}}_{0^{+}}^{\alpha }$${}^c{\mathfrak{D}}_{0^{+}}^{\alpha }f\left(\zeta \right)=f\left(\zeta \right)+c_0+c_1\zeta +c_2{\zeta }^2+\cdots +c_{n-1}{\zeta }^{n-1}$, where $c_i\in R$ for all $i=1,2,...,n$.

Theorem 1

([35]). We let S be a nonempty closed convex bounded subset of a product Banach algebra $E=X\times X$. If the operators $\mathcal{A}:E\to E$, $\mathcal{B}:S\to E$, and $\mathcal{C}:E\to E$ satisfy the following conditions,

• $\mathcal{A}$ and $\mathcal{C}$ are Lipschitzian with respective Lipschitz constants μ and σ,
• $\mathcal{B}$ is completely continuous (compact and continuous),
• If $u=\mathcal{A}u\mathcal{B}v+\mathcal{C}u$, then $u\in S$ for all $v\in S$,
• $\mu \parallel \mathcal{B}\left(S\right)\parallel +\sigma <\rho$ for some $\rho >0$, where $\parallel \mathcal{B}\left(S\right)\parallel =sup\left\{\parallel \mathcal{B}x\parallel :x\in S\right\}$,

then the operator equation $\mathcal{A}u\mathcal{B}v+\mathcal{C}u=u$ has at least one solution in S.

Definition 6.

The solution of the coupled system of hybrid fractional-order differential equations CHFDE (1) is the continuous function $\left(x\left(\zeta \right),y\left(\zeta \right)\right)\in C\left(J\times J,R\right)$ such that $\zeta \to {\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}$ and $\zeta \to {\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}$ are continuous functions for each $x,y\in C\left(J,R\right)$, and $\left(x,y\right)$ satisfies the CHFDE (1).

3. Main Results

In this part, we use the fixed point theory developed by Dhage [35] to study the necessary and sufficient conditions for the existence of coupled solutions for the CHFDE (1). To begin with, we have to consider the following assumptions:

Assumption 1.

The functions $x\to {\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}$ and $y\to {\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}$ are continuous and increasing in R a.e. $\zeta \in J$.

Assumption 2.

The functions $g_i:J\times R\to R\setminus \left\{0\right\}$ and $f_i:J\times R\to R$ for $\left(i=1,2\right)$ are continuous, where there exist positive functions ${\mu }_i\left(\zeta \right)$ and ${\sigma }_i\left(\zeta \right)$ for $\left(i=1,2\right)$ with the respective bounds $\parallel \mu \parallel =max\left\{∥{\mu }_1∥,∥{\mu }_2∥\right\}$ and $\parallel \sigma \parallel =max\left\{∥{\sigma }_1∥,∥{\sigma }_2∥\right\}$, such that

$$|g_i(\zeta ,x)-g_i(\zeta ,y)|\le {\mu }_i\left(\zeta \right)|x-y|,$$

and

$$|f_i(\zeta ,x)-f_i(\zeta ,y)|\le {\sigma }_i\left(\zeta \right)|x-y|,$$

for all $\zeta \in J$ and $x,y\in R$.

Assumption 3.

The functions ${\Im }_i:J\times R\to R$ and $u_i:J\times R\to R$ for $\left(i=1,2\right)$ satisfy Carathéodory conditions. That is, if all the functions ${\Im }_i$ and $u_i$ for $\left(i=1,2\right)$ are measurable in ζ for any $x\in R$ and continuous in x a.e. $\zeta \in$ J, then there exist bounded and measurable functions $\zeta \to {\mathit{a}}_i\left(\zeta \right)$, $\zeta \to {\mathit{b}}_i\left(\zeta \right)$, and $\zeta \to {\mathit{m}}_i\left(\zeta \right)$ such that

$$|{\Im }_i(\zeta ,x)|\le {\mathit{a}}_i\left(\zeta \right)+{\mathit{b}}_i\left(\zeta \right)\left|x\right|,\forall (\zeta ,x)\in J\times R,$$

and

$$|u_i(\zeta ,x)|\le {\mathit{m}}_i\left(\zeta \right),\forall (\zeta ,x)\in J\times R.$$

In addition, for any real numbers $\gamma \le \alpha$ and $c\ge 0$, there exist real numbers ${\mathcal{M}}_i$, ${\mathcal{N}}_i$, and ♭ such that ${\mathcal{I}}_c^{\gamma }{\mathit{a}}_i(.)\le {\mathcal{M}}_i$, and ${\mathcal{I}}_c^{\gamma }{\mathit{m}}_i(.)\le {\mathcal{N}}_i$ with

$$sup\left|{\mathit{a}}_i\left(\zeta \right)\right|\le \mathcal{M},\mathrm{where}\mathcal{M}=max\{{\mathcal{M}}_1,{\mathcal{M}}_2\},$$

$$sup\left|{\mathit{m}}_i\left(\zeta \right)\right|\le \mathcal{N},\mathrm{where}\mathcal{N}=max\{{\mathcal{N}}_1,{\mathcal{N}}_2\},$$

and

$$sup\left|{\mathit{b}}_i\left(\zeta \right)\right|\le ♭.$$

Similar assumption holds for every $y\in R.$

Assumption 4.

The functions $h_i$ and $k_i:J\times R\to R$ (for $i=1,2$) are Lipschitz continuous functions with respective constants ${\omega }_i$ and ${\kappa }_i\in [0,1)$, such that

$$|h_i(\zeta ,x)-h_i(\zeta ,y)|\le {\omega }_i|x-y|,$$

and

$$|k_i(\zeta ,x)-k_i(\zeta ,y)|\le {\kappa }_i|x-y|,$$

for every $\zeta \in J$ and $x,y\in R\times R$ with $\parallel \omega \parallel =max\left\{∥{\omega }_1∥,∥{\omega }_2∥\right\}$ and $\parallel \kappa \parallel =max\left\{∥{\kappa }_1∥,∥{\kappa }_2∥\right\}$.

Remark 1.

From Assumptions 2 and 4, we deduce that for all $i\in \{1,2\}$ and any $\zeta \in J$, we have

$$\begin{array}{cc}\hfill |h_i(\zeta ,x)|& \le |h_i(\zeta ,0)|+{\omega }_i\left|x\right|\le H_i+{\omega }_i\left|x\right|\mathit{where}{H}_i=\underset{\zeta \in J}{sup}\left|h_i(\zeta ,0)\right|,\hfill \\ \hfill |k_i(\zeta ,y)|& \le |k_i(\zeta ,0)|+{\kappa }_i\left|y\right|\le K_i+{\kappa }_i\left|y\right|\mathit{where}{K}_i=\underset{\zeta \in J}{sup}\left|k_i(\zeta ,0)\right|,\hfill \\ \hfill |g_i(\zeta ,x)|& \le |g_i(\zeta ,0)|+{\mu }_i\left|x\right|\le {\breve{G}}_i+{\mu }_i\left|x\right|\mathit{where}{\breve{G}}_i=\underset{\zeta \in J}{sup}\left|g_i(\zeta ,0)\right|,\hfill \end{array}$$

and

$$|f_i(\zeta ,x)|\le |f_i(\zeta ,0)|+{\sigma }_i\left|x\right|\le F_i+{\sigma }_i\left|x\right|\mathit{where}{F}_i=\underset{\zeta \in J}{sup}\left|f_i(\zeta ,0)\right|.$$

Assumption 5.

For any $u=\left(x,y\right)$, there exists a real positive number ρ such that

$$∥u∥\le \rho ,$$

$$\rho \ge \frac{\Gamma \left(\gamma +1\right)\left(1-∥\sigma ∥-∥\mu ∥\aleph \right)-\breve{G}\mathcal{W}-\mathcal{H}∥\mu ∥-∥\mu ∥\mathcal{W}\sqrt{\Delta }}{2∥\mu ∥\mathcal{W}},$$

(2)

and

$$2∥\mu ∥G_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}\right)+∥\sigma ∥<\rho ,$$

(3)

where

$$\Delta =\frac{\begin{array}{c}\Gamma {\left(\gamma +1\right)}^2{(∥\mu ∥\aleph +∥\sigma ∥-1)}^2\\ +{\breve{G}}^2{\mathcal{W}}^2-2\breve{G}\mathcal{W}\left(\Gamma \left(\gamma +1\right)∥\mu ∥\aleph \Gamma \left(\gamma +1\right)\left(1-∥\sigma ∥\right)+\mathcal{H}∥\mu ∥\right)\\ +{\mathcal{H}}^2{∥\mu ∥}^2+2\Gamma \left(\gamma +1\right)∥\mu ∥(\mathcal{H}(∥\mu ∥\aleph +∥\sigma ∥-1)-2\mathcal{WF})\end{array}}{{∥\mu ∥}^2{\mathcal{W}}^2},$$

with $\mathcal{F}=F_1+F_2$, $\breve{G}={\breve{G}}_1+{\breve{G}}_2$, $\aleph =G_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}\right)$, $\mathcal{H}=\left(2H_1+H_2\right)$, and $\mathcal{W}=\left(2{\omega }_1+{\omega }_2\right)$.

Assumption 6.

The functions ${\Im }_i:J\times R\to R$ (for $i=1,2$) are continuous and there exist constants ${\delta }_i\in [0,1)$ such that

$$|{\Im }_i(\zeta ,u\left(\zeta \right))-{\Im }_i(\zeta ,v\left(\zeta \right))|\le {\delta }_i|u-v|.$$

The following lemma connects the hybrid differential equations with quadratic integral equations, offering a simpler representation for solutions. This connection is pivotal for proving the paper’s core results, such as demonstrating solution existence, uniqueness, and exploring Ulam–Hyers stability.

Lemma 2.

We assume that Assumptions 1–4 hold. Then, the solution of the first hybrid differential equation of fractional order $\alpha \in (1,2)$,

$$\left\{\begin{array}{c}{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right)={\Im }_1(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\zeta ,y\left(\zeta \right)),\zeta \in J=[0,T],\\ {\left.{\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right|}_{\zeta =0}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon ,\\ {\left.{\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right|}_{\zeta =T}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_2\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon ,\end{array}\right.$$

(4)

is also a solution of the following quadratic integral equation:

$$x\left(\zeta \right)=v_1(\zeta ,x\left(\zeta \right))+g_1\left(\zeta ,x\left(\zeta \right)\right){\int }_0^{T}G(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon ,$$

(5)

where $v_1\left(\zeta ,x\left(\zeta \right)\right)$ is a continuous function in X given by

$$\begin{array}{cc}\hfill v_1\left(\zeta ,x\left(\zeta \right)\right)& =f_1\left(\zeta ,x\left(\zeta \right)\right)+\frac{g_1\left(\zeta ,x\left(\zeta \right)\right)}{\Gamma \left(\gamma \right)}\left[{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon +\right.\hfill \\ \hfill & \left.\frac{\zeta }{T}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left[h_2\left(\upsilon ,x\left(\upsilon \right)\right)-h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right]d\upsilon \right],\hfill \end{array}$$

and $G\left(\zeta ,\upsilon \right)$ is the Green’s function defined by

$$G(\zeta ,\upsilon )=\left\{\begin{array}{c}{\displaystyle \frac{{(\zeta -\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}}-{\displaystyle \frac{\zeta {(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}}\mathit{for}0\le \upsilon \le \zeta \le T,\hfill \\ -{\displaystyle \frac{\zeta {(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}}\mathit{for}0\le \zeta \le \upsilon \le T.\hfill \end{array}\right.$$

Proof. 

We let $x\in C(J,R)$ be a soltuion of the hybrid differential Equation (4). Then, utilizing Lemma 1 and applying ${\mathcal{I}}_{0^{+}}^{\alpha }$ to both sides of Equation (4), we obtain

$${\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}={\mathcal{I}}_{0^{+}}^{\alpha }{\Im }_1(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\zeta ,y\left(\zeta \right))+c_0+c_1\zeta ,\mathrm{where}{c}_0,c_1\in R.$$

Hence,

$$x\left(\zeta \right)=f_1(\zeta ,x\left(\zeta \right))+g_1\left(\zeta ,x\left(\zeta \right)\right)\left[\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\zeta }{\left(\zeta -\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon +c_0+c_1\zeta \right].$$

Applying the integral boundary conditions of the CHFDE (1), we obtain

$$c_0=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon ,$$

and

$$\begin{array}{cc}\hfill c_1& =\frac{1}{T}\left\{\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left[h_2\left(\upsilon ,x\left(\upsilon \right)\right)-h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right]d\upsilon \right.\hfill \\ \hfill & -\left.\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^T{\left(T-\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right\}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill x\left(\zeta \right)& =f_1\left(\zeta ,x\left(\zeta \right)\right)+g_1\left(\zeta ,x\left(\zeta \right)\right)\left\{\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\zeta }{\left(\zeta -\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right.\hfill \\ \hfill & +\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon \hfill \\ \hfill & +\frac{\zeta }{T}\left\{\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left[h_2\left(\upsilon ,x\left(\upsilon \right)\right)-h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right]d\upsilon \right.\hfill \\ \hfill & -\left.\left.\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^T{\left(T-\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right\}\right\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =f_1\left(\zeta ,x\left(\zeta \right)\right)+\frac{g_1\left(\zeta ,x\left(\zeta \right)\right)}{\Gamma \left(\gamma \right)}\left[{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon \right.\hfill \\ \hfill & +\left.\frac{\zeta }{T}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left[h_2\left(\upsilon ,x\left(\upsilon \right)\right)-h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right]d\upsilon \right]\hfill \\ \hfill & +g_1\left(\zeta ,x\left(\zeta \right)\right)\left[\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\zeta }{\left(\zeta -\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right.\hfill \\ \hfill & -\frac{\zeta }{\Gamma \left(\alpha \right)}\left({\int }_0^{\zeta }{\left(T-\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right.\hfill \\ \hfill & \left.\left.+{\int }_0^T{\left(T-\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right)\right],\hfill \\ \hfill & =v_1(\zeta ,x\left(\zeta \right))+g_1\left(\zeta ,x\left(\zeta \right)\right){\int }_0^T{G}_1(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon .\hfill \end{array}$$

Furthermore, we illustrate that Equation (5) satisfies Problem (4) when it is combined with the boundary conditions from (4), thus finalizing the proof of the lemma. In particular, when we differentiate both sides of (5) with respect to ζ, we obtain the following:

$${}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right)={\Im }_1(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\zeta ,y\left(\zeta \right)),$$

Once again, this refers to Equation (4). Now, since

$$\begin{array}{cc}\hfill x\left(\zeta \right)& =f_1\left(\zeta ,x\left(\zeta \right)\right)+g_1\left(\zeta ,x\left(\zeta \right)\right)\left\{\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\zeta }{\left(\zeta -\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right.\hfill \\ \hfill & +\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon \hfill \\ \hfill & +\frac{\zeta }{T}\left\{\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left[h_2\left(\upsilon ,x\left(\upsilon \right)\right)-h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right]d\upsilon \right.\hfill \\ \hfill & -\left.\left.\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^T{\left(T-\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right\}\right\},\hfill \end{array}$$

then

$$x\left(0\right)=f_1\left(0,x\left(0\right)\right)+g_1\left(0,x\left(0\right)\right)\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon ,$$

which implies that

$${\left.{\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right|}_{\zeta =0}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon .$$

Similarly,

$$\begin{array}{cc}\hfill x\left(T\right)& =f_1\left(T,x\left(T\right)\right)+g_1\left(T,x\left(T\right)\right)\left\{\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^T{\left(T-\upsilon \right)}^{\alpha -1}\Im \left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right.\hfill \\ \hfill & +\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon \hfill \\ \hfill & +\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left[h_2\left(\upsilon ,x\left(\upsilon \right)\right)-h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right]d\upsilon \hfill \\ \hfill & -\left.\left.\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^T{\left(T-\upsilon \right)}^{\alpha -1}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon \right\}\right\}\hfill \\ \hfill & =f_1\left(T,x\left(T\right)\right)+g_1\left(T,x\left(T\right)\right)\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_2\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon ,\hfill \end{array}$$

which implies that

$${\left.{\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right|}_{\zeta =T}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_2\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon .$$

Therefore, the proof is complete. □

Lemma 3.

We assume that Assumptions 1–4 hold. Then, the solution of the second hybrid differential equation of fractional order $\alpha \in (1,2)$,

$$\left\{\begin{array}{c}{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}\right)={\Im }_2(\zeta ,{\mathcal{I}}^{\beta }{u}_2(\zeta ,x\left(\zeta \right)),\zeta \in J=[0,T],\\ {\left.{\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}\right|}_{\zeta =0}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{k}_1\left(\upsilon ,y\left(\upsilon \right)\right)d\upsilon ,\\ {\left.{\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}\right|}_{\zeta =T}=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{k}_2\left(\upsilon ,y\left(\upsilon \right)\right)d\upsilon ,\end{array}\right.$$

(6)

is also a solution of the following quadratic integral equation:

$$y\left(\zeta \right)=v_2(\zeta ,x\left(\zeta \right))+g_2\left(\zeta ,x\left(\zeta \right)\right){\int }_0^{T}G(\zeta ,\upsilon ){\Im }_2\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_2\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon ,$$

(7)

where $v_2\left(\zeta ,x\left(\zeta \right)\right)$ is a continuous function in X given by

$$\begin{array}{cc}\hfill v_2\left(\zeta ,x\left(\zeta \right)\right)& =f_2\left(\zeta ,x\left(\zeta \right)\right)+\frac{g_2\left(\zeta ,x\left(\zeta \right)\right)}{\Gamma \left(\gamma \right)}\left[{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{k}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon +\right.\hfill \\ \hfill & \left.\frac{\zeta }{T}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left[k_2\left(\upsilon ,x\left(\upsilon \right)\right)-k_1\left(\upsilon ,x\left(\upsilon \right)\right)\right]d\upsilon \right],\hfill \end{array}$$

and $G\left(\zeta ,\upsilon \right)$ is the Green’s function defined by

$$G(\zeta ,\upsilon )=\left\{\begin{array}{c}{\displaystyle \frac{{(\zeta -\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}}-{\displaystyle \frac{\zeta {(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}}\mathit{for}0\le \upsilon \le \zeta \le T,\hfill \\ -{\displaystyle \frac{\zeta {(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}}\mathit{for}0\le \zeta \le \upsilon \le T.\hfill \end{array}\right.$$

Proof. 

Similar to that of Lemma 2. □

3.1. Existence of Solutions

In the following section, we prove the essential lemmas and theorems concerning the existence of coupled solutions for CHFDE (1). To begin, we introduce the following assumption:

Assumption 7.

There exists a positive real number $G_0$ such that

$$G_0=sup\left\{G\left(\zeta ,\upsilon \right)\right|\}\mathit{for}\mathit{all}\left(\zeta ,\upsilon \right)\in J\times J.$$

The following theorem establishes the existence of at least one mild solution for the CHFDE (1) based on the stated assumptions. This result is essential for demonstrating the practicality of solving the problem being studied and forms the basis for further analysis in the study.

Theorem 2.

We assume that Assumptions 1–7 hold. Then, the CHFDE (1) has at least one mild solution defined in $J\times J$.

Proof. 

By Lemma 2, the coupled solutions of CHFDE (1) are equivalent to the solutions of the following system of coupled fractional quadratic integral equations:

$$x\left(\zeta \right)=v_1(\zeta ,x\left(\zeta \right))+g_1\left(\zeta ,x\left(\zeta \right)\right){\int }_0^T{G}_1(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y\left(\upsilon \right)\right)\right)d\upsilon ,$$

(8)

and

$$y\left(\zeta \right)=v_2(\zeta ,y\left(\zeta \right))+g_2\left(\zeta ,y\left(\zeta \right)\right){\int }_0^T{G}_2(\zeta ,\upsilon ){\Im }_2\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_2\left(\upsilon ,x\left(\upsilon \right)\right)\right)d\upsilon .$$

(9)

If $X=C(J,R)$ denotes a Banach space, then the Cartesian product space $E=X\times X$ also forms a Banach space. We define the subset $S=S_1\times S_2$ of E as $S=(x,y)\in E:\left|\right(x,y\left)\right|\le \rho$, where $\rho$ satisfies Inequality (2). It is evident that S is a closed, convex, and bounded subset of E.

By combining functions $f_i$, $g_i$, and ${\Im }_i$ for $i=1,2$, we obtain three operators: $\mathcal{A}=({\mathcal{A}}_1,{\mathcal{A}}_2):E\to E$, $\mathcal{B}=({\mathcal{B}}_1,{\mathcal{B}}_2):S\to E$, and $\mathcal{C}=({\mathcal{C}}_1,{\mathcal{C}}_2):E\to E$.

$$\mathcal{A}\left(x,y\right)=\left({\mathcal{A}}_{1}x,{\mathcal{A}}_{2}y\right)=\left(g_1(\zeta ,x\left(\zeta \right)),g_2\left(\zeta ,y\left(\zeta \right)\right)\right),$$

(10)

$$\begin{array}{ccc}\hfill \mathcal{B}\left(x,y\right)& =& \left({\mathcal{B}}_{1}y,{\mathcal{B}}_{2}x\right)\hfill \\ & =& \left({\int }_0^T{G}_1\left(\zeta ,\upsilon \right){\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon ,{\int }_0^T{G}_2\left(\zeta ,\upsilon \right){\Im }_2(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_2(\upsilon ,x\left(\upsilon \right))d\upsilon \right),\hfill \end{array}$$

(11)

and

$$\mathcal{C}\left(x,y\right)=\left({\mathcal{C}}_{1}x,{\mathcal{C}}_{2}y\right)=\left(v_1(\zeta ,x\left(\zeta \right)),v_2(\zeta ,y\left(\zeta \right))\right).$$

(12)

Therefore, the coupled system of quadratic integral Equations (8) and (9) can be expressed as the following operator equation:

$$\begin{array}{cc}\hfill \left(x,y\right)\left(\zeta \right)& =\mathcal{A}\left(x,y\right)\left(\zeta \right)·\mathcal{B}\left(x,y\right)\left(\zeta \right)+\mathcal{C}\left(x,y\right)\left(\zeta \right),\hfill \\ \hfill & =\left({\mathcal{A}}_{1}x\left(\zeta \right).{\mathcal{B}}_{1}y\left(\zeta \right)+{\mathcal{C}}_{1}x\left(\zeta \right),{\mathcal{A}}_{2}y\left(\zeta \right).{\mathcal{B}}_{2}x\left(\zeta \right)+{\mathcal{C}}_{2}x\left(\zeta \right)\right)\forall \zeta \in J,\hfill \end{array}$$

(13)

which is equivalent to the following coupled system:

$$\left\{\begin{array}{c}{\mathcal{A}}_{1}x\left(\zeta \right).{\mathcal{B}}_{1}y\left(\zeta \right)+{\mathcal{C}}_{1}x\left(\zeta \right)=x\left(\zeta \right)\forall \zeta \in J,\\ {\mathcal{A}}_{2}y\left(\zeta \right).{\mathcal{B}}_{2}x\left(\zeta \right)+{\mathcal{C}}_{2}x\left(\zeta \right)=y\left(\zeta \right)\forall \zeta \in J.\end{array}\right.$$

In the following steps, we show that the operators $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ meet all the requirements outlined in Lemma 1.

Step 1: Operators $\mathcal{A}=\left({\mathcal{A}}_1,{\mathcal{A}}_2\right)$ and $\mathcal{C}=\left({\mathcal{C}}_1,{\mathcal{C}}_2\right)$ are Lipschitz defined on E.

We let $(x,y)\in E$. Then, based on Assumption 2 and for $i=1,2$, we can deduce that

$$|{\mathcal{A}}_{i}x\left(\zeta \right)-{\mathcal{A}}_{i}y\left(\zeta \right)|\hspace{0.17em}=\hspace{0.17em}|g_i(\zeta ,x\left(\zeta \right))-g_i(\zeta ,y\left(\zeta \right))|\le {\mu }_i\left(\zeta \right)|x\left(\zeta \right)-y\left(\zeta \right)|.$$

This implies that if the supremum is taken over all $\zeta \in J$ and for all $(x,y)\in E$, then

$$\parallel {\mathcal{A}}_{i}x-{\mathcal{A}}_{i}y\parallel \hspace{0.17em}\le \hspace{0.17em}\parallel {\mu }_i\parallel \parallel x-y\parallel .$$

Hence, ${\mathcal{A}}_i$ represents a Lipschitz operator over X with a Lipschitz constant of $\parallel {\mu }_i\parallel$ for $i=1,2$.

Furthermore, considering Assumptions 2 and 4 along with Remark 1, for any $(x,y)\in E$, we obtain

$$\begin{array}{cc}\hfill & |{\mathcal{C}}_{1}x\left(\zeta \right)-{\mathcal{C}}_{1}y\left(\zeta \right)|\hfill \\ \hfill & =\hspace{0.17em}|v_1(\zeta ,x\left(\zeta \right))-v_1(\zeta ,y\left(\zeta \right))|,\hfill \\ \hfill & =\hspace{0.17em}|f_1\left(\zeta ,x\left(\zeta \right)\right)-f_1\left(\zeta ,y\left(\zeta \right)\right),\hfill \\ \hfill & +\frac{g_1\left(\zeta ,x\left(\zeta \right)\right)}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,x\left(\upsilon \right)\right)d\upsilon -\frac{g_1\left(\zeta ,y\left(\zeta \right)\right)}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{h}_1\left(\upsilon ,y\left(\upsilon \right)\right)d\upsilon \hfill \\ \hfill & +\frac{\zeta }{T}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left[h_2\left(\upsilon ,x\left(\upsilon \right)\right)-h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right]d\upsilon \hfill \\ \hfill & -\frac{\zeta }{T}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left[h_2\left(\upsilon ,y\left(\upsilon \right)\right)-h_1\left(\upsilon ,y\left(\upsilon \right)\right)\right]d\upsilon |,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \hspace{0.17em}|f_1\left(\zeta ,x\left(\zeta \right)\right)-f_1\left(\zeta ,y\left(\zeta \right)\right)|+\frac{|g_1\left(\zeta ,x\left(\zeta \right)\right)-g_1\left(\zeta ,y\left(\zeta \right)\right)|}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left|h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right|d\upsilon \hfill \\ \hfill & +\frac{|g_1\left(\zeta ,y\left(\zeta \right)\right)|}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}|h_1\left(\upsilon ,x\left(\upsilon \right)\right)-h_1\left(\upsilon ,y\left(\upsilon \right)\right)|d\upsilon \hfill \\ \hfill & +\frac{\zeta }{T}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left|h_2\left(\upsilon ,x\left(\upsilon \right)\right)-h_2\left(\upsilon ,y\left(\upsilon \right)\right)\right|d\upsilon \hfill \\ \hfill & +\frac{\zeta }{T}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left|h_1\left(\upsilon ,y\left(\upsilon \right)\right)-h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right|d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\sigma }_1\left(\zeta \right)|x\left(\zeta \right)-y\left(\zeta \right)|+\frac{{\mu }_1\left(\zeta \right)|x\left(\zeta \right)-y\left(\zeta \right)|}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left(H_1+{\omega }_1\left(\zeta \right)\left|x\right|\right)d\upsilon \hfill \\ \hfill & +\frac{{\breve{G}}_1+{\mu }_1\left|y\right|}{\Gamma \left(\gamma \right)}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{\omega }_1\left(\zeta \right)|x\left(\zeta \right)-y\left(\zeta \right)|d\upsilon +\frac{\zeta }{T}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{\omega }_2\left(\zeta \right)|x\left(\zeta \right)-y\left(\zeta \right)|d\upsilon \hfill \\ \hfill & +\frac{\zeta }{T}{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}{\omega }_1\left(\zeta \right)|y\left(\zeta \right)-x\left(\zeta \right)|d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\sigma }_1\left(\zeta \right)|x\left(\zeta \right)-y\left(\zeta \right)|+\frac{{\mu }_1\left(\zeta \right)|x\left(\zeta \right)-y\left(\zeta \right)|}{\Gamma \left(\gamma +1\right)}\left(H_1+{\omega }_1\left(\zeta \right)\left|x\right|\right)+\frac{{\breve{G}}_1+{\mu }_1\left|y\right|}{\Gamma \left(\gamma +1\right)}{\omega }_1\left(\zeta \right)|x\left(\zeta \right)-y\left(\zeta \right)|\hfill \\ \hfill & +\frac{\zeta }{\gamma T}\left({\omega }_2\left(\zeta \right)|x\left(\zeta \right)-y\left(\zeta \right)|+{\omega }_1\left(\zeta \right)|y\left(\zeta \right)-x\left(\zeta \right)|\right).\hfill \end{array}$$

Taking supremum over all $\zeta \in J$, we deduce that

$$\begin{array}{cc}\hfill & |{\mathcal{C}}_{1}x\left(\zeta \right)-{\mathcal{C}}_{1}y\left(\zeta \right)|\hfill \\ \hfill & \le \left(∥{\sigma }_1∥+\frac{∥{\mu }_1∥}{\Gamma \left(\gamma +1\right)}\left(H_1+∥{\omega }_1∥∥x∥\right)+\frac{{\breve{G}}_1+∥{\mu }_1∥∥y∥}{\Gamma \left(\gamma +1\right)}∥{\omega }_1∥+\frac{1}{\gamma }(∥{\omega }_2∥+∥{\omega }_1∥)\right)∥x-y∥.\hfill \end{array}$$

This implies that ${\mathcal{C}}_1$ is a Lipschitz operator with Lipschitz constant

$$c_1=\left(∥{\sigma }_1∥+\frac{∥{\mu }_1∥}{\Gamma \left(\gamma +1\right)}\left(H_1+∥{\omega }_1∥∥x∥\right)+\frac{{\breve{G}}_1+∥{\mu }_1∥∥y∥}{\Gamma \left(\gamma +1\right)}∥{\omega }_1∥+\frac{1}{\gamma }(∥{\omega }_1∥+∥{\omega }_2∥)\right).$$

Similarly, operator ${\mathcal{C}}_2$ is Lipschitz with constant

$$c_2=\left(∥{\sigma }_2∥+\frac{∥{\mu }_2∥}{\Gamma \left(\gamma +1\right)}\left(H_2+∥{\omega }_2∥∥x∥\right)+\frac{{\breve{G}}_2+∥{\mu }_2∥∥y∥}{\Gamma \left(\gamma +1\right)}∥{\omega }_2∥+\frac{1}{\gamma }(∥{\omega }_1∥+∥{\omega }_2∥)\right).$$

Hence, $∥{\mathcal{C}}_{i}x-{\mathcal{C}}_{i}y∥\le ∥c_i∥∥x-y∥$, which implies that ${\mathcal{C}}_i$$\left(i=1,2\right)$ is a Lipschitz operator on E with Lipschitz constant $∥c_i∥$.

In addition, we deduce for all $u=\left(x_1,y_1\right)\in E$ and $v=\left(x_2,y_2\right)\in E$ that

$$\begin{array}{cc}\hfill ∥\mathcal{A}u-\mathcal{A}v∥& =∥\mathcal{A}\left(x_1,y_1\right)-\mathcal{A}\left(x_2,y_2\right)∥,\hfill \\ \hfill & =∥\left({\mathcal{A}}_1{x}_1,{\mathcal{A}}_2{y}_1\right)-\left({\mathcal{A}}_1{x}_2,{\mathcal{A}}_2{y}_2\right)∥,\hfill \\ \hfill & =∥\left({\mathcal{A}}_1{x}_1-{\mathcal{A}}_1{x}_2,{\mathcal{A}}_2{y}_1-{\mathcal{A}}_2{y}_2\right)∥,\hfill \\ \hfill & \le ∥{\mathcal{A}}_1{x}_1-{\mathcal{A}}_1{x}_2∥+∥{\mathcal{A}}_2{y}_1-{\mathcal{A}}_2{y}_2∥,\hfill \\ \hfill & \le \hspace{0.17em}\parallel {\mu }_1\parallel ∥x_1-x_2∥+\parallel {\mu }_2\parallel ∥y_1-y_2∥,\hfill \\ \hfill & \le \parallel \mu \parallel \left(∥x_1-x_2∥+∥y_1-y_2∥\right),\hfill \\ \hfill & \le \parallel \mu \parallel ∥u-v∥.\hfill \end{array}$$

In the same way, it can be obtained that operator $\mathcal{C}$ is Lipschitz on E with Lipschitz constant $∥c∥=max\left\{∥c_1∥,∥c_2∥\right\}$.

Step 2: Operator $\mathcal{B}=\left({\mathcal{B}}_1,{\mathcal{B}}_2\right)$ is compact and continuous.

First, we establish the continuity of the operator $\mathcal{B}$ on E.

Consider a sequence of pairs $u_n=\left(x_n,y_n\right)\in S$ that converges to the pair $u=\left(x,y\right)\in S$, i.e., $\left\{\left(x_n,y_n\right)\right\}\to \left(x,y\right)$ as $n\to \infty$. By applying the Lebesgue Dominated Convergence Theorem, we can conclude that

$$\underset{n\to \infty }{lim}{\mathcal{I}}_{0^{+}}^{\beta }{u}_2(\upsilon ,x_n\left(\upsilon \right))={\mathcal{I}}_{0^{+}}^{\beta }{u}_2(\upsilon ,x\left(\upsilon \right)),$$

and

$$\underset{n\to \infty }{lim}{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y_n\left(\upsilon \right))={\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right)).$$

Furthermore, the continuity of ${\Im }_1(\zeta ,y\left(\zeta \right))$ holds for all $y\in R$. Consequently, through the utilization of properties associated with fractional order integrals and the application of the Lebesgue Dominated Convergence Theorem, we deduce that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{\mathcal{B}}_1{y}_n\left(\zeta \right)& =\underset{n\to \infty }{lim}{\int }_0^T{G}_1\left(\zeta ,\upsilon \right){\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y_n\left(\upsilon \right))d\upsilon ,\hfill \\ \hfill & ={\int }_0^T{G}_1\left(\zeta ,\upsilon \right)\underset{n\to \infty }{lim}{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y_n\left(\upsilon \right))d\upsilon ,\hfill \\ \hfill & ={\int }_0^T{G}_1\left(\zeta ,\upsilon \right){\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon ,\hfill \\ \hfill & ={\mathcal{B}}_{1}y\left(\zeta \right).\hfill \end{array}$$

In a similar manner, ${lim}_{n\to \infty }{\mathcal{B}}_2{x}_n\left(\zeta \right)={\mathcal{B}}_{2}x\left(\zeta \right)$. Thus, operators ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ are continuous on S into S.

Hence,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}\mathcal{B}{u}_n\left(\zeta \right)& =(\underset{n\to \infty }{lim}{\mathcal{B}}_1{y}_n\left(\zeta \right),\underset{n\to \infty }{lim}{\mathcal{B}}_2{x}_n\left(\zeta \right)),\hfill \\ \hfill & =({\mathcal{B}}_{1}y\left(\zeta \right),{\mathcal{B}}_{2}x\left(\zeta \right)),\hfill \\ \hfill & =\mathcal{B}\left(x,y\right)\left(\zeta \right),\hfill \\ \hfill & =\mathcal{B}u\left(\zeta \right).\hfill \end{array}$$

Therefore, $\mathcal{B}{u}_n\to \mathcal{B}u$ as $n\to \infty$ uniformly on $R^{+}$ and $\mathcal{B}$ is a continuous operator on S into S.

Second, we establish the compactness of operator $\mathcal{B}$ on S.

We let $(x,y)\in S$ be arbitrary. Then, according to Assumption 3, it is clear that

$$\begin{array}{cc}\hfill |{\mathcal{B}}_{1}y\left(\zeta \right)|& =\hspace{0.17em}|{\int }_0^T{G}_1\left(\zeta ,\upsilon \right){\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon |,\hfill \\ \hfill & \le {\int }_0^T\left|G_1\left(\zeta ,\upsilon \right)\right|\left|{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon ,\hfill \\ \hfill & \le G_0{\int }_0^T\left[{\mathit{a}}_1\left(\upsilon \right)+{\mathit{b}}_1\left(\upsilon \right){\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|\right]d\upsilon ,\hfill \\ \hfill & \le G_0{\int }_0^T\left|{\mathit{a}}_1\left(\upsilon \right)\right|d\upsilon +G_0{\int }_0^T\left|{\mathit{b}}_1\left(\upsilon \right)\right|{\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon ,\hfill \\ \hfill & \le G_0{\mathcal{M}}_1{\int }_0^{T}d\upsilon +G_0♭{\int }_0^T{\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon ,\hfill \\ \hfill & \le G_0{\mathcal{M}}_1{\int }_0^{T}d\upsilon +G_0♭{\int }_0^T{\mathcal{I}}_{0^{+}}^{\beta }{\mathit{m}}_1\left(\upsilon \right)d\upsilon ,\hfill \\ \hfill & \le G_0{\mathcal{M}}_{1}T+G_0♭T{\mathcal{I}}_{0^{+}}^{\beta -\gamma }{\mathcal{I}}_{0^{+}}^{\gamma }{\mathit{m}}_1\left(\zeta \right),\hfill \\ \hfill & \le G_0{\mathcal{M}}_{1}T+G_0♭T{\mathcal{N}}_1{\int }_0^{\zeta }\frac{{\left(\zeta -\upsilon \right)}^{\beta -\gamma -1}}{\Gamma \left(\beta -\gamma \right)}d\upsilon ,\hfill \\ \hfill & \le G_0\mathcal{M}T+G_{0}T♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma (\beta -\gamma +1)}.\hfill \end{array}$$

Taking supremum over all $\zeta \in \mathcal{I}$, it is acquired that

$$\parallel {\mathcal{B}}_{1}y\parallel \hspace{0.17em}\le \hspace{0.17em}{G}_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma (\beta -\gamma +1)}\right).$$

Thus, operator ${\mathcal{B}}_1$ is uniformly bounded on S.

Similarly,

$$\parallel {\mathcal{B}}_{2}x\parallel \hspace{0.17em}\le \hspace{0.17em}{G}_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma (\beta -\gamma +1)}\right),$$

which implies that operator ${\mathcal{B}}_2$ is also uniformly bounded on $S_2$.

Thus, for any $u=\left(x,y\right)\in S$, we have

$$\begin{array}{cc}\hfill \parallel \mathcal{B}u\parallel & =\hspace{0.17em}\parallel \mathcal{B}\left(x,y\right)\parallel \hspace{0.17em}=\hspace{0.17em}\parallel \left({\mathcal{B}}_{1}y,{\mathcal{B}}_{2}x\right)\parallel \hspace{0.17em}=\hspace{0.17em}\parallel {\mathcal{B}}_{1}y\parallel +\hspace{0.17em}\parallel {\mathcal{B}}_{2}x\parallel ,\hfill \\ \hfill & \le 2G_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma (\beta -\gamma +1)}\right).\hfill \end{array}$$

(14)

Therefore, operator $\mathcal{B}$ is uniformly bounded on S.

To establish that $\mathcal{B}\left(S\right)$ is an equicontinuous set in E, we consider ${\zeta }_1$ and ${\zeta }_2$ in J such that ${\zeta }_1<{\zeta }_2$ and a pair $(x,y)\in S$. Then,

$$\begin{array}{cc}\hfill & \left|{\mathcal{B}}_{1}y\left({\zeta }_2\right)-B_{1}y\left({\zeta }_1\right)\right|\hfill \\ \hfill & =\hspace{0.17em}|{\int }_0^{{\zeta }_2}\left[\frac{{({\zeta }_2-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{{\zeta }_2{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon \hfill \\ \hfill & -{\int }_{{\zeta }_2}^T\left[\frac{{\zeta }_2{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon \hfill \\ \hfill & -{\int }_0^{{\zeta }_1}\left[\begin{array}{c}\frac{{({\zeta }_1-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}\\ -\frac{{\zeta }_1{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\end{array}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon +{\int }_{{\zeta }_1}^T\left[\frac{{\zeta }_1{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon |,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\hspace{0.17em}|{\int }_0^{{\zeta }_1}\left[\frac{{({\zeta }_2-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{{\zeta }_2{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon \hfill \\ \hfill & -{\int }_0^{{\zeta }_1}\left[\frac{{({\zeta }_1-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{{\zeta }_1{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon \hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left[\frac{{({\zeta }_2-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{{\zeta }_2{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon \hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left[\frac{{\zeta }_1{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon +{\int }_{{\zeta }_2}^T\left[\frac{{\zeta }_1{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon \hfill \\ \hfill & -{\int }_{{\zeta }_2}^T\left[\frac{{\zeta }_2{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon |,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =|{\int }_0^{{\zeta }_1}\left[\frac{{({\zeta }_2-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{{\zeta }_2{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}-\frac{{({\zeta }_1-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}+\frac{{\zeta }_1{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon \hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left[\frac{{({\zeta }_2-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{{\zeta }_2{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}+\frac{{\zeta }_1{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon \hfill \\ \hfill & +{\int }_{{\zeta }_2}^T\left[\frac{{\zeta }_1{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}-\frac{{\zeta }_2{(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))d\upsilon |,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\int }_0^{{\zeta }_1}\left[\frac{{({\zeta }_2-\upsilon )}^{\alpha -1}-{({\zeta }_1-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]\left|{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon \hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left[\frac{{({\zeta }_2-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]\left|{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon \hfill \\ \hfill & +{\int }_{{\zeta }_2}^T\left[\frac{\left({\zeta }_1-{\zeta }_2\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]\left|{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon .\hfill \end{array}$$

According to Assumption 3, we deduce that

$$\begin{array}{cc}\hfill & \left|{\mathcal{B}}_{1}y\left({\zeta }_2\right)-B_{1}y\left({\zeta }_1\right)\right|\hfill \\ \hfill & \le {\int }_0^{{\zeta }_1}\left[\frac{{({\zeta }_2-\upsilon )}^{\alpha -1}-{({\zeta }_1-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]\left(\begin{array}{c}\left|{\mathit{a}}_1\left(\upsilon \right)\right|\\ +\left|{\mathit{b}}_1\left(\upsilon \right)\right|{\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|\end{array}\right)d\upsilon \hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left[\frac{{({\zeta }_2-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}-\frac{\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]\left(\left|{\mathit{a}}_1\left(\upsilon \right)\right|+\left|{\mathit{b}}_1\left(\upsilon \right)\right|{\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|\right)d\upsilon \hfill \\ \hfill & +{\int }_{{\zeta }_2}^T\left[\frac{\left({\zeta }_1-{\zeta }_2\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right]\left(\left|{\mathit{a}}_1\left(\upsilon \right)\right|+\left|{\mathit{b}}_1\left(\upsilon \right)\right|{\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|\right)d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le ∥{\mathit{a}}_1∥\left[{\int }_0^{{\zeta }_1}\left(\frac{T\left[{({\zeta }_2-\upsilon )}^{\alpha -1}-{({\zeta }_1-\upsilon )}^{\alpha -1}\right]-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}\right)d\upsilon \right.\hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left(\frac{T{({\zeta }_2-\upsilon )}^{\alpha -1}-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right)d\upsilon \left.+{\int }_{{\zeta }_2}^T\frac{\left({\zeta }_1-{\zeta }_2\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}d\upsilon \right]\hfill \\ \hfill & +∥{\mathit{b}}_1∥\left[{\int }_0^{{\zeta }_1}\left(\frac{T\left[{({\zeta }_2-\upsilon )}^{\alpha -1}-{({\zeta }_1-\upsilon )}^{\alpha -1}\right]-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}\right){\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon \right.\hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left(\frac{T{({\zeta }_2-\upsilon )}^{\alpha -1}-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right){\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon \hfill \\ \hfill & \left.+{\int }_{{\zeta }_2}^T\frac{\left({\zeta }_1-{\zeta }_2\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}{\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon \right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le ∥{\mathit{a}}_1∥\left[\frac{({\zeta }_2^{\alpha }-{\zeta }_1^{\alpha })T-({\zeta }_2-{\zeta }_1)T^{\alpha }}{T\Gamma (\alpha +1)}\right]\hfill \\ \hfill & +∥{\mathit{b}}_1∥\left[{\int }_0^{{\zeta }_1}\left(\frac{T\left[{({\zeta }_2-\upsilon )}^{\alpha -1}-{({\zeta }_1-\upsilon )}^{\alpha -1}\right]-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}\right){\mathcal{I}}_{0^{+}}^{\beta -\gamma }{\mathcal{I}}_{0^{+}}^{\gamma }{\mathit{m}}_1\left(\upsilon \right)d\upsilon \right.\hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left(\frac{T{({\zeta }_2-\upsilon )}^{\alpha -1}-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right){\mathcal{I}}_{0^{+}}^{\beta -\gamma }{\mathcal{I}}_{0^{+}}^{\gamma }{\mathit{m}}_1\left(\upsilon \right)d\upsilon \hfill \\ \hfill & \left.+{\int }_{{\zeta }_2}^T\frac{\left({\zeta }_1-{\zeta }_2\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}{\mathcal{I}}_{0^{+}}^{\beta -\gamma }{\mathcal{I}}_{0^{+}}^{\gamma }{\mathit{m}}_1\left(\upsilon \right)d\upsilon \right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \mathcal{M}\left[\frac{({\zeta }_2^{\alpha }-{\zeta }_1^{\alpha })T-({\zeta }_2-{\zeta }_1)T^{\alpha }}{T\Gamma (\alpha +1)}\right]\hfill \\ \hfill & +♭\mathcal{N}\left[{\int }_0^{{\zeta }_1}\left(\frac{T\left[{({\zeta }_2-\upsilon )}^{\alpha -1}-{({\zeta }_1-\upsilon )}^{\alpha -1}\right]-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}\right){\int }_0^{\upsilon }\frac{{\left(\upsilon -\tau \right)}^{\beta -\gamma -1}}{\Gamma \left(\beta -\gamma \right)}d\tau d\upsilon \right.\hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left(\frac{T{({\zeta }_2-\upsilon )}^{\alpha -1}-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right){\int }_0^{\upsilon }\frac{{\left(\upsilon -\tau \right)}^{\beta -\gamma -1}}{\Gamma \left(\beta -\gamma \right)}d\tau d\upsilon \hfill \\ \hfill & \left.+{\int }_{{\zeta }_2}^T\frac{\left({\zeta }_1-{\zeta }_2\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}{\int }_0^{\upsilon }\frac{{\left(\upsilon -\tau \right)}^{\beta -\gamma -1}}{\Gamma \left(\beta -\gamma \right)}d\tau d\upsilon \right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \mathcal{M}\left[\frac{({\zeta }_2^{\alpha }-{\zeta }_1^{\alpha })T-({\zeta }_2-{\zeta }_1)T^{\alpha }}{T\Gamma (\alpha +1)}\right]\hfill \\ \hfill & +♭\mathcal{N}\left[{\int }_0^{{\zeta }_1}\left(\frac{T\left[{({\zeta }_2-\upsilon )}^{\alpha -1}-{({\zeta }_1-\upsilon )}^{\alpha -1}\right]-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{\Gamma \left(\alpha \right)}\right)\frac{{\upsilon }^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}d\upsilon \right.\hfill \\ \hfill & +{\int }_{{\zeta }_1}^{{\zeta }_2}\left(\frac{T{({\zeta }_2-\upsilon )}^{\alpha -1}-\left({\zeta }_2-{\zeta }_1\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\right)\frac{{\upsilon }^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}d\upsilon \hfill \\ \hfill & \left.+{\int }_{{\zeta }_2}^T\frac{\left({\zeta }_1-{\zeta }_2\right){(T-\upsilon )}^{\alpha -1}}{T\Gamma \left(\alpha \right)}\frac{{\upsilon }^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}d\upsilon \right],\hfill \end{array}$$

$$\le \mathcal{M}\left[\frac{({\zeta }_2^{\alpha }-{\zeta }_1^{\alpha })T-({\zeta }_2-{\zeta }_1)T^{\alpha }}{T\Gamma (\alpha +1)}\right]+♭\mathcal{N}\left[\frac{({\zeta }_2^{\alpha }-{\zeta }_1^{\alpha })T-({\zeta }_2-{\zeta }_1)T^{\alpha }}{T\Gamma (\alpha +1)\Gamma \left(\beta -\gamma +1\right)}\right]T^{\beta -\gamma }.$$

Therefore,

$$|{\mathcal{B}}_{1}y\left({\zeta }_2\right)-{\mathcal{B}}_{1}y\left({\zeta }_1\right)|\hspace{0.17em}\le \left(\frac{({\zeta }_2^{\alpha }-{\zeta }_1^{\alpha })T-({\zeta }_2-{\zeta }_1)T^{\alpha }}{T\Gamma (\alpha +1)}\right)\left(\mathcal{M}+\frac{♭\mathcal{M}{T}^{\beta -\alpha }}{\Gamma \left(\beta -\gamma +1\right)}\right),$$

which is independent on $y\in S_2$. The same argument is used for operator ${\mathcal{B}}_2$.

As a result, for each $i=1,2$, we can infer the existence of a ${\delta }_i>0$ corresponding to any given ${\epsilon }_i>0$, such that when $|{\zeta }_2-{\zeta }_1|<{\delta }_i$, the inequality $|{\mathcal{B}}_{i}y\left({\zeta }_2\right)-{\mathcal{B}}_{i}y\left({\zeta }_1\right)|<{\epsilon }_i$ holds for all ${\zeta }_1,{\zeta }_2\in J$. This enables us to select $\epsilon =max\{{\epsilon }_1,{\epsilon }_2\}$ and $\delta =max\{{\delta }_1,{\delta }_2\}$, guaranteeing that if $|{\zeta }_2-{\zeta }_1|<\delta$, then $|\mathcal{B}y\left({\zeta }_2\right)-\mathcal{B}y\left({\zeta }_1\right)|<\epsilon$. This establishes that the set $\mathcal{B}\left(S\right)$ is equicontinuous in E. By employing the Arzela–Ascoli Theorem, we can deduce that $\mathcal{B}\left(S\right)$ is uniformly bounded and equicontinuous in E, thereby confirming its compactness within E. Consequently, operator $\mathcal{B}$ can be identified as a complete continuous operator on S.

Step 3: Now, in order to demonstrate that operator $u=\mathcal{A}u\mathcal{B}v+\mathcal{C}u$ remains bounded for all $u\in E$, we consider $(x,y)\in S_1\times S_2$ and let $u=(x,y)=({\mathcal{A}}_{1}x{\mathcal{B}}_{1}y+{\mathcal{C}}_{1}x,{\mathcal{A}}_{2}y{\mathcal{B}}_{2}x+{\mathcal{C}}_{2}y)\in E$. Then, for $\zeta \in J$, it can be concluded that

$$\begin{array}{cc}\hfill \left|x\right(\zeta \left)\right|& \le |{\mathcal{A}}_{1}x\left(\zeta \right)\left|\right|{\mathcal{B}}_{1}y\left(\zeta \right)|+|{\mathcal{C}}_{1}x\left(\zeta \right)|,\hfill \\ \hfill & \le {|g}_1(\zeta ,x\left(\zeta \right))|{\int }_0^T\left|G_1\left(\zeta ,\upsilon \right)\right|\left|{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))\right|{d\upsilon +|v}_1(\zeta ,x(\zeta \left)\right)|,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left[\right|g_1(\zeta ,x\left(\zeta \right))-g_1(\zeta ,0)|+|g_1(\zeta ,0)\left|\right]{\int }_0^T\left|G_1\left(\zeta ,\upsilon \right)\right|\left|{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon \hfill \\ \hfill & +\left|v_1\left(\zeta ,x\left(\zeta \right)\right)-v_1\left(\zeta ,0\right)+v_1\left(x,0\right)\right|,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left[\right|g_1(\zeta ,x\left(\zeta \right))-g_1(\zeta ,0)|+|g_1(\zeta ,0)\left|\right]{\int }_0^T\left|G_1\left(\zeta ,\upsilon \right)\right|\left|{\Im }_1(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon \hfill \\ \hfill & +\left|f_1\left(x,\zeta \right)-f_1\left(x,0\right)+f_1\left(x,0\right)\right|+\frac{|g_1(\zeta ,x\left(\zeta \right))|}{\Gamma \left(\gamma \right)}\left[{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left|h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right|d\upsilon \right.\hfill \\ \hfill & +\left.{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left|h_2\left(\upsilon ,x\left(\upsilon \right)\right)-h_1\left(\upsilon ,x\left(\upsilon \right)\right)\right|d\upsilon \right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left({\breve{G}}_1+{\mu }_1\left(\zeta \right)\left|x\right|\right){\int }_0^T{G}_0|{\mathit{a}}_1\left(\upsilon \right)+{\mathit{b}}_1\left(\upsilon \right){\mathcal{I}}_{0^{+}}^{\beta }\left|u_1(\upsilon ,y\left(\upsilon \right))\right|d\upsilon \hfill \\ \hfill & +\left|f\left(x,\zeta \right)-f\left(x,0\right)+f\left(x,0\right)\right|+\frac{\left|g\right(\zeta ,x\left(\zeta \right)\left)\right|}{\Gamma \left(\gamma \right)}\left[{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left(k_1\left|x\right|+H_1\right)d\upsilon \right.\hfill \\ \hfill & +\left.{\int }_0^1{\left(1-\upsilon \right)}^{\gamma -1}\left(k_2|x\left(\upsilon \right)|+H_2+k_1\left|x\left(\upsilon \right)\right|+H_1\right)d\upsilon \right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left({\breve{G}}_1+{\mu }_1\left(\zeta \right)\left|x\right|\right)G_0{\int }_0^T\left(|{\mathit{a}}_1\left(\upsilon \right)+{\mathit{b}}_1\left(\upsilon \right){\mathcal{I}}^{\beta }{\mathit{m}}_1\left(\upsilon \right)\right)d\upsilon +\left(F_1+{\sigma }_1\left(\zeta \right)\left|x\right|\right)\hfill \\ \hfill & +\left({\breve{G}}_1+{\mu }_1\left(\zeta \right)\left|x\right|\right)\left[\frac{2\left(H_1+{\omega }_1\left|x\left(\zeta \right)\right|\right)+\left(H_2+{\omega }_2\left|x\left(\zeta \right)\right|\right)}{\Gamma \left(\gamma +1\right)}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(∥{\sigma }_1∥\left|x\left(\zeta \right)\right|+F_1\right)+\left(∥{\mu }_1∥\left|x\left(\zeta \right)\right|+{\breve{G}}_1\right)G_0{\int }_0^T\left(|{\mathit{a}}_1\left(\upsilon \right)+{\mathit{b}}_1\left(\upsilon \right){\mathcal{I}}_{0^{+}}^{\beta -\gamma }{\mathcal{I}}^{\gamma }{\mathit{m}}_1\left(\upsilon \right)\right)d\upsilon \hfill \\ \hfill & +\left({\breve{G}}_1+{\mu }_1\left(\zeta \right)\left|x\right|\right)\left[\frac{2\left(H_1+{\omega }_1\left|x\left(\zeta \right)\right|\right)+\left(H_2+{\omega }_2\left|x\left(\zeta \right)\right|\right)}{\Gamma \left(\gamma +1\right)}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(∥{\sigma }_1∥\left|x\left(\zeta \right)\right|+F_1\right)+\left(∥{\mu }_1∥\left|x\left(\zeta \right)\right|+{\breve{G}}_1\right)G_0\left(\mathcal{M}T+♭\mathcal{N}T{\mathcal{I}}_{0^{+}}^{\beta -\gamma }\left(\upsilon \right)\right)\hfill \\ \hfill & +\left({\breve{G}}_1+{\mu }_1\left(\zeta \right)\left|x\right|\right)\left[\frac{2\left(H_1+{\omega }_1\left|x\left(\zeta \right)\right|\right)+\left(H_2+{\omega }_2\left|x\left(\zeta \right)\right|\right)}{\Gamma \left(\gamma +1\right)}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(∥{\sigma }_1∥\left|x\left(\zeta \right)\right|+F_1\right)+\left(∥{\mu }_1∥\left|x\left(\zeta \right)\right|+{\breve{G}}_1\right)G_{0}T\left(\mathcal{M}+♭\mathcal{N}{\int }_0^{\upsilon }\frac{{\left(\upsilon -u\right)}^{\beta -\gamma -1}}{\Gamma \left(\beta -\gamma \right)}du\right)\hfill \\ \hfill & +\left({\breve{G}}_1+{\mu }_1\left(\zeta \right)\left|x\right|\right)\left[\frac{2\left(H_1+{\omega }_1\left|x\left(\zeta \right)\right|\right)+\left(H_2+{\omega }_2\left|x\left(\zeta \right)\right|\right)}{\Gamma \left(\gamma +1\right)}\right].\hfill \end{array}$$

By considering the supremum over all $\zeta \in J$, we can conclude that

$$\begin{array}{cc}\hfill ∥x∥& \le \left(∥{\sigma }_1∥\left|x\left(\zeta \right)\right|+F_1\right)+\left(∥{\mu }_1∥\left|x\left(\zeta \right)\right|+{\breve{G}}_1\right)G_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}\right)\hfill \\ \hfill & +\left({\breve{G}}_1+{\mu }_1\left(\zeta \right)\left|x\right|\right)\left[\frac{2\left(H_1+{\omega }_1\left|x\left(\zeta \right)\right|\right)+\left(H_2+{\omega }_2\left|x\left(\zeta \right)\right|\right)}{\Gamma \left(\gamma +1\right)}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(∥{\sigma }_1∥∥x∥+F_1\right)\hfill \\ \hfill & +\left(∥{\mu }_1∥∥x∥+{\breve{G}}_1\right)\left[G_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}\right)+\frac{\left(2H_1+H_2\right)+\left(2{\omega }_1+{\omega }_2\right)∥x∥}{\Gamma \left(\gamma +1\right)}\right].\hfill \end{array}$$

Similarly, for every $\zeta \in J$ and $y\in S_2$ such that $y\left(\zeta \right)={\mathcal{A}}_{2}y{\mathcal{B}}_{2}x+{\mathcal{C}}_{2}y\in S_2$, it is also inferred that

$$\begin{array}{cc}\hfill ∥y∥& \le \left(∥{\sigma }_2∥∥y∥+F_2\right)\hfill \\ \hfill & +\left(∥{\mu }_2∥∥y∥+{\breve{G}}_2\right)\left[G_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}\right)+\frac{\left(2H_1+H_2\right)+\left(2{\omega }_1+{\omega }_2\right)∥y∥}{\Gamma \left(\gamma +1\right)}\right].\hfill \end{array}$$

Therefore, for every $u=\left(x,y\right)\in S$, it is attained that

$$\begin{array}{cc}\hfill ∥\left(x,y\right)∥& =∥x∥+∥y∥,\hfill \\ \hfill & \le ∥\sigma ∥\left(∥x∥+∥y∥\right)+F_1+F_2\hfill \\ \hfill & +\left(∥\mu ∥\left(∥x∥+∥y∥\right)+{\breve{G}}_1+{\breve{G}}_2\right)\left[G_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}\right)\right.\hfill \\ \hfill & \left.+\frac{\left(2H_1+H_2\right)+\left(2{\omega }_1+{\omega }_2\right)\left(∥x∥+∥y∥\right)}{\Gamma \left(\gamma +1\right)}\right].\hfill \end{array}$$

If we consider $\mathcal{F}=F_1+F_2$, $\breve{G}={\breve{G}}_1+{\breve{G}}_2$, $\aleph =G_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma \left(\beta -\gamma +1\right)}\right)$, $\mathcal{H}=\left(2H_1+H_2\right)$, and $\mathcal{W}=\left(2{\omega }_1+{\omega }_2\right)$, we have

$$∥u∥\le ∥\sigma ∥∥u∥+\mathcal{F}+\left(∥\mu ∥∥u∥+\breve{G}\right)\left[\aleph +\frac{\mathcal{H}+\mathcal{W}∥u∥}{\Gamma \left(\gamma +1\right)}\right],$$

which implies that $∥u∥\le \rho$ for all

$$\rho \ge \frac{\Gamma \left(\gamma +1\right)}{2∥\mu ∥\mathcal{W}}\left(1-\frac{\breve{G}\mathcal{W}+\mathcal{H}∥\mu ∥}{\Gamma \left(\gamma +1\right)}-∥\mu ∥\aleph -∥\sigma ∥-\sqrt{\Delta }\right),$$

where $\Delta ={\left(-\frac{\breve{G}\mathcal{W}+\mathcal{H}∥\mu ∥}{\Gamma \left(\gamma +1\right)}-∥\mu ∥\aleph -∥\sigma ∥+1\right)}^2-\frac{4∥\mu ∥\mathcal{W}\left(\mathcal{F}+\frac{\breve{G}\mathcal{H}}{\Gamma \left(\gamma +1\right)}+\breve{G}\aleph \right)}{\Gamma \left(\gamma +1\right)}.$

Step 4: Finally, according to the result obtained in (14), it becomes clear that

$$\parallel \mathcal{B}\left(S\right)\parallel =\underset{u\in S}{sup}\left\{\underset{\zeta \in J}{sup}\left|\mathcal{B}u\left(\zeta \right)\right|\right\}\le 2G_{0}T\left(\mathcal{M}+♭\mathcal{N}\frac{T^{\beta -\gamma }}{\Gamma (\beta -\gamma +1)}\right),$$

which leads to the conclusion that $∥\mu ∥\parallel \mathcal{B}\left(S\right)\parallel +∥\sigma ∥<\rho .$

Hence, all the conditions of Theorem 1 are satisfied, and for every $(u,v)\in S_1\times S_2$, we have $u=\mathcal{A}u\mathcal{B}v+\mathcal{C}u$. Therefore, employing the findings from Dhage’s study [35], we can deduce that the coupled system of hybrid fractional differential equations with integral boundary conditions, denoted as CHFDE (1), possesses at least one mild solution defined over the interval J. Thus, the proof is complete. □

3.2. Uniqueness of the Solution

To investigate the necessary conditions for the uniqueness of solutions to the coupled system of fractional quadratic integral Equations (8) and (9), we have to take into account the following assumption:

Assumption 8.

We assume that for $i=1,2$, the functions ${\Im }_i:[0,T]\times R\to R$ and $u_i:[0,T]\times \mathbb{R}\to \mathbb{R}$ are continuous and satisfy the Lipschitz condition. Additionally, we suppose there exist two positive functions $w_i\left(\zeta \right)$ and ${\theta }_i\left(\zeta \right)$ with norms $\parallel w_i\parallel$ and $\parallel {\theta }_i\parallel$, respectively. Then, the following inequalities hold for $i=1,2$:

$$\begin{array}{cc}\hfill |{\Im }_i(\zeta ,x)-{\Im }_i(\zeta ,y)|& \le w_i\left(\zeta \right)|x-y|\mathit{for}i=1,2,\hfill \\ \hfill |u_i(\zeta ,x)-u_i(\zeta ,y)|& \le {\theta }_i\left(\zeta \right)|x-y|\mathit{for}i=1,2,\hfill \end{array}$$

where ${\tilde{\Im }}_i={sup}_{\zeta \in J}\left|{\Im }_i(\zeta ,0)\right|$ and $U={sup}_{\zeta \in J}\left|u_i(\zeta ,0)\right|$.

The following theorem is essential in our study as it establishes the uniqueness of solutions for the coupled fractional quadratic integral equations in (8) and (9) based on conditions outlined in (15), ensuring that the obtained solutions are not only existent but also unique. This result is fundamental for drawing meaningful conclusions from the analysis conducted in the paper.

Theorem 3.

Assume that Assumptions 1–4, 7 and 8 hold. Then, solution $u=\left(x,y\right)\in J\times J$ of the coupled system of fractional quadratic integral Equations (8) and (9) is unique if

$$\left(\begin{array}{c}∥\sigma ∥+\frac{∥\mu ∥}{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥u∥\right)+\frac{{\breve{G}}_1+∥\mu ∥∥u∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥+\frac{2∥\omega ∥}{\gamma }\\ +∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+\tilde{\Im }T\right]+\left[∥\mu ∥∥u∥+\breve{G}\right]∥w∥∥\theta ∥G_0\frac{T^{\beta +1}}{\Gamma \left(\beta +1\right)}\end{array}\right)<1.$$

(15)

Proof. 

We let $u_1=\left(x_1,y_1\right)$ and $u_2=\left(x_2,y_2\right)$ be two solutions of the coupled system (1), which are equivalent to the solutions of the following system of coupled fractional quadratic integral Equations (8) and (9). Therefore,

$$\begin{array}{cc}\hfill & |x_1\left(\zeta \right)-x_2\left(\zeta \right)|\hfill \\ \hfill & \le \left|v_1(\zeta ,x_1\left(\zeta \right))-v_1(\zeta ,x_2\left(\zeta \right))\right|\hfill \\ \hfill & +|g_1\left(\zeta ,x_1\left(\zeta \right)\right){\int }_0^T\left|G_1(\zeta ,\upsilon )\right|{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)d\upsilon \hfill \\ \hfill & -g_1\left(\zeta ,x_2\left(\zeta \right)\right){\int }_0^T\left|G_1(\zeta ,\upsilon )\right|{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_2\left(\upsilon \right)\right)\right)d\upsilon |,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left|v_1(\zeta ,x_1\left(\zeta \right))-v_1(\zeta ,x_2\left(\zeta \right))\right|\hfill \\ \hfill & +\left|g_1\left(\zeta ,x_1\left(\zeta \right)\right)-g_1\left(\zeta ,x_2\left(\zeta \right)\right)\right|{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left|{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)\right|d\upsilon \hfill \\ \hfill & +\left|g_1\left(\zeta ,x_2\left(\zeta \right)\right)\right|{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left|{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)-{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_2\left(\upsilon \right)\right)\right)\right|d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\sigma }_1\left(\zeta \right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|+\frac{{\mu }_1\left(\zeta \right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|}{\Gamma \left(\gamma +1\right)}\left(H_1+{\omega }_1\left(\zeta \right)\left|x_1\right|\right)\hfill \\ \hfill & +\frac{{\breve{G}}_1+{\mu }_1\left|x_2\right|}{\Gamma \left(\gamma +1\right)}{\omega }_1\left(\zeta \right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|+\frac{\zeta }{\gamma T}\left({\omega }_2\left(\zeta \right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|+{\omega }_1\left(\zeta \right)|x_2\left(\zeta \right)-x_1\left(\zeta \right)|\right)\hfill \\ \hfill & +\left|g_1\left(\zeta ,x_1\left(\zeta \right)\right)-g_1\left(\zeta ,x_2\left(\zeta \right)\right)\right|{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left[\begin{array}{c}\left|{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)-{\Im }_1\left(\upsilon ,0\right)\right|\\ +\left|{\Im }_1\left(\upsilon ,0\right)\right|\end{array}\right]d\upsilon \hfill \\ \hfill & +\left[\begin{array}{c}\left|g_1\left(\zeta ,x_2\left(\zeta \right)\right)-g_1\left(\zeta ,0\right)\right|\\ +\left|g_1\left(\zeta ,0\right)\right|\end{array}\right]{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left[\begin{array}{c}{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)\\ -{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_2\left(\upsilon \right)\right)\right)\end{array}\right]d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\sigma }_1\left(\zeta \right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|+\frac{{\mu }_1\left(\zeta \right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|}{\Gamma \left(\gamma +1\right)}\left(H_1+{\omega }_1\left(\zeta \right)\left|x_1\right|\right)\hfill \\ \hfill & +\frac{{\breve{G}}_1+{\mu }_1\left|x_2\right|}{\Gamma \left(\gamma +1\right)}{\omega }_1\left(\zeta \right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|+\frac{\zeta }{\gamma T}\left({\omega }_2\left(\zeta \right)+{\omega }_1\left(\zeta \right)\right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|\hfill \\ \hfill & +\left|{\mu }_1\left(\zeta \right)\right||x_1\left(\zeta \right)-x_2\left(\zeta \right)|{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left[\left|w_1\left(\upsilon \right)\right|{\mathcal{I}}_{0^{+}}^{\beta }\left|u_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right|+{\tilde{\Im }}_1\right]d\upsilon \hfill \\ \hfill & +\left[\left|{\mu }_1\left(\zeta \right)\right|\left|x_2\left(\zeta \right)\right|+{\breve{G}}_1\right]{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left|w_1\left(\upsilon \right)\right|{\mathcal{I}}_{0^{+}}^{\beta }\left|u_1\left(\upsilon ,y_1\left(\upsilon \right)\right)-u_1\left(\upsilon ,y_2\left(\upsilon \right)\right)\right|d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(\begin{array}{c}{\sigma }_1\left(\zeta \right)+\frac{{\mu }_1\left(\zeta \right)\left(H_1+{\omega }_1\left(\zeta \right)\left|x_1\right|\right)}{\Gamma \left(\gamma +1\right)}\\ +\frac{{\breve{G}}_1+{\mu }_1\left|x_2\right|}{\Gamma \left(\gamma +1\right)}{\omega }_1\left(\zeta \right)+\frac{\zeta }{\gamma T}\left({\omega }_2\left(\zeta \right)+{\omega }_1\left(\zeta \right)\right)\end{array}\right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|\hfill \\ \hfill & +\left|{\mu }_1\left(\zeta \right)\right||x_1\left(\zeta \right)-x_2\left(\zeta \right)|{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left[\left|w_1\left(\upsilon \right)\right|{\mathcal{I}}_{0^{+}}^{\beta -\gamma }{\mathcal{I}}_{0^{+}}^{\gamma }{\mathit{m}}_1\left(\upsilon \right)+{\tilde{\Im }}_1\right]d\upsilon \hfill \\ \hfill & +\left[\left|{\mu }_1\left(\zeta \right)\right|\left|x_2\left(\zeta \right)\right|+{\breve{G}}_1\right]{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left|w_1\left(\upsilon \right)\right|\left|{\theta }_1\left(\zeta \right)\right|{\mathcal{I}}_{0^{+}}^{\beta }\left|y_1\left(\zeta \right)-y_2\left(\zeta \right)\right|d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(\begin{array}{c}{\sigma }_1\left(\zeta \right)+\frac{{\mu }_1\left(\zeta \right)\left(H_1+{\omega }_1\left(\zeta \right)\left|x_1\right|\right)}{\Gamma \left(\gamma +1\right)}\\ +\frac{{\breve{G}}_1+{\mu }_1\left|x_2\right|}{\Gamma \left(\gamma +1\right)}{\omega }_1\left(\zeta \right)+\frac{\zeta }{\gamma T}\left({\omega }_2\left(\zeta \right)+{\omega }_1\left(\zeta \right)\right)\end{array}\right)|x_1\left(\zeta \right)-x_2\left(\zeta \right)|\hfill \\ \hfill & +\left|{\mu }_1\left(\zeta \right)\right||x_1\left(\zeta \right)-x_2\left(\zeta \right)|{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left[\left|w_1\left(\upsilon \right)\right|{\mathcal{I}}_{0^{+}}^{\beta -\gamma }{\mathcal{M}}_1+{\tilde{\Im }}_1\right]d\upsilon \hfill \\ \hfill & +\left[\left|{\mu }_1\left(\zeta \right)\right|\left|x_2\left(\zeta \right)\right|+{\breve{G}}_1\right]{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left|w_1\left(\upsilon \right)\right|\left|{\theta }_1\left(\upsilon \right)\right|{\mathcal{I}}_{0^{+}}^{\beta }\left|y_1\left(\upsilon \right)-y_2\left(\upsilon \right)\right|d\upsilon .\hfill \end{array}$$

Taking supremum for all $\zeta \in J$, we obtain

$$\begin{array}{cc}\hfill & ∥x_1-x_2∥\hfill \\ \hfill & \le \left(\begin{array}{c}∥{\sigma }_1∥+\frac{∥{\mu }_1∥}{\Gamma \left(\gamma +1\right)}\left(H_1+∥{\omega }_1∥∥x_1∥\right)\\ +\frac{{\breve{G}}_1+∥{\mu }_1∥∥x_2∥}{\Gamma \left(\gamma +1\right)}∥{\omega }_1∥+\frac{1}{\gamma }(∥{\omega }_1∥+∥{\omega }_2∥)\end{array}\right)∥x_1-x_2∥\hfill \\ \hfill & +∥{\mu }_1∥∥x_1-x_2∥G_0{\int }_0^T\left[∥w_1∥{\mathcal{M}}_1{\mathcal{I}}^{\beta -\gamma }+{\tilde{\Im }}_1\right]d\upsilon \hfill \\ \hfill & +\left[∥{\mu }_1∥∥x_2∥+{\breve{G}}_1\right]∥w_1∥∥{\theta }_1∥∥y_1-y_2∥G_0{\int }_0^T{\mathcal{I}}_{0^{+}}^{\beta }d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(\begin{array}{c}∥{\sigma }_1∥+\frac{∥{\mu }_1∥}{\Gamma \left(\gamma +1\right)}\left(H_1+∥{\omega }_1∥∥x_1∥\right)\\ +\frac{{\breve{G}}_1+∥{\mu }_1∥∥x_2∥}{\Gamma \left(\gamma +1\right)}∥{\omega }_1∥+\frac{1}{\gamma }(∥{\omega }_1∥+∥{\omega }_2∥)\end{array}\right)∥x_1-x_2∥\hfill \\ \hfill & +∥{\mu }_1∥∥x_1-x_2∥G_0\left[∥w_1∥{\mathcal{M}}_1{\int }_0^T{\int }_0^{\upsilon }\frac{{\left(\zeta -\tau \right)}^{\beta -\gamma -1}}{\Gamma \left(\beta -\gamma \right)}d\tau d\upsilon +{\tilde{\Im }}_1{\int }_0^{T}d\upsilon \right]\hfill \\ \hfill & +\left[∥{\mu }_1∥∥x_2∥+{\breve{G}}_1\right]∥w_1∥∥{\theta }_1∥∥x_1-x_2∥G_0{\int }_0^T{\int }_0^{\upsilon }\frac{{\left(\upsilon -\tau \right)}^{\beta -1}}{\Gamma \left(\beta \right)}d\tau d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(\begin{array}{c}∥{\sigma }_1∥+\frac{∥{\mu }_1∥}{\Gamma \left(\gamma +1\right)}\left(H_1+∥{\omega }_1∥∥x_1∥\right)\\ +\frac{{\breve{G}}_1+∥{\mu }_1∥∥x_2∥}{\Gamma \left(\gamma +1\right)}∥{\omega }_1∥+\frac{1}{\gamma }(∥{\omega }_1∥+∥{\omega }_2∥)\end{array}\right)∥x_1-x_2∥\hfill \\ \hfill & +∥{\mu }_1∥∥x_1-x_2∥G_0\left[∥w_1∥{\mathcal{M}}_1\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+{\tilde{\Im }}_{1}T\right]\hfill \\ \hfill & +\left[∥{\mu }_1∥∥x_2∥+{\breve{G}}_1\right]∥w_1∥∥{\theta }_1∥∥y_1-y_2∥G_0\frac{T^{\beta +1}}{\Gamma \left(\beta +1\right)}.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill & ∥y_1-y_2∥\hfill \\ \hfill & \le \left(\begin{array}{c}∥{\sigma }_1∥+\frac{∥{\mu }_1∥}{\Gamma \left(\gamma +1\right)}\left(H_1+∥{\omega }_1∥∥y_1∥\right)\\ +\frac{{\breve{G}}_1+∥{\mu }_1∥∥y_2∥}{\Gamma \left(\gamma +1\right)}∥{\omega }_1∥+\frac{1}{\gamma }(∥{\omega }_1∥+∥{\omega }_2∥)\end{array}\right)∥y_1-y_2∥\hfill \\ \hfill & +∥{\mu }_1∥∥y_1-y_2∥G_0\left[∥w_1∥{\mathcal{M}}_1\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+{\tilde{\Im }}_{1}T\right]\hfill \\ \hfill & +\left[∥{\mu }_1∥∥y_2∥+{\breve{G}}_1\right]∥w_1∥∥{\theta }_1∥∥x_1-x_2∥G_0\frac{T^{\beta +1}}{\Gamma \left(\beta +1\right)}.\hfill \end{array}$$

Taking $\mathcal{M}=max\left\{{\mathcal{M}}_1,{\mathcal{M}}_2\right\}$, $∥\sigma ∥=max\left\{∥{\sigma }_1∥,∥{\sigma }_1∥\right\}$, $∥\mu ∥=max\left\{∥{\mu }_1∥,∥{\mu }_1∥\right\}$, $∥\omega ∥=max\left\{∥{\omega }_1∥,∥{\omega }_1∥\right\}$, $∥w∥=max\{∥w_1∥,∥w_2∥\}$, $∥\theta ∥=max\left\{∥{\theta }_1∥,∥{\theta }_1∥\right\}$, $\breve{G}=max\{{\breve{G}}_1,{\breve{G}}_2\}$, $\tilde{\Im }=max\{{\tilde{\Im }}_1,{\tilde{\Im }}_2\}$, and $H=max\{H_1,H_2\}$, it is acquired that

$$\begin{array}{cc}\hfill ∥x_1-x_2∥& \le \left(\begin{array}{c}∥\sigma ∥+\frac{∥\mu ∥}{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥x_1∥\right)\\ +\frac{{\breve{G}}_1+∥\mu ∥∥x_2∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥+\frac{2∥\omega ∥}{\gamma }\end{array}\right)∥x_1-x_2∥\hfill \\ \hfill & +∥\mu ∥∥x_1-x_2∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+\tilde{\Im }T\right]\hfill \\ \hfill & +\left[∥\mu ∥∥x_2∥+\breve{G}\right]∥w∥∥\theta ∥∥y_1-y_2∥G_0\frac{T^{\beta +1}}{\Gamma \left(\beta +1\right)},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ∥y_1-y_2∥& \le \left(\begin{array}{c}∥\sigma ∥+\frac{∥\mu ∥}{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥y_1∥\right)\\ +\frac{\breve{G}+∥\mu ∥∥y_2∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥+\frac{2∥\omega ∥}{\gamma }\end{array}\right)∥y_1-y_2∥\hfill \\ \hfill & +∥\mu ∥∥y_1-y_2∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+\tilde{\Im }T\right]\hfill \\ \hfill & +\left[∥\mu ∥∥y_2∥+\breve{G}\right]∥w∥∥\theta ∥∥x_1-x_2∥G_0\frac{T^{\beta +1}}{\Gamma \left(\beta +1\right)}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill ∥u_1-u_2∥& =∥\left(x_1,y_1\right)-\left(x_2,y_2\right)∥=∥x_1-x_2∥+∥y_1-y_2∥,\hfill \\ \hfill & \le \left(\begin{array}{c}∥\sigma ∥+\frac{∥\mu ∥}{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥x_1∥\right)\\ +\frac{{\breve{G}}_1+∥\mu ∥∥x_2∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥+\frac{2∥\omega ∥}{\gamma }\end{array}\right)∥x_1-x_2∥\hfill \\ \hfill & +\left(\begin{array}{c}∥\sigma ∥+\frac{∥\mu ∥}{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥y_1∥\right)\\ +\frac{\breve{G}+∥\mu ∥∥y_2∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥+\frac{2∥\omega ∥}{\gamma }\end{array}\right)∥y_1-y_2∥\hfill \\ \hfill & +∥\mu ∥∥x_1-x_2∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+\tilde{\Im }T\right]\hfill \\ \hfill & +\left[∥\mu ∥∥y_2∥+\breve{G}\right]∥w∥∥\theta ∥∥x_1-x_2∥G_0\frac{T^{\beta +1}}{\Gamma \left(\beta +1\right)},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(\begin{array}{c}∥\sigma ∥+\frac{∥\mu ∥}{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥\left(∥x∥+∥y∥\right)\right)\\ +\frac{{\breve{G}}_1+∥\mu ∥\left(∥x∥+∥y∥\right)}{\Gamma \left(\gamma +1\right)}∥\omega ∥\end{array}\right.\hfill \\ \hfill & +\frac{2∥\omega ∥}{\gamma }+∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+\tilde{\Im }T\right]\hfill \\ \hfill & \left.+\left[∥\mu ∥\left(∥x∥+∥y∥\right)+\breve{G}\right]∥w∥∥\theta ∥G_0\frac{T^{\beta +1}}{\Gamma \left(\beta +1\right)}\right)\left(∥x_1-x_2∥+∥y_1-y_2∥\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left(∥\sigma ∥+\frac{∥\mu ∥}{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥u∥\right)+\frac{{\breve{G}}_1+∥\mu ∥∥u∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥+\frac{2∥\omega ∥}{\gamma }\right.\hfill \\ \hfill & \left.\begin{array}{c}+∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+\tilde{\Im }T\right]\\ +\left[∥\mu ∥∥u∥+\breve{G}\right]∥w∥∥\theta ∥G_0\frac{T^{\beta +1}}{\Gamma \left(\beta +1\right)}\end{array}\right)∥u_1-u_2∥.\hfill \end{array}$$

Therefore,

$$\left[1-\left(\begin{array}{c}∥\sigma ∥+\frac{∥\mu ∥}{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥u∥\right)+\frac{{\breve{G}}_1+∥\mu ∥∥u∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥\\ +\frac{2∥\omega ∥}{\gamma }+∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+\tilde{\Im }T\right]\\ +\left[∥\mu ∥∥u∥+\breve{G}\right]∥w∥∥\theta ∥G_0\frac{T^{\beta +1}}{\Gamma \left(\beta +1\right)}\end{array}\right)\right]∥u_1-u_2∥\le 0,$$

which implies that if $∥u_1-u_2∥=0$, then $u_1=u_2$. This proves the uniqueness of the solution for the coupled system of fractional quadratic integral Equations (8) and (9), and consequently the uniqueness of the solution for the coupled system of hybrid fractional-order differential equations CHFDE (1). □

3.3. Hyers–Ulam Stability of Solutions

We let ${ϵ}_1>0$, ${ϵ}_2>0$, ${\Phi }_1:J\to R^{+}$, and ${\Phi }_2:J\to R^{+}$ be continuous functions and consider the following systems of inequalities:

$$\left\{\begin{array}{c}\left|{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right)-{\Im }_1(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\zeta ,y\left(\zeta \right))\right|\le {ϵ}_1,\zeta \in J,\\ \left|{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}\right)-{\Im }_2(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_2(\zeta ,x\left(\zeta \right))\right|\le {ϵ}_2,\zeta \in J,\end{array}\right.$$

(16)

$$\left\{\begin{array}{c}\left|{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right)-{\Im }_1(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\zeta ,y\left(\zeta \right))\right|\le {\Phi }_1\left(\zeta \right),\zeta \in J,\\ \left|{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}\right)-{\Im }_2(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_2(\zeta ,x\left(\zeta \right))\right|\le {\Phi }_2\left(\zeta \right),\zeta \in J,\end{array}\right.$$

(17)

$$\left\{\begin{array}{c}\left|{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{x\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right)-{\Im }_1(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\zeta ,y\left(\zeta \right))\right|\le {ϵ}_1{\Phi }_1\left(\zeta \right),\zeta \in J,\\ \left|{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{y\left(\zeta \right)-f_2(\zeta ,y\left(\zeta \right))}{g_2(\zeta ,y\left(\zeta \right))}}\right)-{\Im }_2(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_2(\zeta ,x\left(\zeta \right))\right|\le {ϵ}_2{\Phi }_2\left(\zeta \right),\zeta \in J.\end{array}\right.$$

(18)

Definition 7.

The problem CHFDE (1) is said to be Ulam–Hyers stable if there exists a real number $c_f>0$ such that for every $ϵ>0$ and for each solution $u=(x_1,y_1)\in C(J,R)\times C(J,R)$ of inequality (16) there exists a solution $v=(x_2,y_2)\in C(J,R)\times C(J,R)$ of (1) with $|u\left(\zeta \right)-v\left(\zeta \right)|\le ϵc_f$ and $\zeta \in J$, where $ϵ=max\{{ϵ}_1,{ϵ}_2\}$.

Definition 8.

The problem CHFDE (1) is said to be generalized Ulam–Hyers stable if there exists $c_f\in C(R^{+},R^{+})$ with $c_f\left(0\right)=0$ such that for every $ϵ>0$ and for each solution $u=(x_1,y_1)\in C(J,R)\times C(J,R)$ of the inequality (16) there exists a solution $v=(x_2,y_2)\in C(J,R)\times C(J,R)$ of (1) with $|u\left(\zeta \right)-v\left(\zeta \right)|\le c_f\left(ϵ\right)$ and $\zeta \in J$, where $ϵ=max\{{ϵ}_1,{ϵ}_2\}$.

Definition 9.

The problem CHFDE (1) is said to be Ulam–Hyers–Rassias stable with respect to Φ if there exists a real number $c_{f,\Phi }>0$ such that for every $ϵ>0$ and for each solution $u=(x_1,y_1)\in C(J,R)\times C(J,R)$ of the inequality (17) there exists a solution $v=(x_2,y_2)\in C(J,R)\times C(J,R)$ of (1) with $|u\left(\zeta \right)-v\left(\zeta \right)|\le ϵc_{f,\Phi }\Phi \left(\zeta \right)$ and $\zeta \in J$, where $ϵ=max\{{ϵ}_1,{ϵ}_2\}$ and $\Phi \left(\zeta \right)=sup\{{\Phi }_1\left(\zeta \right),{\Phi }_2\left(\zeta \right)\}$.

The problem CHFDE (1) is considered to be generalized Ulam–Hyers–Rassias stable with respect to $\Phi$ if there exists a positive real number $c_{f,\Phi }>0$ such that for every solution $u=(x_1,y_1)\in C(J,\mathbb{R})\times C(J,\mathbb{R})$ of the inequality (18) there exists another solution $v=(x_2,y_2)\in C(J,\mathbb{R})\times C(J,\mathbb{R})$ of (1) with the property that $|u\left(\zeta \right)-v\left(\zeta \right)|\le c_{f,\Phi }\Phi \left(\zeta \right),\zeta \in J$, where $ϵ=max\left\{{ϵ}_1,{ϵ}_2\right\}$ and $\Phi \left(\zeta \right)=sup\left\{{\Phi }_1\left(\zeta \right),{\Phi }_2\left(\zeta \right)\right\}$.

The following theorem establishes the Ulam–Hyers stability of the CHFDE problem (1), which is crucial for demonstrating the robustness and reliability of the proposed mathematical model in capturing and describing real-world phenomena in a stable manner.

Theorem 4.

We suppose that the assumptions of Theorem 3 are satisfied. Then, the problem (CHFDE) (1) is Ulam–Hyers stable.

Proof. 

We assume that ${ϵ}_1>0$, and let $v=\left(z_1\left(\zeta \right),{\xi }_1\left(\zeta \right)\right)\in C(J,R)$ be a function satisfying the first part of inequality (16), i.e.,

$$\left|{}^c{\mathfrak{D}}_{0^{+}}^{\alpha }\left({\displaystyle \frac{z_1\left(\zeta \right)-f_1(\zeta ,x\left(\zeta \right))}{g_1(\zeta ,x\left(\zeta \right))}}\right)-{\Im }_1(\zeta ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1(\zeta ,y\left(\zeta \right))\right|\le {ϵ}_1,\zeta \in J,$$

(19)

and let $u=\left(x_1\left(\zeta \right),y_1\left(\zeta \right)\right)\in C(J,R)\times C(J,R)$ be the unique solution of the CHFDE (1). This solution, as established in Lemma 2, is equivalent to the solution of the following quadratic integral equation of fractional order

$$x_1\left(\zeta \right)=v_1(\zeta ,x_1\left(\zeta \right))+g\left(\zeta ,x_1\left(\zeta \right)\right){\int }_0^{T}G(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)d\upsilon .$$

If ${\mathcal{I}}^{\alpha }$ is applied to both sides of the inequality (19), the resulting inequality is as follows:

$$|z_1\left(\zeta \right)-v_1(\zeta ,z_1\left(\zeta \right))-g_1\left(\zeta ,z_1\left(\zeta \right)\right){\int }_0^T{G}_1(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)d\upsilon |\le \frac{{ϵ}_1{T}^{\alpha }}{\Gamma (\alpha +1)}.$$

(20)

This implies that for every $\zeta \in J$, we have

$$\begin{array}{cc}\hfill & {|z}_1{\left(\zeta \right)-x}_1\left(\zeta \right)|\hfill \\ \hfill & =\hspace{0.17em}|z_1{\left(\zeta \right)-v}_1{(\zeta ,x}_1{\left(\zeta \right))-g}_1\left(\zeta ,x_1\left(\zeta \right)\right){\int }_0^T{G}_1(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)d\upsilon |,\hfill \\ \hfill & =\hspace{0.17em}|z_1\left(\zeta \right)-v_1(\zeta ,z_1\left(\zeta \right))-g_1\left(\zeta ,z_1\left(\zeta \right)\right){\int }_0^T{G}_1(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)d\upsilon \hfill \\ \hfill & +v_1(\zeta ,z_1\left(\zeta \right))+g_1\left(\zeta ,z_1\left(\zeta \right)\right){\int }_0^T{G}_1(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)d\upsilon \hfill \\ \hfill & -v_1(\zeta ,x_1\left(\zeta \right))-g_1\left(\zeta ,x_1\left(\zeta \right)\right){\int }_0^T{G}_1(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)d\upsilon |,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \hspace{0.17em}|z_1\left(\zeta \right)-v_1(\zeta ,z_1\left(\zeta \right))-g_1\left(\zeta ,z_1\left(\zeta \right)\right){\int }_0^T{G}_1(\zeta ,\upsilon ){\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)d\upsilon |\hfill \\ \hfill & +\left|v_1(\zeta ,z_1\left(\zeta \right))-v_1(\zeta ,x_1\left(\zeta \right))\right|\hfill \\ \hfill & +\left|g_1\left(\zeta ,z_1\left(\zeta \right)\right)-g_1\left(\zeta ,x_1\left(\zeta \right)\right)\right|{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left|{\Im }_1\left(\upsilon ,{\mathcal{I}}_{0^{+}}^{\beta }{u}_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right)\right|d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{{ϵ}_1{T}^{\alpha }}{\Gamma (\alpha +1)}+{\sigma }_1\left(\zeta \right)|z_1\left(\zeta \right)-x_1\left(\zeta \right)|+\frac{{\mu }_1\left(\zeta \right)|z_1\left(\zeta \right)-x_1\left(\zeta \right)|}{\Gamma \left(\gamma +1\right)}\left(H_1+{\omega }_1\left(\zeta \right)\left|z_1\right|\right)\hfill \\ \hfill & +\frac{{\breve{G}}_1+{\mu }_1\left|x_1\right|}{\Gamma \left(\gamma +1\right)}{\omega }_1\left(\zeta \right)|z_1\left(\zeta \right)-x_1\left(\zeta \right)|\hfill \\ \hfill & +\frac{\zeta }{\gamma T}\left({\omega }_2\left(\zeta \right)|z_1\left(\zeta \right)-x_1\left(\zeta \right)|+{\omega }_1\left(\zeta \right)|x_1\left(\zeta \right)-z_1\left(\zeta \right)|\right)\hfill \\ \hfill & +\left|{\mu }_1\left(\zeta \right)\right||z_1\left(\zeta \right)-x_1\left(\zeta \right)|{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left[\left|w_1\left(\upsilon \right)\right|{\mathcal{I}}^{\beta }\left|u_1\left(\upsilon ,y_1\left(\upsilon \right)\right)\right|+H_1\right]d\upsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{{ϵ}_1{T}^{\alpha }}{\Gamma (\alpha +1)}+{\sigma }_1\left(\zeta \right)|z_1\left(\zeta \right)-x_1\left(\zeta \right)|+\frac{{\mu }_1\left(\zeta \right)|z_1\left(\zeta \right)-x_1\left(\zeta \right)|}{\Gamma \left(\gamma +1\right)}\left(H_1+{\omega }_1\left(\zeta \right)\left|z_1\right|\right)\hfill \\ \hfill & +\frac{{\breve{G}}_1+{\mu }_1\left|x_1\right|}{\Gamma \left(\gamma +1\right)}{\omega }_1\left(\zeta \right)|z_1\left(\zeta \right)-x_1\left(\zeta \right)|+\frac{\zeta }{\gamma T}\left({\omega }_2\left(\zeta \right)|z_1\left(\zeta \right)-x_1\left(\zeta \right)|+{\omega }_1\left(\zeta \right)|x_1\left(\zeta \right)-z_1\left(\zeta \right)|\right)\hfill \\ \hfill & +\left|{\mu }_1\left(\zeta \right)\right||z_1\left(\zeta \right)-x_1\left(\zeta \right)|{\int }_0^T\left|G_1(\zeta ,\upsilon )\right|\left[\left|w\left(\upsilon \right)\right|{\mathcal{I}}^{\beta -\gamma }{\mathcal{I}}^{\gamma }{\mathit{m}}_1\left(\upsilon \right)+H_1\right]d\upsilon .\hfill \end{array}$$

Taking supremum for all $\zeta \in J$, we obtain

$$\begin{array}{cc}\hfill & {\parallel z}_1{-x}_1\parallel \hfill \\ \hfill & \le \frac{{ϵ}_1{T}^{\alpha }}{\Gamma (\alpha +1)}+∥{\sigma }_1∥{\parallel z}_1{-x}_1\parallel +\frac{∥{\mu }_1∥{\parallel z}_1{-x}_1\parallel }{\Gamma \left(\gamma +1\right)}\left(H_1+∥{\omega }_1∥∥z_1∥\right)\hfill \\ \hfill & +\frac{{\breve{G}}_1+∥{\mu }_1∥∥x_1∥}{\Gamma \left(\gamma +1\right)}∥{\omega }_1∥{\parallel z}_1{-x}_1\parallel +\frac{1}{\gamma }\left(∥{\omega }_2∥{\parallel z}_1{-x}_1\parallel +∥{\omega }_1∥{\parallel x}_1{-z}_1\parallel \right)\hfill \\ \hfill & +∥{\mu }_1∥G_0\left[∥w_1∥{\mathcal{M}}_1\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+H_{1}T\right]{\parallel z}_1{-x}_1\parallel ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{ϵT^{\alpha }}{\Gamma (\alpha +1)}+∥\sigma ∥{\parallel z}_1{-x}_1\parallel +\frac{∥\mu ∥{\parallel z}_1{-x}_1\parallel }{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥z_1∥\right)\hfill \\ \hfill & +\frac{\breve{G}+∥\mu ∥∥x_1∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥{\parallel z}_1{-x}_1\parallel +\frac{2∥\omega ∥}{\gamma }{\parallel z}_1{-x}_1\parallel +∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+HT\right]{\parallel z}_1{-x}_1\parallel .\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill & \parallel {\xi }_1{-y}_1\parallel \hfill \\ \hfill & \le \frac{{ϵ}_2{T}^{\alpha }}{\Gamma (\alpha +1)}+∥{\sigma }_2∥\parallel {\xi }_1{-y}_1\parallel +\frac{∥{\mu }_2∥\parallel {\xi }_1{-y}_1\parallel }{\Gamma \left(\gamma +1\right)}\left(H_2+∥{\omega }_2∥∥{\xi }_1∥\right)\hfill \\ \hfill & +\frac{{\breve{G}}_2+∥{\mu }_2∥∥y_1∥}{\Gamma \left(\gamma +1\right)}∥{\omega }_2∥\parallel {\xi }_1{-y}_1\parallel +\frac{1}{\gamma }\left(∥{\omega }_1∥\parallel {\xi }_1{-y}_1\parallel +∥{\omega }_2∥{\parallel y}_1-{\xi }_1\parallel \right)\hfill \\ \hfill & +∥{\mu }_2∥G_0\left[∥w2∥{\mathcal{M}}_2\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+H_{2}T\right]\parallel {\xi }_1{-y}_1\parallel ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{ϵT^{\alpha }}{\Gamma (\alpha +1)}+∥\sigma ∥\parallel {\xi }_1{-y}_1\parallel +\frac{∥\mu ∥\parallel {\xi }_1{-y}_1\parallel }{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥{\xi }_1∥\right)\hfill \\ \hfill & +\frac{\breve{G}+∥\mu ∥∥y_1∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥\parallel {\xi }_1{-y}_1\parallel +\frac{2∥\omega ∥}{\gamma }\parallel {\xi }_1{-y}_1\parallel +∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+HT\right]\parallel {\xi }_1{-y}_1\parallel .\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill & ∥v-u∥\hfill \\ \hfill & \le ∥\left(z_1,{\xi }_1\right)-\left(x_1,y_1\right)∥=∥z_1-x_1∥+∥{\xi }_1-y_1∥,\hfill \\ \hfill & \le \frac{ϵT^{\alpha }}{\Gamma (\alpha +1)}+∥\sigma ∥{\parallel z}_1{-x}_1\parallel +\frac{∥\mu ∥{\parallel z}_1{-x}_1\parallel }{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥z_1∥\right)\hfill \\ \hfill & +\frac{\breve{G}+∥\mu ∥∥x_1∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥{\parallel z}_1{-x}_1\parallel +\frac{ϵT^{\alpha }}{\Gamma (\alpha +1)}+∥\sigma ∥\parallel {\xi }_1{-y}_1\parallel +\frac{∥\mu ∥\parallel {\xi }_1{-y}_1\parallel }{\Gamma \left(\gamma +1\right)}\left(H+∥\omega ∥∥{\xi }_1∥\right)\hfill \\ \hfill & +\frac{\breve{G}+∥\mu ∥∥y_1∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥\parallel {\xi }_1{-y}_1\parallel +\frac{2∥\omega ∥}{\gamma }{\parallel z}_1{-x}_1\parallel \hfill \\ \hfill & +∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+HT\right]{\parallel z}_1{-x}_1\parallel \hfill \\ \hfill & +\frac{2∥\omega ∥}{\gamma }\parallel {\xi }_1{-y}_1\parallel +∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+HT\right]\parallel {\xi }_1{-y}_1\parallel ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{2ϵT^{\alpha }}{\Gamma (\alpha +1)}+∥\sigma ∥\left({\parallel z}_1{-x}_1\parallel +\parallel {\xi }_1{-y}_1\parallel \right)+\frac{∥\mu ∥\left(H+∥\omega ∥\left(∥u∥+∥v∥\right)\right)}{\Gamma \left(\gamma +1\right)}\left({\parallel z}_1{-x}_1\parallel +\parallel {\xi }_1{-y}_1\parallel \right)\hfill \\ \hfill & +\frac{\breve{G}+∥\mu ∥∥u∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥\left({\parallel z}_1{-x}_1\parallel +\parallel {\xi }_1{-y}_1\parallel \right)+\frac{2∥\omega ∥}{\gamma }\left({\parallel z}_1{-x}_1\parallel +\parallel {\xi }_1{-y}_1\parallel \right)\hfill \\ \hfill & +∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+HT\right]\left({\parallel z}_1{-x}_1\parallel +\parallel {\xi }_1{-y}_1\parallel \right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{2ϵT^{\alpha }}{\Gamma (\alpha +1)}+∥\sigma ∥∥v-u∥+\frac{∥\mu ∥\left(H+∥\omega ∥\left(∥u∥+∥v∥\right)\right)}{\Gamma \left(\gamma +1\right)}∥v-u∥\hfill \\ \hfill & +\frac{\breve{G}+∥\mu ∥∥u∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥∥v-u∥+\frac{2∥\omega ∥}{\gamma }∥v-u∥\hfill \\ \hfill & +∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+HT\right]∥v-u∥.\hfill \end{array}$$

If we consider

$$ϵ=max\{{ϵ}_1,{ϵ}_2\},$$

and

$$\begin{array}{cc}\hfill \delta & =1-∥\sigma ∥-\frac{∥\mu ∥\left(H+∥\omega ∥\left(∥u∥+∥v∥\right)\right)}{\Gamma \left(\gamma +1\right)}-\frac{{\breve{G}}_1+∥\mu ∥∥x_1∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥\hfill \\ \hfill & -\frac{\breve{G}+∥\mu ∥∥u∥}{\Gamma \left(\gamma +1\right)}∥\omega ∥-\frac{2∥\omega ∥}{\gamma }-∥\mu ∥G_0\left[∥w∥\mathcal{M}\frac{T^{\beta -\gamma +1}}{\Gamma \left(\beta -\gamma +1\right)}+HT\right],\hfill \end{array}$$

then we deduce that

$$\parallel v-u\parallel \le \left(\frac{2T^{\alpha }}{\Gamma (\alpha +1)}{\delta }^{-1}\right)ϵ=c_fϵ.$$

Therefore, problem (CHFDE) (1) is Ulam–Hyers stable. This completes the proof. □

Remark 2.

If the function $\Phi \left(ϵ\right)=c_fϵ$ with $\Phi \left(0\right)=0$, then it can be deduced that the coupled system of hybrid fractional-order differential equations CHFDE (1) is generally Ulam–Hyers stable. Moreover, in cases where Φ is a monotonically increasing function and a positive constant ${\lambda }_{\Phi }>0$ exists such that for any $\zeta \in J$, inequality ${\mathcal{I}}_{0^{+}}^{\alpha }\Phi \le {\lambda }_{\Phi }\Phi$ holds, the CHFDE problem (1) demonstrates Ulam–Hyers–Rassias stability with respect to function Φ. Additionally, there exists a real constant $c_{f,\Phi }=\frac{{\lambda }_{\Phi }}{\zeta }$ associated to this stability property.

4. Conclusions

In conclusion, this research contributes a comprehensive analysis of coupled systems of hybrid fractional order differential equations with Caputo differential operators and hybrid integral boundary conditions. By employing a hybrid fixed point theorem under mixed Lipschitz and Carathéodory conditions, the existence and uniqueness of mild solutions are rigorously established. Moreover, the investigation into Ulam–Hyers stability sheds light on the reliability of the solutions. The findings presented in this paper provide valuable insights into the behavior of complex hybrid fractional order systems and their practical implications.

As we look to the future, several promising research directions emerge. First, the exploration of the proposed model’s applicability in diverse scientific and engineering domains holds the potential to yield innovative solutions to real-world problems. Additionally, the verification of our theoretical findings through numerical simulations will enhance the practicality and applicability of our model.

Furthermore, extending our investigations to higher-order hybrid systems and accommodating more general boundary conditions will broaden the scope of our research, making it even more versatile and insightful. Finally, addressing the dynamic behavior of these systems in the presence of noise and uncertainties is a promising avenue that could further enhance the practical relevance of our findings.

In summary, this study not only deepens our understanding of complex hybrid fractional-order systems, but also paves the way for future research endeavors. By embarking on these exciting research directions, we can contribute to the advancement of fractional calculus and its application in modeling intricate real-world phenomena, thus ensuring that this field continues to evolve and make a meaningful impact on scientific and engineering practices.