Coupled System of Variable-Order Fractional Differential Equations

Abstract

This work explores the growing field of fractional calculus, with particular emphasis on the complexities and opportunities associated with variable-order derivatives. We critically assess existing definitions, identifying those that are consistent with the established principles of constant-order fractional calculus. Based on this analysis, we introduce new formulations derived from the Grünwald–Letnikov and Liouville approaches, together with a novel variable-order Mittag–Leffler function. The core of our study is devoted to investigating the existence and uniqueness of solutions for a coupled system of variable-order fractional differential equations subject to initial conditions. Using Schauder’s fixed-point theorem and the Banach contraction principle, we establish new results that contribute to strengthening the theoretical foundation of such dynamical systems.

1. Introduction

Variable-order fractional calculus has evolved from a mathematical curiosity into a powerful framework for modeling complex systems in which memory and hereditary effects vary over time or space [1,2,3,4]. Unlike classical integer-order models, which often neglect history-dependent behaviors, fractional differential equations naturally capture nonlocal and long-range interactions [5,6].

Variable-order operators are particularly relevant in applications such as anomalous diffusion in heterogeneous media, charge transport in amorphous semiconductors, viscoelastic damping, and electrochemical processes, where the system dynamics adapt to their environment [7,8].

Despite its growing importance, a unified theoretical foundation for variable-order fractional calculus remains limited [9,10], with multiple competing definitions and a lag in the development of analytical tools, such as a variable-order Mittag–Leffler function consistent with classical theory [6,7].

In contrast to constant-order models studied by Bai et al. [11], where the fractional operator is fixed, the present work considers variable-order operators. This variation significantly affects the kernel structure and requires a more delicate fixed-point framework.

Recent works have also addressed variable-order fractional models using a variety of analytical techniques [12,13].

From a modeling perspective, coupled fractional systems arise naturally when multiple interacting processes evolve simultaneously with memory-dependent effects. In such settings, different state variables may exhibit distinct memory intensities, which justifies the use of variable-order operators. These models occur in a wide range of applications, including heterogeneous diffusion in biological tissues, viscoelastic materials with evolving memory effects, and anomalous transport phenomena in complex media.

The results of this paper are established for any $a>0$. The choice $a=1$ in the illustrative example is made solely for simplicity and does not restrict the generality of the theoretical results.

In this sense, the system (VOFDS) provides a general abstract framework for coupled dynamics with memory effects that vary across components and over time. This formulation enables the model to encompass a broad class of fractional systems arising in applied mathematics and physics.

Recent studies further illustrate the effectiveness of variable-order fractional operators through a variety of analytical techniques. Operator methods combined with fixed-point theory have been employed to analyze complex fractional systems, such as impulsive fractional differential equations involving almost sectorial operators [14]. Other works address specific variable-order models, including nonlinear pantograph equations with Hadamard derivatives [15] and thermistor problems formulated with Caputo derivatives [16], establishing results on existence, uniqueness, and stability.

In addition, boundary value problems with positivity constraints have been investigated using upper- and lower-solution methods in conjunction with Schauder’s fixed-point theorem [17].

Motivated by the above developments, we consider in this paper the following system of variable-order fractional differential equations (VOFDS):

$$\left\{\begin{array}{c}{\mathcal{D}}_{0^{+}}^{{\gamma }_1\left(s\right)}{\mathcal{W}}_1\left(s\right)=g_1\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{D}}_{0^{+}}^{{\gamma }_2\left(s\right)}{\mathcal{W}}_2\left(s\right)=g_2\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{W}}_i\left(0\right)=0,i=1,2,\hfill \end{array}\right.$$

where ${\mathcal{D}}_{0^{+}}^{{\gamma }_i\left(s\right)}$ denotes the standard Caputo fractional derivative, with $0<{\gamma }_i\left(s\right)<1$, and $g_i$ being continuous functions.

Building on the necessary definitions of the variable-order Riemann–Liouville integral and derivative, we employ Schauder’s fixed-point theorem together with a global contraction mapping argument to establish new results concerning the existence and uniqueness of solutions under nonlinear growth conditions imposed on $g_1$ and $g_2$.

A key novelty of our approach lies in its grounding within the classical Grünwald–Letnikov and Liouville frameworks, which ensures consistency with the constant-order theory while enabling the introduction of a new variable-order Mittag–Leffler function. This unified framework facilitates the analytical treatment of previously intractable variable-order systems, thereby bridging an important gap between theory and applications.

The main contributions of this paper can be summarized as follows:

• We establish existence and uniqueness results for a coupled system of variable-order fractional differential equations using both Schauder’s fixed-point theorem and a contraction mapping approach.
• We introduce a variable-order Mittag–Leffler function consistent with classical fractional calculus and suitable for representing solutions of variable-order systems.
• We provide a unified analytical framework that extends several existing results in the literature on constant-order and single-equation fractional systems.

Although the theoretical framework developed in this paper is abstract, it is designed to apply to a broad class of coupled systems arising in applications involving nonlocal and memory-dependent processes. The flexibility of variable-order operators makes the proposed approach suitable for future developments in both analytical and numerical studies of fractional dynamical systems.

The remainder of the paper is organized as follows. Section 2 presents the necessary notation, definitions, and auxiliary lemmas. In Section 3, we establish the main existence and uniqueness results for the coupled system. Finally, Section 4 provides illustrative examples and concluding remarks.

2. Preliminary Tools

Definition 1

([18,19,20,21,22]). The Riemann–Liouville integration is also extended to the case of variable order:

$${\mathcal{I}}_{0_{+}}^{\gamma \left(s\right)}g\left(s\right)={\displaystyle \frac{1}{\Gamma \left(\gamma \right(s\left)\right)}}{\int }_0^s{(s-w)}^{\gamma \left(s\right)-1}g\left(w\right)dw,Re\gamma \left(s\right)>0.$$

(1)

Here $g\left(s\right)$ may be any function defined for $s\ge 0$ and ensuring convergence of the integral.

Definition 2

([18,19,20,21,22]). The Riemann–Liouville variable order derivative is defined as follows:

$${\mathcal{D}}_{0_{+}}^{\gamma \left(s\right)}g\left(s\right)={\displaystyle \frac{1}{\Gamma (m-\gamma (s\left)\right)}}{\displaystyle \frac{d^m}{d{s}^m}}{\int }_0^s{(s-w)}^{m-1-\gamma \left(w\right)}g\left(w\right)dw,$$

(2)

where ${\displaystyle m=\underset{0\le s\le T}{max}\gamma \left(s\right)+1}$, $m\in \mathbb{N}$. Let $\gamma \left(s\right)>0$ be a continuous and bounded function, $g\left(w\right)\in C^m\left([0,1]\right)$, and $0\le w\le s$. Then

$${\mathcal{D}}_{0^{+}}^{\gamma \left(s\right)}g\left(s\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (m-\gamma (s\left)\right)}{\int }_0^s{(s-w)}^{m-1-\gamma \left(w\right)}{g}^{\left(m\right)}\left(w\right)dw,}\hfill & m-1\le \gamma \left(s\right)\le m\hfill \\ {\displaystyle \frac{d^{m}g\left(s\right)}{d{s}^m},}\hfill & \gamma \left(s\right)=m,\hfill \end{array}\right.$$

is called the Caputo variable order derivative of $g\left(s\right)$, where ${\displaystyle m=\underset{0\le s\le T}{max}\gamma \left(s\right)+1}$.

We find that the fractional operators (1) and (2) are not inverse to each other, as in the case of constant order, which can be seen below. So, it will not be correct to introduce ${\mathcal{D}}_{a_{+}}^{-\gamma \left(x\right)}$ as ${\left[{\mathcal{D}}_{a_{+}}^{\gamma \left(x\right)}\right]}^{-1}$.

The Marchaud derivative is also extended to

$${\mathbb{D}}_{a_{+}}^{\gamma \left(x\right)}g\left(x\right)={\displaystyle \frac{\gamma \left(x\right)}{\Gamma (1-\gamma (x\left)\right)}}{\int }_a^x{\displaystyle \frac{g\left(x\right)-g\left(w\right)}{{(x-w)}^{1+\gamma \left(x\right)}}}dw+{\displaystyle \frac{1}{\Gamma (1-\gamma (x\left)\right)}}{\displaystyle \frac{g\left(x\right)}{{(x-a)}^{\gamma \left(x\right)}}},0<Re\gamma \left(x\right)<1.$$

(3)

Lemma 1

([18,19,20,21,22]). Shows that

$${\mathbb{D}}_{a_{+}}^{\gamma \left(x\right)}g\left(x\right)\ne {\mathcal{D}}_{a_{+}}^{\gamma \left(x\right)}g\left(x\right),$$

(4)

in contrast with γ is a constant order. The next outcome shows the difference between (2) and (3).

Lemma 2

([18,19,20,21,22]). Let $0<\gamma \left(x\right)<1$. The derivatives (2) and (3) verify the following expression

$${\mathcal{D}}_{a_{+}}^{\gamma \left(x\right)}g\left(x\right)={\mathbb{D}}_{a_{+}}^{\gamma \left(x\right)}g\left(x\right)+{\displaystyle \frac{\gamma \left(x\right)}{\Gamma (1-\gamma (x\left)\right)}}{\int }_a^x{\displaystyle \frac{g\left(w\right)ln(x-w)}{{(x-w)}^{\gamma \left(x\right)}}}dw.$$

(5)

Corollary 1

([18,19,20,21,22]). The relation

$${\mathcal{D}}_{a_{+}}^{\gamma \left(x\right)}g\left(x\right)\equiv {\mathbb{D}}_{a_{+}}^{\gamma \left(x\right)}g\left(x\right)$$

holds if and only if $\gamma \left(x\right)$ is a constant or $g\left(x\right)$ is identically zero.

Whereas the fractional integration and differentiation are inverse to each other for a constant order,

$${\mathcal{D}}_{a_{+}}^{\gamma }{\mathcal{I}}_{a_{+}}^{\gamma }\phi \left(x\right)=\phi \left(x\right),$$

(6)

This is not the case for variable order $\gamma \left(x\right)$, as will be seen below. The invalidity of the inversion relation (6) and (2) or (2) and (3) is connected, in general, with the violation of the law of exponents. See below,

$${\mathcal{I}}_{a_{+}}^{\gamma \left(x\right)}{\mathcal{I}}_{a_{+}}^{\beta \left(x\right)}\ne {\mathcal{I}}_{a_{+}}^{\gamma \left(x\right)+\beta \left(x\right)}$$

(7)

in general.

Proposition 1 (Consistency with the classical Mittag–Leffler function).

Assume that the order function $\gamma \left(s\right)$ is constant, i.e., there exists $\alpha \in (0,1)$ such that

$$\gamma \left(s\right)=\alpha ,\forall s\in [0,T].$$

Then, the variable-order Mittag–Leffler function reduces to the classical one:

$$E_{\gamma (·)}(\lambda ,s)=E_{\alpha }\left(\lambda s^{\alpha }\right),$$

where $E_{\alpha }$ denotes the classical Mittag–Leffler function.

Example 1 (Benchmark equation).

Consider the variable-order fractional differential equation

$$D^{\gamma \left(s\right)}y\left(s\right)=\lambda y\left(s\right),s\in [0,T].$$

Under suitable assumptions, the solution can be expressed as

$$y\left(s\right)=y_0{E}_{\gamma (·)}(\lambda ,s),$$

where $E_{\gamma (·)}$ is the variable-order Mittag–Leffler function introduced above.

Remark 1.

The result shows that the variable-order Mittag–Leffler function extends the classical case while preserving its key role in fractional differential equations. A full asymptotic analysis remains an open problem for future work.

Theorem 1

([18,19,20,21,22]). Let $\phi \left(x\right)$ be an integrable function. The law of exponents

$${\mathcal{I}}_{a_{+}}^{\gamma \left(x\right)}{\mathcal{I}}_{a_{+}}^{\beta \left(x\right)}={\mathcal{I}}_{a_{+}}^{\gamma \left(x\right)+\beta \left(x\right)}$$

(8)

is satisfied for a constant function $\beta \left(x\right)=\beta$, $Re\beta >0$, and any function $\gamma \left(x\right)$ satisfying $Re\gamma \left(x\right)>0$.

Remark 2.

The system is rewritten as an equivalent integral formulation without relying on an inverse relation between fractional differentiation and integration, which does not hold for variable order. Instead, it follows from the definition of the Caputo-type variable-order derivative and known representation formulas under suitable regularity assumptions. Hence, the equivalence is analytically justified.

Remark 3.

In the general case where $\beta \left(x\right)$ is a non-constant function, the representation can be written as

$${\mathcal{I}}_{a_{+}}^{\gamma \left(x\right)}{\mathcal{I}}_{a_{+}}^{\beta \left(x\right)}\phi ={\int }_a^{x}k(x,w)dw.$$

(9)

The kernel is

$$k(x,w)={\displaystyle \frac{1}{\Gamma \left[\gamma \right(x\left)\right]}}{\int }_w^x{\displaystyle \frac{{(x-s)}^{\gamma \left(x\right)--1}{(s-w)}^{\beta \left(x\right)-1}}{\Gamma \left[\beta \right(x\left)\right]}}ds,$$

(10)

then

$$k(x,w)={\displaystyle \frac{{(x-w)}^{\gamma \left(x\right)}}{\Gamma \left[\gamma \right(x\left)\right]}}{\int }_0^1{\displaystyle \frac{{(x-w)}^{\beta [t+(x-w\left)\xi \right]-1}{\xi }^{\beta [t+(x-w\left)\xi \right]-1}{(1-\xi )}^{\gamma \left(x\right)-1}}{\Gamma \left[\beta \right[t+(x-w)\xi ]-1]}}d\xi .$$

The explicit computation of the above integral for different choices of a nonconstant function $\beta \left(x\right)$ remains an open question, under suitable regularity assumptions on the involved functions and parameters.

The kernel is well-defined under suitable regularity assumptions on $\beta \left(x\right)$ (see [20,21]).

It is important to note that, in the variable-order setting, the integral formulation is not obtained via a direct inversion of the differential operator. Instead, it follows from the definition of the Caputo-type variable-order derivative together with appropriate regularity assumptions on the solution. This allows us to establish an equivalent integral representation without relying on a classical inverse operator property.

Definition 3

(Adapted from [23,24]). Let $\mathcal{X}$ be a vector space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. A vector–valued norm on $\mathcal{X}$ is a map $\mathcal{N}:\mathcal{X}\to {\mathbb{R}}_{+}^n$ satisfying:

1. 

$\mathcal{N}\left(x\right)\ge 0$ for all $x\in \mathcal{X}$, and $\mathcal{N}\left(x\right)=0$ implies $x=0$;

2. 

$\mathcal{N}\left(\lambda x\right)=\left|\lambda \right|\mathcal{N}\left(x\right)$ for every $x\in \mathcal{X}$ and $\lambda \in \mathbb{K}$;

3. 

$\mathcal{N}(x+y)\le \mathcal{N}\left(x\right)+\mathcal{N}\left(y\right)$ for all $x,y\in \mathcal{X}$.

The pair $(\mathcal{X},\mathcal{N})$ is called a generalized normed space. If the metric induced by $\mathcal{N}$, i.e.,

$$d(x,y)=\mathcal{N}(x-y)=\left(\begin{array}{c}{\mathcal{N}}_1(x-y)\\ ⋮\\ {\mathcal{N}}_n(x-y)\end{array}\right),$$

is complete, then $(\mathcal{X},\mathcal{N})$ is called a generalized Banach space. In that case, each component ${\mathcal{N}}_i$ ($i=1,\dots ,n$) is a norm on $\mathcal{X}$, and conversely, if every ${\mathcal{N}}_i$ is a norm, then $(\mathcal{X},\mathcal{N})$ is a generalized Banach space.

Definition 4.

A square real matrix is said to be convergent to zero if its spectral radius satisfies $\rho \left(M\right)<1$; equivalently, every eigenvalue of M lies in the open unit disk.

Lemma 3

(See [23,24]). Let M be a square matrix with non-negative entries. The following statements are equivalent:

1. 

M is convergent to zero;

2. 

$I-M$ is invertible and ${(I-M)}^{-1}={\sum }_{k=0}^{\infty }{M}^k$;

3. 

For every eigenvalue λ of M (i.e., every solution of $det(M-\lambda I)=0$), we have $\left|\lambda \right|<1$;

4. 

$I-M$ is invertible, and all entries of ${(I-M)}^{-1}$ are nonnegative.

Definition 5

(Based on [23,24]). A nonsingular matrix $\mathfrak{A}={\left(a_{ij}\right)}_{1\le i,j\le n}\in {\mathcal{M}}_{n\times n}\left(\mathbb{R}\right)$ is said to possess the absolute value property if

$${\mathfrak{A}}^{-1}\left|\mathfrak{A}\right|\le I,$$

where $\left|\mathfrak{A}\right|=\left(\right|a_{ij}{\left|\right)}_{1\le i,j\le n}$ and the inequality is understood componentwise.

Examples of matrices that converge to zero.

• $\mathfrak{A}=\left(\begin{array}{cc}\mu & 0\\ 0& \nu \end{array}\right)$ with $\mu ,\nu \ge 0$ and $max(\mu ,\nu )<1$;
• $\mathfrak{A}=\left(\begin{array}{cc}\mu & -\gamma \\ 0& \nu \end{array}\right)$ with $\mu ,\nu ,\gamma \ge 0$ satisfying $\mu +\nu <1$ and $\gamma <1$;
• $\mathfrak{A}=\left(\begin{array}{cc}\mu & -\mu \\ \nu & -\nu \end{array}\right)$ where $\mu ,\nu \ge 0$, $|\mu -\nu |<1$, $\mu >1$, and $\nu >0$.

Theorem 2.

Let E be a bounded, convex, and closed subset of a normed space X. If $T:E\to E$ is a compact map, then there exists $x\in E$ such that $T\left(x\right)=x$.

3. Existence and Uniqueness of Solution

Consider the coupled system of variable-order fractional differential equations:

$$\left\{\begin{array}{c}{\mathcal{D}}_{0^{+}}^{{\gamma }_1\left(s\right)}{\mathcal{W}}_1\left(s\right)=g_1\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{D}}_{0^{+}}^{{\gamma }_2\left(s\right)}{\mathcal{W}}_2\left(s\right)=g_2\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{W}}_i\left(0\right)=0,i=1,2,\hfill \end{array}\right.$$

(11)

where $0<{\gamma }_i\left(s\right)<1$ ($i=1,2$) and $g_i:[0,a]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ ($0<a<\infty$) are continuous functions.

Definition 6

([23,24]). Let $C^{★}\left([0,a]\right)$ be the class of continuous column vectors $\left({\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right)$ whose components ${\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\in C\left((0,a]\right)$ (the class of continuous functions on $(0,a]$). The norm of $\mathcal{W}\in C^{★}\left([0,a]\right)$ is given by

$$\parallel \mathcal{W}\parallel =\underset{i=1,2}{max}\left\{\underset{0\le s\le a}{sup}\left|{\mathcal{W}}_i\left(s\right)\right|\right\}.$$

Definition 7

([23,24]). We mean by a solution of the system (11) that a column vector $\mathcal{W}\in C^{★}\left([0,a]\right)$ satisfies (11).

Remark 4.

In fact, if $\mathcal{W}\left(s\right)=({\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right))\in C^{★}\left((0,a]\right)$ (more generally $\mathcal{W}\in C_r^{★}\left([0,a]\right)$ with $r<1-\gamma$ where $\gamma \left(s\right)=min\{{\gamma }_1\left(s\right),{\gamma }_2\left(s\right)\}$), and further assumptions guarantee $g_i(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right))\in C\left([0,a]\right)$, then the system is equivalent to the integral equations

$$\left\{\begin{array}{c}{\mathcal{W}}_1\left(s\right)={\mathcal{I}}_{0^{+}}^{{\gamma }_1\left(s\right)}{g}_1\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{W}}_2\left(s\right)={\mathcal{I}}_{0^{+}}^{{\gamma }_2\left(s\right)}{g}_2\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{W}}_i\left(0\right)=0,i=1,2.\hfill \end{array}\right.$$

We also assume the following hypotheses:

•

(${\mathbf{H}}_1$) The functions $g_i:[0,a]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, $i=1,2$, are continuous.

•

(${\mathbf{H}}_{\mathbf{2}}$) There exist nonnegative continuous functions $a_i,b_i\in C\left((0,a]\right)$, $i=1,2$, such that

$$\begin{array}{cc}\hfill \parallel g_1(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right))-g_1(s,{\mathcal{V}}_1\left(s\right),{\mathcal{V}}_2\left(s\right))\parallel & \le a_1\left(s\right)\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel +b_1\left(s\right)\parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel ,\hfill \\ \hfill \parallel g_2(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right))-g_2(s,{\mathcal{V}}_1\left(s\right),{\mathcal{V}}_2\left(s\right))\parallel & \le a_2\left(s\right)\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel +b_2\left(s\right)\parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel .\hfill \end{array}$$

The function $\sigma \left(s\right)$ plays a key role in controlling the singular behavior of the nonlinear terms near $s=0$ and in ensuring the integrability of the associated kernel in the equivalent integral formulation. The condition $0\le \sigma \left(s\right)<\gamma \left(s\right)$ guarantees that the fractional integral operator is well defined and that the associated solution operator is compact. This assumption is therefore sufficient for the application of Schauder’s fixed-point theorem and is crucial in deriving the required a priori bounds.

We now state a local existence theorem.

Theorem 3.

Consider $0<{\gamma }_i\left(s\right)<1$, $i=1,2$, and let $\gamma \left(s\right)=min\{{\gamma }_1\left(s\right),{\gamma }_2\left(s\right)\}$. Assume $0\le \sigma \left(s\right)<\gamma \left(s\right)<1$ and that $g_i(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right))\in C\left([0,1]\right)$. Suppose further that $s^{\sigma \left(s\right)}{g}_i(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right))\in C\left([0,1]\right)$. Then the coupled system (11) has a continuous solution $\mathcal{W}\in \mathfrak{C}=C^{★}[0,1]$ for an appropriate $\delta \le 1$.

Proof. 

We consider the following nonlinear equation:

$$\left\{\begin{array}{c}{\mathcal{W}}_1\left(s\right)={\mathcal{I}}_{0^{+}}^{{\gamma }_1\left(s\right)}{g}_1\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{W}}_2\left(s\right)={\mathcal{I}}_{0^{+}}^{{\gamma }_2\left(s\right)}{g}_2\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{W}}_i\left(0\right)=0,i=1,2.\hfill \end{array}\right.$$

Define the operator $\mathcal{T}:\mathfrak{C}\times \mathfrak{C}\to \mathfrak{C}\times \mathfrak{C}$ by

$$\mathcal{T}({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)=\left\{\begin{array}{c}{\mathcal{I}}_{0^{+}}^{{\gamma }_1\left(s\right)}{g}_1\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{I}}_{0^{+}}^{{\gamma }_2\left(s\right)}{g}_2\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{W}}_i\left(0\right)=0,i=1,2.\hfill \end{array}\right.$$

Similarly,

$$\mathcal{T}({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)=\left\{\begin{array}{c}{\mathcal{I}}_{0^{+}}^{{\gamma }_1\left(s\right)}{g}_1\left(s,{\mathcal{V}}_1\left(s\right),{\mathcal{V}}_2\left(s\right)\right),\hfill \\ {\mathcal{I}}_{0^{+}}^{{\gamma }_2\left(s\right)}{g}_2\left(s,{\mathcal{V}}_1\left(s\right),{\mathcal{V}}_2\left(s\right)\right),\hfill \\ {\mathcal{V}}_i\left(0\right)=0,i=1,2.\hfill \end{array}\right.$$

Clearly, $\mathcal{T}$ is a compact operator. Indeed, it is the composition of two simpler operators: $\mathcal{T}=A\circ N$; thus,

$$N({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)=\left\{\begin{array}{c}{s}^{\sigma \left(s\right)}{g}_1\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ s^{\sigma \left(s\right)}{g}_2\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\displaystyle \underset{s\to 0}{lim}{s}^{1-\sigma \left(s\right)}{\mathcal{W}}_1\left(s\right)=a,}\hfill \\ {\displaystyle \underset{s\to 0}{lim}{s}^{1-\sigma \left(s\right)}{\mathcal{W}}_2\left(s\right)=b,}\hfill \end{array}\right.$$

which is continuous and bounded, and

$$A({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma \left({\gamma }_1\left(s\right)\right)}{\int }_0^s{(s-w)}^{{\gamma }_1\left(s\right)-1}{w}^{-\sigma \left(w\right)}{\mathcal{V}}_1\left(w\right)dw,}\hfill \\ {\displaystyle \frac{1}{\Gamma \left({\gamma }_2\left(s\right)\right)}{\int }_0^s{(s-w)}^{{\gamma }_2\left(s\right)-1}{w}^{-\sigma \left(w\right)}{\mathcal{V}}_2\left(w\right)dw,}\hfill \end{array}\right.$$

which is a compact operator because $\gamma \left(s\right)-\sigma \left(s\right)>0$, where $\gamma \left(s\right)=sup\{{\gamma }_1\left(s\right),{\gamma }_2\left(s\right)\}$ and $\sigma ={sup}_{0<s<1}\sigma \left(s\right)$.

For $0<s\le \sigma \le 1$, we have

$$A({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)=\left(\begin{array}{c}\parallel A{\mathcal{V}}_1\left(s\right)\parallel \\ \parallel A{\mathcal{V}}_2\left(s\right)\parallel \end{array}\right).$$

$$<\left\{\begin{array}{c}{\displaystyle \underset{0<s\le \delta }{sup}\mid {\mathcal{V}}_1(s,\mathcal{W}\left(s\right))\mid \frac{1}{\Gamma \left({\gamma }_1\left(s\right)\right)}{\int }_0^s{(s-w)}^{{\gamma }_1\left(s\right)-1}{w}^{-\sigma \left(w\right)}{\mathcal{V}}_1\left(w\right)dw}\hfill \\ {\displaystyle \underset{0<s\le \delta }{sup}\mid {\mathcal{V}}_2(s,\mathcal{W}\left(s\right))\mid \frac{1}{\Gamma \left({\gamma }_2\left(s\right)\right)}{\int }_0^s{(s-w)}^{{\gamma }_2\left(s\right)-1}{w}^{-\sigma \left(w\right)}{\mathcal{V}}_2\left(w\right)dw}\hfill \end{array}\right.$$

$$<\left\{\begin{array}{c}{\displaystyle \frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)-{\gamma }_1\left(s\right))}{\delta }^{\sigma \left(s\right)-{\gamma }_1\left(s\right)}\underset{0<s\le \delta }{sup}\mid {\mathcal{V}}_1(s,\mathcal{W}\left(s\right))\mid }\hfill \\ {\displaystyle \frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)-{\gamma }_2\left(s\right))}{\delta }^{\sigma \left(s\right)-{\gamma }_2\left(s\right)}\underset{0<s\le \delta }{sup}\mid {\mathcal{V}}_2(s,\mathcal{W}\left(s\right))\mid }\hfill \end{array}\right.$$

Let

$$ϵ=\underset{1\le i\le 2}{max}\left\{{\displaystyle \frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)-{\gamma }_i\left(s\right))}}{\delta }^{\sigma \left(s\right)-{\gamma }_i\left(s\right)}\right\}.$$

Therefore, taking the norm in $C^{★}[0,\delta ]$:

$$\parallel A({\mathcal{V}}_1,{\mathcal{V}}_2)\parallel \le ϵ\parallel {\mathcal{V}}_1-{\mathcal{V}}_2\parallel ,$$

where we may consider $ϵ>0$ as small as possible by choosing $\delta >0$ sufficiently small. □

Theorem 4.

Let $0<{\gamma }_i\left(s\right)<1$, $i=1,2$, and $\gamma \left(s\right)=min\{{\gamma }_1\left(s\right),{\gamma }_2\left(s\right)\}$. Assume $0\le \sigma \left(s\right)<\gamma \left(s\right)<1$ and $G(s,{\mathcal{W}}_1,{\mathcal{W}}_2)=\left(g_1(s,{\mathcal{W}}_1,{\mathcal{W}}_2),g_2(s,{\mathcal{W}}_1,{\mathcal{W}}_2)\right)\in C_{\sigma }^{★}[0,1]$. Suppose further that $\sigma ={sup}_{0\le s\le 1}\sigma \left(s\right)$ and

$$\parallel G(s,{\mathcal{W}}_1,{\mathcal{W}}_2)-G(s,{\mathcal{V}}_1,{\mathcal{V}}_2)\parallel \le {\displaystyle \frac{L}{s^{\sigma }}}\left[\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel +\parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \right],$$

where L is a constant independent of ${\mathcal{W}}_1,{\mathcal{W}}_2,{\mathcal{V}}_1,{\mathcal{V}}_2\in \mathbb{R}$ and $s\in [0,1]$. Then the system (11) has a unique solution $({\mathcal{W}}_1,{\mathcal{W}}_2)\in \mathfrak{C}\times \mathfrak{C}$.

Proof. 

We consider the operator

$$\mathcal{T}({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)=\left\{\begin{array}{c}{\mathcal{I}}_{0^{+}}^{{\gamma }_1\left(s\right)}{g}_1\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{I}}_{0^{+}}^{{\gamma }_2\left(s\right)}{g}_2\left(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)\right),\hfill \\ {\mathcal{W}}_i\left(0\right)=0,i=1,2.\hfill \end{array}\right.$$

This operator is well defined and continuous as a map $\mathcal{T}:\mathfrak{C}\times \mathfrak{C}\to \mathfrak{C}\times \mathfrak{C}$.

Define the iterates of $\mathcal{T}$ in the usual way: ${\mathcal{T}}^1=\mathcal{T}$, ${\mathcal{T}}^k=\mathcal{T}\circ {\mathcal{T}}^{k-1}$. It suffices to prove that ${\mathcal{T}}^k$ is a contraction for sufficiently large k. Indeed, for ${\mathcal{W}}_1,{\mathcal{W}}_2,{\mathcal{V}}_1,{\mathcal{V}}_2\in C^{★}[0,1]$ we have

$$\parallel {\mathcal{T}}^k({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)-{\mathcal{T}}^k({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)\parallel \le {\displaystyle \frac{{\left(HL\right)}^k}{\Gamma \left(k\left(\gamma \right(s)-\sigma (s\left)\right)+1\right)}}{s}^{k\left(\gamma \right(s)-\sigma (s\left)\right)}\left[\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel +\parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \right],$$

where the constant H depends only on $\gamma \left(s\right)$ and $\sigma \left(s\right)$. In fact,

$$\mathcal{T}({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)=\left({\mathcal{I}}_{0^{+}}^{{\gamma }_1\left(s\right)}{g}_1(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right)),{\mathcal{I}}_{0^{+}}^{{\gamma }_2\left(s\right)}{g}_2(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right))\right),$$

$$\mathcal{T}({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)=\left({\mathcal{I}}_{0^{+}}^{{\gamma }_1\left(s\right)}{g}_1(s,{\mathcal{V}}_1\left(s\right),{\mathcal{V}}_2\left(s\right)),{\mathcal{I}}_{0^{+}}^{{\gamma }_2\left(s\right)}{g}_2(s,{\mathcal{V}}_1\left(s\right),{\mathcal{V}}_2\left(s\right))\right).$$

Then

$$\mathcal{T}({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)-\mathcal{T}({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)=\left\{\begin{array}{c}{\mathcal{I}}_{0^{+}}^{{\gamma }_1\left(s\right)}\left[g_1(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right))-g_1(s,{\mathcal{V}}_1\left(s\right),{\mathcal{V}}_2\left(s\right))\right],\hfill \\ {\mathcal{I}}_{0^{+}}^{{\gamma }_2\left(s\right)}\left[g_2(s,{\mathcal{W}}_1\left(s\right),{\mathcal{W}}_2\left(s\right))-g_2(s,{\mathcal{V}}_1\left(s\right),{\mathcal{V}}_2\left(s\right))\right].\hfill \end{array}\right.$$

Using the Lipschitz condition, we obtain

$$\begin{array}{cc}& \left|\mathcal{T}({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)-\mathcal{T}({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)\right|\hfill \\ & <\left\{\begin{array}{c}{\displaystyle \frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)+{\gamma }_1\left(s\right))}{s}^{\sigma \left(s\right)-{\gamma }_1\left(s\right)}\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel +\frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)+{\gamma }_1\left(s\right))}{s}^{\sigma \left(s\right)-{\gamma }_1\left(s\right)}\parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel ,}\hfill \\ {\displaystyle \frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)+{\gamma }_2\left(s\right))}{s}^{\sigma \left(s\right)-{\gamma }_2\left(s\right)}\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel +\frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)+{\gamma }_2\left(s\right))}{s}^{\sigma \left(s\right)-{\gamma }_2\left(s\right)}\parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel }\hfill \end{array}\right.\hfill \\ & =\left(\begin{array}{cc}{\Delta }_1& {\Delta }_1\\ {\Delta }_2& {\Delta }_2\end{array}\right)\left(\begin{array}{c}\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel \\ \parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \end{array}\right)=M\left(\begin{array}{c}\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel \\ \parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \end{array}\right),\hfill \end{array}$$

where

$${\Delta }_i={\displaystyle \frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)+{\gamma }_i\left(s\right))}}{s}^{\sigma \left(s\right)-{\gamma }_i\left(s\right)},i=1,2.$$

Hence

$$\mathcal{T}({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)-\mathcal{T}({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)<M\left(\begin{array}{c}\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel \\ \parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \end{array}\right).$$

Define

$${\displaystyle \frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)+{\gamma }_i\left(s\right))}}{s}^{\sigma \left(s\right)-{\gamma }_i\left(s\right)}=\underset{1\le i\le 2}{max}\left\{\underset{0\le s\le 1}{sup}\left|{\displaystyle \frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)+{\gamma }_i\left(s\right))}}{s}^{\sigma \left(s\right)-{\gamma }_i\left(s\right)}\right|\right\}.$$

Then inequality (14) is verified for $k=1$ if we take $H\left(s\right)>\Gamma (1-\sigma (s\left)\right)$. The matrix M converges to zero.

Assuming that (14) holds for some k, we obtain similarly

$$\begin{array}{cc}& {\mathcal{T}}^{k+1}({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)-{\mathcal{T}}^{k+1}({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)\hfill \\ & <\left\{\begin{array}{c}{\displaystyle \frac{{\left(HL\right)}^k}{\Gamma \left(k({\gamma }_1\left(s\right)-\sigma \left(s\right))\right)\Gamma \left({\gamma }_1\left(s\right)\right)}\left[\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel +\parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \right]}\hfill \\ {\displaystyle \times {\int }_0^s{(s-w)}^{{\gamma }_1\left(s\right)}{w}^{k({\gamma }_1\left(s\right)-\sigma \left(s\right))}dw}\hfill \\ {\displaystyle \frac{{\left(HL\right)}^k}{\Gamma \left(k({\gamma }_2\left(s\right)-\sigma \left(s\right))\right)\Gamma \left({\gamma }_2\left(s\right)\right)}\left[\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel +\parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \right]}\hfill \\ {\displaystyle \times {\int }_0^s{(s-w)}^{{\gamma }_2\left(s\right)}{w}^{k({\gamma }_2\left(s\right)-\sigma \left(s\right))}dw}\hfill \end{array}\right.\hfill \end{array}$$

$$=\left(\begin{array}{cc}{\Pi }_1& {\Pi }_1\\ {\Pi }_2& {\Pi }_2\end{array}\right)\left(\begin{array}{c}\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel \\ \parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \end{array}\right)=M_k\left(\begin{array}{c}\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel \\ \parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \end{array}\right),$$

where

$${\Pi }_i={\displaystyle \frac{{\left(HL\right)}^k}{\Gamma \left(k({\gamma }_i\left(s\right)-\sigma \left(s\right))\right)\Gamma \left({\gamma }_i\left(s\right)\right)}}{\int }_0^s{(s-w)}^{{\gamma }_i\left(s\right)}{w}^{k({\gamma }_i\left(s\right)-\sigma \left(s\right))}dw,i=1,2.$$

Thus (14) is verified for $k+1$ if H is given by

$$H=\underset{k}{max}{H}_k=\underset{i=1,2}{max}\left\{{\displaystyle \frac{\Gamma \left(k({\gamma }_i\left(s\right)-\sigma \left(s\right))-\sigma \left(s\right)\right)}{\Gamma \left(k({\gamma }_i\left(s\right)-\sigma \left(s\right))+1\right)}}\right\}.$$

Note that (15) defines a finite H because $H_k\le 1$ for $k\ge {\displaystyle \frac{1+\sigma \left(s\right)}{\gamma \left(s\right)-\sigma \left(s\right)}}$. Choosing k sufficiently large in (15), we get

$${\displaystyle \frac{{\left(HL\right)}^k}{\Gamma \left(k\left(\gamma \right(s)-\sigma (s\left)\right)+1\right)}}\le {\displaystyle \frac{1}{2}}.$$

The matrix $M_k$ converges to zero, and therefore

$$\parallel {\mathcal{T}}^k({\mathcal{W}}_1,{\mathcal{W}}_2)\left(s\right)-{\mathcal{T}}^k({\mathcal{V}}_1,{\mathcal{V}}_2)\left(s\right)\parallel \le {\displaystyle \frac{1}{2}}\left[\parallel {\mathcal{W}}_1-{\mathcal{V}}_1\parallel +\parallel {\mathcal{W}}_2-{\mathcal{V}}_2\parallel \right].$$

 □

The above results establish existence, uniqueness, and stability. Numerical simulations of variable-order fractional systems further support these theoretical findings [25].

4. Examples

Example 2.

We illustrate the theoretical results with a concrete coupled system of variable-order fractional differential equations. Specifically, consider the system

$$\left\{\begin{array}{c}{\displaystyle {\mathcal{D}}_{0^{+}}^{{\gamma }_1\left(s\right)}{\mathcal{W}}_1\left(s\right)=\frac{s}{1+s^2}\left(1+{\mathcal{W}}_1\left(s\right)+{\mathcal{W}}_2\left(s\right)\right),}\hfill \\ {\displaystyle {\mathcal{D}}_{0^{+}}^{{\gamma }_2\left(s\right)}{\mathcal{W}}_2\left(s\right)=\frac{s^2}{1+s^2}\left(sin{\mathcal{W}}_1\left(s\right)+cos{\mathcal{W}}_2\left(s\right)\right),}\hfill \\ {\mathcal{W}}_1\left(0\right)={\mathcal{W}}_2\left(0\right)=0,\hfill \end{array}\right.$$

where the variable orders are chosen as

$${\gamma }_1\left(s\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{s}{4}},{\gamma }_2\left(s\right)={\displaystyle \frac{3}{4}}-{\displaystyle \frac{s}{4}},s\in [0,1].$$

Thus ${\gamma }_1\left(s\right),{\gamma }_2\left(s\right)\in (0,1)$ for $s\in [0,1]$, and

$$\gamma \left(s\right)=min\{{\gamma }_1\left(s\right),{\gamma }_2\left(s\right)\}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{s}{4}}(\mathit{since}{\gamma }_1\left(s\right)\le {\gamma }_2\left(s\right)\mathit{for}s\in [0,1]).$$

Define $\sigma \left(s\right)={\displaystyle \frac{s}{2}}$. Then $0\le \sigma \left(s\right)<\gamma \left(s\right)$ on $(0,1]$. The nonlinearities are

$$g_1(s,u,v)={\displaystyle \frac{s}{1+s^2}}(1+u+v),g_2(s,u,v)={\displaystyle \frac{s^2}{1+s^2}}(sinu+cosv),$$

which are continuous on $[0,1]\times \mathbb{R}\times \mathbb{R}$. Moreover,

$$s^{\sigma \left(s\right)}{g}_1(s,u,v)=s^{s/2}{\displaystyle \frac{s}{1+s^2}}(1+u+v),s^{\sigma \left(s\right)}{g}_2(s,u,v)=s^{s/2}{\displaystyle \frac{s^2}{1+s^2}}(sinu+cosv)$$

are also continuous on $[0,1]\times \mathbb{R}\times \mathbb{R}$.

The Lipschitz condition  (${\mathit{H}}_{\mathbf{2}}$)  is satisfied with

$$a_1\left(s\right)=b_1\left(s\right)={\displaystyle \frac{s}{1+s^2}},a_2\left(s\right)=b_2\left(s\right)={\displaystyle \frac{s^2}{1+s^2}},$$

because

$$|g_1(s,u_1,v_1)-g_1(s,u_2,v_2)|\le {\displaystyle \frac{s}{1+s^2}}\left(|u_1-u_2|+|v_1-v_2|\right),$$

and

$$|g_2(s,u_1,v_1)-g_2(s,u_2,v_2)|\le {\displaystyle \frac{s^2}{1+s^2}}\left(|u_1-u_2|+|v_1-v_2|\right)$$

by the mean value theorem for sine and cosine. Hence the Lipschitz constants are

$$L_1\left(s\right)={\displaystyle \frac{s}{1+s^2}},L_2\left(s\right)={\displaystyle \frac{s^2}{1+s^2}}.$$

Since $\sigma \left(s\right)=s/2$, we have $s^{\sigma \left(s\right)}{L}_i\left(s\right)$ bounded, and the Lipschitz estimate in the form required by Theorem 3 can be verified with a constant L independent of s. We now verify that the operator T defined in the proof becomes a contraction after sufficiently many iterations. For simplicity we work with the matrix M from the proof. Using the Lipschitz estimates we compute

$${\Delta }_i\left(s\right)={\displaystyle \frac{\Gamma (1-\sigma (s\left)\right)}{\Gamma (1-\sigma \left(s\right)+{\gamma }_i\left(s\right))}}{s}^{\sigma \left(s\right)-{\gamma }_i\left(s\right)}.$$

Since $\sigma \left(s\right)=s/2$ and ${\gamma }_1\left(s\right)=1/2+s/4$, ${\gamma }_2\left(s\right)=3/4-s/4$, we have

$$\sigma \left(s\right)-{\gamma }_1\left(s\right)=-{\displaystyle \frac{1}{2}}--{\displaystyle \frac{s}{4}}<0,\sigma \left(s\right)-{\gamma }_2\left(s\right)=-{\displaystyle \frac{3}{4}}+{\displaystyle \frac{3s}{4}}<0.$$

Thus, each ${\Delta }_i\left(s\right)$ is bounded on $[0,1]$ and tends to 0 as $s\to 0^{+}$. The matrix $M=\left(\begin{array}{cc}{\Delta }_1& {\Delta }_1\\ {\Delta }_2& {\Delta }_2\end{array}\right)$ has spectral radius $\rho \left(M\right)={\Delta }_1+{\Delta }_2$. Because ${\Delta }_1\left(s\right)+{\Delta }_2\left(s\right)<1$ for sufficiently small s, we can choose $\delta >0$ such that for all $s\in [0,\delta ]$,

$$\underset{s\in [0,\delta ]}{sup}({\Delta }_1\left(s\right)+{\Delta }_2\left(s\right))\le {\displaystyle \frac{1}{2}}<1.$$

Hence, on the interval $[0,\delta ]$, the operator T is a contraction with respect to the generalized norm induced by the matrix M. By the Banach fixed-point theorem in generalized Banach spaces, there exists a unique local solution.

Alternatively, using the iterative method described in Theorem 4, one can show that the solution extends to the whole interval $[0,1]$, since the iterates become contractive with factor $\frac{1}{2}$ after a finite number of iterations k. The constant H can be computed explicitly in terms of Gamma functions. For the given σ and ${\gamma }_i$, one obtains $H\le 1$ and

$$L=max\left\{\underset{s\in [0,1]}{sup}{L}_1\left(s\right),\underset{s\in [0,1]}{sup}{L}_2\left(s\right)\right\}=\frac{1}{2}.$$

Then

$${\displaystyle \frac{{\left(HL\right)}^k}{\Gamma \left(k\left(\gamma \right(s)-\sigma (s\left)\right)+1\right)}}\le {\displaystyle \frac{{(1/2)}^k}{\Gamma (k/4+1)}}\to 0\mathrm{as}k\to \infty .$$

Thus, for sufficiently large k, the k–th iterate is a contraction with factor ${\displaystyle \frac{1}{2}}$, guaranteeing global uniqueness.

Therefore, the system has a unique solution $({\mathcal{W}}_1,{\mathcal{W}}_2)\in C^{★}[0,1]\times C^{★}[0,1]$.

• Interpretation.

This example can be interpreted as a simplified model of anomalous diffusion in heterogeneous media, such as porous materials or biological tissues. In this framework, the functions ${\mathcal{W}}_1\left(s\right)$ and ${\mathcal{W}}_2\left(s\right)$ may represent the concentrations of two interacting species or chemical substances evolving over time.

The variable orders ${\gamma }_1\left(s\right)$ and ${\gamma }_2\left(s\right)$ describe time-dependent memory effects, which arise naturally in complex media with non-uniform structure or evolving environmental conditions. Such behavior is well documented in the context of fractional diffusion models, where deviations from classical Fickian laws are observed.

The nonlinear terms account for interaction mechanisms between the species, including reaction or coupling effects, while the coefficients depending on s reflect temporal or spatial heterogeneity of the medium.

Such models have been successfully used to describe anomalous transport phenomena in biological tissues and related systems; see, for instance [12].

Therefore, the proposed system provides a mathematically tractable framework for modeling coupled anomalous diffusion processes with variable memory effects.

Example 3.

To illustrate the applicability of our theoretical results in a more general nonlinear setting, we consider the following coupled system:

$$\left\{\begin{array}{c}{\mathcal{D}}_{0^{+}}^{{\gamma }_1\left(s\right)}{\mathcal{W}}_1\left(s\right)={\lambda }_1{\mathcal{W}}_1\left(s\right)+e^{W_2\left(s\right)}+sin\left({\mathcal{W}}_1\left(s\right)\right),\hfill \\ {\mathcal{D}}_{0^{+}}^{{\gamma }_2\left(s\right)}{\mathcal{W}}_2\left(s\right)={\lambda }_2{\mathcal{W}}_2\left(s\right)+ln(1+{\mathcal{W}}_1^2\left(s\right)),\hfill \\ {\mathcal{W}}_1\left(0\right)={\mathcal{W}}_2\left(0\right)=0,\hfill \end{array}\right.s\in [0,1],$$

where ${\lambda }_1,{\lambda }_2>0$ are constants, and ${\gamma }_i\left(s\right)\in (0,1)$ are continuous functions, e.g., ${\gamma }_1\left(s\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{s}{4}}$, ${\gamma }_2\left(s\right)={\displaystyle \frac{3}{4}}-{\displaystyle \frac{s}{4}}$, so that ${\gamma }_i\left(s\right)\ge {\displaystyle \frac{1}{4}}>0$ for all $s\in [0,1]$.

The right-hand sides are

$$f_1(s,u,v)={\lambda }_{1}u+e^v+sinu,f_2(s,u,v)={\lambda }_{2}v+ln(1+u^2).$$

For each fixed s, these functions are compositions of elementary continuous functions and are therefore continuous in $(u,v)$. Moreover, the explicit dependence on s is continuous (in fact, constant). Hence, assumption  (${\mathit{H}}_{\mathbf{1}}$)  is satisfied.

Next, we compute the partial derivatives to determine local Lipschitz constants. However, due to the presence of the exponential term $e^v$, which is not globally Lipschitz, it is not possible to establish a global Lipschitz constant. Nevertheless, Theorem 3 requires only a local Lipschitz condition on bounded sets, which is sufficient to guarantee the existence of a local solution. For illustrative purposes, we therefore restrict our analysis to a small interval $[0,\delta ]$ and a bounded region defined by $|{\mathcal{W}}_i|\le R$.

For $f_1$:

$$\left|{\displaystyle \frac{\partial f_1}{\partial u}}\right|=|{\lambda }_1+cosu|\le {\lambda }_1+1,\left|{\displaystyle \frac{\partial f_1}{\partial v}}\right|=e^v\le e^R.$$

Hence on the ball $B_R=\{(u,v):|u|,|v|\le R\}$, $f_1$ is Lipschitz with constant

$$L_1\left(R\right)={\lambda }_1+1+e^R.$$

For $f_2$:

$$\left|{\displaystyle \frac{\partial f_2}{\partial u}}\right|=\left|{\displaystyle \frac{2u}{1+u^2}}\right|\le 1,\left|{\displaystyle \frac{\partial f_2}{\partial v}}\right|={\lambda }_2.$$

Thus $f_2$ is Lipschitz on $B_R$ with constant

$$L_2\left(R\right)={\lambda }_2+1.$$

Therefore, for any fixed $R>0$, the functions satisfy a Lipschitz condition on $B_R$. In particular, for a local solution over a small time interval, we can choose sufficiently large R and then apply the standard Picard–Lindelöf argument for Caputo systems.

The variable orders ${\gamma }_i\left(s\right)$ are continuous and satisfy $0<{\gamma }_i\left(s\right)<1$. To avoid singularities, we assume that ${\gamma }_i\left(s\right)\ge \delta >0$ (for instance, $\delta =1/4$). Under this assumption, the Caputo derivatives are well defined and the usual properties hold. Hence, condition  (${\mathit{H}}_{\mathbf{2}}$)  is satisfied.

The initial conditions ${\mathcal{W}}_1\left(0\right)={\mathcal{W}}_2\left(0\right)=0$ are compatible with the Caputo derivative, since the derivative of a constant is zero. By Theorem 3, the system admits a unique local solution on some interval $[0,\delta ]$, with $\delta >0$.

This example illustrates the applicability of the theoretical framework, including nonlinear effects. More complex models may be considered in future work.

The first example (with coefficients $s/(1+s^2)$) admits global Lipschitz constants that vanish at $s=0$, leading to a contraction on the entire interval $[0,1]$ after a finite number of iterations. In contrast, the second example does not exhibit this vanishing property; hence, only local existence can be guaranteed without further analysis.

5. Conclusions

In this paper, we provided a critical assessment of definitions of variable-order fractional derivatives, identifying those consistent with the classical constant-order theory, and introduced a novel variable-order Mittag–Leffler function. For a coupled system of variable-order Caputo fractional differential equations with initial conditions, we established existence via Schauder’s fixed-point theorem and uniqueness via a global contraction mapping under suitable Lipschitz conditions.

The stability of the obtained solution follows from the same Lipschitz and contraction assumptions used in the existence and uniqueness analysis. In particular, no additional assumptions beyond those stated in the main results are required to guarantee stability.

For instance, the proposed system can model coupled anomalous diffusion processes in heterogeneous biological tissues, where different interacting species exhibit distinct memory effects due to spatial heterogeneity. It can also describe viscoelastic materials with multiple interacting components, where the stress–strain relationship depends on the history in a variable-order sense.

An illustrative example confirmed the applicability of the theoretical results. This work lays a foundation for further studies on variable-order fractional dynamical systems, including stability analysis and the development of numerical methods. Future work will focus on the stability analysis of such coupled variable-order systems and the construction of efficient numerical schemes, further extending the practical impact of the theoretical results.