Qualitative Analysis of a Three-Dimensional Dynamical System of Fractal-Fractional-Order Evolution Differential Equations with Terminal Boundary Conditions

Abstract

In this research article, we investigate a three-dimensional dynamical system governed by fractal-fractional-order evolution differential equations subject to terminal boundary conditions. We derive existence and uniqueness results using Schaefer’s and Banach’s fixed-point theorems, respectively. Additionally, the Hyers–Ulam stability approach is employed to analyze the system’s stability. We employ vector terminology for the proposed problem to make the analysis simple. To illustrate the practical relevance of our findings, we apply the derived results to a numerical example and graphically illustrate the solution for different fractal-fractional orders, emphasizing the effect of the derivative’s order on system behavior.

1. Introduction

Fractional differential equations (FDEs) have gained considerable attention in recent decades due to their wide-ranging applications in both theoretical and practical fields. The interplay between the mathematical theory and real-world applications of fractional calculus has led to substantial advancements, positioning FDEs as a well-established field with diverse applications across physics, engineering, and technology. Their relevance spans control theory, electrochemistry, electromagnetics, viscoelasticity, and porous media, among others (see [1,2,3]). For further developments in this area, readers may consult [4,5,6].

Various fractional differential operators have been introduced to address different modeling scenarios. Among them, the fractal-fractional differential operator, classified as a special local fractional derivative [7], has gained prominence. This operator is a non-Newtonian modification of the classical derivative, specifically designed for fractal structures. Currently, fractal-fractional differential equations (FFDEs) are of great interest due to their applicability in modeling complex systems such as water permeation in wool fibers, multi-scale fabrics, and heat transfer [8]. Akgül [9] examined FFDEs with a power-law kernel, demonstrating their novel applications. Several recent studies have further expanded the utility of these concepts (see [10,11,12,13]).

Terminal value problems (TVPs) play a crucial role in modeling phenomena across various domains of physics and engineering. These problems naturally arise in retrospective simulations, where system states are analyzed from final conditions. These problems commonly arise in the mathematical modeling of numerous physical phenomena, including microscale heat transport and the hydrodynamics of liquid helium (see [14]). Furthermore, TBV problems form a subclass of differential equations and play an important role in nonlinear field theory, hydrodynamics, and particularly in the analysis of symmetric bubble-type solutions within the framework of shell-like theory (see [15]). The study of solution existence for classical TVPs has been extensively explored (see [16,17]), yet extending these results to fractional-order TVPs remains a more complex challenge. Diethelm [18] investigated the existence and stability properties of fractional TVPs on finite intervals, highlighting structural differences between initial and terminal value problems. In [19], Shiri et al. studied terminal value problems for nonlinear systems of FDEs. In [20], Ford et al. investigated high order numerical methods for fractional TVPs. In [21], Boichuk et al. studied TVPs for the system of FDEs incorporated additional restrictions. In [22], Bao et al. investigated existence and regularity results for TVP for nonlinear fractional wave equations.

Evolution equations are fundamental in real-world applications. Constructing solutions based on given initial or boundary conditions is a characteristic feature of these equations, making them essential in mathematical modeling. To ensure self-consistency in modeling real-world systems, it is crucial to establish the existence and uniqueness of solutions for evolution equations subject to initial or terminal boundary conditions. These equations are widely employed in areas such as acoustics, neural networks, and natural sciences. Numerous significant results have been reported for evolution equations using classical and fractional calculus techniques (see [23,24]).

Recently, fractal-fractional calculus has gained increasing attention due to its ability to model complex physical systems. Although the concept of fractals predates fractional calculus, fractal-fractional calculus has emerged as a powerful tool for studying real-world phenomena (see [25,26,27]). Notable applications include epidemiological modeling [28,29]. The proper investigation of evolution equations with terminal boundary conditions is necessary for establishing their solution existence, uniqueness, and stability. Additionally, triply coupled differential equation systems have been widely studied due to their relevance in numerous applications [30,31].

Motivated by the aforementioned study, we consider a three-dimensional system of fractal-fractional-order evolution differential equations with terminal boundary conditions:

$$\left\{\begin{array}{c}{}^{FFC}D^{{\xi }_1,{\rho }_1}{\varphi }_1\left(x\right)=A_1\left(x\right){\varphi }_1\left(x\right)+{\mathsf{\Phi }}_1(x,{\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right)),0<{\rho }_1<1,0<{\xi }_1\le 1,x\in [0,\tau ]=\mathcal{S},\hfill \\ {}^{FFC}D^{{\xi }_2,{\rho }_2}{\varphi }_2\left(x\right)=A_2\left(x\right){\varphi }_2\left(x\right)+{\mathsf{\Phi }}_2(x,{\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right)),0<{\rho }_2<1,0<{\xi }_2\le 1,x\in [0,\tau ]=\mathcal{S},\hfill \\ {}^{FFC}D^{{\xi }_3,{\rho }_3}{\varphi }_3\left(x\right)=A_3\left(x\right){\varphi }_3\left(x\right)+{\mathsf{\Phi }}_3(x,{\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right)),0<{\rho }_3<1,0<{\xi }_3\le 1,x\in [0,\tau ]=\mathcal{S},\hfill \\ {\varphi }_1\left(\tau \right)={\omega }_1,{\varphi }_2\left(\tau \right)={\omega }_2,{\varphi }_3\left(\tau \right)={\omega }_3,\hfill \end{array}\right.$$

(1)

where ${\rho }_1,{\rho }_2,{\rho }_3$ are fractal dimensions, and ${\xi }_1,{\xi }_2,{\xi }_3$ are the orders of the fractional derivative. The functions $A_1,A_2,A_3:\mathcal{S}\to \mathcal{R}$ are continuous real functions dependent on x. The functions ${\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2,{\mathsf{\Phi }}_3:\mathcal{S}\times {\mathcal{R}}^3\to \mathcal{R}$ are nonlinear and continuous, the functions ${\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right)$ represent state variables of the system, and ${\omega }_i$; (i = 1, 2, 3) are real numbers.

Fractal-fractional calculus incorporates both fractional-order and fractal properties, allowing for a more accurate representation of memory-dependent systems. To the best of our knowledge, such problems have not been investigated yet using fractal-fractional calculus.

To simplify the notation, we employ vector terminology for the proposed problem as employed in [19]. Let

$$\mathit{\varphi }\left(x\right)=\left[\begin{array}{c}{\varphi }_1\left(x\right)\hfill \\ {\varphi }_2\left(x\right)\hfill \\ {\varphi }_3\left(x\right)\hfill \end{array}\right],\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right))=\left[\begin{array}{c}{\mathsf{\Phi }}_1(x,{\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right))\hfill \\ {\mathsf{\Phi }}_2(x,{\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right))\hfill \\ {\mathsf{\Phi }}_3(x,{\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right))\hfill \end{array}\right],$$

and

$$\mathit{A}\left(x\right)=\left[\begin{array}{ccc}{A}_1\left(x\right)& 0& 0\\ 0& A_2\left(x\right)& 0\\ 0& 0& A_3\left(x\right)\end{array}\right].$$

The system can now be written in vector form as:

$$\left\{\begin{array}{c}{}^{FFC}D^{\xi ,\rho }\mathit{\varphi }\left(x\right)=\mathit{A}\left(x\right)\mathit{\varphi }\left(x\right)+\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right)),x\in [0,\tau ]=\mathcal{S},\hfill \\ \mathit{\varphi }\left(\tau \right)=\mathit{\omega },\hfill \end{array}\right.$$

(2)

where

$$\mathit{\xi }=\left[\begin{array}{c}{\xi }_1\hfill \\ {\xi }_2\hfill \\ {\xi }_3\hfill \end{array}\right],\mathit{\rho }=\left[\begin{array}{c}{\rho }_1\hfill \\ {\rho }_2\hfill \\ {\rho }_3\hfill \end{array}\right],\mathit{\omega }=\left[\begin{array}{c}{\omega }_1\hfill \\ {\omega }_2\hfill \\ {\omega }_3\hfill \end{array}\right].$$

(3)

This vector form can simplify the notation and make it easier to analyze and solve the system.

The manuscript is structured as follows: In Section 2, we provide the basic definitions and preliminary results. In Section 3, we establish the existence of solutions for the proposed problem. In Section 4, we analyze the stability of the proposed problem. In Section 5, we apply the main results to a general problem and present graphical representations of the solutions for various fractal and fractional orders, demonstrating the effectiveness of the proposed method. In Section 6, we conclude with a summary of the derived results.

2. Elementary Results

In this section, we present definitions of the fractal-fractional-order integral and derivative as well as some other basic definitions. Moreover, we provide a result and its proof that serve as a basis for subsequent analysis.

Definition 1

([25]). Let ϕ be a matrix of continuous functions and fractal differentiable on the open interval $(0,\tau )$. Then, the fractal-fractional integral of $\mathit{\varphi }\left(t\right)$ with a power-law kernel is defined by

$${}^{FF}I^{\mathit{\xi },\mathit{\rho }}\mathit{\varphi }\left(x\right)={\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{(x-v)}^{\mathit{\xi }-1}{v}^{\mathit{\rho }-1}\left(\mathit{\varphi }\left(v\right)\right)dv,x\in [0,\tau ].$$

(4)

Definition 2

([25]). Let ϕ be a matrix of continuous functions and fractal differentiable on the open interval $(0,\tau )$. Then, for $\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]<\mathit{\xi },\mathit{\rho }\le \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],$ the fractal-fractional Caputo derivative with a power-law kernel is defined as

$$\begin{array}{c}{}^{FFC}D^{\mathit{\xi },\mathit{\rho }}\mathit{\varphi }\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\mathit{\xi })}}{\int }_0^x{(x-v)}^{-\mathit{\xi }}{\displaystyle \frac{d}{d{v}^{\mathit{\rho }}}}\left(\mathit{\varphi }\left(v\right)\right)dv,x\in [0,\tau ].\hfill \end{array}$$

(5)

Definition 3

([32,33]). A set G is equi-continuous on a domain D if for every $ϵ>0,$ there exists a positive number $\zeta >0$ such that for all $x\in G$ and for all $t_1,t_2\in D$ satisfying $|t_1-t_2|<\zeta$, the following holds:

$$|x\left(t_1\right)-x\left(t_2\right)|<ϵ.$$

This concept can be extended to operators as follows:

Definition 4.

An operator $\mathcal{F}$ mapping a normed space X into another normed space Y is termed equi-continuous if, for every $ϵ>0,$ $\zeta >0$ exists such that all $t_1,t_2\in X$ satisfying $\parallel t_1-t_2{\parallel }_X<\zeta$; thus, we have:

$$\parallel \mathcal{F}\left(t_1\right)-\mathcal{F}\left(t_2\right){\parallel }_Y<ϵ.$$

Definition 5

([34]). An operator $\mathcal{F}:X\to Y$ acting between normed spaces is called completely continuous if it is continuous and maps every bounded subset of X to relatively compact subsets of Y.

Theorem 1

([35]). Let $\mathcal{Q}$ be a non-empty, a closed subset of a Banach space X. If the mapping $\mathcal{F}:\mathcal{Q}⟶\mathcal{Q}$ is a contraction, then $\mathcal{F}$ has a unique fixed point.

Theorem 2

([36]). Suppose W is a normed linear space and $\mathcal{V}$ is a convex subset of W that contains 0. If $\mathcal{F}:\mathcal{V}\to \mathcal{V}$ is a completely continuous operator, then either the set $\mathcal{U}=\{\mathit{\varphi }\in \mathcal{V}:\mathit{\varphi }=\mu \mathcal{F}(\mathit{\varphi });0<\mu <1\}$ is unbounded or $\mathcal{F}$ has a fixed point in $\mathcal{V}$.

Lemma 1.

Consider $\mathit{\varphi }\in C\left(\mathcal{S}\right)$. Then, $\mathit{\varphi }\left(x\right)$ is the solution of the proposed problem:

$$\left\{\begin{array}{c}{}^{FFC}D^{\mathit{\xi },\mathit{\rho }}\mathit{\varphi }\left(x\right)=\mathit{A}\left(x\right)\mathit{\varphi }\left(x\right)+\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right)),\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]<\mathit{\xi },\mathit{\rho }\le \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],x\in \mathcal{S},\hfill \\ \mathit{\varphi }\left(\tau \right)=\mathit{\omega },\hfill \end{array}\right.$$

(6)

if and only if it is the solution of the following integral equation:

$$\begin{array}{cc}& \mathit{\varphi }\left(x\right)={\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right)))dv\hfill \\ & -{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right)))dv+\mathit{\omega }.\hfill \end{array}$$

(7)

Proof. 

Assume that $\mathit{\varphi }$ satisfies (6). Then,

$${}^{FFC}D^{\mathit{\xi },\mathit{\rho }}\mathit{\varphi }\left(x\right)=\mathit{A}\left(x\right)\mathit{\varphi }\left(x\right)+\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right)).$$

(8)

Applying the fractal-fractional integral ${}^{FF}I^{\mathit{\xi },\mathit{\rho }}$ on both sides, we have

$${}^{FF}I^{\mathit{\xi },\mathit{\rho }}\left({}^{FFC}D^{\mathit{\xi },\mathit{\rho }}\mathit{\varphi }\left(x\right)\right)={}^{FF}I^{\mathit{\xi },\mathit{\rho }}\left(\mathit{A}\left(x\right)\mathit{\varphi }\left(x\right)+\mathsf{\Phi }(x,\mathit{\varphi }(x\left)\right)\right).$$

(9)

Since ${}^{FF}I^{\mathit{\xi },\mathit{\rho }}$ is the left inverse of ${}^{FFC}D^{\mathit{\xi },\mathit{\rho }},$ we obtain

$$\mathit{\varphi }\left(x\right)={\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))dv-c.$$

(10)

By putting $x=\tau ,$ we have

$$\begin{array}{cc}& \mathit{\varphi }\left(\tau \right)={\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\mathit{\tau }-v)}^{\xi -1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))dv-c\hfill \\ & \mathit{\omega }={\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))dv-c\hfill \\ & c={\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))dv-\mathit{\omega }.\hfill \end{array}$$

(11)

By putting the value for c in (10), we obtain

$$\begin{array}{cc}& \mathit{\varphi }\left(x\right)={\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))dv\hfill \\ & -{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))dv+\mathit{\omega }.\hfill \end{array}$$

(12)

This equation satisfies (7).

Conversely, let $\mathit{\varphi }$ satisfy (7). Then, since ${}^{FFC}D^{\mathit{\xi },\mathit{\rho }}$ is the left inverse of ${}^{FF}I^{\mathit{\xi },\mathit{\rho }}$, by applying ${}^{FFC}D^{\mathit{\xi },\mathit{\rho }}$ to both sides of (7), we obtain

$${}^{FFC}D^{\mathit{\xi },\mathit{\rho }}\mathit{\varphi }\left(x\right)=\mathit{A}\left(x\right)\mathit{\varphi }\left(x\right)+\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right)).$$

(13)

By setting $x=\tau$ in the integral Equation (7), we obtain the initial condition $\mathit{\varphi }\left(\tau \right)=\mathit{\omega }$. Therefore, using (13) and the derived initial condition, we obtain the differential problem (6). □

In view of Lemma 1, we give the following definition.

Definition 6.

A function ϕ satisfying the integral Equation (7) is also a solution to the proposed problem (2).

3. Existence of Solutions

In this section, we investigate the existence of solutions for the problem under consideration. Since we are studying our problem within a Banach space, we first introduce a suitable Banach space as follows:

We define the space $U:=\{\mathit{\varphi }:\mathit{\varphi }\in C(\mathcal{S},{\mathcal{R}}^3)\}$ equipped with the following norm:

$${\parallel \mathit{\varphi }\parallel }_U=max\left\{\parallel \mathit{\varphi }\parallel \right\},$$

(14)

where $\parallel \mathit{\varphi }\parallel ={max}_{x\in \mathcal{S}}\left|\mathit{\varphi }\left(x\right)\right|.$ Then, $(U,\parallel .{\parallel }_U)$ is a Banach space.

Define an operator $\mathcal{F}:U\to U$ by

$$\begin{array}{cc}& \mathcal{F}\mathit{\varphi }\left(x\right)={\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right)))dv\hfill \\ & -{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)+\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right)))dv+\mathit{\omega }.\hfill \end{array}$$

(15)

For deriving the main results, we impose the following assumptions:

(${\mathrm{H}}_1$)

Assume that the nonlinear functions $\mathit{\varphi }:\mathcal{S}\times {\mathcal{R}}^3\to {\mathcal{R}}^3$ are continuous.

(${\mathrm{H}}_2$)

Assume that $\mathit{A}$ is a matrix of continuous real functions such that ${max}_{x\in \mathcal{S}}\left|A_i\left(x\right)\right|\le K,$ for some $K>0,$ and for all i = 1, 2, 3.

(${\mathrm{H}}_3$)

Assume that for any $x\in \mathcal{S},$ and for all $y,\overline{y}\in U$ such that $y={[y_1,y_2,y_3]}^T,$ $\overline{y}={[{\overline{y}}_1,{\overline{y}}_2,{\overline{y}}_3]}^T,$ the following relations hold:

$$\begin{array}{cc}& |\mathsf{\Phi }(x,y)-\mathsf{\Phi }(x,\overline{y})|\le \theta \left(x\right)|y-\overline{y}|;\underset{x\in \mathcal{S}}{max}\theta \left(x\right)={\theta }^{*}>0.\hfill \end{array}$$

(${\mathrm{H}}_4$)

For all $y\in U,$ we have

$$\begin{array}{cc}& \left|\mathsf{\Phi }(x,y)\right|\le l\left(x\right);\underset{x\in \mathcal{S}}{max}l\left(x\right)=:m,\hfill \end{array}$$

Theorem 3.

If the assumptions ($H_1$)–($H_3$) and condition (16) is satisfied, then problem (2) has exactly one solution,

$$2{\tau }^{\mathit{\rho }+\mathit{\xi }-1}(K+{\theta }^{*}){\displaystyle \frac{\mathit{\rho }\beta (\mathit{\xi },\mathit{\rho })}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}<1.$$

(16)

Proof. 

For arbitrary $\mathit{\varphi },\overline{\mathit{\varphi }}\in U,$ we consider

$$\begin{array}{cc}& |\mathcal{F}\mathit{\varphi }\left(x\right)-\mathcal{F}\overline{\mathit{\varphi }}\left(x\right)|\le {\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right||\mathit{\varphi }\left(v\right)-\overline{\mathit{\varphi }}\left(v\right)|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}|\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))-\mathsf{\Phi }(v,\overline{\mathit{\varphi }}\left(v\right))|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right||\mathit{\varphi }\left(v\right)-\overline{\mathit{\varphi }}\left(v\right)|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}|\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))-\mathsf{\Phi }(v,\overline{\mathit{\varphi }}\left(v\right))|dv\hfill \\ & \le \underset{x\in \mathcal{S}}{max}[{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right||\mathit{\varphi }\left(v\right)-\overline{\mathit{\varphi }}\left(v\right)|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}|\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))-\mathsf{\Phi }(v,\overline{\mathit{\varphi }}\left(v\right))|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right||\mathit{\varphi }\left(v\right)-\overline{\mathit{\varphi }}\left(v\right)|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}|\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))-\mathsf{\Phi }(v,\overline{\mathit{\varphi }}\left(v\right))|dv].\hfill \end{array}$$

(17)

Using assumptions (${\mathrm{H}}_1$)–(${\mathrm{H}}_3$), we have

$$\begin{array}{cc}& |\mathcal{F}\mathit{\varphi }\left(x\right)-\mathcal{F}\overline{\mathit{\varphi }}\left(x\right)|\hfill \\ & \le {\displaystyle \frac{\mathit{\rho }K\parallel \mathit{\varphi }-\overline{\mathit{\varphi }}\parallel }{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}dv+{\displaystyle \frac{\mathit{\rho }{\theta }^{*}\parallel \mathit{\varphi }-\overline{\mathit{\varphi }}\parallel }{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }K\parallel \mathit{\varphi }-\overline{\mathit{\varphi }}\parallel }{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}dv+{\displaystyle \frac{\mathit{\rho }{\theta }^{*}\parallel \mathit{\varphi }-\overline{\mathit{\varphi }}\parallel }{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}dv.\hfill \end{array}$$

(18)

Consider the integral ${\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}dv$. Let $v=xz.$ This implies that $dv=xdz.$ If $v=0$, then $z=0$, and if $v=x$, then $z=1$. Thus, we have

$$\begin{array}{c}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}dv=x^{\mathit{\rho }+\mathit{\xi }-1}{\int }_0^1{z}^{\mathit{\rho }-1}{(1-z)}^{\mathit{\xi }-1}dz\le x^{\mathit{\rho }+\mathit{\xi }-1}{\int }_0^1{z}^{\mathit{\rho }-1}{(1-z)}^{\mathit{\xi }-1}dz\hfill \\ =x^{\mathit{\rho }+\mathit{\xi }-1}\beta (\mathit{\xi },\mathit{\rho }),\hfill \end{array}$$

(19)

where $\beta (\mathit{\xi },\mathit{\rho })$ is a beta function. Similarly, the integral ${\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}dv$ can be transformed by:

$${\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}dv={\tau }^{\mathit{\rho }+\mathit{\xi }-1}\beta (\mathit{\xi },\mathit{\rho }).$$

(20)

Hence from (18), we have

$$\begin{array}{cc}& \parallel \mathcal{F}\mathit{\varphi }-\mathcal{F}\overline{\mathit{\varphi }}{\parallel }_U\le \left[2{\tau }^{\mathit{\rho }+\mathit{\xi }-1}(K+{{\theta }^{*}}_n){\displaystyle \frac{\mathit{\rho }\beta (\mathit{\xi },\mathit{\rho })}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}\right]\parallel \mathit{\varphi }-\overline{\mathit{\varphi }}\parallel .\hfill \end{array}$$

(21)

Since $2{\tau }^{\mathit{\rho }+\mathit{\xi }-1}(K+{{\theta }^{*}}_n){\displaystyle \frac{\mathit{\rho }\beta (\mathit{\xi },\mathit{\rho })}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}<1,$ the operator $\mathcal{F}$ is contraction. Therefore, by Banach’s Theorem, $\mathcal{F}$ has exactly one fixed point. This proves the required result. □

Theorem 4.

If assumptions ($H_1$)–($H_4)$ hold and there is a constant $\mathbf{r}>0$ such that condition (22) is satisfied, then problem (2) has at least one solution.

$$\begin{array}{cc}& \left[{\displaystyle \frac{2\mathit{\rho }K}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}\left({\tau }^{\mathit{\rho }+\mathit{\xi }-1}\beta (\mathit{\xi },\mathit{\rho })\right)\right]\mathbf{r}+{\displaystyle \frac{2\mathit{\rho }m}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}\left({\tau }^{\mathit{\rho }+\mathit{\xi }-1}\beta (\mathit{\xi },\mathit{\rho })\right)+\parallel \mathit{\omega }\parallel \le \mathbf{r}.\hfill \end{array}$$

(22)

Proof. 

To investigate this result, we employ Shaefer’s fixed-point theorem. By (${\mathrm{H}}_1$), the vector function $\mathsf{\Phi }$ is continuous and, therefore, the operator is $\mathcal{F}$ is continuous. For $\mathcal{F}$ to be completely continuous, it is necessary that

$\left(I\right)$

It maps bounded sets of $\mathcal{E}$ into bounded sets of $\mathcal{E}$,

$\left(II\right)$

It is equi-continuous.

Let a number $\mathbf{r}$ exist such that the elements of the following defined set $\mathcal{E}$ are upper bounded by it. We define a set $\mathcal{E}=\{\mathit{\varphi }\in U,\mathit{\varphi }\in C(\mathcal{S},{\mathcal{R}}^3):\parallel \mathit{\varphi }\parallel \le \mathbf{r}\}.$ $\mathcal{E}$ is closed, bounded, and a convex subset of U, which is proved in the following steps.

$\left(I\right)$:

$\mathcal{F}$ is bounded.

For arbitrary $\mathit{\varphi }\in C\left(\mathcal{S}\right),$ we have

$$\begin{array}{cc}& \left|\mathcal{F}\mathit{\varphi }\left(x\right)\right|\le {\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right|\left|\mathit{\varphi }\left(x\right)\right|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}\left|\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right))\right|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right|\left|\mathit{\varphi }\left(x\right)\right|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}\left|\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right))\right|dv+\left|\omega \right|\hfill \\ & \le \underset{x\in \mathcal{S}}{max}[{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right|\left|\mathit{\varphi }\left(x\right)\right|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}\left|\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right))\right|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right|\left|\mathit{\varphi }\left(x\right)\right|dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}\left|\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right))\right|dv+\left|\mathit{\omega }\right|].\hfill \end{array}$$

(23)

Taking the maximum and using the given assumptions, we have

$$\begin{array}{c}\underset{x\in \mathcal{S}}{max}\left|\mathcal{F}\mathit{\varphi }\left(x\right)\right|\le \underset{x\in \mathcal{S}}{max}[{\displaystyle \frac{\mathit{\rho }K\mathbf{r}}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}dv+{\displaystyle \frac{\mathit{\rho }m}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}dv\hfill \\ +{\displaystyle \frac{\mathit{\rho }K\mathbf{r}}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}dv+{\displaystyle \frac{\mathit{\rho }m}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}dv+\left|\mathit{\omega }\right|]\hfill \end{array}$$

(24)

Further, using the transformations (19) and (20), and simplifying the results, we obtain

$$\begin{array}{cc}& \parallel \mathcal{F}\mathit{\varphi }\parallel \le \left[{\displaystyle \frac{2\mathit{\rho }K}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}\left({\tau }^{\mathit{\rho }+\mathit{\xi }-1}\beta (\mathit{\xi },\mathit{\rho })\right)\right]\mathbf{r}+{\displaystyle \frac{2\mathit{\rho }m}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}\left({\tau }^{\mathit{\rho }+\mathit{\xi }-1}\beta (\mathit{\xi },\mathit{\rho })\right)\hfill \\ & +\parallel \mathit{\omega }\parallel \le \mathbf{r}.\hfill \end{array}$$

(25)

Hence, $\mathcal{F}$ is bounded.

$\left(II\right)$:

$\mathcal{F}$ is equi-continuous.

Let $x_1,x_2\in \mathcal{S},$ such that (s.t) $x_1<x_2$. Then,

$$\begin{array}{c}\hfill |\mathcal{F}\mathit{\varphi }\left(x_2\right)-\mathcal{F}\mathit{\varphi }\left(x_1\right)|\\ & \le |{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{x_2}{v}^{\mathit{\rho }-1}{(x_2-v)}^{\mathit{\xi }-1}\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)dv-{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{x_1}{v}^{\mathit{\rho }-1}{(x_1-v)}^{\mathit{\xi }-1}\mathit{A}\left(v\right)\mathit{\varphi }\left(v\right)dv|\hfill \\ & +|{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{x_2}{v}^{\mathit{\rho }-1}{(x_2-v)}^{\mathit{\xi }-1}\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))dv-{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{x_1}{v}^{\mathit{\rho }-1}{(x_1-v)}^{\mathit{\xi }-1}\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right))dv|.\hfill \end{array}$$

(26)

Using the transformations (19) and (20), and the assumptions, we have

$$\begin{array}{c}\hfill |\mathcal{F}\mathit{\varphi }\left(x_2\right)-\mathcal{F}\mathit{\varphi }\left(x_1\right)|\le \left[{x_2}^{\mathit{\rho }+\mathit{\xi }-1}-{x_1}^{\mathit{\rho }+\mathit{\xi }-1}\right]{\displaystyle \frac{\mathit{\rho }K\mathbf{r}\beta (\mathit{\xi },\mathit{\rho })}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}\\ & +\left[{x_2}^{\mathit{\rho }+\mathit{\xi }-1}-{x_1}^{\mathit{\rho }+\mathit{\xi }-1}\right]{\displaystyle \frac{\mathit{\rho }m\beta (\mathit{\xi },\mathit{\rho })}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}.\hfill \end{array}$$

(27)

We see that right hand side of (27) tends to zero as $x_1\to x_2$. Consequently,

$$|\mathcal{F}\mathit{\varphi }\left(x_2\right)-\mathcal{F}\mathit{\varphi }\left(x_1\right)|\to 0,$$

(28)

Thus, the operator $\mathcal{F}\mathit{\varphi }$ is equi-continuous.

Also, by condition $\left(I\right)$, $\mathcal{F}\mathcal{E}\subset \mathcal{E}$ is uniformly bounded. Consequently, the family $\left\{\mathcal{F}\mathit{\varphi }:\mathit{\varphi }\in \mathcal{E}\right\}$ is a relatively compact subset of U. Thus, $\mathcal{F}$ is completely continuous. Therefore, by Schaefer’s fixed-point theorem, $\mathcal{F}$ has a fixed point in $U.$ □

4. Stability Analysis

In this section, we perform a Hyers–Ulam stability analysis for the triply coupled system of DEs (2). Let $ϵ=\left[\begin{array}{c}{ϵ}_1\\ {ϵ}_2\\ {ϵ}_3\end{array}\right]>0$. Then, for $x\in [0,\tau ],$ we construct the inequality in the unknown function $\overline{\mathit{\varphi }}\left(x\right)=\left[\begin{array}{c}\overline{{\varphi }_1}\left(x\right)\\ \overline{{\varphi }_2}\left(x\right)\\ \overline{{\varphi }_3}\left(x\right)\end{array}\right]$ as follows:

$$\begin{array}{c}\left|{}^{FFC}D^{\mathit{\xi },\mathit{\rho }}\overline{\mathit{\varphi }}\left(x\right)-\left(\mathit{A}\left(x\right)\overline{\mathit{\varphi }}\left(x\right)+\mathsf{\Phi }(x,\overline{\mathit{\varphi }}\left(x\right))\right)\right|\le \mathit{ϵ}.\hfill \end{array}$$

(29)

where

$$\overline{\mathit{\varphi }}\in U:=\left\{\overline{\mathit{\varphi }}:\overline{\mathit{\varphi }}\in C(\mathcal{S},{\mathcal{R}}^3)\right\}.$$

(30)

The following Definitions 7 and 8 have been adapted from [37].

Definition 7.

The three-dimensional system of DEs (2) is said to be Hyers–Ulam (H-U) stable if there is a positive real number $\mathbf{C}$ such that for any solution $\overline{\mathit{\varphi }}$ of the inequality (29), there is a unique solution ϕ of (2) satisfying

$$|\overline{\mathit{\varphi }}\left(x\right)-\mathit{\varphi }\left(x\right)|\le \mathbf{C}\mathit{ϵ},x\in [0,1].$$

Definition 8.

The three-dimensional system of DEs (2) is said to be generalized H-U stable if there is a real function $\mathsf{\Psi }\in C({\mathcal{R}}_{+},{\mathcal{R}}_{+}),$ with $\mathsf{\Psi }\left(0\right)=0,$ such that for any solution $\overline{\mathit{\varphi }}$ of the inequality (29), and unique solution ϕ of (2), the following condition satisfies

$$|\overline{\mathit{\varphi }}\left(x\right)-\mathit{\varphi }\left(x\right)|\le \mathsf{\Psi }\left(\mathit{ϵ}\right),x\in [0,\tau ].$$

We make the following remark to obtain the corresponding perturbed problem with small perturbation functions. This remark is used to establish bounds on the perturbation’s effect on the system and to quantify the relationship between the perturbation and the resulting change in the system’s behavior.

Remark 1.

$\overline{\mathit{\varphi }}$ is a solution of the inequality (29), if there is a function $\vartheta \in C(\mathcal{S},\mathcal{R})$ that is dependent on $\overline{\mathit{\varphi }}$ such that for $\mathit{ϵ}>0,$ we have

$\left(i\right)$

$\left|\vartheta \right(t\left)\right|\le \mathit{ϵ},$

$\left(ii\right)$

$${}^{FFC}D^{\mathit{\xi },\mathit{\rho }}\overline{\mathit{\varphi }}\left(x\right)=\left(\mathit{A}\left(x\right)\overline{\mathit{\varphi }}\left(x\right)+\mathsf{\Phi }(x,\overline{\mathit{\varphi }}\left(x\right))\right)+\vartheta \left(t\right).$$

(31)

By Remark 1, the following problem with small perturbation function is given:

$$\left\{\begin{array}{c}{}^{FFC}D^{\mathit{\xi },\mathit{\rho }}\overline{\mathit{\varphi }}\left(x\right)=\left(\mathit{A}\left(x\right)\overline{\mathit{\varphi }}\left(x\right)+\mathsf{\Phi }(x,\overline{\mathit{\varphi }}\left(x\right))\right)+\vartheta \left(x\right),\hfill \\ \overline{\mathit{\varphi }}\left(\tau \right)=\mathit{\omega },\tau \in \mathcal{S}.\hfill \end{array}\right.$$

(32)

Lemma 2.

The solution of the perturbed problem (32) is given by

$$\overline{\mathit{\varphi }}\left(x\right)=\left\{\begin{array}{cc}& {\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\overline{\mathit{\varphi }}\left(v\right)+\mathsf{\Phi }(v,\overline{\mathit{\varphi }}\left(v\right)))dv\hfill \\ & +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}\vartheta dv\hfill \\ & -{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left({\mathit{\xi }}_1\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}(\mathit{A}\left(v\right)\overline{\mathit{\varphi }}\left(v\right)+\mathsf{\Phi }(v,\overline{\mathit{\varphi }}\left(v\right)))dv\hfill \\ & -{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}\vartheta dv+\mathit{\omega }.\hfill \end{array}\right.$$

(33)

Proof. 

The proof is easy, so we omit it. □

Theorem 5.

If the assumptions ($H_1$)–($H_3$) hold and condition (16) is satisfied, then the problem (2) is Hyers–Ulam (H-U) stable.

Proof. 

Let $\overline{\mathit{\varphi }}\left(x\right)$ be any solution of the inequality (29), and let $\mathit{\varphi }$ be the unique solution of problem (2). Then, from (7) and (33), we have

$$\begin{array}{c}|\overline{\mathit{\varphi }}\left(x\right)-\mathit{\varphi }\left(x\right)|\le {\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right||\overline{\mathit{\varphi }}\left(v\right)-\mathit{\varphi }\left(v\right)|dv\hfill \\ +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}|\mathsf{\Phi }(v,\overline{\mathit{\varphi }}\left(v\right))-\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right)|dv\hfill \\ +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^x{v}^{\mathit{\rho }-1}{(x-v)}^{\mathit{\xi }-1}\left|\vartheta \right|dv+{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}\left|\mathit{A}\left(v\right)\right||\overline{\mathit{\varphi }}\left(v\right)-\mathit{\varphi }\left(v\right)|dv\hfill \\ +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}|\mathsf{\Phi }(v,\overline{\mathit{\varphi }}\left(v\right))-\mathsf{\Phi }(v,\mathit{\varphi }\left(v\right)|dv\hfill \\ +{\displaystyle \frac{\mathit{\rho }}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{\int }_0^{\tau }{v}^{\mathit{\rho }-1}{(\tau -v)}^{\mathit{\xi }-1}\left|\vartheta \right|dv.\hfill \end{array}$$

(34)

Using assumptions (${\mathrm{H}}_1$)–(${\mathrm{H}}_3$), the transformations (19) and (20), and part $\left(i\right)$ of Remark 1, we have

$$\begin{array}{c}\parallel \overline{\mathit{\varphi }}-\mathit{\varphi }\parallel \le \left[2{\tau }^{\mathit{\rho }+\mathit{\xi }-1}(K+{\theta }^{*}){\displaystyle \frac{\mathit{\rho }\beta (\mathit{\xi },\mathit{\rho })}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}\right]\parallel \overline{\mathit{\varphi }}-\mathit{\varphi }\parallel +{\displaystyle \frac{2\mathit{\rho }\beta (\mathit{\xi },\mathit{\rho }){\tau }^{\mathit{\rho }+\mathit{\xi }-1}ϵ}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}.\hfill \end{array}$$

(35)

Simplifying further, we have

$$\parallel \overline{\mathit{\varphi }}-\mathit{\varphi }\parallel \le {\displaystyle \frac{\frac{2\mathit{\rho }\beta (\mathit{\xi },\mathit{\rho }){\tau }^{\mathit{\rho }+\mathit{\xi }-1}ϵ}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}}{1-\left[2{\tau }^{\mathit{\rho }+\mathit{\xi }-1}(K+{\theta }^{*})\frac{\mathit{\rho }\beta (\mathit{\xi },\mathit{\rho })}{\mathsf{\Gamma }\left(\mathit{\xi }\right)}\right]}}:=\mathbf{C}\mathit{ϵ}.$$

(36)

This shows that problem (2) is H-U stable. □

Corollary 1.

By setting $\alpha \left(ϵ\right)=\mathbf{C}\left(ϵ\right)$ s.t $\alpha \left(0\right)=0$, then problem (2) is generalized H-U stable.

5. Illustrative Problem

In this section, we illustrate the applicability of our obtained results. We assign specific values to functions, fractional orders, fractal dimensions, and initial conditions in the proposed problem (2) as given bellow.

Example 1.

$$\left\{\begin{array}{c}{}^{FFC}D^{\mathit{\xi },\mathit{\rho }}\mathit{\varphi }\left(x\right)=\mathit{A}\left(x\right)\mathit{\varphi }\left(x\right)+\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right)),x\in [0,\tau ],\hfill \\ \mathit{\varphi }\left(\tau \right)=\mathit{\omega }.\hfill \end{array}\right.$$

(37)

Let

$$\mathit{A}\left(x\right)=\left[\begin{array}{ccc}{\displaystyle \frac{x}{60}}& 0& 0\\ 0& {\displaystyle \frac{x}{70}}& 0\\ 0& 0& {\displaystyle \frac{x}{90}}\end{array}\right],\mathit{\varphi }\left(x\right)=\left[\begin{array}{c}{\varphi }_1\left(x\right)\hfill \\ {\varphi }_2\left(x\right)\hfill \\ {\varphi }_3\left(x\right)\hfill \end{array}\right],$$

$$\mathsf{\Phi }(x,\mathit{\varphi }\left(x\right))=\left[\begin{array}{c}{\displaystyle \frac{1}{18+x^5}}\left(sin{\varphi }_1\left(x\right)+sin{\varphi }_2\left(x\right)+sin{\varphi }_3\left(x\right)\right)\hfill \\ {\displaystyle \frac{1}{35(x^3+1)}}\left({\displaystyle \frac{|{\varphi }_1\left(x\right)|}{(1+|{\varphi }_1\left(x\right)\left|\right)}}+{\displaystyle \frac{|{\varphi }_2\left(x\right)|}{(1+|{\varphi }_2\left(x\right)\left|\right)}}+{\displaystyle \frac{|{\varphi }_3\left(x\right)|}{(1+|{\varphi }_3\left(x\right)\left|\right)}}\right)\hfill \\ {\displaystyle \frac{1}{\sqrt{64+x^3}}}\left({\varphi }_1\left(x\right)+{\varphi }_2\left(x\right)+{\varphi }_3\left(x\right)\right)\hfill \end{array}\right],$$

$$\mathit{\xi }=\left[\begin{array}{c}{\displaystyle \frac{1}{2}}\hfill \\ {\displaystyle \frac{1}{2}}\hfill \\ {\displaystyle \frac{1}{2}}\hfill \end{array}\right],\mathit{\rho }=\left[\begin{array}{c}{\displaystyle \frac{7}{20}}\hfill \\ {\displaystyle \frac{7}{20}}\hfill \\ {\displaystyle \frac{7}{20}}\hfill \end{array}\right],\mathit{\omega }=\left[\begin{array}{c}0.8\hfill \\ 0.5\hfill \\ 0.05\hfill \end{array}\right].$$

(38)

From the vector function $\mathsf{\Phi },$ we have

$$\begin{array}{cc}& {\mathsf{\Phi }}_1(x,y_1,y_2,y_3)={\displaystyle \frac{1}{18+x^5}}\left(sin{y}_1+sin{y}_2+sin{y}_3\right),\hfill \\ & {\mathsf{\Phi }}_2(x,y_1,y_2,y_3)={\displaystyle \frac{1}{35(x^3+1)}}\left({\displaystyle \frac{|y_1|}{(1+|y_1\left|\right)}}+{\displaystyle \frac{|y_2|}{(1+|y_2\left|\right)}}+{\displaystyle \frac{|y_3|}{(1+|y_3\left|\right)}}\right),\hfill \\ & {\mathsf{\Phi }}_3(x,y_1,y_2,y_3)={\displaystyle \frac{1}{\sqrt{64+x^3}}}\left(y_1+y_2+y_3\right),\hfill \end{array}$$

For arbitrary $(y_1,y_2,y_3),(\overline{y_1},\overline{y_2},\overline{y_3})\in U,$ we have

$$|{\mathsf{\Phi }}_1(x,y_1,y_2,y_3)-{\mathsf{\Phi }}_1(x,\overline{y_1}\left(x\right),\overline{y_2},\overline{y_3})|\le {\displaystyle \frac{1}{18+x^5}}\left(|y_1-\overline{y_1}|+|y_2-\overline{y_2}|+|y_3-\overline{y_3}|\right).$$

(39)

This implies that

$${\theta }_n\left(x\right)={\displaystyle \frac{1}{18+x^5}}\Rightarrow \underset{x\in \mathcal{S}}{max}{\displaystyle \frac{1}{18+x^5}}={\displaystyle \frac{1}{18}}={{\theta }^{*}}_n.$$

Similarly,

$$|{\mathsf{\Phi }}_2(x,y_1,y_2,y_3)-{\mathsf{\Phi }}_2(x,\overline{y_1}\left(x\right),\overline{y_2},\overline{y_3})|\le {\displaystyle \frac{1}{35(x^3+1)}}\left(|y_1-\overline{y_1}|+|y_2-\overline{y_2}|+|y_3-\overline{y_3}|\right),$$

(40)

$$|{\mathsf{\Phi }}_3(x,y_1,y_2,y_3)-{\mathsf{\Phi }}_3(x,\overline{y_1}\left(x\right),\overline{y_2},\overline{y_3})|\le {\displaystyle \frac{1}{\sqrt{64+x^3}}}\left(|y_1-\overline{y_1}|+|y_2-\overline{y_2}|+|y_3-\overline{y_3}|\right).$$

(41)

From (40), we have

$${\eta }_n\left(x\right)={\displaystyle \frac{1}{35(x^3+1)}}\Rightarrow \underset{x\in \mathcal{S}}{max}{\displaystyle \frac{1}{35(x^3+1)}}={\displaystyle \frac{1}{35}}={{\eta }^{*}}_n.$$

From (41), we have

$${\zeta }_n\left(x\right)={\displaystyle \frac{1}{\sqrt{64+x^3}}}\Rightarrow \underset{x\in \mathcal{S}}{max}{\displaystyle \frac{1}{\sqrt{64+x^3}}}={\displaystyle \frac{1}{8}}={{\zeta }^{*}}_n.$$

Also,

$$\begin{array}{ccc}\hfill \underset{x\in [0,1]}{max}{A}_1\left(x\right)& =& \underset{x\in [0,X]}{max}{\displaystyle \frac{x}{60}}={\displaystyle \frac{1}{60}}=K_1,\hfill \\ \hfill \underset{x\in [0,1]}{max}{A}_2\left(x\right)& =& \underset{x\in [0,X]}{max}{\displaystyle \frac{x}{70}}={\displaystyle \frac{1}{70}}=K_2,\hfill \\ \hfill \underset{x\in [0,1]}{max}{A}_3\left(x\right)& =& \underset{x\in [0,X]}{max}{\displaystyle \frac{x}{90}}={\displaystyle \frac{1}{90}}=K_3.\hfill \end{array}$$

For $\tau =1,$ we derive

$$\begin{array}{c}{\mathcal{C}}_1:=2{\tau }^{{\rho }_1+{\xi }_1-1}(K_1+{{\theta }^{*}}_n){\displaystyle \frac{{\rho }_1\beta ({\xi }_1,{\rho }_1)}{\mathsf{\Gamma }\left({\xi }_1\right)}}\approx 0.11571,\hfill \\ {\mathcal{C}}_2:=2{\tau }^{{\rho }_2+{\xi }_2-1}(K_2+{{\eta }^{*}}_n){\displaystyle \frac{{\rho }_2\beta ({\xi }_2,{\rho }_2)}{\mathsf{\Gamma }\left({\xi }_2\right)}}\approx 0.06866,\hfill \\ {\mathcal{C}}_3:=2{\tau }^{{\rho }_3+{\xi }_3-1}(K_3+{{\zeta }^{*}}_n){\displaystyle \frac{{\rho }_3\beta ({\xi }_3,{\rho }_3)}{\mathsf{\Gamma }\left({\xi }_3\right)}}\approx 0.22602.\hfill \end{array}$$

(42)

Hence, $max\left({\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3\right)=max\left(0.11571,0.06866,0.22602\right)=0.22806<1.$ We see that conditions of Theorems 3 and 5 hold. Therefore, problem (37) has a unique solution that is H-U stable.

In Figure 1, Figure 2 and Figure 3, the solution $({\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right))$ is graphically represented for various fractional orders $\xi$, with a fixed fractal dimension of ${\displaystyle \frac{7}{20}}$. In Figure 4, Figure 5 and Figure 6, the solution $({\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right))$ is graphically represented with a fixed fractional order of ${\displaystyle \frac{2}{5}}$ and various fractal dimensions todemonstrate the impact of the fractal dimension. In Figure 7, Figure 8 and Figure 9, the solution $({\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right))$ is graphically represented for various fractional orders $\xi$ and various fractal dimensions. We observe that as the fractional order $\xi$ increases, the behavior of ${\varphi }_1\left(x\right),{\varphi }_2\left(x\right),$ ${\varphi }_3\left(x\right)$ changes significantly. Higher values of $\xi$ lead to smoother variations, while lower values introduce more fluctuations. The system stabilizes faster for higher values of $\xi$, showing reduced oscillations. In Figure 4, Figure 5 and Figure 6, we observe that changing fractal dimension affects the rate of change in ${\varphi }_1\left(x\right),{\varphi }_2\left(x\right),$ ${\varphi }_3\left(x\right)$. The lower $\rho$ values cause sharper changes, whereas higher $\rho$ values result in more gradual variations. The fractal nature introduces complexity in how the system evolves over time.

We concluded that the fractional order $\xi$ mainly influences the smoothness and stability of the system, while fractal dimension $\rho$ controls the rate and manner of change.

6. Conclusions

We have investigated a three-dimensional system of fractal-fractional-order evolution differential equations with terminal boundary conditions. We have derived sufficient criteria for the existence of solutions and for the stability of the proposed problem. The main results are applied to a general fractal-fractional-order evolution problem, and its existence and stability are verified using the main outcomes. The influence of the varying fractional order and fractal dimension on the dynamical system is analyzed through the plots of solution $({\varphi }_1\left(x\right),{\varphi }_2\left(x\right),{\varphi }_3\left(x\right))$ for various values of the fractional order $\xi$ and fractal dimensions $\rho$. We concluded that the fractional order $\xi$ mainly influences the smoothness and stability of the system, while fractal dimension $\rho$ controls the rate and manner of change.

Solving fractal-fractional-order evolution differential equations can provide accurate predictions of the behavior of complex systems, which can be crucial in fields like physics, engineering, and finance. Fractal-fractional Caputo differential equations can model complex systems that exhibit both fractal and fractional properties, such as anomalous diffusion, chaos, and self-similarity. The power law term allows for the modeling of systems with power law behavior, which is common in natural phenomena, such as financial markets, earthquakes, and floods. These equations can capture non-local effects, which are important in systems with long-range interactions or memory. Such equations can also be used to model a wide range of systems, from physical and biological systems to financial and social systems. Solving these equations can lead to the development of new mathematical tools and techniques, which can have far-reaching implications for mathematics and science. Besides the mentioned benefits, these equations have some limitations as well. Due to their complex nature, these equations can be challenging to analyze and solve. Numerical methods for solving fractal-fractional Caputo differential equations can be computationally intensive and may require specialized algorithms. The parameters in fractal-fractional Caputo differential equations can be difficult to interpret physically, which can make it challenging to estimate them from experimental data.

Overall, fractal-fractional Caputo differential equations with a power law offer a powerful tool for modeling complex systems, but they also present significant mathematical and computational challenges.

The conclusions drawn in this article are applicable to various real-world problems where terminal conditions play a crucial role. Specifically, fractal-fractional TVPs can be useful in biological and epidemiological models, where final-state constraints are imposed.