Hyers–Ulam Stability Analysis of Nonlinear Volterra–Fredholm Integro-Differential Equation with Caputo Derivative

Abstract

The main aim of this study is to examine the Hyers–Ulam stability of fractional derivatives in Volterra–Fredholm integro-differential equations using Caputo fractional derivatives. We explore the existence and uniqueness of solutions for the proposed integro-differential equation using Banach and Krasnoselskii’s fixed-point techniques. Furthermore, we examine the Hyers–Ulam stability of the equation under the Caputo fractional derivative by deriving suitable sufficient conditions. We analyze the graphical behavior of the obtained results to demonstrate the efficiency of the analytical method, highlighting its ability to deliver accurate and precise approximate numerical solutions for fractional differential equations. Finally, numerical applications are presented to validate the stability of the proposed integro-differential equation.

1. Introduction

Fractional calculus primarily involves the study of integral and derivative operators with fractional orders, having a history in mathematics as old as that of ordinary differential calculus [1,2]. Leibniz introduced fractional derivatives, distinct from their integer-order counterparts, in 1695. Fractional calculus has grown into a significant method for modeling and analyzing complex phenomena across a wide range of scientific and engineering domains. It involves the calculus of integrals and derivatives of non-integer orders. Its popularity has grown rapidly in recent years because of its applications in areas such as biology [3], economics [4], control theory [5] and mechanics [6]. Additionally, fractional derivatives have also been studied in the context of Navier–Stokes equations [7] and reaction–subdiffusion equations [8].

In 1926, Vito Volterra introduced Volterra integral equations and Volterra integro-differential equations. Since then, they have been widely utilized in science and engineering [9,10]. These equations appear in various physical applications, such as nano-hydrodynamics, heat transfer, diffusion processes in general neutron diffusion, and the interaction of biological species with fluctuating generation rates. They also describe wind ripples in deserts.

Furthermore, the widespread use of fractional calculus, along with integro-differential equations, has led to their incorporation. Many researchers contribute to its development. For instance, the authors in [11,12] explored the existence and uniqueness of solutions for fractional integro-differential equations. The collocation method utilizing Bell polynomials is used to solve the fractional integro-differential equation, as studied in [13]. In [14], the authors examined this equation using the Hahn wavelets collocation method. Fractional integro-differential equation was additionally employed to obtain the numerical solution in [15]. The authors in [16] studied a second-order novel explicit fast numerical scheme for the Cauchy problem, incorporating integro-differential equations. Fractional-order Volterra–Fredholm integro-differential equation was studied using the iterative method [17] and the second-order numerical method [18].

The investigation into the stability theory began with a question posed by Ulam regarding the stability of group homomorphisms [19]. In 1941, Hyers provided a positive answer to Ulam’s question specifically for the Cauchy equation in Banach spaces [20]. Consequently, Ulam stability has become increasingly important within the framework of fractional calculus. To illustrate, the authors in [21] examined the Hyers–Ulam stability for some fractional dynamic equations. The Hyers–Ulam stability of various fractional differential equations was explored in [22,23]. In [24], the authors investigated the Hyers–Ulam stability analysis of fractional integro-differential equations. Additionally, the existence and controllability results for fractional integro-differential equations have been established using the Caputo [25] and Atangana–Baleanu [26] derivatives.

In [27], the authors investigated the existence and uniqueness of solutions for the following integro-differential equation:

$$\begin{array}{cc}\hfill & D^{\mu }u(\ell )=g\left(\ell ,u(\ell ),Iu\left(\phi \right)\right),0<\mu <1,\hfill \\ \hfill & g\left(0\right)+u\left(0\right)=g_0.\hfill \end{array}$$

where $\ell \in P:[0,1],g:P\times X\times X⟶X,u:C(P,X)⟶X$ defined as $Iu(\ell )={\int }_0^{\ell }N(\ell ,s,u\left(s\right))ds.$ Here, $(X,∥.∥)$ is a Banach space and $C=C(P,X)$ represents the Banach space of all bounded continuous functions from P into X provided with the norm $∥.∥C.$

Using Pachpatte’s inequality, the authors in [28] studied the existence and uniqueness of the solutions for the following fractional-order integro-differential equation:

$$\begin{array}{cc}\hfill {}^{C}D^{\mu }u(\ell )& =\lambda u(\ell )+g(\ell ,u(\ell ),Iu(\ell \left)\right),\ell \in [a,b]\hfill \\ \hfill u\left(a\right)& =u_a\in \mathcal{R},\hfill \end{array}$$

where $\mu \in (0,1)$ and $Iu(\ell )={\int }_a^{\ell }N(\ell ,\phi )u\left(\phi \right)d\phi .$

In [29], the authors applied the modified Adomian decomposition method to obtain numerical solutions for the following problem:

$$\begin{array}{cc}\hfill {}^{C}D^{\mu }u(\ell )& =\mathcal{A}(\ell )u(\ell )+{\int }_0^{\ell }N(\ell ,\phi )F\left(u\left(\phi \right)\right)d\phi +f\left(t\right)\hfill \\ \hfill u\left(0\right)& =u_0.\hfill \end{array}$$

Additionally, the authors also explored the existence, uniqueness, and convergence of the solution to the above problem.

Motivated by the aforementioned discussion, the primary objective of this work is to provide new insights into the field of fractional derivatives, along with offering a graphical representation and theoretical applications of the problem under consideration. We aim to analyze the Hyers–Ulam stability of the following nonlinear Volterra–Fredholm integro-differential equations using fixed-point techniques:

$$\begin{array}{cc}\hfill {}^{C}D^{\mu }u(\ell )& =\mathcal{A}u(\ell )+\mathcal{B}h(\ell )+g(\ell ,u(\ell ),Iu(\phi \left)\right)\hfill \\ \hfill u\left({\ell }_0\right)& =u_0,\ell \in P=[{\ell }_0,b],\hfill \end{array}$$

(1)

where $Iu\left(\phi \right)={\int }_{{\ell }_0}^{\ell }N(\ell ,s,u\left(s\right))ds,{}^{C}D^{\mu }$ represents the Caputo fractional derivative with order $\mu ,$ where $0<\mu <1,$ and $\mathcal{A}$ from $D\left(\mathcal{A}\right)\subset X$ to X as a closed linear operator on a Banach space $X.$$u(\ell )$ assumes the values in Banach space $X.$$u(.)\in L^2[P,U]$ is the control function with U as a Banach function and operator B from U to X is bounded and linear. Functions $N:\delta \times X\to X$ and $g:P\times X\times X\to X$, where $\delta =\left\{(\ell ,s):0\le {\ell }_0\le s\le \ell \le 1\right\}.$

To the best of our knowledge, a gap exists in the research concerning the Hyers–Ulam stability analysis of nonlinear Volterra–Fredholm integro-differential Equation (1) involving fractional derivative operators. This gap underscores the necessity for further investigation. To bridge this gap, the present study utilizes fractional derivatives in the Caputo sense to examine the Hyers–Ulam stability of nonlinear Volterra–Fredholm integro-differential Equation (1). Driven by the above considerations, this paper seeks to bridge the identified research gap. The primary contributions of this work are outlined as follows:

$\left(i\right)$

There is a lack of sufficient research on the analysis of Hyers-Ulam stability, which has created a gap in the existing literature. The main focus of this study is to analyze the Hyers–Ulam stability of the fractional-order Volterra–Fredholm integro-differential equation.

$\left(ii\right)$

Utilizing Banach and Krasnoselskii’s fixed-point techniques, the existence and uniqueness of solutions for the integro-differential equation are established and the Hyers–Ulam stability is derived with the Caputo fractional derivative.

$\left(iii\right)$

A numerical example is provided to showcase the practical effectiveness of the theoretical findings, demonstrating how the results can be applied in real-world scenarios.

$\left(iv\right)$

In addition, two numerical applications are presented to further illustrate the practical implementation of the numerical findings, highlighting their relevance and utility in solving actual problems.

The remaining of the paper is structured as follows: In Section 2, some basic definitions regarding Caputo fractional derivative, theorems and hypotheses are presented. Section 3 discusses the conditions for the existence and uniqueness of solutions for the proposed integro-differential equation. Section 4 explores the conditions for Hyers–Ulam stability. Furthermore, a numerical result and applications for the proposed equation are provided in Section 5. Some general conclusions are presented in Section 6.

2. Preliminaries

This section presents some fundamental definitions and theorems of fractional calculus. The most widely used definitions of fractional-order derivatives include the Grünwald–Letnikov, Riemann–Liouville and Caputo definitions. Since the Caputo fractional-order derivative is more practical and widely applicable than the others, this paper primarily uses the Caputo definition.

Definition 1 

([1,30]). The Riemannn–Liouville fractional integral of function g with order $\mu >0$ is defined by

$$\begin{array}{c}\hfill {}^{RL}I^{\mu }g(\ell )={\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_0^{\ell }{(\ell -\phi )}^{(\mu -1)}g\left(\phi \right)d\phi ,for\ell >0,\mu \in {\mathbb{R}}^{+},\end{array}$$

where ${}^{RL}I^{0}g(\ell )=g(\ell )$ and ${\mathbb{R}}^{+}$ refers to the set of all positive real numbers.

Definition 2 

([30]). The Caputo fractional derivative of the function g with order $0<\mu <1$ is defined by

$$\begin{array}{c}\hfill {}^{C}D^{\mu }g(\ell )={\displaystyle \frac{1}{\Gamma (m-\mu )}}{\int }_0^{\ell }{(\ell -\phi )}^{m-\mu -1}{g}^{\left(m\right)}\left(\phi \right)d\phi ,m-1<\mu <m,\end{array}$$

where $m\in \mathbb{N}$ and $g^{\left(m\right)}$ denotes the $m^{th}$ derivative of $g.$

Lemma 1 

([30]). (Banach’s fixed-point theorem) If $\mathcal{G}$ is a nonemply closed subset of Banach space X, then for any contraction mapping $\mathcal{T}$ from $\mathcal{G}$ to itself, there exists a unique fixed point.

Lemma 2 

([30,31]). (Arzela–Ascoli theorem) We let $\mathcal{G}$ be a compact Hausdorff metric spaces. Then $\mathcal{K}\in C\left(\mathcal{G}\right)$ is relatively compact if and only if $\mathcal{K}$ is uniformly bounded and equicontinuous.

Lemma 3 

([31,32]). (Krasnoselskii fixed-point theorem) We let η be a nonempty closed, bounded and convex subset of Banach space X and let $\mathcal{U}$ and $\mathcal{V}$ be the two functions satisfying the following conditions:

$\left(i\right)$

$\mathcal{U}\left(\eta \right)+\mathcal{V}\left(\eta \right)\subset \eta$;

$\left(ii\right)$

$\mathcal{V}$ is continuous on η and $\mathcal{V}\left(\eta \right)$ is a relatively compact subset of X;

$\left(iii\right)$

$\mathcal{U}$ is a contraction on $\eta ,$ i.e., there exists $k\in [0,1)$ such that $\left|\right|\mathcal{U}\left(u\right)+\mathcal{U}\left(v\right)\left|\right|\le k\left|\right|s-l\left|\right|$ for every $u,v\in \eta$.

Then, there exists $u\in \eta$ such that $\mathcal{U}u+\mathcal{V}u=u$.

Hypotheses

In this section, we present some essential hypotheses [24,32] for the proposed Problem (1).

${\mathbf{H}}_{\mathbf{1}}$:

We let $u\in C[0,1]$. We suppose $g\in C[0,1]\times P\times P$ is a continuous function and there exists a positive constant ${\lambda }_1,{\lambda }_2$ and $\lambda$ such that

$$\begin{array}{c}\hfill ∥g(\ell ,u_1,v_1)-g(\ell ,u_2,v_2)∥\le {\lambda }_1(∥u_1-u_2∥+∥v_1+v_2∥)\end{array}$$

for every $u_1,v_1\in Y,$${\lambda }_2=ma{x}_{\ell \in P}∥g(\ell ,0,0)∥$ and $\lambda =max\left\{{\lambda }_1,{\lambda }_2\right\}$. We let $Y=C[P,X]$ be the continuous functions on P with values in Banach spaces X.

${\mathbf{H}}_{\mathbf{2}}$:

There exist non-negative constants $n_1,n_2$ and n such that

$$\begin{array}{c}\hfill ∥I(\ell ,r,u_1)-I(\ell ,r,u_2)∥\le n_1∥u_1-u_2∥\end{array}$$

$\forall u_1,u_2\in Y,$ $n_2=ma{x}_{(\ell ,r)\in D}∥I(\ell ,r,0)∥$ and $n=max\left\{n_1,n_2\right\}$.

${\mathbf{H}}_{\mathbf{3}}$:

We let $B_r=\left\{u\in (C[0,1],P):∥u∥\le \gamma \right\}$; then $B_r$ is a closed, convex and bounded subset in $\left(C\right[0,1],P)$. We consider $v=\left(\lambda (∥u∥+n\ell ∥u∥)\right)$.

3. Existence and Uniqueness of Solutions

In this section, we present the existence and uniqueness of solutions for the proposed Equation (1).

Lemma 4. 

We let $u_0(\ell )\in C(P,X).$ Then $u(\ell )\in C(P,X)$ constitutes a solution to Problem (1) if and only if it complies with the following conditions:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u(\ell )& =u_0+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi ,\hfill \end{array}\end{array}$$

(2)

for $\ell \in P$.

Proof. 

By applying Integral Operator 1 to both sides of Equation (1), we can readily derive integral Equation (2). □

To establish our results, we provide the following uniqueness theorem with suitable fixed-point technique:

Theorem 1. 

We assume that hypotheses $\left[H_1\right]$–$\left[H_3\right]$ are satisfied with $\left({\displaystyle \frac{∥\mathcal{A}∥+v}{\Gamma (\mu +1)}}\right)b^{\mu }=\beta$ and $|u_0|+\left({\displaystyle \frac{∥\mathcal{B}∥∥h∥}{\Gamma (\mu +1)}}\right)b^{\mu }=(1-\beta )\gamma .$ Then Problem (1) has a unique solution.

Proof. 

We define operator $T:C(P,X)\to C(P,X)$ as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left(Tu\right)(\ell )& =u_0+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi ,\hfill \end{array}\end{array}$$

Step 1 Our objective is to demonstrate that operator T preserves the functions within the set $B_r$. Based on the given hypotheses, for any function u belonging to $B_r$ and for all ℓ in the interval P, the following statement can be made:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|\right(Tu\left)\right(\ell \left)\right|& \le |u_0|+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\left|A\right|\left|u\left(\phi \right)\right|d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\left|B\right|\left|h\left(\phi \right)\right|d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\left|g(\phi ,u\left(\phi \right),Iu\left(\phi \right))\right|d\phi \hfill \\ \hfill & \le |u_0|+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }(\ell -\phi )∥\mathcal{A}∥∥u∥d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}∥\mathcal{B}∥∥h∥d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\lambda (∥u∥+n\ell ∥u∥)d\phi \hfill \\ \hfill & \le |u_0|+{\displaystyle \frac{∥\mathcal{A}∥b^{\mu }∥u∥}{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }∥\mathcal{B}∥∥h∥}{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}v∥u∥\hfill \\ \hfill & \le |u_0|+{\displaystyle \frac{∥\mathcal{A}∥b^{\mu }\gamma }{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }∥\mathcal{B}∥∥h∥}{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}v\gamma \hfill \\ \hfill & \le |u_0|+b^{\mu }\left({\displaystyle \frac{∥\mathcal{B}∥∥h∥}{\Gamma (\mu +1)}}\right)+b^{\mu }\gamma \left({\displaystyle \frac{∥\mathcal{A}∥+v}{\Gamma (\mu +1)}}\right)\hfill \\ \hfill & \le (1-\beta )\gamma +\beta \gamma \hfill \\ \hfill & =\gamma .\hfill \end{array}\end{array}$$

Since $b=\ell -{\ell }_0$, $∥Tu∥\le \gamma$, which implies that $Tu\in B_r,$ that is, $T{B}_r$ is a subset of $B_r.$

Step 2 Now we prove that operator T is a contraction mapping. Let us consider $u_1$ and $u_2$ belonging to $B_r$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |T{u}_1(\ell )-T{u}_2(\ell )|& \le {\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\left|\mathcal{A}\right||u_1\left(\phi \right)-u_2\left(\phi \right)|d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}|g(\phi ,u_1\left(\phi \right),Iu\left(\phi \right))-g(\phi ,u_2\left(\phi \right),I{u}_2\left(\phi \right))|d\phi \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\left|\mathcal{A}\right||u_1\left(\phi \right)-u_2\left(\phi \right)|d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\left(\lambda (∥u_1-u_2∥+n\ell ∥u_1-u_2∥)\right)d\phi \hfill \\ \hfill & \le {\displaystyle \frac{∥\mathcal{A}∥b^{\mu }}{\Gamma (\mu +1)}}∥u_1-u_2∥+{\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}v∥u_1-u_2∥\hfill \\ \hfill & \le \rho ∥u_1-u_2∥,\hfill \end{array}\end{array}$$

where $\rho =\left({\displaystyle \frac{∥\mathcal{A}∥b^{\mu }}{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }v}{\Gamma (\mu +1)}}\right)<1.$ Hence, $∥T{u}_1(\ell )-T{u}_2(\ell )∥\le \rho ∥u_1-u_2∥.$ This shows that operator T satisfies as a contraction mapping. Consequently, according to the Banach fixed-point theorem, there exists a fixed point, denoted as u, such that $Tu=u$. □

Next, we investigate the existence of solutions for the proposed Problem (1) by using Krasnoselskii’s fixed-point theorem.

Theorem 2. 

If hypotheses $\left[H_1\right]-\left[H_3\right]$ are satisfied, then problem (1) has at least one solution over the interval $[{\ell }_0,b]$.

Proof. 

For any positive constant r and $u\in B_r$, we define two operators $T_1$ and $T_2$ on $B_r$ as follows:

$$\begin{array}{cc}{T}_{1}u(\ell )\hfill & =u_0+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi \hfill \end{array}$$

(3)

$$\begin{array}{cc}{T}_{2}u(\ell )& ={\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi .\hfill \end{array}$$

(4)

Obviously, u is a solution of Equation (1) if and only if operator $T_{1}u+T_{2}u=u$ has solution $u\in B_r.$ Our proof is divided into four steps.

Step 1. The norm of the sum of $T_{1}u$ and $T_{2}u$ is an element of the ball $B_r$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ∥T_{1}u(\ell )+T_{2}u(\ell )∥& \le ∥u_0∥+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}∥\mathcal{A}u\left(\phi \right)∥d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}∥\mathcal{B}h\left(\phi \right)∥d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}∥g(\phi ,u(\phi ),Iu(\phi \left)\right)∥d\phi \hfill \\ \hfill & \le (1-\beta )\gamma +\beta \gamma =\gamma \hfill \end{array}\end{array}$$

Hence, $∥T_{1}u+T_{2}u∥\le \gamma .$ That is $T_{1}u+T_{2}u\in B_r.$

Step 2. Operator $T_1$ is contraction on ball $B_r$

For any $u,y\in B_r.$ According to $\left[H_3\right]$ and (3), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ∥T_{1}u(\ell )-T_{1}y(\ell )∥& \le ∥u_0+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi -y_0-{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi ∥\hfill \\ \hfill & =L∥u_0-y_0∥,\hfill \end{array}\end{array}$$

which implies that $∥T_{1}u(\ell )-T_{1}y(\ell )∥\le L∥u_0-y_0∥.$ Since $L=1$, operator $T_1$ is a contraction mapping.

Step 3. Operator $T_2$ is continuous

We let ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ be a sequence of converging to u to u∈$B_r.$ Then, by $\left[H_1\right]$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ∥T_2{u}_n(\ell )-T_{2}u(\ell )∥& \le {\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}∥\mathcal{A}∥∥u_n-u∥d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}∥g(\phi ,u_n\left(\phi \right),I{u}_n\left(\phi \right))-g(\phi ,u\left(\phi \right),Iu\left(\phi \right))∥d\phi \hfill \\ \hfill & \le {\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}∥\mathcal{A}∥∥u_n-u∥+{\displaystyle \frac{∥g(\phi ,u_n\left(\phi \right),I{u}_n\left(\phi \right))-g(\phi ,u\left(\phi \right),Iu\left(\phi \right))∥}{\Gamma (\mu +1)}}{b}^{\mu }.\hfill \end{array}\end{array}$$

Since sequence $\left\{u_n\right\}$ converges to u and function g is continuous, $∥T_2{u}_n(\ell )-T_{2}u(\ell )∥\to 0$ as $n\to \infty$. It shows that operator $T_2$ is continuous.

Step 4. Compactness of operator $T_2$

It is important to underscore that operator $T_2$ remains uniformly bounded on $B_r$ to establish the compactness of the operator, as demonstrated below:

$$\begin{array}{c}\hfill ∥T_{2}u∥\le {\displaystyle \frac{∥\mathcal{A}∥∥u∥}{\Gamma (\mu +1)}}{b}^{\mu }+{\displaystyle \frac{\lambda (∥u∥+n\ell ∥u∥)}{\Gamma (\mu +1)}}{b}^{\mu }\end{array}$$

Now, we demonstrate the equicontinuous of operator $T_2,$ for ${\ell }_1,{\ell }_2\in [{\ell }_0,b],$ where ${\ell }_2>{\ell }_1,$ and for any function $u\in B_r$.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ∥T_{2}u\left({\ell }_1\right)-T_{2}u\left({\ell }_2\right)∥& =\parallel {\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{{\ell }_1}{({\ell }_1-\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{{\ell }_1}{({\ell }_1-\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi \hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{{\ell }_2}{({\ell }_2-\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{{\ell }_2}{({\ell }_2-\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi \parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\mu \right)}}\parallel {\int }_{{\ell }_2}^{{\ell }_1}{({\ell }_1-\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi +{\int }_{{\ell }_2}^{{\ell }_1}{({\ell }_1-\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi \hfill \\ \hfill & -{\int }_{{\ell }_0}^{{\ell }_2}{({\ell }_2-\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi -{\int }_{{\ell }_0}^{{\ell }_2}{({\ell }_2-\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi \hfill \\ \hfill & +{\int }_{{\ell }_0}^{{\ell }_2}{({\ell }_1-\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi +{\int }_{{\ell }_0}^{{\ell }_2}{({\ell }_1-\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi \parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\mu \right)}}\parallel {\int }_{{\ell }_2}^{{\ell }_1}{({\ell }_1-\phi )}^{(\mu -1)}\left[\mathcal{A}u\left(\phi \right)+g(\phi ,u(\phi ),Iu(\phi \left)\right)\right]d\phi \hfill \\ \hfill & -{\int }_{{\ell }_0}^{{\ell }^2}\left[{({\ell }_2-\phi )}^{(\mu -1)}-{({\ell }_1-\phi )}^{(\mu -1)}\right]\mathcal{A}u\left(\phi \right)d\phi \hfill \\ \hfill & -{\int }_{{\ell }_0}^{{\ell }_2}\left[{({\ell }_2-\phi )}^{(\mu -1)}-{({\ell }_1-\phi )}^{(\mu -1)}\right]g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi \parallel \hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \le {\displaystyle \frac{∥\mathcal{A}∥∥u∥}{\Gamma (\mu +1)}}|2{({\ell }_1-{\ell }_2)}^{\mu }+{\ell }_2^{\mu }-{\ell }_1^{\mu }|+{\displaystyle \frac{∥g(\phi ,u(\phi ),Iu(\phi \left)\right)∥}{\Gamma (\mu +1)}}|2{({\ell }_1-{\ell }_2)}^{\mu }+{\ell }_2^{\mu }-{\ell }_1^{\mu }|\hfill \\ \hfill & \le {\displaystyle \frac{∥\mathcal{A}∥∥u∥}{\Gamma (\mu +1)}}|2{({\ell }_1-{\ell }_2)}^{\mu }+{\ell }_2^{\mu }-{\ell }_1^{\mu }|+{\displaystyle \frac{\lambda (∥u(\ell )∥+n\ell ∥u(\ell )∥)|}{\Gamma (\mu +1)}}2{({\ell }_1-{\ell }_2)}^{\mu }+{\ell }_2^{\mu }-{\ell }_1^{\mu }|,\hfill \end{array}\end{array}$$

since $∥T_{2}u\left({\ell }_1\right)-T_{2}u\left({\ell }_2\right)∥\to 0$ as ${\ell }_1\to {\ell }_2$. Thus, operator $T_2$ is a equicontinuous on $B_r.$ That is, $T_2$ is uniformly bounded and equicontinuous. This implies that $T_2$ is relatively compact on $B_r$. Consequently, by the Arzela–Ascoli theorem, operator $T_2$ is compact on $B_r$. All the requirements of Krasnoselskii’s fixed-point theorem are fulfilled, ensuring that the problem (1) has at least one solution. This concludes the proof of the theorem. □

4. Hyers–Ulam Stability

In this section, we present Hyers–Ulam stability analysis of the proposed fractional-order Volterra–Fredholm integro-differential Equation (1) using the Caputo derivative.

Definition 3. 

Equations (1) are said to be Hyers–Ulam stable. If there exists a constant $\mathcal{M}>0$ such that for every $ϵ>0$ and for each solution $z\in C(P,X)$ to inequality

$$\begin{array}{c}\hfill |{}^{C}D^{\mu }z(\ell )-\mathcal{A}z(\ell )-\mathcal{B}h(\ell )-g(\ell ,z(\ell ),Iz(\ell ))|\le ϵ,\end{array}$$

(5)

there exists a unique solution $u\in C(P,X)$ of Equation (1) such that $\left|z\right(\ell )-u(\ell \left)\right|\le \mathcal{M}ϵ$, $\ell \in P$.

Remark 1. 

A function $z\in C(P,X)$ satisfies the above inequality if and only if there exists a function $\Omega \in C(P,X)$ such that

$\left(i\right)$

$|\Omega (\ell \left)\right|<ϵ,$ $\ell \in P$;

$\left(ii\right)$

${}^{C}D^{\mu }z(\ell )=\mathcal{A}z(\ell )+\mathcal{B}h(\ell )+g(\ell ,z(\ell ),Iz(\ell ))+\Omega (\ell ),$ $\ell \in P$.

Theorem 3. 

If all the hypotheses and condition $\rho =\left({\displaystyle \frac{∥\mathcal{A}∥b^{\mu }}{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}v\right)<1$ are satisfied, then problem (1) exhibits Hyers–Ulam stability.

Proof. 

Let us consider that $ϵ>0$ and $z\in C(P,X)$ fulfill Inequality (5). Furthermore, we let $u\in C(P,X)$ be the unique solution of Problem (1). Then, we have

$$\begin{array}{cc}\hfill u(\ell )& =u_0+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi .\hfill \end{array}$$

From Inequality (5) and for each $\ell \in P,$ we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |z(\ell )-z_0& -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}z\left(\phi \right)d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,z\left(\phi \right),Iz\left(\phi \right))d\phi |\le {\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}ϵ.\hfill \end{array}\end{array}$$

For $\ell \in C(P,X)$, we let $u(\ell )$ be the unique solution of the proposed Problem (1) satisfying $u_0=z_0.$ Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|z\right(\ell )-u(\ell \left)\right|& =|z(\ell )-z_0-{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}y\left(\phi \right)d\phi \hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi |\hfill \\ \hfill & =|z(\ell )-z_0-{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}z\left(\phi \right)d\phi -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi \hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,z\left(\phi \right),Iz\left(\phi \right))d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}z\left(\phi \right)d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,z\left(\phi \right),Iz\left(\phi \right))d\phi \hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{B}h\left(\phi \right)d\phi \hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi |\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \le {\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}ϵ+|{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}z\left(\phi \right)d\phi +{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\hfill \\ \hfill & \times g(\phi ,z\left(\phi \right),Iz\left(\phi \right))d\phi -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}\mathcal{A}u\left(\phi \right)d\phi \hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{{\ell }_0}^{\ell }{(\ell -\phi )}^{(\mu -1)}g(\phi ,u\left(\phi \right),Iu\left(\phi \right))d\phi |\hfill \\ \hfill & \le {\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}ϵ+{\displaystyle \frac{∥\mathcal{A}∥b^{\mu }}{\Gamma (\mu +1)}}∥z-u∥+{\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}\left(\lambda (∥z-u∥+n\ell ∥|z-u∥)\right)\hfill \\ \hfill & \le {\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}ϵ+\left(∥|z-u∥\right)\left({\displaystyle \frac{∥\mathcal{A}∥b^{\mu }}{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}v\right)\hfill \\ \hfill & \le {\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)(1-\rho )}}ϵ=\mathcal{M}ϵ.\hfill \end{array}\end{array}$$

where $\rho ={\displaystyle \frac{∥\mathcal{A}∥b^{\mu }}{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)}}v$ and $\mathcal{M}={\displaystyle \frac{b^{\mu }}{\Gamma (\mu +1)(1-\rho )}}$. Hence, we conclude that Problem (1) exhibits Hyers–Ulam stability. □

5. Numerical Example

This section provides a numerical example to illustrate the effectiveness of the proposed solution. In this section, we develop a finite difference method to address Equation (1). The method is applied to Equation (1) over a domain that uses a uniform mesh as $0={\ell }_1<{\ell }_2<....<{\ell }_N=1.$

Example 1. 

We consider the following nonlinear fractional-order integro-differential equation:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^{C}D^{\mu }u(\ell )& =\mathcal{A}u(\ell )+\mathcal{B}h(\ell )+g(\ell ,u(\ell ),Iu(\phi \left)\right)\hfill \\ \hfill u\left(0\right)& =u_0,\ell \in P=[0,b]\hfill \end{array}\end{array}$$

By employing the Caputo fractional derivative, we have the following equation:

$$\begin{array}{c}\hfill u(\ell )=u_0+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_0^{\ell }{(\ell -\phi )}^{(\mu -1)}\left[\mathcal{A}u\left(\phi \right)+\mathcal{B}h\left(\phi \right)+g(\phi ,u(\phi ),Iu(\phi \left)\right)\right]d\phi \end{array}$$

Let us take $\ell ={\ell }_{j+1}$ in the above equation. Then, the equation becomes

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u\left({\ell }_{j+1}\right)=u_0+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_0^{{\ell }_{j+1}}{({\ell }_{j+1}-\phi )}^{(\mu -1)}\left[\mathcal{A}u\left(\phi \right)+\mathcal{B}h\left(\phi \right)+g(\phi ,u(\phi ),Iu(\phi \left)\right)\right]d\phi \end{array}\end{array}$$

Using the finite difference technique, the previous equation can be approximated as

$$\begin{array}{cc}\hfill u\left({\ell }_{j+1}\right)& =u_0+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}\sum _{k=0}^j{\int }_{{\ell }_k}^{{\ell }_{k+1}}{({\ell }_{j+1}-\phi )}^{(\mu -1)}[\mathcal{A}u\left(\phi \right)+\mathcal{B}h\left(\phi \right)+g(\phi ,u\left(\phi \right),Iu\left(\phi \right))]d\phi \hfill \\ \hfill & \approx u_0+{\displaystyle \frac{1}{\Gamma \left(\mu \right)}}\sum _{k=0}^j{\displaystyle \frac{1}{2}}(\mathcal{A}(u\left({\ell }_k\right)+u\left({\ell }_{k+1}\right))+\mathcal{B}(h\left({\ell }_k\right)+h\left({\ell }_{k+1}\right))+(g({\ell }_k,u\left({\ell }_k\right),Iu\left({\ell }_k\right))\hfill \\ \hfill & +g({\ell }_{k+1},u\left({\ell }_{k+1}\right),Iu\left({\ell }_{k+1}\right))\left)\right){\int }_{{\ell }_k}^{{\ell }_{k+1}}{({\ell }_{j+1}-\phi )}^{(\mu -1)}d\phi \hfill \\ \hfill & =u_0+{\displaystyle \frac{1}{2\Gamma (\mu +1)}}\sum _{k=0}^j(\mathcal{A}(u\left({\ell }_k\right)+u\left({\ell }_{k+1}\right))+\mathcal{B}(h\left({\ell }_k\right)+h\left({\ell }_{k+1}\right))+(g({\ell }_k,u\left({\ell }_k\right),Iu\left({\ell }_k\right))\hfill \\ \hfill & +g({\ell }_{k+1},u\left({\ell }_{k+1}\right),Iu\left({\ell }_{k+1}\right))\left)\right)\times \left[{({\ell }_{j+1}-{\ell }_k)}^{\mu }-{({\ell }_{j+1}-{\ell }_{k+1})}^{\mu }\right]\hfill \end{array}$$

Using MATLAB 2021a software, the graphical representation of the proposed fractional-order integro-differential Equation (1) is presented in Figure 1. The figure demonstrates that the finite difference scheme applied to the fractional integro-differential Problem (1) reaches stability when the parameters are set as $\mathcal{A}={\displaystyle \frac{1}{16}},\mathcal{B}={\displaystyle \frac{1}{8}},h=1$ and function g is defined by integral expression $g={\int }_0^{\ell }{e}^{-\phi }\phi u\left(\phi \right)d\phi .$ This demonstrates the critical influence of these parameter values in ensuring the stability of the numerical solution to the fractional integro-differential equation.

Application

In this section, we provide two examples to validate the effectiveness of the proposed problem.

Example 2. 

We consider the following fractional-order Volterra–Fredholm integro-differential equation with order $\mu =0.5$:

$$\begin{array}{c}\hfill {}^{C}D^{0.5}u(\ell )={\displaystyle \frac{1}{16}}+{\displaystyle \frac{1}{32}}u(\ell )+{\int }_0^{\ell }{e}^{-2\phi }\phi u\left(\phi \right)d\phi ,\ell \in [0,2],\end{array}$$

(6)

with initial condition $u(\ell )=0.$

• From Equation (6), we observe that $\mathcal{A}={\displaystyle \frac{1}{32}},\mathcal{B}={\displaystyle \frac{1}{16}},h(\ell )=1$ and $g(\ell ,u(\ell ),Iu(\ell ))={\int }_0^{\ell }{e}^{-2\phi }\phi u\left(\phi \right)d\phi .$ One can see that function g is continuous on $[0,2]$. Also, function g satisfying the following inequality with $v={\displaystyle \frac{1}{4}}$ and for $u_1,u_2\in C\left(\right[0,2],X)$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |g(\phi ,u_1\left(\phi \right),I{u}_1\left(\phi \right))-g(\phi ,u_2\left(\phi \right),I{u}_2\left(\phi \right))|& =|{\int }_0^{\ell }{e}^{-2\phi }\phi u_1\left(\phi \right)d\phi -{\int }_0^{\ell }{e}^{-2\phi }\phi u_2\left(\phi \right)d\phi |\hfill \\ \hfill & \le {\int }_0^{\ell }\left|e^{-2\phi }\phi \right||u_1\left(\phi \right)-u_2\left(\phi \right)|d\phi \hfill \\ \hfill & \le |-{\displaystyle \frac{1}{2}}\ell e^{-2\ell }-{\displaystyle \frac{1}{4}}{e}^{-2\ell }+{\displaystyle \frac{1}{4}}|∥u_1-u_2∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{4}}∥u_1-u_2∥.\hfill \end{array}\end{array}$$

Since $v={\displaystyle \frac{1}{4}},$ we also see that the uniqueness condition becomes

$$\begin{array}{c}\hfill \rho =\left({\displaystyle \frac{∥\mathcal{A}∥b^{\mu }}{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }v}{\Gamma (\mu +1)}}\right)=\left({\displaystyle \frac{\frac{1}{32}{\left(2\right)}^{0.5}}{\Gamma (0.5+1)}}+{\displaystyle \frac{{\left(2\right)}^{0.5}\frac{1}{4}}{\Gamma (0.5+1)}}\right)=0.4489<1\end{array}$$

Thus, function g satisfies the hypotheses and the condition of Theorem 3: the initial value Problem (6) has a unique solution.

• Next, we consider the following inequality to check the Hyers–Ulam stability for any $ϵ>0:$

$$\begin{array}{c}\hfill |{}^{C}D^{0.5}u(\ell )-{\displaystyle \frac{1}{16}}-{\displaystyle \frac{1}{32}}u(\ell )-{\int }_0^{\ell }{e}^{-2\phi }\phi u\left(\phi \right)d\phi |\le ϵ,\mathit{for}\mathit{all}\ell \in [0,2]\end{array}$$

By using Theorem 4, it is easy to prove that Problem (6) is Hyers–Ulam stable. Since u and z are two solutions of Equation (6),

$$\begin{array}{c}\hfill \left|u\right(\ell )-z(\ell \left)\right|\le \mathcal{M}ϵ,\end{array}$$

where $\mathcal{M}={\displaystyle \frac{2^{0.5}}{\Gamma (0.5+1)}}\times {\displaystyle \frac{1}{1-0.4489}}.$

• Hence, the initial value Problem (6) is Hyers–Ulam stable.

Example 3. 

We consider the following fractional-order nonlinear Volterra–Fredholm integro-differential equation with order $\mu =0.5$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^{C}D^{0.5}u(\ell )& ={\displaystyle \frac{{\ell }^2{e}^{\ell }}{5}}u(\ell )+{\displaystyle \frac{1}{16}}{\int }_0^{\ell }{e}^{-2(\phi +\ell )}u\left(\phi \right)d\phi +{\displaystyle \frac{1}{16}}{\int }_0^{\ell }\ell e^{-2(\phi +\ell )}u\left(\phi \right)d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{16}}{\int }_0^{\ell }sin(\ell -\phi )u\left(\phi \right)d\phi ,\ell \in [0,1],\hfill \end{array}\end{array}$$

(7)

with initial condition $u\left(l\right)=0.$

• From Equation (7), we notice that $\mathcal{A}u(\ell )={\displaystyle \frac{{\ell }^2{e}^{\ell }}{5}}u(\ell ),\mathcal{B}={\displaystyle \frac{1}{16}}{\int }_0^{\ell }{e}^{-2(\phi +\ell )}u\left(\phi \right)d\phi$ and $g(\ell ,u(\ell ),Iu(\ell ))={\displaystyle \frac{1}{16}}{\int }_0^{\ell }\ell e^{-2(\phi +\ell )}u\left(\phi \right)d\phi +{\displaystyle \frac{1}{16}}{\int }_0^{\ell }sin(\ell -\phi )u\left(\phi \right)d\phi .$ Function g is continuous on interval $[0,1]$. Furthermore, function g satisfying the following inequality with $v={\displaystyle \frac{1}{16}}$ and for $u_1,u_2\in C\left(\right[0,1],X)$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |g(\phi ,u_1\left(\phi \right),I{u}_1\left(\phi \right))-g(\phi ,u_2\left(\phi \right),I{u}_2\left(\phi \right))|& =|{\displaystyle \frac{1}{16}}{\int }_0^{\ell }\ell e^{-2(\phi +\ell )}{u}_1\left(\phi \right)d\phi +{\displaystyle \frac{1}{16}}{\int }_0^{\ell }sin(\ell -\phi )u_1\left(\phi \right)d\phi \hfill \\ \hfill & -{\displaystyle \frac{1}{16}}{\int }_0^{\ell }\ell e^{2(\phi -\ell )}{u}_2\left(\phi \right)d\phi -{\displaystyle \frac{1}{16}}{\int }_0^{\ell }sin(\ell -\phi )u_2\left(\phi \right)d\phi |\hfill \\ \hfill & \le {\displaystyle \frac{1}{16}}{\int }_0^{\ell }|\ell e^{-2(\phi +\ell )}\phi ||u_1\left(\phi \right)-u_2\left(\phi \right)|d\phi \hfill \\ \hfill & +{\displaystyle \frac{1}{16}}{\int }_0^{\ell }|sin(\ell -\phi )||u_1\left(\phi \right)-u_2\left(\phi \right)|d\phi \hfill \\ \hfill & \le |{\displaystyle \frac{-\ell e^{-4\ell }}{16}}+{\displaystyle \frac{\ell e^{-2\ell }}{16}}|∥u_1-u_2∥+\left|{\displaystyle \frac{1}{16}}\right|∥u_1-u_2∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{16}}∥u_1-u_2∥.\hfill \end{array}\end{array}$$

Since $v={\displaystyle \frac{1}{16}}$ and $su{p}_{t\in P}{\int }_0^{\ell }\left|\right|sin(\ell -\phi )\left|\right|d\phi \le 1,$ it can be observed that the uniqueness condition is given as

$$\begin{array}{c}\hfill \rho =\left({\displaystyle \frac{∥\mathcal{A}∥b^{\mu }}{\Gamma (\mu +1)}}+{\displaystyle \frac{b^{\mu }v}{\Gamma (\mu +1)}}\right)=\left({\displaystyle \frac{\left(0.5437\right){\left(1\right)}^{0.5}}{\Gamma (0.5+1)}}+{\displaystyle \frac{{\left(1\right)}^{0.5}\left(\frac{1}{16}\right)}{\Gamma (0.5+1)}}\right)\approx 0.6840<1\end{array}$$

Thus, function g satisfies the hypotheses and the condition of Theorem 3, ensuring that the initial value Problem (7) has a unique solution.

• Next, we consider the following inequality to verify the Hyers–Ulam stability for any $ϵ>0:$

$$\begin{array}{cc}\hfill |{}^{C}D^{0.5}u(\ell )& -{\displaystyle \frac{{\ell }^2{e}^{\ell }}{3}}u(\ell )1{\int }_0^{\ell }{e}^{2(\phi -\ell )}u\left(\phi \right)d\phi -{\int }_0^{\ell }t{e}^{2(\phi -\ell )}u\left(\phi \right)d\phi -{\int }_0^{\ell }sin(\ell -\phi )u\left(\phi \right)d\phi |\le ϵ,\hfill \\ \hfill & \mathit{for}\mathit{all}\ell \in [0,1]\hfill \end{array}$$

Using Theorem 4, it is straightforward to demonstrate that Problem (7) is Hyers–Ulam stable, as u and z are two solutions of Equation (7). That is,

$$\begin{array}{c}\hfill \left|u\right(\ell )-z(\ell \left)\right|\le \mathcal{M}ϵ,\end{array}$$

where $\mathcal{M}={\displaystyle \frac{1^{0.5}}{\Gamma (0.5+1)}}\times {\displaystyle \frac{1}{1-0.6840}}<1.$

• Hence, the initial value Problem (7) is Hyers–Ulam stable.

6. Conclutions

This paper explores the Hyers–Ulam stability of an integro-differential equation involving a Caputo derivative. By applying Banach and Krasnoselskii’s fixed-point theorems, the existence and uniqueness of a solution are proven under appropriate hypotheses. Additionally, sufficient conditions for Hyers–Ulam stability are provided. The obtained numerical results indicate that the proposed method is efficient and accurate for this kind of equations. Finally, we provide an example to demonstrate the application of the results. This study provides a significant improvement to the field by enhancing the theoretical foundations and practical methodologies for modeling and analyzing complex systems through fractional calculus and integro-differential equations. In the future, this work will act as a cornerstone for exploring fractional differential equations and their solutions across various fields. It can inspire advancements in areas like control systems, signal processing and biomedical engineering. By addressing complex systems with memory effects, it enables the development of more accurate models. This research opens new pathways for both theoretical studies and practical applications.