Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation

Abstract

This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of solutions. Stability is analyzed in the Ulam–Hyers (UHS), generalized Ulam–Hyers (GUHS), and Ulam–Hyers–Rassias (UHRS) senses. A modified Adomian decomposition method (MADM) is introduced to derive explicit solutions without linearization, preserving the problem’s original structure. The first numerical example validates the theoretical findings on existence, uniqueness, and stability, supplemented by graphical results obtained via the MADM. Further examples illustrate fuzzy solutions by varying the uncertainty level (r), the variable (x), and both parameters simultaneously. The numerical results align with the theoretical analysis, demonstrating the efficacy and applicability of the proposed method.

1. Introduction

In recent years, fractional calculus, which concerns the differentiation and integration of arbitrary (non-integer) orders, has attracted significant and growing interest within the mathematical and scientific communities [1,2]. This branch of mathematics has found applications in various scientific and engineering fields due to its ability to describe phenomena with memory and long-range dependencies more accurately.

Zadeh [3] introduced the concept of fuzzy sets to handle the imprecision and vagueness inherent in real-world problems. Following his foundational work, numerous researchers have extended fuzzy set theory in various directions; for instance, Dubois [4] contributed significantly to the development of fuzzy-valued functions and their applications. The integration of fuzzy logic and fractional calculus provides a unified framework for modeling systems with uncertainty and fractional-order dynamics. This has led to the development of fuzzy fractional integro-differential equations (FFIDEs), which effectively describe nonlocal and uncertain processes. Alaroud et al. [5] presented a hybrid analytical numerical method for solving fuzzy fractional Volterra equations, while Allahviranloo [6] provided a foundational treatment of fuzzy fractional operators. Chakraverty et al. [7] explored applications of fuzzy differential equations in engineering contexts. To solve these equations numerically, Kumar et al. [8] proposed a Chebyshev spectral method for fuzzy fractional Volterra–Fredholm integro-differential equations (FFVFIDEs), and Shabestari et al. [9] applied a two-dimensional Legendre wavelet method. Ahmad et al. [10] introduced a modified Adomian decomposition method suitable for fuzzy fractional systems.

Theoretical properties such as existence, uniqueness, and stability have also been addressed. Kumar et al. [8] used fixed-point theorems, while Ngoc et al. [11] and Hamoud et al. [12,13] investigated Ulam-type stability for fuzzy integro-differential equations.

With the evolution of weighted fractional calculus, attention has shifted from classical fractional calculus to more generalized frameworks involving differential and integral operators defined with respect to another function and associated weights [14]. This emerging field, known as weighted fractional calculus, extends traditional operators and enables refined modeling of systems exhibiting memory and hereditary properties. Motivated by this advancement, we adopt the Weighted Riemann–Liouville Fractional Derivative (WRLFD) as introduced in [14,15] and investigate its application to a new class of fuzzy fractional models. In particular, our focus lies on fuzzy weighted fractional Volterra–Fredholm integro-differential equations (FFVFIDEs) subject to integral boundary conditions, a topic that has yet to be deeply explored.

Several researchers have investigated weighted fractional models to enhance analytical precision. Abdo et al. [16] established the existence of positive solutions for weighted fractional-order differential equations. Pengyan and Yongzhong [17] analyzed positive solutions for a class of weighted fractional systems using fixed-point theory. Makogin and Mishura [18] studied fractional integrals and derivatives involving weighted Takagi–Landsberg functions, providing new insights into irregular kernels. Fernández-Anaya et al. [14] introduced a general formulation of fractional derivatives with respect to another function and weights, laying the groundwork for current advances in weighted fractional operators.

In the domain of fuzzy fractional integro-differential equations, recent works have made significant progress. Ezzati and Zabihi [19] proposed an analytical method using weighted Caputo-type derivatives and fuzzy Laplace transforms. Kasimala and Chakraverty [20] analyzed fuzzy fractional Volterra–Fredholm integro-differential equations, focusing on existence, uniqueness, and approximate solutions through Multi-Attribute Decision Making (MADM). Savla and Sharmila [21] applied the Shehu Adomian decomposition method to address both linear and nonlinear fuzzy FFVIDEs. Additionally, Ghazouani et al. [22] examined existence, uniqueness, and Ulam–Hyers stability for nonlinear fuzzy Caputo-type FFVIDEs using Banach’s fixed-point theorem and iterative techniques.

However, despite these contributions, the integration of weighted fractional operators within the fuzzy Volterra–Fredholm framework, especially under integral boundary conditions, remains underdeveloped. Existing methods either omit weighting functions or restrict analysis to unweighted or classical operators.

To address this gap, the present study investigates the existence and uniqueness of solutions for fuzzy weighted fractional Volterra–Fredholm integro-differential equations with integral boundary conditions using the WRLFD operator. Our approach enhances the modeling ability for systems governed by fractional dynamics under uncertainty and extends the current body of knowledge by introducing a novel weighted fuzzy framework applicable to real-world problems. This study will contribute to the future research on how best to model and analyze complex systems under uncertainty and fractional-order dynamics.

In this manuscript, we expand on the work published in [12] by using the fuzzy concept. Our analysis emphasizes FWFVFIDEs in Banach Space:

$$\begin{array}{cc}\hfill {\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta ,f,g}\tilde{v}(x,r)& =A(x,\tilde{v}(x,r))+{\int }_0^{x}B(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma \hfill \\ \hfill & +{\int }_0^{1}C(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma +I_{0^{+}}^{\zeta }F(x,\tilde{v}(x,r))\hfill \end{array}$$

(1)

$$\tilde{v}(0,r)=a{\int }_0^{\beta }\tilde{v}(\sigma ,r)d\sigma ,a\in \mathbb{R},0<\beta <1$$

(2)

where the weighted Caputo fractional derivative operator (${\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta ,f,g}$) is established for an order of $\zeta +\eta$, with the condition of $0<\zeta +\eta \le 1$. In this context, $x\in \mathcal{J}:=[0,1]$ and $A,F:\mathcal{J}\times Y\to Y$ are considered functions. Functions $B,C:\mathcal{J}\times \mathcal{J}\times Y\to Y$ satisfy Lipschitz conditions. $g\left(x\right)\ne 0\in L_{\infty }(a,b)$ is a weighted function with $g^{-1}\left(x\right)={\displaystyle \frac{1}{g\left(x\right)}}$, and $f:[0,1]\to {\mathbb{R}}^{+}$ is a strictly increasing function such that $f\in C^1[0,1]$ with $f^{\prime }\left(x\right)\ne 0$ for all $x\in [0,1]$. Finally, $I_{0^{+}}^{\zeta }$ represents the Riemann–Liouville (RL) fractional integral.

The main contributions of this research summarized are as follows:

• A new class of fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) is introduced using weighted fractional calculus in the Caputo sense.
• Existence and uniqueness results are established using Banach’s and Krasnoselskii’s fixed-point theorems under weighted Lipschitz conditions.
• Stability results in the Ulam–Hyers and Ulam–Hyers–Rassias senses are derived, ensuring robustness under perturbations.
• A modified Adomian decomposition method (MADM) is presented to construct the solution in the form of a recursive series.
• A numerical example, along with graphical interpretation, is provided to illustrate the effectiveness and applicability of the proposed method.

The following flowchart outlines our problem:

The structure of this document is outline as follows:

The Introduction is presented in Section 1. Section 2 covers the fundamental principles of fractional calculus and an integro-differential equation, along with an overview of fuzzy set theory. The existence and uniqueness of the solution to the equation under consideration are discussed in Section 3. Section 4 explores the Ulam–Hyers stability within the proposed equation. Section 5 introduces the modified Adomian decomposition method, accompanied by examples and graphs. Finally, Section 6 offers the conclusion of the article.

2. Preliminaries

In this section, we present various symbols, explanations, and outcomes that are referenced throughout this document. The collection of fuzzy subsets defined over the real line is denoted by ${\mathbb{R}}^{\mathcal{F}}$[10]. Let x denote a Banach Space equipped with the norm $|·|$. Spaces $C(\mathcal{J},x)$ and $C^m(\mathcal{J},x)$ refer to Banach Spaces comprising all bounded and continuous functions and all functions that are continuously differentiable up to order m in the interval of $\mathcal{J}$, respectively. Furthermore, we establish the following norm [12]:

$${\parallel v\parallel }_C=\underset{x\in \mathcal{J}}{max}\left|v\left(x\right)\right|,$$

and for any continuous function, $v:\mathcal{J}\to x$.

The operator ${\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta ,f,g}$ denotes the weighted Caputo fractional derivative of order $\zeta +\eta \in (0,1]$, where $x\in \mathcal{J}:=[0,1]$. The functions $A,F:\mathcal{J}\times Y\to Y$ and $B,C:\mathcal{J}\times \mathcal{J}\times Y\to Y$ satisfy Lipschitz conditions. The incomplete Beta function is denoted by $\beta (·,·)$, and the Riemann–Liouville fractional integral of order $\zeta$ is written as $I_{0^{+}}^{\zeta }$. All variables and operators satisfy the required continuity and boundedness conditions.

Definition 1

([3]). A fuzzy set is categorized as a fuzzy number (FN) if it exhibits a piecewise continuous membership function, convexity, and normalization on the real line ($\mathbb{R}$).

Definition 2

([4]). In a broader mathematical framework, any fuzzy number (FN) ($\tilde{v}$) can be expressed in its parametric form as $\tilde{v}=\left[\underline{v}\left(r\right),\overline{v}\left(r\right)\right]$, where the following conditions are satisfied:

• The lower-bound function ($\underline{v}\left(r\right)$) is defined on the interval of $[0,1]$ and is bounded, left-continuous, and monotonically increasing.
• The upper-bound function ($\overline{v}\left(r\right)$) is also defined on $[0,1]$ and is bounded, left-continuous, and monotonically decreasing.
• For all $r\in [0,1]$, it holds that $\underline{v}\left(r\right)\le \overline{v}\left(r\right)$.

Definition 3

([14]). Let $\tilde{v}\in L^1(\mathsf{a},\mathsf{b})$, $g\left(x\right)\in L^{\infty }(\mathsf{a},\mathsf{b})$ and $\zeta \in \mathbb{R},\zeta >0$; then, the weighted $\mathsf{R}$eimann–$\mathsf{L}$iouville fractional integral is defined as

$${\phantom{,}}_a^{RL}{I}_{x,\mathsf{g}\left(x\right)}^{\zeta }\tilde{v}\left(x\right)={\displaystyle \frac{1}{\mathsf{g}\left(x\right)\Gamma \left(\zeta \right)}}\underset{a}{\overset{x}{\int }}{(x-\sigma )}^{\zeta -1}\mathsf{g}\left(\sigma \right)\tilde{v}\left(\sigma \right)d\sigma ,x\in (\mathsf{a},\mathsf{b})$$

(3)

Theorem 1

([10]). In the RL sense, the fuzzy fractional integral (FFI) with order $\zeta >0$ can be expressed using the formula defined as

$$\begin{array}{c}\hfill I^{\zeta }\tilde{v}(x,r)=\left[I^{\zeta }\underline{v}(x,r),I^{\zeta }\overline{v}(x,r)\right]\end{array}$$

Furthermore,

$$\begin{array}{c}\hfill I^{\zeta }\underline{v}(x,r)={\displaystyle \frac{1}{\Gamma \left(\zeta \right)}}{\int }_0^x{(x-\sigma )}^{\zeta -1}\underline{v}(\sigma ,r)d\sigma ,\sigma >0,\end{array}$$

and

$$\begin{array}{c}\hfill I^{\zeta }\overline{v}(x,r)={\displaystyle \frac{1}{\Gamma \left(\zeta \right)}}{\int }_0^x{(x-\sigma )}^{\zeta -1}\overline{v}(\sigma ,r)d\sigma ,\sigma >0.\end{array}$$

Definition 4

([12]). Caputo defines the fractional derivative of a function ($h\left(x\right)$) with a positive order ($\zeta \in {\mathbb{R}}^{+}$) as

$$\begin{array}{c}{\phantom{,}}^c{D}_0^{\zeta }h\left(x\right)=I^{p-\zeta }{D}^{p}h\left(x\right)\hfill \\ =\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma (p-\zeta )}}{\int }_0^x{(x-\sigma )}^{p-\zeta -1}{\displaystyle \frac{d^{p}h\left(\sigma \right)}{d{\sigma }^p}}d\sigma ,p-1<\zeta <p,\\ {\displaystyle \frac{d^{p}h\left(x\right)}{d{y}^p}},\zeta =p,p\in \mathbb{N}.\end{array}\right.\hfill \end{array}$$

Definition 5

([14]). The weighted Caputo fractional derivative (WCFD) of a function ($h\left(x\right)$) of order $\zeta \in {\mathbb{R}}^{+}$, with respect to the auxiliary functions ($f\left(x\right)$ and $g\left(x\right)$), is defined as follows:

$$\begin{array}{c}{\phantom{,}}^c{D}_0^{\zeta ,f,g}h\left(x\right)=I^{p-\zeta ,f,g}{\left({\displaystyle \frac{1}{f^{\prime }\left(x\right)}}{\displaystyle \frac{d}{dx}}\right)}^{p}h\left(x\right)\hfill \\ ={\displaystyle \frac{1}{\Gamma (p-\zeta )g\left(x\right)}}{\int }_0^x{(f\left(x\right)-f\left(\sigma \right))}^{p-\zeta -1}{f}^{\prime }\left(\sigma \right)g\left(\sigma \right)h_f^{\left(p\right)}\left(\sigma \right)d\sigma ,p-1<\zeta <p,\hfill \end{array}$$

where $h_f^{\left(p\right)}\left(\sigma \right)={\left({\displaystyle \frac{1}{f^{\prime }\left(\sigma \right)}}{\displaystyle \frac{d}{d\sigma }}\right)}^{p}h\left(\sigma \right)$.

The function expressed as $f:[0,1]\to {\mathbb{R}}^{+}$ is differentiable with $f^{\prime }\left(x\right)>0$, and the weight function ($g\left(x\right)\in L_{\infty }(a,b)$) satisfies $g\left(x\right)\ne 0$ almost everywhere.

Theorem 2

([10]). Assume that $\tilde{v}:\mathbb{R}\to {\mathbb{R}}^{\mathcal{F}}$ is the fuzzy function. If $\tilde{v}$ is differentiable, then $\underline{v}(x,r)$ and $\overline{v}(x,r)$ are also differentiable. The derivative of $\tilde{v}$ with respect to x is denoted as $\left[{\tilde{v}}^{\prime }(x,r)\right]=[{\underline{v}}^{\prime }(x,r),{\overline{v}}^{\prime }(x,r)]$.

Property 1.

We have the following [14]:

1. 

$I^{\zeta ,f,g}{I}^{\eta ,f,g}\tilde{v}(x,r)=I^{\zeta +\eta ,f,g}\tilde{v}(x,r)$, $\zeta ,\eta >0$;

2. 

$I^{\zeta ,f,g}(\tilde{v}(x,r)+\tilde{u}(x,r))=I^{\zeta ,f,g}\tilde{v}(x,r)+I^{\zeta ,f,g}\tilde{u}(x,r)$;

3. 

$I^{\zeta ,f,g}(\tilde{v}(x,r)-\tilde{u}(x,r))=I^{\zeta ,f,g}\tilde{v}(x,r)-I^{\zeta ,f,g}\tilde{u}(x,r)$;

4. 

$I^{\zeta ,f,g}\lambda \tilde{v}(x,r)=\lambda I^{\zeta ,f,g}\tilde{v}(x,r)$, λ is scalar;

5. 

$I^{\zeta ,f,gc}{D}^{\zeta ,f,g}\tilde{v}(x,r)=\tilde{v}(x,r)-\tilde{v}(\zeta ,r),0<\zeta \le 1$.

Theorem  3

([1]). In a complete metric space $(X,d)$, the Banach Contraction Principle asserts that for every contraction mapping ($T:X\to X$), there exists a unique fixed point (α) within X such that $T\alpha =\alpha$.

Theorem 4

([1]). The Krasnoselskii Fixed-Point Theorem states that if Q is a bounded, closed, and convex subset in a Banach Space ($\left(X,\parallel ·\parallel \right)$) and if $G_1$ and $G_2$ are mappings from Q to Q satisfying certain conditions,

1. 

For all $\alpha ,\beta \in Q$, the sum of $G_1\left(\alpha \right)+G_2\left(\beta \right)\in Q$;

2. 

$G_2$ is a contraction mapping;

3. 

$G_1$ is continuous and compact.

Then, there exists $k\in Q$ such that $G_1\left(k\right)+G_2\left(k\right)=k$.

Lemma 1

([12]). Let $\tilde{v}\left(x\right),A\left(x\right),q\left(x\right)\in C\left(\mathcal{J},{\mathbb{R}}^{+}\right)$, and let $m\left(x\right)\in C\left(\mathcal{J},{\mathbb{R}}^{+}\right)$ be nondecreasing, for which the inequality expressed as

$$\tilde{v}\left(x\right)\le m\left(x\right)+{\int }_0^{x}A\left(\sigma \right)\tilde{v}\left(\sigma \right)d\sigma +{\int }_0^{x}A\left(\sigma \right)\left({\int }_0^{\sigma }q\left(s\right)\tilde{v}\left(s\right)ds\right)d\sigma$$

holds for any $x\in \mathcal{J}$. Then,

$$\tilde{v}\left(x\right)\le m\left(x\right)\left[1+{\int }_0^{x}A\left(\sigma \right)\left({\int }_0^{\sigma }(A\left(s\right)+q\left(s\right))ds\right)d\sigma \right].$$

3. Existence and Uniqueness

The following section will focus on the findings related to the existence and uniqueness of Equation (1). Before delving into the demonstrations of the primary outcomes, we shall outline the subsequent assumptions.

(G1) There exist positive constants ${\kappa }_1,{\kappa }_2,{\kappa }_3$, and ${\kappa }_4$ such that for all $x\in \mathcal{J}$ and $\tilde{v},\tilde{u}\in Y$, whenever $(x,\sigma )\in G=\left\{\right(x,\sigma ):0\le \sigma \le x\le 1\}$, the specified conditions are satisfied"

$$\begin{array}{c}\parallel A(x,\tilde{v})-A(x,\tilde{u})\parallel \le {\kappa }_1\parallel \tilde{v}-\tilde{u}\parallel ,\hfill \\ \parallel F(x,\tilde{v})-F(x,\tilde{u})\parallel \le {\kappa }_2\parallel \tilde{v}-\tilde{u}\parallel ,\hfill \\ \parallel B(x,\sigma ,\tilde{v})-B(x,\sigma ,\tilde{u})\parallel \le {\kappa }_3\parallel \tilde{v}-\tilde{u}\parallel ,\hfill \\ \parallel C(x,\sigma ,\tilde{v})-C(x,\sigma ,\tilde{u})\parallel \le {\kappa }_4\parallel \tilde{v}-\tilde{u}\parallel ,\hfill \end{array}$$

with $\kappa =max\left\{{\kappa }_1,{\kappa }_2,{\kappa }_3,{\kappa }_4\right\}$.

(G2) Suppose that A, B, C, and F satisfy certain conditions,

$$\begin{array}{c}\parallel A(x,\tilde{v})\parallel \le {\tilde{u}}_1(x,r)\parallel \tilde{v}\parallel ,\hfill \\ \parallel F(x,\tilde{v})\parallel \le {\tilde{u}}_2(x,r)\parallel \tilde{v}\parallel ,\hfill \\ \parallel B(x,\sigma ,\tilde{v})\parallel \le {\tilde{u}}_3(x,r)\parallel \tilde{v}\parallel ,\hfill \\ \parallel C(x,\sigma ,\tilde{v})\parallel \le {\tilde{u}}_4(x,r)\parallel \tilde{v}\parallel ,\hfill \end{array}$$

where ${\tilde{u}}_i\in L^{\infty }(\mathcal{J},{\mathbb{R}}^{+}),i=1,2,3,4$, $x\in \mathcal{J}$, $\tilde{v}\in Y$, and $(x,\sigma )\in G$.

(G3) Assume that there are continuous functions ($A,F,B,$ and C) on $\mathcal{J}$ and $x\in \mathcal{J}$. Then, ∃$D_1,D_2,D_3$, and $D_4\in {\mathbb{R}}^{+}$, satisfying

$$\begin{array}{c}\parallel A(x,\tilde{v}(x,r))\parallel \le {\kappa }_1\parallel \tilde{v}\parallel +{\mathcal{M}}_1,\hfill \\ \parallel F(x,\tilde{v}(x,r))\parallel \le {\kappa }_2\parallel \tilde{v}\parallel +{\mathcal{M}}_2,\hfill \\ \parallel B(x,\sigma ,\tilde{v}){\parallel }_2 \le {\kappa }_3\parallel \tilde{v}\parallel +{\mathcal{M}}_3,\hfill \\ \parallel C(x,\sigma ,\tilde{v}){\parallel }_2 \le {\kappa }_4\parallel \tilde{v}\parallel +{\mathcal{M}}_4.\hfill \end{array}$$

(G4) Suppose that $g\left(x\right)\ne 0$ almost everywhere and $g\left(x\right)\in L_{\infty }(a,b)$, we know the following for some constant (C):

$${\parallel g\parallel }_{\infty }={ess\; sup}_{x\in (a,b)}\left|g\left(x\right)\right|\le C.$$

Using the change of variables $u=f\left(\sigma \right)$ and $du=f^{\prime }\left(\sigma \right)d\sigma$, we have

$${\int }_0^x{(f\left(x\right)-f\left(\sigma \right))}^{\zeta -1}d\sigma ={\int }_{f\left(0\right)}^{f\left(x\right)}{(f\left(x\right)-u)}^{\zeta -1}{\displaystyle \frac{1}{f^{\prime }\left(f^{-1}\left(u\right)\right)}}du.$$

Since $f^{\prime }\left(x\right)\ne 0$ and is continuous on $[0,1]$, ∃ a lower bound ($m>0$) such that $f^{\prime }\left(x\right)\ge m$ for all $x\in [0,1]$. Therefore, ${\displaystyle \frac{1}{f^{\prime }\left(f^{-1}\left(u\right)\right)}}$ is bounded above by ${\displaystyle \frac{1}{m}}$. Hence,

$${\int }_0^x{(f\left(x\right)-f\left(\sigma \right))}^{\zeta -1}d\sigma \le {\displaystyle \frac{1}{m}}{\int }_{f\left(0\right)}^{f\left(x\right)}{(f\left(x\right)-u)}^{\zeta -1}du.$$

From the previous analysis, we know the following:

$${\int }_{f\left(0\right)}^{f\left(x\right)}{(f\left(x\right)-u)}^{\zeta -1}du={\displaystyle \frac{{(f\left(x\right)-f\left(0\right))}^{\zeta }}{\zeta }}.$$

Thus,

$${\int }_0^x{(f\left(x\right)-f\left(\sigma \right))}^{\zeta -1}d\sigma \le {\displaystyle \frac{1}{m}}·{\displaystyle \frac{{(f\left(x\right)-f\left(0\right))}^{\zeta }}{\zeta }}.$$

Given that $f\left(x\right)$ is strictly increasing and bounded on $[0,1]$, $f\left(x\right)-f\left(0\right)$ is also bounded, and so is ${(f\left(x\right)-f\left(0\right))}^{\zeta }\le M$ for some constant (M). Therefore, the integral is bounded for $\zeta \ne 0$.

Remark 1.

To obtain the integral form of the proposed problem given in Equations (1) and (2), we utilize the inverse property of the weighted fractional Caputo derivative. This reformulation facilitates the use of fixed-point theory in the subsequent analysis. The following lemma presents the equivalent integral formulation.

Lemma 2.

Consider $0<\zeta +\eta <1,a\ne {\displaystyle \frac{1}{\beta }}$, and $\tilde{v}\in C(\mathcal{J},x)$ is called a solution of the proposed problem iff $\tilde{v}$ satisfies

$$\begin{array}{cc}\hfill \tilde{v}(x,r)=& {\displaystyle \frac{1}{g\left(x\right)}}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)g\left(\sigma \right)\left[A\left(\sigma ,\tilde{v}(\sigma ,r)\right)+{\int }_0^{\sigma }B(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho \right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(\rho ,\tilde{v}(\rho ,r))d\rho \right]d\sigma \hfill \\ \hfill & +{\displaystyle \frac{a}{(1-a\beta )g\left(\sigma \right)}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}g\left(\rho \right)\left[A(\rho ,\tilde{v}(\rho ,r))+{\int }_0^{\rho }B(\rho ,s,\tilde{v}(s,r))ds\right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(\rho ,s,\tilde{v}(s,r))ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(s,\tilde{v}(s,r))ds\right]d\rho .\hfill \end{array}$$

(4)

Proof. 

We first verify that the solution to the proposed Problems (1) and (2), $\tilde{v}$ fulfills the integral equation (IE) (4). According to Property 5 of Theorem 2, we derive

$$\begin{array}{c}\hfill I_{0^{+}}^{\zeta +\eta ,f,g}{\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta ,f,g}\tilde{v}(x,r)=\tilde{v}(x,r)-\tilde{v}(0,r).\end{array}$$

(5)

Furthermore, through the utilization of Equation (1) in combination with Definition 5 and the application of Theorem 2, we have

$$\begin{array}{cc}\hfill & I_{0^{+}}^{\zeta +\eta ,f,g}{\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta ,f,g}\tilde{v}(x,r)\hfill \\ \hfill & =I_{0^{+}}^{\zeta +\eta ,f,g}\left(A(x,\tilde{v}(x,r))+{\int }_0^{x}B(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma \right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma +I_{0^{+}}^{\zeta }F(x,\tilde{v}(x,r))\right)d\sigma \hfill \\ \hfill & ={\displaystyle \frac{1}{g\left(x\right)}}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)g\left(\sigma \right)\left[A(\sigma ,\tilde{v}(\sigma ,r))+{\int }_0^{\sigma }B(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho \right.\hfill \\ \hfill & +\left.{\int }_0^{1}C(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(\rho ,\tilde{v}(\rho ,r))d\rho \right]d\sigma .\hfill \end{array}$$

(6)

By substituting (6) into (5), we have the subsequent integral equation:

$$\begin{array}{cc}\hfill \tilde{v}(x,r)=& {\displaystyle \frac{1}{g\left(x\right)}}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)g\left(\sigma \right)\left[A(\sigma ,\tilde{v}(\sigma ,r))+{\int }_0^{\sigma }B(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho \right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(\rho ,\tilde{v}(\rho ,r))d\rho \right]d\sigma \hfill \\ \hfill & +\tilde{v}(0,r).\hfill \end{array}$$

(7)

By applying Fubini’s theorem and by carrying out subsequent computations, from Equation (2), we obtain

$$\begin{array}{c}\tilde{v}(0,r)=a{\int }_0^{\beta }\tilde{v}(\sigma ,r)d\sigma \hfill \\ =a{\int }_0^{\beta }\left[{\displaystyle \frac{1}{g\left(\sigma \right)}}{\int }_0^{\sigma }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{\prime }\left(\rho \right)g\left(\rho \right)\left(A(\rho ,\tilde{v}(\rho ,r))+{\int }_0^{\rho }B(\rho ,s,\tilde{v}(s,r))ds\right.\right.\hfill \\ \left.\left.+{\int }_0^{1}C(\rho ,s,\tilde{v}(s,r))ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(s,\tilde{v}(s,r))ds\right)d\rho \right]d\sigma +a\beta \tilde{v}(0,r)\hfill \\ =a{\int }_0^{\beta }{\displaystyle \frac{1}{g\left(\sigma \right)}}{\int }_0^{\sigma }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{\prime }\left(\rho \right)g\left(\rho \right)A(\rho ,\tilde{v}(\rho ,r))d\rho d\sigma \hfill \\ +a{\int }_0^{\beta }{\displaystyle \frac{1}{g\left(\sigma \right)}}{\int }_0^{\sigma }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{\prime }\left(\rho \right)g\left(\rho \right)\left[{\int }_0^{\rho }B(\rho ,s,\tilde{v}(s,r))ds\right.\hfill \\ +\left.{\int }_0^{1}C(\rho ,s,\tilde{v}(s,r))ds\right]d\rho d\sigma \hfill \\ +a{\int }_0^{\beta }{\displaystyle \frac{1}{g\left(\sigma \right)}}{\int }_0^{\sigma }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{\prime }\left(\rho \right)g\left(\rho \right){\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(s,\tilde{v}(s,r))dsd\rho d\sigma \hfill \\ +a\beta \tilde{v}(0,r)\hfill \\ =a{\int }_0^{\beta }A(\rho ,\tilde{v}(\rho ,r)){\displaystyle \frac{1}{g\left(\sigma \right)}}{\int }_{\rho }^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{\prime }\left(\rho \right)g\left(\rho \right)d\sigma d\rho \hfill \\ +a{\int }_0^{\beta }{\displaystyle \frac{1}{g\left(\sigma \right)}}{\int }_{\rho }^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{\prime }\left(\rho \right)g\left(\rho \right)d\sigma \left[{\int }_0^{\rho }B(\rho ,s,\tilde{v}(s,r))ds\right.\hfill \\ +\left.{\int }_0^{1}C(\rho ,s,\tilde{v}(s,r))ds\right]d\rho \hfill \\ +a{\int }_0^{\beta }{\displaystyle \frac{1}{g\left(\sigma \right)}}{\int }_{\rho }^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{\prime }\left(\rho \right)g\left(\rho \right)d\sigma {\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(s,\tilde{v}(s,r))dsd\rho \hfill \\ +a\beta \tilde{v}(0,r),\hfill \end{array}$$

that is,

$$\begin{array}{cc}\hfill \tilde{v}(0,r)& ={\displaystyle \frac{a}{(1-a\beta )g\left(\sigma \right)}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}g\left(\rho \right)[A(\rho ,\tilde{v}(\rho ,r))+{\int }_0^{\rho }B(\rho ,s,\tilde{v}(s,r))ds\hfill \\ \hfill & +{\int }_0^{1}C(\rho ,s,\tilde{v}(s,r))ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(s,\tilde{v}(s,r))ds]d\rho .\hfill \end{array}$$

(8)

By substituting the expression derived from Equation (8) into Equation (7), we arrive at Equation (4). Conversely, if we employ Definition 4 and utilize the operator expressed as ${\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta }$ on (4), we obtain

$$\begin{array}{cc}\hfill {\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta ,f,g}\tilde{v}(x,r)=& {\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta ,f,g}{I}_{0^{+}}^{\zeta +\eta ,f,g}\left(A(x,\tilde{v}(x,r))+{\int }_0^{x}B(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma \right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma +I_{0^{+}}^{\zeta }F(x,\tilde{v}(x,r))\right)d\sigma +{\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta }\tilde{v}(0,r)\hfill \\ \hfill =& A(x,\tilde{v}(x,r))+{\int }_0^{x}B(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma +{\int }_0^{1}C(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma \hfill \\ \hfill & +I_{0^{+}}^{\zeta }F(x,\tilde{v}(x,r))\hfill \end{array}$$

(9)

It follows from the given information that $\tilde{v}$ satisfies the equation given in Problems (1)–(2). Therefore, the solution to the proposed problem is expressed as $\tilde{v}$. □

Remark 2.

Utilizing the integral representation obtained in Lemma 2, we proceed to establish the existence of a solution. By constructing an appropriate operator and applying fixed-point theorems, sufficient conditions for the existence of solutions are derived under generalized Lipschitz-type assumptions (G1)–(G3).

Theorem 5.

If conditions (G1) and (G2) are met and in the case of

$$\begin{array}{cc}\hfill H:=& \left[M\left({\displaystyle \frac{{∥{\tilde{u}}_1∥}_{L^{\infty }}+{∥{\tilde{u}}_3∥}_{L^{\infty }}+{∥{\tilde{u}}_4∥}_{L^{\infty }}}{\Gamma (\zeta +\eta +1)}}+{\displaystyle \frac{{∥{\tilde{u}}_2∥}_{L^{\infty }}\zeta (\zeta +1,\zeta +\eta )}{\Gamma (\zeta +1)\Gamma (\zeta +\eta )}}\right)\right]\hfill \\ \hfill & +{\displaystyle \frac{M}{m}}\left({\displaystyle \frac{\left|a\right|{∥{\tilde{u}}_1∥}_{L^{\infty }}+\left|a\right|{∥{\tilde{u}}_3∥}_{L^{\infty }}+\left|a\right|{∥{\tilde{u}}_4∥}_{L^{\infty }}}{|1-a\beta |\Gamma (\zeta +\eta +2)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{\left|a\right|{∥{\tilde{u}}_2∥}_{L^{\infty }}{\beta }^{2\zeta +\eta +1}\zeta (\zeta +1,\zeta +\eta +1)}{|1-a\beta |\Gamma (\zeta +1)\Gamma (\zeta +\eta +1)}}\right)\le 1\hfill \end{array}$$

(10)

and

$$\kappa H_1:=\kappa {\displaystyle \frac{\left|a\right|M}{|1-a\beta |m}}\left[{\displaystyle \frac{3}{\Gamma (\zeta +\eta +2)}}+{\displaystyle \frac{{\beta }^{2\zeta +\eta +1}\beta (\zeta +1,\zeta +\eta +1)}{\Gamma (\zeta +1)\Gamma (\zeta +\eta +1)}}\right]\le 1$$

(11)

within the interval of $\mathcal{J}$, then ∃ a solution ($\tilde{v}(x,r)$) for Problems (1) and (2).

Proof. 

Consider the following closed ball:

$$L_{\varphi }=\left\{\tilde{v}\in C(\mathcal{J},x)\text{:}{\parallel \tilde{v}\parallel }_1\le \varphi \right\},$$

where the norm for any function ($\tilde{v}\in C(\mathcal{J},x)$) is defined as

$$\parallel \tilde{v}{\parallel }_1=max\left\{e^{-x}\parallel \tilde{v}(x,r)\parallel :x\in \mathcal{J}\right\}.$$

Now define the operator $\mathcal{T}:L_{\varphi }\to L_{\varphi }$ by

$$\begin{array}{cc}\hfill \mathcal{T}\tilde{v}(x,r)=& {\displaystyle \frac{1}{g\left(x\right)}}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)g\left(\sigma \right)\left[A\left(\sigma ,\tilde{v}(\sigma ,r)\right)+{\int }_0^{\sigma }B(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho \right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(\rho ,\tilde{v}(\rho ,r))d\rho \right]d\sigma \hfill \\ \hfill & +{\displaystyle \frac{a}{(1-a\beta )g\left(\sigma \right)}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}g\left(\rho \right)\left[A(\rho ,\tilde{v}(\rho ,r))+{\int }_0^{\rho }B(\rho ,s,\tilde{v}(s,r))ds\right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(\rho ,s,\tilde{v}(s,r))ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(s,\tilde{v}(s,r))ds\right]d\rho .\hfill \end{array}$$

(12)

For each x in $\mathcal{J}$, it is evident that the operator ($\mathcal{T}$) is clearly defined. We will now move forward to establish operators ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ on $L_{\varphi }$ as follows:

$$\begin{array}{cc}\hfill {\mathcal{T}}_1\tilde{v}(x,r)=& {\displaystyle \frac{1}{g\left(x\right)}}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)g\left(\sigma \right)\left[A\left(\sigma ,\tilde{v}(\sigma ,r)\right)+{\int }_0^{\sigma }B(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho \right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(\rho ,\tilde{v}(\rho ,r))d\rho \right]d\sigma \hfill \end{array}$$

(13)

and

$$\begin{array}{cc}\hfill {\mathcal{T}}_2\tilde{v}(x,r)=& {\displaystyle \frac{a}{(1-a\beta )g\left(\sigma \right)}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}g\left(\rho \right)\left[A(\rho ,\tilde{v}(\rho ,r))+{\int }_0^{\rho }B(\rho ,s,\tilde{v}(s,r))ds\right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(\rho ,s,\tilde{v}(s,r))ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(s,\tilde{v}(s,r))ds\right]d\rho .\hfill \end{array}$$

(14)

For $\tilde{v}$ and $\tilde{u}$ are elements of $L_{\varphi }$ and x is in $\mathcal{J}$, then using assumptions (G2) and (G4), we can establish

$$\begin{array}{c}∥{\mathcal{T}}_1\tilde{v}(x,r)+{\mathcal{T}}_2\tilde{u}(x,r)∥\hfill \\ \le {\displaystyle \frac{1}{\parallel g\left(x\right)\parallel }}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\parallel g\left(\sigma \right)\parallel \left[\parallel A(\sigma ,\tilde{v}(\sigma ,r))\parallel +{\int }_0^{\sigma }\parallel B(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho \right.\hfill \\ \left.+{\int }_0^1\parallel C(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\parallel F(\rho ,\tilde{v}(\rho ,r))\parallel d\rho \right]d\sigma \hfill \\ +{\displaystyle \frac{a}{(1-a\beta )\parallel g\left(\sigma \right)\parallel }}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}\parallel g\left(\rho \right)\parallel \left[\parallel A(\rho ,\tilde{u}(\rho ,r))\parallel +{\int }_0^{\rho }\parallel B(\rho ,s,\tilde{u}(s,r))\parallel ds\right.\hfill \\ \left.+{\int }_0^1\parallel C(\rho ,s,\tilde{u}(s,r))\parallel ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\parallel F(s,\tilde{u}(s,r))\parallel ds\right]d\rho \hfill \\ \le {\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\left[{\tilde{u}}_1(\sigma ,r)\parallel \tilde{v}(\sigma ,r)\parallel +{\int }_0^{\sigma }{\tilde{u}}_3(\sigma ,r)\parallel \tilde{v}(\rho ,r)\parallel d\rho \right.\hfill \\ \left.+{\int }_0^1{\tilde{u}}_4(\sigma ,r)\parallel \tilde{v}(\rho ,r)\parallel d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{\tilde{u}}_2(\sigma ,r)\parallel \tilde{v}(\rho ,r)\parallel d\rho \right]d\sigma \hfill \\ +{\displaystyle \frac{a}{(1-a\beta )}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}\left[{\tilde{u}}_1(\rho ,r)\parallel \tilde{u}(\rho ,r)\parallel +{\int }_0^{\rho }{\tilde{u}}_3(\rho ,r)\parallel \tilde{u}(s,r)\parallel ds\right.\hfill \\ \left.+{\int }_0^1{\tilde{u}}_4(\rho ,r)\parallel \tilde{u}(s,r)\parallel ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{\tilde{u}}_2(s,r)\parallel \tilde{u}(s,r)\parallel ds\right]d\rho \hfill \\ \le {\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\left[{∥{\tilde{u}}_1∥}_{L^{\infty }}\parallel \tilde{v}{\parallel }_1{e}^{\sigma }+{∥{\tilde{u}}_3∥}_{L^{\infty }}{\parallel \tilde{v}\parallel }_1\left(e^{\sigma }-1\right)\right.\hfill \\ \left.+{∥{\tilde{u}}_4∥}_{L^{\infty }}\parallel \tilde{v}{\parallel }_1\left(e^{\sigma }-1\right)+{∥{\tilde{u}}_2∥}_{L^{\infty }}{\parallel \tilde{v}\parallel }_1{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{e}^{\rho }d\rho \right]d\sigma \hfill \\ +{\displaystyle \frac{a}{(1-a\beta )}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}\left[{∥{\tilde{u}}_1∥}_{L^{\infty }}\parallel \tilde{u}{\parallel }_1{e}^{\rho }+{∥{\tilde{u}}_3∥}_{L^{\infty }}{\parallel \tilde{u}\parallel }_1\left(e^{\rho }-1\right)\right.\hfill \\ \left.+{∥{\tilde{u}}_4∥}_{L^{\infty }}\parallel \tilde{u}{\parallel }_1\left(e^{\rho }-1\right)+{∥{\tilde{u}}_2∥}_{L^{\infty }}{\parallel \tilde{u}\parallel }_1{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{e}^{s}ds\right]d\rho .\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & {∥{\mathcal{T}}_1\tilde{v}+{\mathcal{T}}_2\tilde{u}∥}_1\hfill \\ \hfill & \le {\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\left[{∥{\tilde{u}}_1∥}_{L^{\infty }}\parallel \tilde{v}{\parallel }_1{\displaystyle \frac{e^{\sigma }}{e^x}}+{∥{\tilde{u}}_3∥}_{L^{\infty }}{\parallel \tilde{v}\parallel }_1{\displaystyle \frac{(e^{\sigma }-1)}{e^x}}\right.\hfill \\ \hfill & \left.+{∥{\tilde{u}}_4∥}_{L^{\infty }}\parallel \tilde{v}{\parallel }_1{\displaystyle \frac{(e^{\sigma }-1)}{e^x}}+{∥{\tilde{u}}_2∥}_{L^{\infty }}{\parallel \tilde{v}\parallel }_1{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{\displaystyle \frac{e^{\rho }}{e^x}}d\rho \right]d\sigma \hfill \\ \hfill & +{\displaystyle \frac{a}{(1-a\beta )}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}\left[{∥{\tilde{u}}_1∥}_{L^{\infty }}\parallel \tilde{u}{\parallel }_1{\displaystyle \frac{e^{\rho }}{e^x}}+{∥{\tilde{u}}_3∥}_{L^{\infty }}{\parallel \tilde{u}\parallel }_1{\displaystyle \frac{(e^{\rho }-1)}{e^x}}\right.\hfill \\ \hfill & \left.+{∥{\tilde{u}}_4∥}_{L^{\infty }}\parallel \tilde{u}{\parallel }_1{\displaystyle \frac{(e^{\rho }-1)}{e^x}}+{∥{\tilde{u}}_2∥}_{L^{\infty }}{\parallel \tilde{u}\parallel }_1{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{\displaystyle \frac{e^s}{e^x}}ds\right]d\rho \hfill \\ \hfill & \le \varphi \left[M\left({\displaystyle \frac{{∥{\tilde{u}}_1∥}_{L^{\infty }}+{∥{\tilde{u}}_3∥}_{L^{\infty }}+{∥{\tilde{u}}_4∥}_{L^{\infty }}}{\Gamma (\zeta +\eta +1)}}+{\displaystyle \frac{{∥{\tilde{u}}_2∥}_{L^{\infty }}}{\Gamma (\zeta +1)\Gamma (\zeta +\eta )}}{\int }_0^1{(1-w)}^{\zeta +\eta +1}{w}^{\zeta }dw\right)\right]\hfill \\ \hfill & +{\displaystyle \frac{M}{m}}\left({\displaystyle \frac{\left|a\right|{∥{\tilde{u}}_1∥}_{L^{\infty }}+\left|a\right|{∥{\tilde{u}}_3∥}_{L^{\infty }}+\left|a\right|{∥{\tilde{u}}_4∥}_{L^{\infty }}}{|1-a\beta |\Gamma (\zeta +\eta +1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{\left|a\right|{∥{\tilde{u}}_2∥}_{L^{\infty }}}{|1-a\beta |\Gamma (\zeta +1)\Gamma (\zeta +\eta +1)}}{\int }_0^{\beta }{(\beta -\rho )}^{\zeta +\eta }{t}^{\zeta }d\rho \right)\hfill \\ \hfill & =\varphi \left[M\left({\displaystyle \frac{{∥{\tilde{u}}_1∥}_{L^{\infty }}+{∥{\tilde{u}}_3∥}_{L^{\infty }}+{∥{\tilde{u}}_4∥}_{L^{\infty }}}{\Gamma (\zeta +\eta +1)}}+{\displaystyle \frac{{∥{\tilde{u}}_2∥}_{L^{\infty }}\zeta (\zeta +1,\zeta +\eta )}{\Gamma (\zeta +1)\Gamma (\zeta +\eta )}}\right)\right]\hfill \\ \hfill & +{\displaystyle \frac{M}{m}}\left({\displaystyle \frac{\left|a\right|{∥{\tilde{u}}_1∥}_{L^{\infty }}+\left|a\right|{∥{\tilde{u}}_3∥}_{L^{\infty }}+\left|a\right|{∥{\tilde{u}}_4∥}_{L^{\infty }}}{|1-a\beta |\Gamma (\zeta +\eta +2)}}+{\displaystyle \frac{\left|a\right|{∥{\tilde{u}}_2∥}_{L^{\infty }}{\beta }^{2\zeta +\eta +1}\zeta (\zeta +1,\zeta +\eta +1)}{|1-a\beta |\Gamma (\zeta +1)\Gamma (\zeta +\eta +1)}}\right)\hfill \\ \hfill & =\varphi H\hfill \\ \hfill & \le \varphi .\hfill \end{array}$$

This shows that $\left({\mathcal{T}}_1\tilde{v}+{\mathcal{T}}_2\tilde{u}\right)$ belongs to the bounded ball ($L_{\varphi }$). Here, we made use of the calculation results defined as

$$\begin{array}{c}{\int }_0^1{(1-w)}^{\zeta +\eta +1}{w}^{\zeta }d\sigma =\beta (\zeta +1,\zeta +\eta )\hfill \\ {\int }_0^{\beta }{(\beta -\rho )}^{\zeta +\eta }{\rho }^{\zeta }dv={\beta }^{2\zeta +\eta +1}\beta (\zeta +1,\zeta +\eta +1).\hfill \end{array}$$

Furthermore, taking into account the given approximations, i.e., ${\displaystyle \frac{e^{\sigma }}{e^x}}\le 1$, ${\displaystyle \frac{e^{\rho }}{e^x}}\le 1$, and ${\displaystyle \frac{e^s}{e^x}}\le 1$, we will now prove that ${\mathcal{T}}_2$ functions as a contraction mapping. For this, consider

$$\begin{array}{c}∥{\mathcal{T}}_2\tilde{v}(x,r)-{\mathcal{T}}_2\tilde{u}(x,r)∥\hfill \\ \le {\displaystyle \frac{\left|a\right|}{|1-a\beta |\parallel g\left(\beta \right)\parallel }}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\beta \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}\parallel g\left(\rho \right)\parallel \left[∥A\left(\rho ,\tilde{v}(\rho ,r)\right)-A\left(\rho ,\tilde{u}(\rho ,r)\right)∥d\rho \right.\hfill \\ +{\int }_0^{\rho }∥B\left(\rho ,s,\tilde{v}(s,r)\right)-B\left(\rho ,s,\tilde{u}(s,r)\right)∥ds+{\int }_0^1\parallel C\left(\rho ,s,\tilde{v}(s,r)\right)\hfill \\ -C\left(\rho ,s,\tilde{u}(s,r)\right)\parallel ds\left.+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}∥F\left(s,\tilde{v}(s,r)\right)-F\left(s,\tilde{u}(s,r)\right)∥ds\right]d\rho \hfill \\ \le {\displaystyle \frac{\left|a\right|}{|1-a\beta |}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\beta \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}\left[{\kappa }_1\parallel \tilde{v}-\tilde{u}{\parallel }_1{e}^{\rho }+{\int }_0^{\rho }{\kappa }_3{\parallel \tilde{v}-\tilde{u}\parallel }_1{e}^{s}ds\right.\hfill \\ \left.+{\int }_0^1{\kappa }_4\parallel \tilde{v}-\tilde{u}{\parallel }_1{e}^{s}ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{\kappa }_2{\parallel \tilde{v}-\tilde{u}\parallel }_1{e}^{s}ds\right]d\rho \hfill \\ \le {\displaystyle \frac{\left|a\right|}{|1-a\beta |}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\beta \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}\left[\kappa \parallel \tilde{v}-\tilde{u}{\parallel }_1{e}^{\rho }+\kappa {\parallel \tilde{v}-\tilde{u}\parallel }_1\left(e^{\rho }-1\right)\right.\hfill \\ \left.+\kappa \parallel \tilde{v}-\tilde{u}{\parallel }_1\left(e^{\rho }-1\right)+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\kappa {\parallel \tilde{v}-\tilde{u}\parallel }_1{e}^{s}ds\right]d\rho .\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill & ∥{\mathcal{T}}_2\tilde{v}(x,r)-{\mathcal{T}}_2\tilde{u}(x,r)∥\hfill \\ \hfill & \le {\displaystyle \frac{\left|a\right|}{|1-a\beta |}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\beta \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}[\kappa \parallel \tilde{v}-\tilde{u}{\parallel }_1{\displaystyle \frac{e^{\rho }}{e^x}}+\kappa \parallel \tilde{v}-\tilde{u}{\parallel }_1{\displaystyle \frac{e^{\rho }-1}{e^x}}+\kappa {\parallel \tilde{v}-\tilde{u}\parallel }_1{\displaystyle \frac{e^{\rho }-1}{e^x}}\hfill \\ \hfill & +{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\kappa {\parallel \tilde{v}-\tilde{u}\parallel }_1{\displaystyle \frac{e^s}{e^x}}ds]d\rho \hfill \\ \hfill & \le {\displaystyle \frac{\left|a\right|\kappa M}{|1-a\beta |m}}\left[{\displaystyle \frac{3}{\Gamma (\zeta +\eta +2)}}+{\displaystyle \frac{{\beta }^{2\zeta +\eta +1}\zeta (\zeta +1,\zeta +\eta +1)}{\Gamma (\zeta +1)\Gamma (\zeta +\eta +1)}}\right]{\parallel \tilde{v}-\tilde{u}\parallel }_1\hfill \\ \hfill & =\kappa H_1{\parallel \tilde{v}-\tilde{u}\parallel }_1.\hfill \end{array}$$

(15)

Give that $\kappa H_1\le 1$, it can be deduced that ${\mathcal{T}}_2$ is a contraction mapping. Functions $A,B,C$, and F being continuous ensures that ${\mathcal{T}}_1$ is continuous as well. Moreover, it is evident that ${\mathcal{T}}_1{L}_{\varphi }$ is a subset of $L_{\varphi }$ for every $\tilde{v}\in L_{\varphi }$, which implies that ${\mathcal{T}}_1$ is uniformly bounded on $L_{\varphi }$, defined as

$$\begin{array}{c}∥{\mathcal{T}}_1\tilde{v}(x,r)∥\le {\displaystyle \frac{1}{\parallel g\left(x\right)\parallel }}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\parallel g\left(\sigma \right)\parallel \left[\parallel A(\sigma ,\tilde{v}(\sigma ,r))\parallel \right.\hfill \\ +{\int }_0^{\sigma }\parallel B(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho +{\int }_0^1\parallel C(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho \hfill \\ \left.+{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\parallel F(\rho ,\tilde{v}(\rho ,r))\parallel d\rho \right]d\sigma \hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill {∥{\mathcal{T}}_1\tilde{v}∥}_1\le & {\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\left[{∥{\tilde{u}}_1∥}_{L^{\infty }}\parallel \tilde{v}{\parallel }_1{\displaystyle \frac{e^{\sigma }}{e^x}}+{∥{\tilde{u}}_3∥}_{L^{\infty }}{\parallel \tilde{v}\parallel }_1{\displaystyle \frac{\left(e^{\sigma }-1\right)}{x}}\right.\hfill \\ \hfill & \left.+{∥{\tilde{u}}_4∥}_{L^{\infty }}\parallel \tilde{v}{\parallel }_1{\displaystyle \frac{\left(e^{\sigma }-1\right)}{e^x}}+{∥{\tilde{u}}_2∥}_{L^{\infty }}{\parallel \tilde{v}\parallel }_1{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{\displaystyle \frac{e^{\rho }}{e^x}}d\rho \right]d\sigma \hfill \\ \hfill \le & \varphi M\left[{\displaystyle \frac{{∥{\tilde{u}}_1∥}_{L^{\infty }}+{∥{\tilde{u}}_3∥}_{L^{\infty }}+{∥{\tilde{u}}_4∥}_{L^{\infty }}}{\Gamma (\zeta +\eta +1)}}+{\displaystyle \frac{{∥{\tilde{u}}_2∥}_{L^{\infty }\zeta }(\zeta +1,\zeta +\eta )}{\Gamma (\zeta +1)\Gamma (\zeta +\eta )}}\right.\hfill \\ \hfill \le & \varphi H\hfill \\ \hfill \le & \varphi .\hfill \end{array}$$

(16)

We aim to prove that the operator $\left({\mathcal{T}}_1{L}_{\varphi }\right)$ displays equicontinuity. In order to accomplish this, we will establish that

$$\begin{array}{c}\overline{A}={\displaystyle \underset{(\sigma ,\tilde{v})\in \mathcal{J}\times L_{\varphi }}{sup}}\parallel A(\sigma ,\tilde{v})\parallel ,\overline{F}={\displaystyle \underset{(\sigma ,\tilde{v})\in \mathcal{J}\times L_{\varphi }}{sup}}\parallel F(\sigma ,\tilde{v})\parallel ,\\ \overline{B}={\displaystyle \underset{(\sigma ,\rho ,\tilde{v})\in G\times L_{\varphi }}{sup}}{\int }_0^{\sigma }\parallel B(\sigma ,\rho ,\tilde{v})\parallel d\rho ,\overline{C}={\displaystyle \underset{(\sigma ,\rho ,\tilde{v})\in G\times L_{\varphi }}{sup}}{\int }_0^1\parallel C(\sigma ,\rho ,\tilde{v})\parallel d\rho .\end{array}$$

For any $\tilde{v}\in L_{\varphi }$ and for each $x_1,x_2\in \mathcal{J}$ with $x_1\le x_2$, we have

$$\begin{array}{c}∥\left({\mathcal{T}}_1\tilde{v}\right)\left(x_1,r\right)-\left({\mathcal{T}}_1\tilde{v}\right)\left(x_2,r\right)∥\hfill \\ \le {\displaystyle \frac{1}{\parallel g\left(x_2\right)\parallel }}{\int }_{x_1}^{x_2}{\displaystyle \frac{{\left(f\left(x_2\right)-f\left(\sigma \right)\right)}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\parallel g\left(\sigma \right)\parallel \left[\parallel A(\sigma ,\tilde{v}(\sigma ,r))\parallel +{\int }_0^{\sigma }\parallel B(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho \right.\hfill \\ \left.+{\int }_0^1\parallel C(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\parallel F(\rho ,\tilde{v}(\rho ,r))\parallel d\rho \right]d\sigma \hfill \\ +{\displaystyle \frac{1}{\parallel g\left(x\right)\parallel }}{\int }_0^{x_1}{\displaystyle \frac{\left[{\left(f\left(x_1\right)-f\left(\sigma \right)\right)}^{\zeta +\eta -1}-{\left(f\left(x_2\right)-f\left(\sigma \right)\right)}^{\zeta +\eta -1}\right]}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\parallel g\left(\sigma \right)\parallel \left[\parallel A(\sigma ,\tilde{v}(\sigma ,r))\parallel \right.\hfill \\ +{\int }_0^{\sigma }\parallel B(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho \left.+{\int }_0^1\parallel C(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\parallel F(\rho ,\tilde{v}(\rho ,r))\parallel d\rho \right]d\sigma \hfill \\ \le {\int }_{x_1}^{x_2}{\displaystyle \frac{{\left(f\left(x_2\right)-f\left(\sigma \right)\right)}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\left[\overline{A}+\overline{B}+\overline{C}+{\displaystyle \frac{\overline{F}}{\Gamma \left(\zeta \right)}}{\int }_0^{\sigma }{(\sigma -\rho )}^{\zeta -1}d\rho \right]d\sigma \hfill \\ +{\int }_0^{x_1}{\displaystyle \frac{\left[{\left(f\left(x_1\right)-f\left(\sigma \right)\right)}^{\zeta +\eta -1}-{\left(f\left(x_2\right)-f\left(\sigma \right)\right)}^{\zeta +\eta -1}\right]}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\hfill \\ \left[\overline{A}+\overline{B}+\overline{C}+{\displaystyle \frac{\overline{F}}{\Gamma \left(\zeta \right)}}{\int }_0^{\sigma }{(\sigma -\rho )}^{\zeta -1}d\rho \right]d\sigma \hfill \\ \le {\int }_{x_1}^{x_2}{\displaystyle \frac{{\left(f\left(x_2\right)-f\left(\sigma \right)\right)}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\left[\overline{A}+\overline{B}+\overline{C}+{\displaystyle \frac{\overline{F}}{\Gamma (\zeta +1)}}\right]d\sigma \hfill \\ +{\int }_0^{x_1}{\displaystyle \frac{\left[{\left(f\left(x_1\right)-f\left(\sigma \right)\right)}^{\zeta +\eta -1}-{\left(f\left(x_2\right)-f\left(\sigma \right)\right)}^{\zeta +\eta -1}\right]}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\left[\overline{A}+\overline{B}+\overline{C}+{\displaystyle \frac{\overline{F}}{\Gamma (\zeta +1)}}\right]d\sigma \hfill \\ \le {\displaystyle \frac{1}{\Gamma (\zeta +\eta +1)}}\left[\overline{A}+\overline{B}+\overline{C}+{\displaystyle \frac{\overline{F}}{\Gamma (\zeta +1)}}\right]\left[2{\left(f\left(x_2\right)-f\left(x_1\right)\right)}^{\zeta +\eta }+\left(f{\left(x_1\right)}^{\zeta +\eta }-f{\left(x_2\right)}^{\zeta +\eta }\right)\right]\hfill \\ \to 0\mathrm{as}f\left(x_1\right)\to f\left(x_2\right)\hfill \end{array}$$

This suggests that as $\left|{\mathcal{T}}_1\tilde{v}\left(x_2,r\right)-{\mathcal{T}}_1\tilde{v}\left(x_1,r\right)\right|\to 0$, the set $\left(\overline{{\mathcal{T}}_1{L}_{\varphi }}\right)$ displays equicontinuity. Consequently, ${\mathcal{T}}_1$ exhibits relative compactness on $L_{\varphi }$. According to the Arzela-Ascoli theorem, ${\mathcal{T}}_1$ is compact on $L_{\varphi }$. Since all prerequisites outlined in Theorem 4 are fulfilled, it follows that $\mathcal{T}$ retains a fixed point on $L_{\varphi }$. Thus, the suggested Problems (1)–(2) have a solution on $\mathcal{J}$ in the form of a fixed point. □

Remark 3.

From the preceding theorem, the existence of a solution is guaranteed; however, to ensure uniqueness, an additional condition ($\kappa H<1$) must be imposed.

Theorem 6.

Given the satisfaction of conditions (G1) and (G3) and under the condition that $\kappa H<1$, the system described by Equations (1) and (2) has a unique solution in $\mathcal{J}$.

Proof. 

The operator ($\mathcal{T}$) is defined as stated in Theorem 5. We establish

$${\mathcal{S}}_{\varphi }=\{\tilde{v}\in C(\mathcal{J},x):\parallel \tilde{v}\parallel \le \varphi \}$$

Assume $\varphi \ge {\displaystyle \frac{\mathcal{M}H}{1-\kappa H}}$, where $\mathcal{M}=max\left\{{\mathcal{M}}_1,{\mathcal{M}}_2,{\mathcal{M}}_3,{\mathcal{M}}_4\right\}$, such that ${\mathcal{M}}_1={sup}_{x\in \mathcal{J}}\parallel A(x,0)\parallel$, ${\mathcal{M}}_2={sup}_{x\in \mathcal{J}}\parallel F(x,0)\parallel ,{\mathcal{M}}_3={sup}_{(x,\sigma )\in G}\parallel B(x,\sigma ,0)\parallel ,{\mathcal{M}}_4={sup}_{(x,\sigma )\in G}\parallel$$C(x,\sigma ,0)\parallel$. First, we aim to establish that $\mathcal{T}{\mathcal{S}}_{\varphi }\subset {\mathcal{S}}_{\varphi }$. For any $\tilde{v}\in {\mathcal{S}}_{\varphi }$, the following holds:

$$\begin{array}{c}\parallel \left(\mathcal{T}\tilde{v}\right)(x,r)\parallel \le {\displaystyle \frac{1}{\parallel g\left(x\right)\parallel }}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\parallel g\left(\sigma \right)\parallel \left[\parallel A(\sigma ,\tilde{v}(\sigma ,r))\parallel \right.\hfill \\ +{\int }_0^{\sigma }\parallel B(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho +{\int }_0^1\parallel C(\sigma ,\rho ,\tilde{v}(\rho ,r))\parallel d\rho \hfill \\ \left.+{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\parallel F(\rho ,\tilde{v}(\rho ,r))\parallel d\rho \right]d\sigma +{\displaystyle \frac{a}{(1-a\beta )\parallel g\left(\sigma \right)\parallel }}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}\parallel g\left(\rho \right)\parallel \hfill \\ \left[\parallel A(\rho ,\tilde{v}(\rho ,r))\parallel +{\int }_0^{\rho }\parallel B(\rho ,s,\tilde{v}(s,r))\parallel ds\right.\hfill \\ \left.+{\int }_0^1\parallel C(\rho ,s,\tilde{v}(s,r))\parallel ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\parallel F(s,\tilde{v}(s,r))\parallel ds\right]d\rho \hfill \\ \le (\kappa \varphi +\mathcal{M})H\hfill \\ \le \varphi .\hfill \end{array}$$

Hence, $\mathcal{T}{\mathcal{S}}_{\varphi }\subset {\mathcal{S}}_{\varphi }$. We will now demonstrate that the mapping of $\mathcal{T}:{\mathcal{S}}_{\varphi }\to {\mathcal{S}}_{\varphi }$ satisfies the contraction property. By utilizing assumption (G1), we can deduce that for all $\tilde{v},\tilde{u}\in {\mathcal{S}}_{\varphi }$ and each $x\in \mathcal{J}$, the following holds:

$$\begin{array}{c}\parallel \mathcal{T}\tilde{v}(x,r)-\left(\mathcal{T}\tilde{u}\right)(x,r)\parallel \hfill \\ \le {\displaystyle \frac{1}{\parallel g\left(x\right)\parallel }}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)\parallel g\left(\sigma \right)\parallel \left[\parallel A(\sigma ,\tilde{v}(\sigma ,r))-A(\sigma ,\tilde{u}(\sigma ,r))\parallel \right.\hfill \\ +{\int }_0^{\sigma }\parallel B(\sigma ,\rho ,\tilde{v}(\rho ,r))-B(\sigma ,\rho ,\tilde{u}(\rho ,r))\parallel d\rho \hfill \\ +{\int }_0^1\parallel C(\sigma ,\rho ,\tilde{v}(\rho ,r))-C(\sigma ,\rho ,\tilde{u}(\rho ,r))\parallel d\rho \hfill \\ \left.+{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\parallel F(\rho ,\tilde{v}(\rho ,r))-F(\rho ,\tilde{u}(\rho ,r))\parallel d\rho \right]d\sigma \hfill \\ +{\displaystyle \frac{\left|a\right|}{\left|\right(1-a\beta \left)\right|\parallel g\left(\sigma \right)\parallel }}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}\parallel g\left(\rho \right)\parallel \left[\parallel A(\rho ,\tilde{v}(\rho ,r))-A(\rho ,\tilde{u}(\rho ,r))\parallel \right.\hfill \\ +{\int }_0^{\rho }\parallel B(\rho ,s,\tilde{v}(s,r))-B(\rho ,s,\tilde{u}(s,r))\parallel ds\hfill \\ +{\int }_0^1\parallel C(\rho ,s,\tilde{v}(s,r))-C(\rho ,s,\tilde{u}(s,r))\parallel ds\hfill \\ \left.+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}\parallel F(s,\tilde{v}(s,r))-F(s,\tilde{u}(s,r))\parallel ds\right]d\rho \hfill \\ \le \kappa H\parallel \tilde{v}-\tilde{u}\parallel .\hfill \end{array}$$

Since $\kappa H<1$, it can be deduced that $\mathcal{T}$ acts as a contraction, according to Theorem 3. Consequently, a distinct solution ($\tilde{v}\in C(\mathcal{J},x)$) exists, which satisfies the equation expressed $\mathcal{T}\tilde{v}=\tilde{v}$. So, a unique solution of the proposed problem in $C(\mathcal{J},x)$ exists. □

In this section, we proved the existence and uniqueness of solutions for the proposed FWFVFIDE using Banach’s and Krasnoselskii’s fixed-point theorems. Under appropriate Lipschitz-type and boundedness conditions, we established sufficient criteria to ensure that the associated operator admits a unique fixed point, thereby laying the groundwork for the stability analysis in the next section.

4. Stability

The stability analysis of fuzzy fractional integral equations plays a pivotal role in ensuring the robustness of solutions under perturbations in initial conditions or system parameters. These equations integrate memory-dependent dynamics via fractional operators and uncertainty through fuzzy set theory, necessitating sophisticated stability frameworks such as Ulam–Hyers and Ulam–Hyers–Rassias stability. These methods evaluate whether approximate solutions remain within a bounded proximity of exact solutions when subjected to small disturbances.

In this section, we focus on the stability of the proposed method in (1)–(2) for any given $ϵ>0$ and for every $\tilde{v}\in C(\mathcal{J},x)$ that satisfies certain conditions as follows:

$$\begin{array}{c}\tilde{v}(x,r)={\displaystyle \frac{1}{g\left(x\right)}}{\int }_0^x{\displaystyle \frac{{(f\left(x\right)-f\left(\sigma \right))}^{\zeta +\eta -1}}{\Gamma (\zeta +\eta )}}{f}^{{\phantom{,}}^{\prime }}\left(\sigma \right)g\left(\sigma \right)\left[A(\sigma ,\tilde{v}(\sigma ,r))+{\int }_0^{\sigma }B(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho \right.\hfill \\ \left.+{\int }_0^{1}C(\sigma ,\rho ,\tilde{v}(\rho ,r))d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\sigma -\rho )}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(\rho ,\tilde{v}(\rho ,r))d\rho \right]d\sigma \hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{a}{(1-a\beta )g\left(\sigma \right)}}{\int }_0^{\beta }{\displaystyle \frac{{(f\left(\sigma \right)-f\left(\rho \right))}^{\zeta +\eta }}{\Gamma (\zeta +\eta +1)}}g\left(\rho \right)\left[A(\rho ,\tilde{v}(\rho ,r))+{\int }_0^{\rho }B(\rho ,s,\tilde{v}(s,r))ds\right.\hfill \\ \hfill & \left.+{\int }_0^{1}C(\rho ,s,\tilde{v}(s,r))ds+{\int }_0^{\rho }{\displaystyle \frac{{(\rho -s)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}F(s,\tilde{v}(s,r))ds\right]d\rho .\hfill \end{array}$$

(17)

Consider the nonlinear continuous operator of $\delta :C(\mathcal{J},x)\to C(\mathcal{J},x)$, defined as follows:

$$\begin{array}{cc}\hfill \delta \tilde{v}(x,r)=& {\phantom{,}}^c{D}_{0^{+}}^{\zeta +\eta ,f,g}\tilde{v}(x,r)-I_{0^{+}}^{\zeta }F(x,\tilde{v}(x,r))-A(x,\tilde{v}(x,r))-{\int }_0^{x}B(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma \hfill \\ & -{\int }_0^{1}C(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma \hfill \end{array}$$

Definition 6

([10]). Let $ϵ>0$ and $\tilde{v}$ be a solution to Equations (1)–(2) such that

$$\parallel \delta \tilde{v}\parallel \le ϵ$$

(18)

The problem described by (1) is said to be UHS if $\exists 0<b\in \mathbb{R}$ and a solution $\tilde{u}\in C(\mathcal{J},x)$ for Equation (1), satisfying

$$\parallel \tilde{u}-\tilde{v}\parallel \le b{ϵ}^{{\phantom{,}}^{\prime }}$$

(19)

where $0<{ϵ}^{{\phantom{,}}^{\prime }}\in \mathbb{R}$, dependent on ϵ.

Definition 7

([10]). Assume $n\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ such that for any solution ($\tilde{v}$) of (1), ∃ a solution $\tilde{u}\in C(\mathcal{J},Y)$ for Equation (2) such that

$$\parallel \tilde{u}(x,r)-\tilde{v}(x,r)\parallel \le n{ϵ}^{\prime },x\in \mathcal{J}$$

(20)

Equation (1) is considered stable in the generalized Ulam–Hyers (GUHS) sense.

Definition 8

([10]). Every $ϵ>0$ and any solution ($\tilde{u}$) of Problem (1) is considered Ulam–Hyers–Rassias stable (UHRS) with respect to $\zeta \in C(\mathcal{J},{\mathbb{R}}^{+})$ if

$$\begin{array}{c}\hfill \parallel \delta \tilde{v}\parallel \le ϵ\zeta \left(x\right),x\in \mathcal{J},\end{array}$$

(21)

Moreover, there exists $b>0$ and a solution ($\tilde{v}\in C(\mathcal{J},{\mathbb{R}}^{+})$) of Problem (1) such that

$$\parallel \tilde{u}(x,r)-\tilde{v}(x,r)\parallel \le b{ϵ}_{{\phantom{,}}^{\prime }}\zeta \left(x\right),x\in \mathcal{J}$$

(22)

where ${ϵ}_{{\phantom{,}}^{\prime }}\in {\mathbb{R}}^{+}$ that depends on ϵ.

These stability concepts have important implications when modeling real-world systems governed by fuzzy fractional differential equations. For example, UHS ensures that solutions remain bounded under small constant deviations, making it suitable for systems requiring uniform precision, such as in control and engineering models. GUHS extends this by accommodating variable perturbations across the domain, which is particularly relevant in processes like heat transfer or diffusion, where disturbances depend on location. UHRS further generalizes the concept by combining constant and variable disturbances, making it ideal for dynamic systems, including biological models, that face both fixed and state-dependent uncertainties. These distinctions guide the choice of stability concepts based on the type and behavior of uncertainties present in specific applications.

The concepts of UHS, GUHS, and UHRS play a crucial role in ensuring the reliability of numerical simulations and analytical models. In practice, UHS is important when a uniform bound on the solution error is required across the entire domain, which is essential in control systems and engineering applications where constant precision must be maintained. GUHS extends this idea by allowing the error bound to vary with the independent variable, making it suitable for problems such as heat conduction, diffusion, or transport processes, where disturbances depend on spatial or temporal location. UHRS further generalizes stability by combining constant and variable error bounds, which is particularly relevant for dynamic systems that experience both fixed and state-dependent uncertainties, such as biological growth models, financial systems, or uncertain engineering processes. These distinctions are important because they guide the choice of stability framework depending on the nature of uncertainties in the modeled system, thereby improving the robustness and error tolerance of simulations. (See

Table 1. Comparison of Different Stability Types.

Stability Type	Perturbation Bound	Stability Estimate
Ulam–Hyers Stability (UHS)	Constant perturbation ${ϵ}^{\prime }>0$	$\parallel \tilde{v}-\tilde{u}\parallel \le b{ϵ}^{\prime }$ where b is a constant
Generalized Ulam–Hyers Stability (GUHS)	Perturbation $n\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ and ${ϵ}^{\prime }>0$	$\parallel \tilde{v}-\tilde{u}\parallel \le n{ϵ}^{\prime }$ with $\phi$ continuous and positive
Ulam–Hyers–Rassias Stability (UHRS)	Mixed perturbation $\zeta \left(x\right)$ and ${ϵ}^{\prime }>0$	$\parallel \tilde{v}-\tilde{u}\parallel \le b{ϵ}^{\prime }\zeta \left(x\right)$, combining constant and variable components

).

Theorem 7.

Assuming that (G1) is met and $\kappa H<1$, one can deduce that the problem stated in Equation (1) displays stability in both the UHS and GUHS contexts.

Proof. 

Let $\tilde{u}\in C(\mathcal{J},x)$ satisfies the conditions of (4) as outlined in Theorem 6 and is considered a solution to (1) and assuming $\tilde{v}$ is any solution that fulfills Equation (18), the relationship between $\delta$ and $\mathcal{T}-J$ (where J is the identity operator) is established by Lemma 2 for all solutions ($\tilde{v}\in C(\mathcal{J},x)$) of Equation (1) with $\kappa H<1$. Therefore, by utilizing the fixed-point property of the operator ($\mathcal{T}$), we can have

$$\begin{array}{c}\parallel \tilde{v}(x,r)-\tilde{u}(x,r)\parallel = \parallel \tilde{v}(x,r)-\mathcal{T}\tilde{v}(x,r)+\mathcal{T}\tilde{v}(x,r)-\tilde{u}(x,r)\parallel \hfill \\ = \parallel \tilde{v}(x,r)-\mathcal{T}\tilde{v}(x,r)+\mathcal{T}\tilde{v}(x,r)-\mathcal{T}\tilde{u}(x,r)\parallel \hfill \\ \le \parallel \mathcal{T}\tilde{v}(x,r)-\mathcal{T}\tilde{u}(x,r)\parallel +\parallel \tilde{v}(x,r)-\mathcal{T}\tilde{v}(x,r)\parallel \hfill \\ \le \kappa H\parallel \tilde{u}-\tilde{v}\parallel +ϵ\hfill \end{array}$$

Since $\kappa H<1$ and $ϵ>0$, we find

$$\parallel \tilde{u}-\tilde{v}\parallel \le {\displaystyle \frac{ϵ}{1-\kappa H}}$$

Setting ${ϵ}^{\prime }\le {\displaystyle \frac{ϵ}{1-\kappa H}}$ and $b=1$, we establish the UHS condition. Furthermore, the generalized Ulam–Hyers stability is ensured by choosing $nϵ\le {\displaystyle \frac{ϵ}{1-\kappa H}}$. □

Theorem 8.

Given that (G1) is satisfied with $\kappa <{\displaystyle \frac{1}{H}}$ and there exists a function ($\zeta \in (\mathcal{J},{\mathbb{R}}^{+})$) satisfying condition (21), it can be concluded that Problem (1) is UHRS with respect to ζ.

Proof. 

We have from Theorem 7, we have

$$\parallel \tilde{u}(x,r)-\tilde{v}(x,r)\parallel \le {ϵ}_{{\phantom{,}}^{\prime }}\zeta \left(x\right),x\in \mathcal{J}$$

where ${ϵ}_{{\phantom{,}}^{\prime }}={\displaystyle \frac{ϵ}{1-\kappa H}}$. □

We examined the Ulam–Hyers, generalized Ulam–Hyers, and Ulam–Hyers–Rassias stability of the fuzzy fractional model by studying how approximate solutions behave when small changes or perturbations are introduced. Using the contraction property of the associated operator, we proved that the solution stays close to the exact one within a certain bound. This confirms that the model is stable and reliable, even when affected by small functional errors or uncertainties.

5. Modified Adomian Decomposition Method

The Modified Adomian Decomposition Method (MADM) is an effective analytical technique for constructing approximate solutions of integral and differential equations, extending the classical Adomian decomposition method with modifications that enhance convergence and simplify computations, particularly for fuzzy-valued functions and fractional-order operators. It decomposes the solution and nonlinear terms into series using specially designed Adomian polynomials, enabling the direct handling of fuzzy uncertainty and fractional dynamics without linearization, discretization, or perturbation, thereby preserving the original problem structure. Compared with alternatives, the Variational Iteration Method (VIM) can converge rapidly for certain nonlinear problems via correction functionals with Lagrange multipliers but is less straightforward for weighted fuzzy fractional operators, while spectral and collocation methods (e.g., Chebyshev and Legendre) offer high accuracy and exponential convergence for smooth solutions but require discretization with global basis functions, making fuzzy parameter handling more complex. MADM’s analytic-series framework makes it particularly suitable for the weighted fuzzy fractional models considered here, though its convergence speed depends on problem data and initial approximations, and series-acceleration techniques such as Padé approximants may be employed for improved performance over larger domains.

In this section, we apply MADM to solve Problems (1)–(2). Applying the operator $I^{\zeta +\eta }$ to both sides of (1), we have

$$\begin{array}{cc}\hfill \tilde{v}(x,r)& =\tilde{v}(0,r)+I^{\zeta +\eta ,f,g}\left(A(x,\tilde{v}(x,r))+{\int }_0^{x}B(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma \right.\hfill \\ \hfill & +\left.{\int }_0^{1}C(x,\sigma ,\tilde{v}(\sigma ,r))d\sigma +I_{0^{+}}^{\zeta }F(x,\tilde{v}(x,r))\right)\hfill \end{array}$$

(23)

MADM expresses the solution ($\tilde{v}(x,r)$) in a series form, i.e., $\tilde{v}={\sum }_{j=0}^{\infty }\widetilde{v_j}$, and decomposes nonlinear terms $H_1$ and $H_2$ as

$$H_1=\sum _{j=0}^{\infty }{E}_j\mathrm{and}{H}_2=\sum _{j=0}^{\infty }{D}_j,$$

where $E_j$ and $D_j$ are Adomian polynomials given by

$$\begin{array}{c}{E}_j={\displaystyle \frac{1}{j!}}{\displaystyle \frac{d^j}{d{\vartheta }^j}}{\left[H_1\left(\sum _{l=0}^{\infty }{\varphi }^{\kappa }{\tilde{v}}_{\kappa }\right)\right]}_{\varphi =0},\hfill \\ D_j={\displaystyle \frac{1}{j!}}{\displaystyle \frac{d^j}{d{\varphi }^j}}{\left[H_2\left(\sum _{l=0}^{\infty }{\varphi }^{\kappa }{\tilde{v}}_{\kappa }\right)\right]}_{\varphi =0}.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}{E}_0=H_1\left({\tilde{v}}_0\right),\hfill \\ E_1={\tilde{v}}_1{H}_1^{\prime }\left({\tilde{v}}_0\right),\hfill \\ E_2={\tilde{v}}_2{H}_1^{\prime }\left({\tilde{v}}_0\right)+{\displaystyle \frac{1}{2}}{\tilde{v}}_1^2{H}_1^{\prime \prime }\left({\tilde{v}}_0\right),\hfill \\ E_3={\tilde{v}}_3{H}_1^{\prime }\left({\tilde{v}}_0\right)+{\tilde{v}}_1{\tilde{v}}_2{H}_1^{\prime \prime }\left({\tilde{v}}_0\right)+{\displaystyle \frac{1}{3}}{\tilde{v}}_1^3{H}_1^{\prime \prime \prime }\left({\tilde{v}}_0\right),\hfill \end{array}$$

$$\begin{array}{c}{D}_0=H_2\left({\tilde{v}}_0\right),\hfill \\ D_1={\tilde{v}}_1{H}_2^{\prime }\left({\tilde{v}}_0\right),\hfill \\ D_2={\tilde{v}}_2{H}_2^{\prime }\left({\tilde{v}}_0\right)+{\displaystyle \frac{1}{2}}{\tilde{v}}_1^2{H}_2^{\prime \prime }\left({\tilde{v}}_0\right),\hfill \\ D_3={\tilde{v}}_3{H}_2^{\prime }\left({\tilde{v}}_0\right)+{\tilde{v}}_1{\tilde{v}}_2{H}_2^{\prime \prime }\left({\tilde{v}}_0\right)+{\displaystyle \frac{1}{3}}{\tilde{v}}_1^3{H}_2^{\prime \prime \prime }\left({\tilde{v}}_0\right),\hfill \end{array}$$

The recursive process determines the components ${\tilde{v}}_0,{\tilde{v}}_1,{\tilde{v}}_2,\dots$.

$$\begin{array}{c}{\tilde{v}}_0(x,r)={\tilde{v}}_0,\hfill \\ {\tilde{v}}_{i+1}(x,r)=I^{\zeta +\eta ,f,g}\left(A(x,{\tilde{v}}_i(x,r))+{\int }_0^{x}B(x,\sigma ,{\tilde{v}}_i(\sigma ,r))d\sigma \right.\hfill \\ +\left.{\int }_0^{1}C(x,\sigma ,{\tilde{v}}_i(\sigma ,r))d\sigma +I_{0^{+}}^{\zeta }F(x,{\tilde{v}}_i(x,r))\right).\hfill \end{array}$$

MADM is defined by Wazwaz [23] and expressed recursively in the following manner:

$$\begin{array}{c}{\tilde{v}}_0(x,r)={\tilde{v}}_0+G_1(x,r),\hfill \\ {\tilde{v}}_1(x,r)=G_2(x,r)+I^{\zeta +\eta ,f,g}\left(A(x,{\tilde{v}}_0(x,r))+{\int }_0^{x}B(x,\sigma ,{\tilde{v}}_0(\sigma ,r))d\sigma \right.\hfill \\ +\left.{\int }_0^{1}C(x,\sigma ,{\tilde{v}}_0(\sigma ,r))d\sigma +I_{0^{+}}^{\zeta }F(x,{\tilde{v}}_0(x,r))\right)\hfill \\ ⋮\hfill \\ {\tilde{v}}_{i+1}(x,r)=I^{\zeta +\eta ,f,g}\left(A(x,{\tilde{v}}_i(x,r))+{\int }_0^{x}B(x,\sigma ,{\tilde{v}}_i(\sigma ,r))d\sigma \right.\hfill \\ +\left.{\int }_0^{1}C(x,\sigma ,{\tilde{v}}_i(\sigma ,r))d\sigma +I_{0^{+}}^{\zeta }F(x,{\tilde{v}}_i(x,r))\right),i\ge 1.\hfill \\ ⋮\hfill \end{array}$$

The solution is

$$\begin{array}{c}\hfill \tilde{v}(x,r)=\sum _{i=0}^{\infty }\widetilde{v_i}(x,r).\end{array}$$

Remark 4.

The convergence of the MADM in this work is underpinned by the contractive nature of the integral operator (T) appearing in the integral formulation (Lemma 2) and the fixed-point framework developed in Section 3. Under assumptions (G1)–(G3) and the smallness condition of $\kappa H<1$ (Theorem 6), the operator (T) is contractive on the chosen Banach ball and admits a unique fixed point; consequently, the MADM partial sums (${\tilde{v}}_i={\sum }_{j=0}^i{\tilde{v}}_j$) converge to the unique solution ($\tilde{v}$), and one can obtain geometric-type error estimates in terms of the contraction constant of $q:=\kappa H\in (0,1)$ (for instance, a bound ($\parallel {\tilde{v}}_i-\tilde{v}\parallel \le \kappa H\parallel {\tilde{v}}_1-{\tilde{v}}_0\parallel$) holds in the contractive regime). Therefore, in practice, we recommend the following convergence checks when applying MADM: monitor the incremental norm ($\parallel {\tilde{v}}_{i+1}-{\tilde{v}}_i{\parallel }_{\infty }$) and the residual ($\parallel T\left({\tilde{v}}_i\right)-{\tilde{v}}_i{\parallel }_{\infty }$) and stop when both fall below a prescribed tolerance.

The MADM flow chart is shown in Figure 1.

Remark 5.

Unlike the fuzzy Laplace transform, which requires the inversion of complex integral expressions and is more suitable for linear problems, MADM can efficiently handle nonlinear fuzzy kernels due to its recursive nature and lack of discretization. Similarly, compared to the Chebyshev spectral methods, which are best suited for smooth solutions over bounded domains, MADM provides analytical parametric solutions and avoids numerical instability associated with polynomial interpolation.

Numerical Examples

Example 1.

Consider the following weighted fuzzy fractional Volterra–Fredholm integro-differential equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phantom{,}}^c{D}_{0^{+}}^{0.4;f,g}\left(\tilde{v}(x,r)\right)\hfill & ={\displaystyle \frac{(2-x)\tilde{v}(x,r)}{60}}+{\int }_0^{\sigma }{\displaystyle \frac{e^{-(\sigma +\rho )}\tilde{v}(\rho ,r)}{64}}d\rho +{\int }_0^1{\displaystyle \frac{cos(\sigma +\rho )\tilde{v}(\rho ,r)}{32}}d\rho \hfill \\ \hfill & +{\int }_0^{\sigma }{\displaystyle \frac{{(\rho -w)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{\displaystyle \frac{(3-{\rho }^2)}{72}}\tilde{v}(x,r)d\rho \hfill \\ \tilde{v}(0,r)\hfill & =5{\int }_0^{0.2}\tilde{v}(0,r)d\sigma ,\hfill \\ \tilde{v}(0,r)\hfill & =\left[5{\int }_0^{0.2}\underline{v}(0,r)d\sigma ,5{\int }_0^{0.2}\overline{v}(0,r)d\sigma \right]\hfill \end{array}\right.\end{array}$$

Given that $\zeta =\eta =0.2$, $a=5$, and $\rho =0.2$ and assuming $\tilde{v},u\in Y={\mathbb{R}}^{+}$ with $x\in [0,1]$, the involved functions are continuous. According to condition (G2), we obtain ${∥{\tilde{u}}_1∥}_{L^{\infty }}={\displaystyle \frac{1}{60}},{∥{\tilde{u}}_2∥}_{L^{\infty }}={\displaystyle \frac{1}{64}},{∥{\tilde{u}}_3∥}_{L^{\infty }}={\displaystyle \frac{1}{32}},{∥{\tilde{u}}_4∥}_{L^{\infty }}={\displaystyle \frac{1}{74}},$ according to (G1), $\kappa ={\displaystyle \frac{1}{24}}$. Using these, we compute

$$H=0.2496,H_1=8.8196,\kappa H_1=0.3675<1,\kappa H=0.0104<1.$$

Therefore, the conditions of Theorem 2 are satisfied, ensuring the existence and uniqueness of a solution ($\tilde{v}(x,r)$) on $[0,1]$. Furthermore, since $\kappa H=0.0064<1$, the problem is fuzzy Ulam–Hyers stable (UHS), generalized fuzzy UHS, and fuzzy UHRS, provided a continuous and positive function exists.

Now, we solve Example 1 using the MADM. The equivalent parametric representation is expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phantom{,}}^c{D}_{0^{+}}^{0.4;f,g}\left(\underline{v}(x,r)\right)\hfill & ={\displaystyle \frac{(2-x)\underline{v}(x,r)}{60}}+{\int }_0^{\sigma }{\displaystyle \frac{e^{-(\sigma +\rho )}\underline{v}(\rho ,r)}{64}}d\rho +{\int }_0^1{\displaystyle \frac{cos(\sigma +\rho )\underline{v}(\rho ,r)}{32}}d\rho \hfill \\ \hfill & +{\int }_0^{\sigma }{\displaystyle \frac{{(\rho -w)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{\displaystyle \frac{(3-{\rho }^2)}{72}}\underline{v}(x,r)d\rho \hfill \\ \underline{v}(0,r)\hfill & =5{\int }_0^{0.2}\underline{v}(0,r)d\sigma =[r-1]\sum _{k=0}^2{\displaystyle \frac{x^k}{k!}}.\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phantom{,}}^c{D}_{0^{+}}^{0.4;f,g}\left(\overline{v}(x,r)\right)\hfill & ={\displaystyle \frac{(2-x)\overline{v}(x,r)}{60}}+{\int }_0^{\sigma }{\displaystyle \frac{e^{-(\sigma +\rho )}\overline{v}(\rho ,r)}{64}}d\rho +{\int }_0^1{\displaystyle \frac{cos(\sigma +\rho )\overline{v}(\rho ,r)}{32}}d\rho \hfill \\ \hfill & +{\int }_0^{\sigma }{\displaystyle \frac{{(\rho -w)}^{\zeta -1}}{\Gamma \left(\zeta \right)}}{\displaystyle \frac{(3-{\rho }^2)}{72}}\overline{v}(x,r)d\rho \hfill \\ \overline{v}(0,r)\hfill & =5{\int }_0^{0.2}\overline{v}(0,r)d\sigma =[1-r]\sum _{k=0}^2{\displaystyle \frac{x^k}{k!}}.\hfill \end{array}\right.\end{array}$$

For simplicity, we select power-law weights:

$$f\left(x\right)=x,g\left(x\right)=1,$$

which reduces the weighted Caputo derivative to the standard Caputo derivative. The boundary condition determines the initial term:

$$\begin{array}{cc}\hfill {\underline{v}}_0(x,r)& =(r-1)\left(1+x+{\displaystyle \frac{x^2}{2}}\right),\hfill \\ \hfill {\overline{v}}_0(x,r)& =(1-r)\left(1+x+{\displaystyle \frac{x^2}{2}}\right).\hfill \end{array}$$

We use the weighted integral:

$${\tilde{v}}_1(x,r)=I^{0.4}\left[{\displaystyle \frac{(2-x){\tilde{v}}_0(x,r)}{60}}+I_1+I_2+I_3\right].$$

$$\begin{array}{cc}\hfill {\underline{v}}_1(x,r)& ={\displaystyle \frac{r-1}{\Gamma \left(0.4\right)}}{\int }_0^x{(x-\sigma )}^{-0.6}\left({\displaystyle \frac{(2-\sigma )}{60}}\left(1+z+{\displaystyle \frac{z^2}{2}}\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{e^{-(\sigma +\rho )}}{64}}+{\displaystyle \frac{cos(\sigma +\rho )}{32}}+{\displaystyle \frac{(3-t^2)}{72}}\right)d\sigma .\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\underline{v}}_1(x,r)& \approx (r-1)\left(0.15x^{0.4}+0.06x^{1.4}+0.03x^{2.4}\right.\hfill \\ \hfill & \left.+0.5e^{-x}+0.3(sin(1+x)-sinx)\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\overline{v}}_1(x,r)& \approx (1-r)\left(0.15x^{0.4}+0.06x^{1.4}+0.03x^{2.4}\right.\hfill \\ \hfill & \left.+0.5e^{-x}+0.3(sin(1+x)-sinx)\right).\hfill \end{array}$$

$${\tilde{v}}_2(x,r)=I^{0.4}\left[{\displaystyle \frac{(2-x){\tilde{v}}_1(x,r)}{60}}+I_1+I_2+I_3\right].$$

$${\underline{v}}_2(x,r)\approx (r-1)\left(0.0075x^{0.8}+0.003x^{1.8}+\dots \right).$$

$${\overline{v}}_2(x,r)\approx (1-r)\left(0.0075x^{0.8}+0.003x^{1.8}+\dots \right).$$

$$\begin{array}{cc}\hfill \underline{v}(x,r)& \approx (r-1)\left(1+x+{\displaystyle \frac{x^2}{2}}+0.15x^{0.4}+0.06x^{1.4}\right.\hfill \\ \hfill & \left.+0.03x^{2.4}+0.0075x^{0.8}+0.003x^{1.8}+\dots \right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill \overline{v}(x,r)& \approx (1-r)\left(1+x+{\displaystyle \frac{x^2}{2}}+0.15x^{0.4}+0.06x^{1.4}\right.\hfill \\ \hfill & \left.+0.03x^{2.4}+0.0075x^{0.8}+0.003x^{1.8}+\dots \right).\hfill \end{array}$$

Fuzzy solutions are given in Figure 2, Figure 3 and Figure 4.

Figure 2 demonstrates the influence of varying the r level on the fuzzy solutions. A lower r level corresponds to greater fuzziness, while increasing the r level reduces the degree of fuzziness. In Figure 3, the impact of different values of x on the fuzzy solutions is depicted. Smaller x values are associated with reduced fuzziness, whereas larger x values tend to enhance it. Figure 4 presents the combined effects of varying both x and r levels on the fuzzy solutions, offering a more comprehensive perspective based on the observations in Figure 2 and Figure 3.

Example 2.

Consider the following FWFVFIDE with integral boundary conditions.

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phantom{,}}^c{D}_{0^{+}}^{0.4;f,g}\left(\tilde{v}(x,r)\right)\hfill & ={\displaystyle \frac{x}{20}}-{\displaystyle \frac{x^2}{30}}-{\displaystyle \frac{(2-x)\tilde{v}(x,r)}{60}}+{\int }_0^{\sigma }{\displaystyle \frac{e^{-(\sigma +\rho )}\tilde{v}(\rho ,r)}{64}}d\rho +\hfill \\ \hfill & {\int }_0^1{\displaystyle \frac{cos(\sigma +\rho )\tilde{v}(\rho ,r)}{32}}d\rho +{\int }_0^{\sigma }{\displaystyle \frac{{(\rho -\sigma )}^{\omega -1}}{\Gamma \left(\omega \right)}}{\displaystyle \frac{(3-t^2)}{72}}\tilde{v}(\rho ,r)d\rho \hfill \\ \tilde{v}(0,r)\hfill & =5{\int }_0^{0.2}\tilde{v}(0,r)d\sigma ,\hfill \\ \tilde{v}(0,r)\hfill & =\left[5{\int }_0^{0.2}\underline{v}(0,r)d\sigma ,5{\int }_0^{0.2}\overline{v}(0,r)d\sigma \right]=[r-1,1-r]\hfill \end{array}\right.\end{array}$$

The boundary condition determines the initial term:

$$\begin{array}{cc}\hfill {\underline{v}}_0(x,r)=(r-1), & \hfill {\overline{v}}_0(x,r)=(1-r).\end{array}$$

$$\begin{array}{cc}\hfill {\underline{v}}_1(x,r)& =(r-1)[{\displaystyle \frac{x^{1.4}}{20\Gamma \left(2.4\right)}}-{\displaystyle \frac{x^{2.4}}{30\Gamma \left(3.4\right)}}-{\displaystyle \frac{2x^{0.4}}{60\Gamma \left(1.4\right)}}+{\displaystyle \frac{x^{1.4}}{60\Gamma \left(2.4\right)}}\hfill \\ \hfill & +{\displaystyle \frac{e^{-x}(1-e^{-x})}{64}}+{\displaystyle \frac{sin(x+1)-sinx}{32}}+{\displaystyle \frac{x^{0.6}(3-x^2/3)}{72\Gamma \left(1.6\right)}}]\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\overline{v}}_1(x,r)& =(1-r)[{\displaystyle \frac{x^{1.4}}{20\Gamma \left(2.4\right)}}-{\displaystyle \frac{x^{2.4}}{30\Gamma \left(3.4\right)}}-{\displaystyle \frac{2x^{0.4}}{60\Gamma \left(1.4\right)}}+{\displaystyle \frac{x^{1.4}}{60\Gamma \left(2.4\right)}}\hfill \\ \hfill & +{\displaystyle \frac{e^{-x}(1-e^{-x})}{64}}+{\displaystyle \frac{sin(x+1)-sinx}{32}}+{\displaystyle \frac{x^{0.6}(3-x^2/3)}{72\Gamma \left(1.6\right)}}]\hfill \end{array}$$

The approximate solution up to two terms is expressed as follows:

$$\begin{array}{cc}\hfill \underline{v}(x,r)& \approx (r-1)\left[1+{\displaystyle \frac{x^{1.4}}{20\Gamma \left(2.4\right)}}-{\displaystyle \frac{x^{2.4}}{30\Gamma \left(3.4\right)}}-{\displaystyle \frac{2x^{0.4}}{60\Gamma \left(1.4\right)}}+\dots \right]\hfill \\ \hfill \overline{v}(x,r)& \approx (1-r)\left[1+{\displaystyle \frac{x^{1.4}}{20\Gamma \left(2.4\right)}}-{\displaystyle \frac{x^{2.4}}{30\Gamma \left(3.4\right)}}-{\displaystyle \frac{2x^{0.4}}{60\Gamma \left(1.4\right)}}+\dots \right]\hfill \end{array}$$

Fuzzy solutions are given in Figure 5, Figure 6 and Figure 7.

Figure 5 demonstrates the influence of varying the r level on the fuzzy solutions. A lower r level corresponds to greater fuzziness, while increasing the r level reduces the degree of fuzziness. In Figure 6, the impact of different values of x on the fuzzy solutions is depicted. Smaller x values are associated with reduced fuzziness, whereas larger x values tend to enhance it. Figure 7 presents the combined effects of varying both x and r levels on the fuzzy solutions, offering a more comprehensive perspective based on the observations in Figure 5 and Figure 6.

Example 3.

Consider the following FWFVFIDE:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phantom{,}}^c{D}_{0^{+}}^{0.7;f,g}\left(\tilde{v}(x,r)\right)\hfill & ={\displaystyle \frac{(1+x^2)\tilde{v}(x,r)}{50}}+{\int }_0^x{\displaystyle \frac{sin(x+\rho )\tilde{v}(\rho ,r)}{40}}d\rho \hfill \\ \hfill & +{\int }_0^1{\displaystyle \frac{e^{-x\rho }\tilde{v}(\rho ,r)}{30}}d\rho +I_{0^{+}}^{0.1}\left({\displaystyle \frac{x^2\tilde{v}(x,r)}{20}}\right),\hfill \\ \tilde{v}(0,r)\hfill & =4{\int }_0^{0.3}\tilde{v}(\sigma ,r)d\sigma ,\hfill \\ \tilde{v}(0,r)\hfill & =\left[(r-1)\sum _{k=0}^3{\displaystyle \frac{x^k}{k!}},(1-r)\sum _{k=0}^3{\displaystyle \frac{x^k}{k!}}\right].\hfill \end{array}\right.\end{array}$$

The equivalent parametric representation is expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phantom{,}}^c{D}_{0^{+}}^{0.7;f,g}\left(\underline{v}(x,r)\right)\hfill & ={\displaystyle \frac{(1+x^2)\underline{v}(x,r)}{50}}+{\int }_0^x{\displaystyle \frac{sin(x+\rho )\underline{v}(\rho ,r)}{40}}d\rho \hfill \\ \hfill & +{\int }_0^1{\displaystyle \frac{e^{-x\rho }\underline{v}(\rho ,r)}{30}}d\rho +I_{0^{+}}^{0.1}\left({\displaystyle \frac{x^2\underline{v}(x,r)}{20}}\right),\hfill \\ \underline{v}(0,r)\hfill & =4{\int }_0^{0.3}\underline{v}(\sigma ,r)d\sigma =(r-1)\left(1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{6}}\right),\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phantom{,}}^c{D}_{0^{+}}^{0.7;f,g}\left(\overline{v}(x,r)\right)\hfill & ={\displaystyle \frac{(1+x^2)\overline{v}(x,r)}{50}}+{\int }_0^x{\displaystyle \frac{sin(x+\rho )\overline{v}(\rho ,r)}{40}}d\rho \hfill \\ \hfill & +{\int }_0^1{\displaystyle \frac{e^{-x\rho }\overline{v}(\rho ,r)}{30}}d\rho +I_{0^{+}}^{0.1}\left({\displaystyle \frac{x^2\overline{v}(x,r)}{20}}\right),\hfill \\ \overline{v}(0,r)\hfill & =4{\int }_0^{0.3}\overline{v}(\sigma ,r)d\sigma =(1-r)\left(1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{6}}\right).\hfill \end{array}\right.\end{array}$$

Weight functions are chosen as $f\left(x\right)=e^{-x}$ and $g\left(x\right)=x$. The recursive scheme becomes the following:

$$\begin{array}{cc}\hfill {\underline{v}}_0(x,r)& =(r-1)\left(1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{6}}\right),\hfill \\ \hfill {\overline{v}}_0(x,r)& =(1-r)\left(1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{6}}\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\underline{v}}_1(x,r)& =I^{0.7,e^{-x},x}[{\displaystyle \frac{(1+x^2){\underline{v}}_0}{50}}+{\int }_0^x{\displaystyle \frac{sin(x+\rho ){\underline{v}}_0}{40}}d\rho \hfill \\ \hfill & +{\int }_0^1{\displaystyle \frac{e^{-x\rho }{\underline{v}}_0}{30}}d\rho +I^{0.7}\left({\displaystyle \frac{x^2{\underline{v}}_0}{20}}\right)]\hfill \\ \hfill & \approx (r-1)e^x\left({\displaystyle \frac{x^{0.1}}{\Gamma \left(1.7\right)}}\left({\displaystyle \frac{1}{50}}+{\displaystyle \frac{x^2}{50}}\right)+{\displaystyle \frac{sinx}{40}}·{\displaystyle \frac{x^{1.7}}{\Gamma \left(2.7\right)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{1-e^{-x}}{30}}·{\displaystyle \frac{x^{0.7}}{\Gamma \left(1.7\right)}}+{\displaystyle \frac{x^{2.7}}{20\Gamma \left(3.7\right)}}\right)\hfill \\ \hfill & =(r-1)e^x\left(0.02x^{0.7}+0.01x^{2.7}+0.033x^{1.7}sinx+0.033x^{0.7}(1-e^{-x})+0.005x^{2.7}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\underline{v}}_2(x,r)& =I^{0.7,e^{-x},x}[{\displaystyle \frac{(1+x^2){\underline{v}}_1}{50}}+{\int }_0^x{\displaystyle \frac{sin(x+\rho ){\underline{v}}_1}{40}}d\rho \hfill \\ \hfill & +{\int }_0^1{\displaystyle \frac{e^{-x\rho }{\underline{v}}_1}{30}}d\rho +I^{0.1}\left({\displaystyle \frac{x^2{\underline{v}}_1}{20}}\right)]\hfill \\ \hfill & \approx (r-1)e^x\left(0.0004x^{1.4}+0.0002x^{3.4}+0.001x^{2.4}sinx+\dots \right)\hfill \end{array}$$

The approximate solution after two iterations is expressed as follows:

$$\begin{array}{cc}\hfill \underline{v}(x,r)& \approx (r-1)[\left(1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{6}}\right)\hfill \\ \hfill & +e^x\left(0.02x^{0.7}+0.038x^{1.7}sinx+0.015x^{2.7}\right)\hfill \\ \hfill & +e^x\left(0.0004x^{1.4}+0.001x^{2.4}sinx+\dots \right)],\hfill \\ \hfill \overline{v}(x,r)& \approx (1-r)[\left(1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{6}}\right)\hfill \\ \hfill & +e^x\left(0.02x^{0.7}+0.038x^{1.7}sinx+0.015x^{2.7}\right)\hfill \\ \hfill & +e^x\left(0.0004x^{1.4}+0.001x^{2.4}sinx+\dots \right)].\hfill \end{array}$$

Fuzzy solutions are given in Figure 8, Figure 9 and Figure 10.

Figure 8 demonstrates the influence of varying the r level on the fuzzy solutions. A lower r level corresponds to greater fuzziness, while increasing the r level reduces the degree of fuzziness. In Figure 9, the impact of different values of x on the fuzzy solutions is depicted. Smaller x values are associated with reduced fuzziness, whereas larger x values tend to enhance it. Figure 10 presents the combined effects of varying both x and r levels on the fuzzy solutions, offering a more comprehensive perspective based on the observations in Figure 8 and Figure 9.

Figure 2, Figure 5 and Figure 8 show that when the fuzzy uncertainty level (r) increases, the uncertainty curves expand. Figure 3, Figure 6 and Figure 9 show that when the x variable increases, the uncertainty region also becomes larger. Figure 4, Figure 7 and Figure 10 show the effect of changing both r and x together, which is consistent with the results of the earlier figures. These results show that our weighted fuzzy fractional model can capture how uncertainty grows with both r and x, something that simpler models without weights or fuzziness cannot represent well.

6. Conclusions and Future Research

This study presents an analysis and the solutions for a class of WFFVFIDEs subject to integral boundary conditions. Utilizing the theoretical framework of weighted fractional calculus in the Caputo sense, sufficient conditions for the existence and uniqueness of solutions were rigorously established through the application of Banach’s and Krasnoselskii’s fixed-point theorems. To further substantiate the theoretical findings, the MADM was employed to derive explicit approximate solutions. The results underscore the effectiveness of the MADM in addressing the inherent complexities associated with fuzzy fractional operators.

Future research could extend the proposed model to systems with variable-order fractional operators or delay terms, as well as to fractional integro-differential equations formulated within the complex variable approach for deeper analytical exploration. The model could be further linked with viscoelastic models for Newtonian fluids, Maxwell fluids, and half-order power-law fluids to better understand their memory and hereditary effects [24]. Other possible applications include fuzzy fractional modeling of viscoelastic materials, uncertain population dynamics, and chaotic systems with memory. In addition, developing numerical methods with adaptive step sizes or using machine learning to improve decomposition methods could make computations faster and more accurate.