Generalized Incommensurate Fractional Differential Systems: Commensurate and Incommensurate Weight Analyses, Existence-Uniqueness, HU Stability, and Neural Network Applications

Abstract

Generalized incommensurate fractional differential systems (GIFDSs) unify classical fractional frameworks via weight functions, capturing non-uniform multicomponent system dynamics. This paper fills a critical research gap by analyzing GIFDSs for both commensurate and incommensurate weight functions. For commensurate weights ($w_i\left(t\right)=w\left(t\right)$), classical IFDS equivalence is established via state transformation. Linear homogeneous mild solutions are derived using the incommensurate Mittag–Leffler function. Existence and uniqueness of nonlinear solutions are proved under continuity and Lipschitz assumptions. Hyers–Ulam stability is verified for linear non-homogeneous systems. For incommensurate weights (distinct $w_i\left(t\right))$, a novel framework is developed: by the integral bound lemma and Picard iteration, local existence (existence on $[a,t_1]$) is established, then it is extended to the full interval. The global uniqueness is obtained by Gronwall-type inequality via combined substitution. These results are applied to Hopfield Neural Networks, showing that one-layer HNNs with tanh or sigmoid activations admit unique mild solutions under GIFDS dynamics.

1. Introduction

Multicomponent dynamical systems, pervasive in natural phenomena from biological ecosystems to engineering circuits, necessitate mathematical frameworks that capture their intrinsic complexity. Fractional differential equations (FDEs) have emerged as a powerful tool for this purpose, as they inherently model the memory effects and non-local dynamics absent in integer-order systems [1,2]. A critical extension of FDEs is incommensurate fractional differential systems (IFDSs), where subsystems possess distinct rates of change ($0<{\beta }_i<1$). These systems more accurately reflect real-world non-uniformity than their commensurate counterparts (single fractional order) [3,4,5].

The historical development of IFDSs traces a trajectory from theoretical formulation to widespread application. Early works focused on distinguishing IFDSs from commensurate systems, with the term “multi-order” for non-uniform derivative orders introduced in [6] and IFDSs linked to pharmacokinetic multicompartmental models in [1].

However, the field remained nascent until the 2010s, when a surge in applications highlighted its practical value: Chaotic attractors in IFDSs were identified in [7], a breakthrough that expanded their relevance to nonlinear dynamics; synchronization studies for IFDSs were pioneered in [8], laying the groundwork for control theory applications. In epidemiology, IFDS-based COVID-19 models for uneven population transmission rates were developed in [9,10]; in energy systems, IFDSs were used for lithium-ion battery state-of-charge estimation in [11]; in neuroscience, IFDS-based adaptive exponential integrate-and-fire models simulated neuronal activity with disparate firing rates in [12,13,14,15]. Bounded-input bounded-output (BIBO) stability analysis is a key requirement for control applications. Its role in multi-agent system consensus is studied in [16,17].

Despite these advances, classical IFDS analysis lacks a framework incorporating weight functions, a critical limitation, as weight functions unify diverse fractional derivative frameworks (e.g., Hadamard and Katugampola) and enable flexible modeling of complex physical phenomena [18,19,20].

Generalized incommensurate fractional differential systems (GIFDSs) address this limitation by incorporating a weight function $w\left(t\right)$ into the fractional operator, extending classical IFDSs to a more general framework. Early work on generalized fractional operators focused on single-component systems or trivial weight functions (e.g., $w\left(t\right)=t$) [18,21,22], and Shiri et al. [23,24] established well-posedness for non-generalized IFDSs and delayed IFDSs in continuous function spaces. However, three critical gaps remain in the literature: (1) existing analysis of GIFDSs is restricted to commensurate weight functions (all components sharing $w\left(t\right)$), with no rigorous results for the more general incommensurate weight function case (distinct $w_i\left(t\right)$ for each component); (2) linear IFDSs lack explicit closed-form solutions due to the failure of fractional operator-matrix commutation [25,26] (${\overrightarrow{I}}^{\beta }A\overrightarrow{y}\ne A{\overrightarrow{I}}^{\beta }\overrightarrow{y}$), a barrier that has not been addressed for generalized linear homogeneous GIFDSs; (3) the practical applicability of GIFDSs to real-world engineering systems (e.g., neural networks) has not been formally established, despite the prevalence of fractional dynamics in neuroscience. Additionally, high-dimensional or time-delayed IFDSs lack efficient solvers [27,28], and their uncertainty handling relies on fragmented interval or fuzzy analysis [29,30,31], underscoring the need for rigorous study.

This paper addresses all three gaps with a comprehensive, unified analysis of GIFDSs for both commensurate and incommensurate weight functions, and establishes direct practical applications to neural networks. A key distinction in our work is the systematic treatment of incommensurate weight functions, a case that requires a fundamentally different analytical approach than commensurate weights, as the state transformation equivalence for commensurate weights no longer holds.

We first analyze GIFDSs with commensurate weight functions (all components share the same weight function $w_i(t)=w(t)$), where $w\left(t\right)$ satisfies strict regularity conditions ($C^1$ smoothness, strict monotonicity, bijectivity, and non-vanishing derivative [21]). For this case, we prove a state transformation $y_i\left(t\right)=x_i\left(w^{-1}\left(t\right)\right)$ that converts GIFDSs to classical IFDSs (Theorem 6), enabling the extension of well-developed IFDS theory to the generalized framework. We then derive an explicit mild solution for linear homogeneous GIFDSs using the incommensurate Mittag–Leffler function [32], prove existence and uniqueness of mild solutions for nonlinear GIFDSs under continuity and Lipschitz conditions, and verify Hyers–Ulam (HU) stability for linear non-homogeneous GIFDSs. HU stability is critical for numerical convergence analysis in practical applications [33].

We then consider the more general and challenging case of incommensurate weight functions, where each component is associated with a distinct $w_i\left(t\right)$. This anisotropic weight setting introduces further complexity to high-dimensional systems relative to the commensurate case, yet it also endows the generalized framework with substantial robustness: diverse weight functions can be tailored to individual components, and the system’s temporal complexity is modulated by these weight functions rather than fixed scalar values (the orders of fractional operators). This added flexibility not only enriches the modeling capacity for complex real-world dynamical systems but also opens the door to meaningful applications to neural networks, a promising direction for future exploration.

For the incommensurate weight functions, we develop a novel analytical framework that has not been previously established for GIFDSs: (1) we prove a key auxiliary lemma for weighted fractional integral bounds, which quantifies the growth of fractional integrals for incommensurate weights and forms the foundation of all subsequent existence results; (2) we establish local existence of mild solutions on a subinterval $[a,t_1]$ via Picard iteration, using uniform convergence of fractional integrals and the completeness of the continuous function space $C[a,t_1]$; (3) we extend this local result to the full interval $[a,b]$ through a step-by-step continuation argument, where we redefine the system on successive subintervals $[t_1,t_1+T]$, $[t_1+T,t_1+2T]$, etc., and apply the local existence result iteratively; (4) we prove global uniqueness of mild solutions by combining two substitutions to reduce the problem to a weakly singular Gronwall-type inequality for incommensurate orders [32], leveraging existing integral inequality results to show that any two solutions coincide on $[a,b]$.

Finally, we bridge our theoretical results to practical engineering applications by analyzing Hopfield Neural Networks (HNNs) governed by GIFDSs, a critical application in neuroscience and machine learning. We show that one-layer HNNs with standard activation functions (hyperbolic tangent (tanh) and sigmoid) satisfy our continuity and Lipschitz assumptions, and thus admit unique mild solutions by our existence and uniqueness results for incommensurate weight GIFDSs. This result formalizes the applicability of generalized fractional systems to neural network modeling, a key open problem in fractional neuroscience.

A $\nu$-dimensional GIFDS with commensurate weights, formally defined in component-wise form as follows:

$${}_a\mathfrak{D}_t^{{\alpha }_i,w\left(t\right)}{x}_i\left(t\right)\left(t\right)=f_i\left(t,x_1\left(t\right),\dots ,x_{\nu }\left(t\right)\right)$$

(1)

for $i=1,\cdots ,\nu ,$ where $t\in [a,b]$, and ${}_a\mathfrak{D}_t^{{\alpha }_i,w\left(t\right)}{x}_i\left(t\right)\left(t\right)$ denotes the generalized Caputo fractional derivative of order $0<\mathrm{Re}\left({\alpha }_i\right)<1$ for the i-th state variable $x_i\left(t\right)$ (with respect to weight $w\left(t\right)$). The generalized Caputo derivative is defined as follows:

$${}_a\mathfrak{D}_t^{\alpha ,w\left(t\right)}x\left(t\right)\left(s\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_a^s{\left(w\left(s\right)-w\left(\tau \right)\right)}^{-\alpha }{x}^{\prime }\left(\tau \right)d\tau ,s\in [a,b],$$

(2)

with $\mathrm{\Gamma }(·)$ representing the Gamma function [18]. The incommensurate weight case arises when the common weight w is replaced by component-dependent weights $w_i$.

Special cases of generalized fractional operators have long been recognized under various names and widely applied in diverse fields. For instance, the Hadamard fractional operators, introduced in 1892 [19], correspond to the generalized fractional derivative with $w\left(t\right)=\log \left(t\right)$. Additionally, the Katugampola fractional derivative, proposed in 2011 [20], is equivalent to the generalized fractional operator when substituting $w\left(t\right)=t^p$ ($p\ne 0$). This derivative has gained significant traction in both pure and applied research. Examples include applications in quantum mechanics [34], identification problems [35] and population growth [36].

The compact form of the GIFDS with incommensurate weights is

$${}_a\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha },\overrightarrow{w}\left(t\right)}\overrightarrow{x}\left(t\right)\left(t\right)=\overrightarrow{f}(t,\overrightarrow{x}\left(t\right)).$$

(3)

where $\overrightarrow{\alpha }={[{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\nu }]}^T\in {(0,1)}^{\nu }$ is vector of incommensurate fractional orders, $\overrightarrow{x}\left(t\right)={[x_1\left(t\right),x_2\left(t\right),\dots ,x_{\nu }\left(t\right)]}^T$ is state vector-valued function, mapping $[a,b]\to {\mathbb{R}}^{\nu }$, $\overrightarrow{w}\left(t\right)={[w_1\left(t\right),w_2\left(t\right),\dots ,w_{\nu }\left(t\right)]}^T$ is an incommensurate weight vector,

$$\overrightarrow{f}(t,\overrightarrow{x})={[f_1(t,\overrightarrow{x}),f_2(t,\overrightarrow{x}),\dots ,f_{\nu }(t,\overrightarrow{x})]}^T$$

is source vector-valued function, mapping $[a,b]\times {\mathbb{R}}^{\nu }\to {\mathbb{R}}^{\nu }$ and

$${}_a\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha },\overrightarrow{w}\left(t\right)}\overrightarrow{x}\left(t\right)\left(t\right)={\left[{}_a\mathfrak{D}_t^{{\alpha }_1,w\left(t\right)}{x}_1\left(t\right)\left(t\right),\dots ,{}_a\mathfrak{D}_t^{{\alpha }_{\nu },w\left(t\right)}{x}_{\nu }\left(t\right)\left(t\right)\right]}^T$$

(4)

is a vector of generalized Caputo derivatives.

This work makes six distinct, original contributions to the theory of generalized incommensurate fractional differential systems, addressing critical gaps in the existing literature:

• Commensurate Weight Equivalence: Prove a state transformation that converts GIFDSs with commensurate weights to classical IFDSs, unifying generalized and non-generalized frameworks and extending well-developed IFDS theory to GIFDSs.
• Explicit Linear Solution: Derive the first explicit mild solution for linear homogeneous GIFDSs with commensurate weights using the incommensurate Mittag–Leffler function, overcoming the operator-matrix commutation barrier for incommensurate orders.
• Nonlinear Well-Posedness: Prove existence and uniqueness of mild solutions for nonlinear GIFDSs with commensurate weights under continuity and Lipschitz conditions, leveraging the transformation equivalence to classical IFDSs.
• Hyers-Ulam Stability: Verify HU stability for linear non-homogeneous GIFDSs with commensurate weights, a result pivotal for numerical convergence analysis in practical applications.
• Incommensurate Weight Analysis: Develop the first rigorous analytical framework for GIFDSs with incommensurate weights, including a key integral bound lemma, local/full-interval existence via Picard iteration/continuation, and global uniqueness via Gronwall-type inequalities.
• Neural Network Application: Establish the first formal link between GIFDSs and Hopfield Neural Networks, proving that one-layer HNNs with tanh/sigmoid activation functions admit unique mild solutions.

The rest of this paper is structured as follows: Section 2 establishes preliminaries, including generalized fractional integrals/derivatives, weighted function spaces, and incommensurate operator algebra. Section 3 proves the equivalence between GIFDSs with commensurate weights and classical IFDSs via state transformation. Section 4 derives the explicit mild solution for linear homogeneous GIFDSs using the incommensurate Mittag–Leffler function. Section 5 establishes the existence and uniqueness of mild solutions for nonlinear GIFDSs with commensurate weights. Section 6 analyzes HU stability for linear non-homogeneous GIFDSs with commensurate weights. Section 7 presents our novel analysis of GIFDSs with incommensurate weights, including the auxiliary integral bound lemma, local/full-interval existence, global uniqueness, and HU stability. Section 8 demonstrates practical applicability to Hopfield Neural Networks.

2. Preliminaries

2.1. Necessary Condition on Weight Function w and the Definition of Generalized Operators

Throughout this paper, suppose $w:[a,b]\to \mathbb{R}$ is a weight function that satisfies the following conditions:

Cond. 1:

$w\in C^1[a,b]$;

Cond. 2:

w is strictly increasing;

Cond. 3:

w is one-to-one;

Cond. 4:

$w^{\prime }\left(x\right)\ne 0,$$\forall x\in [a,b].$

These conditions are minimal and standard for generalized fractional operators [21,22]; they guarantee a continuous, well-defined inverse $w^{-1}\left(t\right)$, a critical requirement for our subsequent analysis.

Remark 1.

Conditions 1–4 imply

Cond. 5:

$w^{-1}:\mathrm{Range}(w)\to [a,b]$ exists and $\mathrm{Range}\left(w\right)=w\left(\right[a,b\left]\right)\subset \left[w\right(a),w(b\left)\right].$

Cond. 6:

$w^{-1}$ is continuous by Continuous Inverse Theorem and hence $w\left(\right[a,b\left]\right)=\left[w\right(a),w(b\left)\right].$

Cond. 7:

${\left(w^{-1}\right)}^{\prime }$ exists by Cond. 4 and using Inverse Function Theorem for Differentiability.

Definition 1

([21,22]). Let $x\in C[a,b],$$a,b\in \mathbb{R}.$ The generalized fractional integral Riemann-Liouville (RL) integral operator ${\mathfrak{J}}^{\alpha ,w\left(t\right)}$ of order $\alpha \in \mathbb{C},\mathrm{Re}\left(\alpha \right)>0$ with respect to the weight function w is defined as

$$\begin{array}{cc}\hfill {}_a\mathfrak{J}_s^{\alpha ,w\left(t\right)}x\left(t\right)\left(s\right)& =\left({}_a\mathfrak{J}_s^{\alpha ,w\left(t\right)}x\left(t\right)\right)\left(s\right)\hfill \\ \hfill & ={\int }_a^s{(w\left(s\right)-w\left(\tau \right))}^{\alpha -1}{w}^{\prime }\left(\tau \right)x\left(\tau \right)d\tau ,s\in [a,b].\hfill \end{array}$$

(5)

Remark 2.

Due to the application of transforms and scaling to both input functions and variables, a modified notation is introduced for fractional integrals (and derivatives, subsequent to this definition) to mitigate ambiguity. Specifically, to distinguish the input variable of $x\left(t\right)$ (i.e., t) from the output variable of the integral operator (i.e., s, where $y\left(s\right)={}_a\mathfrak{J}_s^{\alpha ,w\left(t\right)}x\left(t\right)\left(s\right)$), a non-conventional two-parameter notation is adopted. We emphasize that in the expression $\left(t\right)\left(s\right)$, the first parenthetical term denotes the input variable, while the second specifies the output variable. When the input and output variables coincide, the standard notation is retained:

$${}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}x\left(t\right)\left(t\right)={}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}x\left(t\right).$$

Definition 2

([21]). The weighted continuous space $C_w[a,b]$, with respect to the weight function w, is defined by

$$C_w[a,b]=\{x:[a,b]\to \mathbb{R}|x\circ w^{-1}\in C[w\left(a\right),w\left(b\right)]\}$$

(6)

and the weighted absolutely continuous functions with respect to the weight function w is defined by

$${\mathrm{AC}}_w[a,b]=\{x:[a,b]\to \mathbb{R}|x\circ w^{-1}\in \mathrm{AC}[w\left(a\right),w\left(b\right)]\}.$$

(7)

They are equipped with the norm

$${\parallel x\parallel }_w=\parallel x\left(w^{-1}\right)\parallel =\underset{t\in \left[w\right(a),w(b\left)\right]}{\sup }x\left(w^{-1}\left(t\right)\right)=\parallel x\parallel .$$

Remark 3.

We note that $C_w[a,b]=C[a,b],$ but this may not hold for $\mathrm{AC}.$ Consider $w\left(t\right)=t^2$ and $x\left(t\right)=t.$ Then $x\notin {\mathrm{AC}}_w[0,1],$ since $x\circ w^{-1}\left(t\right)=\sqrt{t}.$ But $x\in \mathrm{AC}[0,1].$ We note that w satisfies Conditions 1–3. In the next theorem we provide the conditions on which $\mathrm{AC}[a,b]={\mathrm{AC}}_w[a,b].$

Theorem 1.

Let w be Lipschitz equivalent, i.e., there exists $L_1$ and $L_2$ such that for every $t_1,t_2\in [a,b]$ we have

$$L_1|t_1-t_2|\le \left|w\right(t_1)-w\left(t_2\right)|\le L_2|t_1-t_2|$$

(8)

Then, ${\mathrm{AC}}_w[a,b]=\mathrm{AC}[a,b].$

Proof. 

Let $x\in {\mathrm{AC}}_w[a,b],$ and $ϵ>0.$ Thus, $x\circ w^{-1}\in \mathrm{AC}[w\left(a\right),w\left(b\right)]$ there exist $\delta >0$ such that for every finite pairwise disjoint sub-intervals $({\widehat{t}}_i,{\widehat{s}}_i)\subset [w\left(a\right),w\left(b\right)]$ such that

$$\sum _{i=1}^n|{\widehat{s}}_i-{\widehat{t}}_i|<\delta$$

we have

$$\sum _{i=1}^n|x\circ w^{-1}({\widehat{s}}_i)-x\circ w^{-1}({\widehat{t}}_i)|<ϵ.$$

(9)

Let $(t_i,s_i)\subset (a,b)$ be finite pairwise disjoint sub-intervals such that

$$\sum _{i=1}^n|s_i-t_i|<\delta /L_2$$

Then,

$$\sum _{i=1}^n|w\left(s_i\right)-w\left(t_i\right)|\le L_2\sum _{i=1}^n|s_i-t_i|<\delta$$

Let ${\widehat{t}}_i=t_i$ and ${\widehat{s}}_i=s_i,$ Then from (9)

$$\sum _{i=1}^n|x\left(s_i\right)-x\left(t_i\right)|=\sum _{i=1}^n|x\circ w^{-1}({\widehat{s}}_i)-x\circ w^{-1}({\widehat{t}}_i)|<ϵ.$$

(10)

Since $t_i<s_i,$ and $ϵ$ was arbitrary, $x\in \mathrm{AC}[a,b],$ and ${\mathrm{AC}}_w[a,b]\subset \mathrm{AC}[a,b].$ Similarly, we can prove $\mathrm{AC}[a,b]\subset {\mathrm{AC}}_w[a,b]$ which completes the proof. □

Remark 4.

If w satisfies Conditions 1–4 and the following additional condition:

Cond. 8:

$w^{-1}\in C^1[w\left(a\right),w\left(b\right)],$

then, Equation (8) holds via the mean-value theorem.

Definition 3

([21]). Let $x\in \mathrm{AC}[a,b],a,b\in \mathbb{R}.$ Suppose w satisfies Conds. 1–4 and Cond. 8. The generalized fractional Caputo derivative operator ${}_a\mathfrak{D}_t^{\alpha ,w\left(t\right)}$ of order $0<\mathrm{Re}\left(\alpha \right)<1$ with respect to the weight function w is defined by (2).

The main properties of the generalized Caputo fractional derivative and RL integral are summarized in the following theorem.

Theorem 2

([21]). Suppose w satisfies Conds. 1–4. Then,

1. 

${}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}x\left(w\left(t\right)\right)\left(t\right)={}_{w\left(a\right)}\mathfrak{J}_t^{\alpha ,t}x\left(t\right)\left(w\left(t\right)\right),x\in C\left([a,b]\right),\mathrm{Re}\left(\alpha \right)>0;$

2. 

${}_a\mathfrak{D}_t^{\alpha ,w\left(t\right)}x\left(w\left(t\right)\right)\left(t\right)={}_{w\left(a\right)}\mathfrak{D}_t^{\alpha ,t}x\left(t\right)\left(w\left(t\right)\right),x\in C\left([a,b]\right),1>\mathrm{Re}\left(\alpha \right)>0;$

3. 

${}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}\left({(w\left(t\right)-w\left(a\right))}^{\beta }\right)\left(t\right)={\displaystyle \frac{\mathrm{\Gamma }(\beta +1)}{\mathrm{\Gamma }(\beta +\alpha +1)}}{(w\left(t\right)-w\left(a\right))}^{\beta +\alpha },\beta >0;$

4. 

${}_a\mathfrak{D}_t^{\alpha ,w\left(t\right)}\left({(w\left(t\right)-w\left(a\right))}^{\beta }\right)\left(t\right)={\displaystyle \frac{\mathrm{\Gamma }(\beta +1)}{\mathrm{\Gamma }(\beta -\alpha +1)}}{(w\left(t\right)-w\left(a\right))}^{\beta -\alpha },\beta >0;$

5. 

${}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}\left({}_a\mathfrak{D}_t^{\alpha ,w}x\left(t\right)\left(t\right)\right)\left(t\right)=x\left(t\right)-x\left(a\right),x\in \mathrm{AC}\left([a,b]\right),1>\mathrm{Re}\left(\alpha \right)>0.$

Remark 5.

For the case where $w\left(t\right)=t$, the operators ${}_a\mathfrak{J}_t^{\alpha ,w}$ and ${}_a\mathfrak{D}_t^{\alpha ,w}$ are equivalent to the classical Riemann-Liouville fractional integral and Caputo fractional derivative, respectively. Consequently, the following relationships hold:

$${}_a\mathfrak{D}_t^{\alpha ,t}={}_a\mathfrak{D}_t^{\alpha },{}_a\mathfrak{J}_t^{\alpha ,t}={}_a\mathfrak{J}_t^{\alpha }$$

(11)

Theorem 3

(Time Shift Properties). Let $a,b\in \mathbb{R}$ with $a<b$, $c\in \mathbb{R}$, and $\mathrm{Re}\left(\alpha \right)\in (0,1)$.

(1) 

If ${}_a\mathfrak{D}_t^{\alpha ,t}x\left(t\right)$ is well-defined for $t\in [a,b]$, then

$${}_a\mathfrak{D}_{t\pm c}^{\alpha }x\left(t\right)(t\pm c)={}_{a\mp c}\mathfrak{D}_t^{\alpha }x(t\pm c)\left(t\right),t\in [a\mp c,b\mp c].$$

(12)

(2) 

If ${}_a\mathfrak{J}_t^{\alpha ,t}x\left(t\right)$ is well-defined for $t\in [a,b]$, then

$${}_a\mathfrak{J}_{t\pm c}^{\alpha }x\left(t\right)(t\pm c)={}_{a\mp c}\mathfrak{J}_t^{\alpha }x(t\pm c)\left(t\right),t\in [a\mp c,b\mp c].$$

(13)

Proof. 

We prove assertion (1); the proof of assertion (2) follows similarly. By the definition of ${}_a\mathfrak{D}_t^{\alpha ,t}x\left(t\right)$, we have

$${}_a\mathfrak{D}_{t+c}^{\alpha }x\left(t\right)(t+c):={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_a^{t+c}{(t+c-\tau )}^{-\alpha }{x}^{\prime }\left(\tau \right)d\tau .$$

Substitute $\tau =z+c$ (which implies $d\tau =dz$). Substituting into the integral, we obtain

$${}_a\mathfrak{D}_{t+c}^{\alpha }x\left(t\right)(t+c)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_{a-c}^t{(t-z)}^{-\alpha }{x}^{\prime }(z+c)dz.$$

By the definition of ${}_{a-c}\mathfrak{D}_t^{\alpha }x(t+c)\left(t\right)$, the right-hand side of the above equation equals ${}_{a-c}\mathfrak{D}_t^{\alpha }x(t+c)\left(t\right)$. Thus, assertion (1) holds. □

Putting $c=a$ in Theorem 3, we obtain

$${}_a\mathfrak{D}_{t+a}^{\alpha }x\left(t\right)(t+a)={}_0\mathfrak{D}_t^{\alpha }x(t+a)\left(t\right),t\in [0,b-a],$$

and

$${}_a\mathfrak{J}_{t+a}^{\alpha }x\left(t\right)(t+a)={}_0\mathfrak{J}_t^{\alpha }x(t+a)\left(t\right),t\in [0,b-a],$$

which we frequently use in our analysis.

In our analysis, we also require the interchange of limit operations and fractional integration. This interchange is valid for uniformly convergent sequences in $C[a,b]$, as established by the following theorem.

Theorem 4.

Let ${\left\{x_i\right\}}_{i\in \mathbb{N}}\subset C[a,b]$ converge uniformly to $y\in C[a,b]$, and let the weight function w satisfy Conditions 1–4 and Condition 8. Then,

$$\underset{i\to \infty }{\lim }{}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}{x}_i\left(t\right)={}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}y\left(t\right),$$

(14)

where the convergence holds uniformly on $[a,b]$.

Proof. 

For all $t\in [a,b]$, we have

$$\begin{array}{cc}\hfill \left|{}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}{x}_i\left(t\right)-{}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}y\left(t\right)\right|& \le {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_a^t{\left(w\left(t\right)-w\left(\tau \right)\right)}^{\alpha -1}\left|x_i\left(\tau \right)-y\left(\tau \right)\right|w^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & \le {\displaystyle \frac{\parallel x_i{-y\parallel }_{\infty }}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_a^t{\left(w\left(t\right)-w\left(\tau \right)\right)}^{\alpha -1}{w}^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & \le \parallel x_i{-y\parallel }_{\infty }·{}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}\left[{\left(w\left(t\right)-w\left(a\right)\right)}^0\right]\left(t\right)\hfill \\ \hfill & \le \parallel x_i{-y\parallel }_{\infty }·{\displaystyle \frac{{\left(w\left(t\right)-w\left(a\right)\right)}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}.\hfill \end{array}$$

(15)

Since $w\left(t\right)\in C^1[a,b]$, w is bounded on $[a,b]$, and thus there exists a constant $M>0$ such that

$$\underset{t\in [a,b]}{\sup }\left|{}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}{x}_i\left(t\right)-{}_a\mathfrak{J}_t^{\alpha ,w\left(t\right)}y\left(t\right)\right|\le M{\parallel x_i-y\parallel }_{\infty }.$$

(16)

The right-hand side tends to zero as $i\to \infty$ by uniform convergence, which completes the proof. □

2.2. Algebra of Incommensurate Operations

It is known that the vector operator for the generalized fractional derivative (cf. Equation (4)) does not commute for incommensurate-order derivatives when dealing with the multiplication of a matrix by a vector-valued function, i.e.,

$${}_a\overrightarrow{\mathfrak{D}}_t^{{\alpha }_i,w\left(t\right)}(A\overrightarrow{x}\left(t\right))\left(t\right)\ne A{}_a\overrightarrow{\mathfrak{D}}_t^{{\alpha }_i,w\left(t\right)}\overrightarrow{x}\left(t\right)\left(t\right)$$

(17)

for a matrix A of dimension $\nu \ge 2$. Similarly, the vector of the generalized Riemann-Liouville (RL) integral

$${}_a\overrightarrow{\mathfrak{J}}_t^{\overrightarrow{\alpha },\overrightarrow{w}\left(t\right)}\overrightarrow{x}\left(t\right)\left(t\right)={\left[{}_a\mathfrak{J}_t^{{\alpha }_1,w\left(t\right)}{x}_1\left(t\right)\left(t\right),\dots ,{}_a\mathfrak{J}_t^{{\alpha }_{\nu },w\left(t\right)}{x}_{\nu }\left(t\right)\left(t\right)\right]}^T$$

(18)

fails to satisfy the commutativity property. This necessitates the introduction of a new algebra. In the present work, we do not delve into such an algebra and instead refer the reader to the relevant literature [2,23,32,33].

We recall Gronwall’s inequality for incommensurate weakly singular Volterra integral inequalities [32], which we use to establish the uniqueness result.

Theorem 5.

Assume $F\in {\mathbb{R}}_{+}^{\nu }$, $A\in {\mathbb{R}}_{+}^{\nu \times \nu }$, $\overrightarrow{\alpha }\in {(0,\infty )}^{\nu }$, and there exists a continuous non-negative vector-valued function $\overrightarrow{y}\in {\left(C_{+}[0,\mathcal{T}]\right)}^{\nu }$ such that

$$\overrightarrow{y}\left(t\right)\le F+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\overrightarrow{\alpha }\right)}}\circ {\int }_0^t{(t-s)}^{\overrightarrow{\alpha }-\mathbf{1}}\circ \left(A\overrightarrow{y}\left(s\right)\right)ds,$$

where the inequality ≤ is understood element-wise, and ∘ denotes the Hadamard (element-wise) product. Then,

$$\overrightarrow{y}\left(t\right)\le E_{\infty ,\overrightarrow{\alpha }}(A,t)F.$$

Proof. 

The result follows directly from Theorem 3.1 in [32]. □

3. Converting GIFDS into IFDS with Commensurate Weight

Generalized incommensurate fractional differential systems (GIFDSs) with identical commensurate weight functions represent a practically meaningful subclass, as they allow transformation into classical IFDS. The equivalence between classical IFDSs and GIFDSs with commensurate weights relies on three key foundations: (1) the strict monotonicity and smoothness of the weight function w(t) (Conditions 1–4), which guarantees a well-defined inverse $w^{-1}\left(t\right)$; (2) the fundamental composition property of generalized fractional operators (Theorem 2 Case 5); and (3) valid variable substitution in fractional integrals, which preserves the mild solution structure. This conversion simplifies the analysis of solution existence, uniqueness, and stability by leveraging well-established results for standard fractional differential equations. The following theorem formalizes this transformation.

Theorem 6.

Suppose $w_i=w$ for all $i=1,\dots ,\nu$ (commensurate weight function), where $w\in C^1([a,b],\mathbb{R})$ satisfying Conds. 1–4 and Cond. 8. Let $\overrightarrow{x}:[a,b]\to {\mathbb{R}}^{\nu }$ be a mild solution of the GIFDS (1) with initial conditions $\overrightarrow{x}\left(a\right)={\overrightarrow{x}}_a\in {\mathbb{R}}^{\nu }$. Then, $\overrightarrow{y}:[w\left(a\right),w\left(b\right)]\to {\mathbb{R}}^{\nu }$ defined by $y_i\left(t\right)=x_i\left(w^{-1}\left(t\right)\right)$ is the mild solution of the classical IFDS

$${}_{w\left(a\right)}\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha }}\overrightarrow{y}\left(t\right)\left(t\right)=\overrightarrow{f}(w^{-1}\left(t\right),\overrightarrow{y}\left(t\right)),t\in [w\left(a\right),w\left(b\right)],$$

(19)

with initial condition $\overrightarrow{y}\left(w\left(a\right)\right)={\overrightarrow{x}}_a.$ Conversely, if $\overrightarrow{y}$ is the mild solution of IFDS (19), then $\overrightarrow{x}$ with $x_i\left(t\right)=y_i\left(w\left(t\right)\right)$ is the mild solution of GIFDS (1).

Proof. 

First, we convert Equation (1) to an integral equation by applying ${}_a\mathfrak{I}_t^{{\alpha }_1,w_1}$ to both side of the equation, and we obtain the following:

$$\begin{array}{cc}\hfill x_i\left(t\right)-x_i\left(a\right)& ={}_a\mathfrak{I}_t^{{\alpha }_i,w\left(t\right)}\left(f_i(t,x_1\left(t\right),\dots ,x_{\nu }\left(t\right))\right)\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\int }_a^t{(w\left(t\right)-w\left(\tau \right))}^{{\alpha }_i-1}{f}_i(\tau ,x_1\left(\tau \right),\dots ,x_{\nu }\left(\tau \right))w^{\prime }\left(\tau \right)d\tau .\hfill \end{array}$$

(20)

The solution of the integral equation described by Equation (20) is recognized as a mild/weak solution of the original problem (1) under the stated conditions. By substituting $z=w\left(\tau \right)$ (where $\tau =w^{-1}\left(z\right)$ and $dz=w^{\prime }\left(z\right)d\tau$), we obtain the following:

$$x_i\left(t\right)-x_i\left(a\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{w\left(a\right)}^{w\left(t\right)}{(w_i\left(t\right)-z)}^{\alpha -1}{f}_i(w^{-1}\left(z\right),x_1\left(w^{-1}\left(z\right)\right),\dots ,x_{\nu }\left(w^{-1}\left(z\right)\right))dz.$$

(21)

We note that $w^{-1}\left(z\right)$ is well-defined by Conds. 1–4. Next, by changing the time variable by $t=w^{-1}\left(s\right),$ (with $s\in \left[w\right(a),w(b\left)\right]$), we derive the following:

$$x_i\left(w^{-1}\left(s\right)\right)-x_i\left(a\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{w\left(a\right)}^s{(s-z)}^{\alpha -1}{f}_i(w^{-1}\left(z\right),x_1\left(w^{-1}\left(z\right)\right),\dots ,x_{\nu }\left(w^{-1}\left(z\right)\right))dz.$$

(22)

Subsequently, using scaled-states variables $y_i\left(s\right)=x_i\left(w^{-1}\left(s\right)\right)$, we rewrite Equation (22) as follows:

$$y_i\left(s\right)-x_i\left(a\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{w\left(a\right)}^s{(s-z)}^{\alpha -1}{f}_i(w^{-1}\left(z\right),y_1\left(z\right),\dots ,y_{\nu }\left(z\right))dz.$$

(23)

Thus, the solution of Equation (23) is a mild solution of the following classical (non-general) IFDS, defined on $\left[w\right(a),w(b\left)\right]$ with initial condition $y_i\left(w\left(a\right)\right)=x_i\left(a\right):$

$$\begin{array}{cc}\hfill {}_{w\left(a\right)}\mathfrak{D}_t^{{\alpha }_1}{y}_1\left(t\right)\left(t\right)& =f_1(w^{-1}\left(t\right),y_1\left(t\right),\dots ,y_{\nu }\left(t\right)),\hfill \\ \hfill & ⋮\hfill \\ \hfill {}_{w\left(a\right)}\mathfrak{D}_t^{{\alpha }_{\nu }}{y}_{\nu }\left(t\right)\left(t\right)& =f_{\nu }(w^{-1}\left(t\right),y_1\left(t\right),\dots ,y_{\nu }\left(t\right)).\hfill \end{array}$$

(24)

This result extends to the reverse direction through the inverse of the above substitutions, with the proof proceeding similarly. □

This bidirectional mapping (GIFDS ⇔ classical IFDS) confirms the systems are mathematically equivalent, as the mild solution and fractional operator structure are fully preserved under the transformation.

4. Solution of Linear Homogeneous GIFDS with a Commensurate Weight Function

A linear homogeneous GIFDS

$${}_a\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha },\overrightarrow{w}\left(t\right)}\overrightarrow{x}\left(t\right)\left(t\right)=A\overrightarrow{x}\left(t\right),\overrightarrow{x}\left(a\right)={\overrightarrow{x}}_a,$$

(25)

where $A\in {\mathbb{R}}^{\nu \times \nu },$ also known as relaxation equation, has diverse applications. Relaxation refers to the return of a perturbed system to its equilibrium state. Fractional relaxation equations play a crucial role in modeling perturbed systems in material science [37]. For the case $w\left(t\right)=t$ (non-general IFDSs), the solution to this system was recently derived in [32] using a new type of Mittag–Leffler function termed the incommensurate Mittag–Leffler function.

Definition 4

(Incommensurate Mittag–Leffler function [32]). Let $\mathrm{Re}\left({\alpha }_j\right)>0$ for all $j=1,\dots ,\nu .$ The incommensurate Mittag–Leffler function of matrix $A=\left(a_{rj}\right)$ is defined by

$${\left(E_{\infty ,\overrightarrow{\alpha }}(A,t)\right)}_{rj}={\delta }_{rj}+{\displaystyle \frac{t^{{\alpha }_r}{a}_{rj}}{\mathrm{\Gamma }({\alpha }_r+1)}}+\sum _{i=2}^{\infty }\sum _{\begin{array}{c}{k}_1=1\\ \dots \\ k_{i-1}=1\end{array}}^{\nu }{\displaystyle \frac{t^{{\alpha }_r+{\sum }_{s=1}^{i-1}{\alpha }_{k_s}}{a}_{r{k}_1}{a}_{k_1{k}_2}\cdots a_{k_{i-1}j}}{\mathrm{\Gamma }({\alpha }_r+{\sum }_{s=1}^{i-1}{\alpha }_{k_s}+1)}}$$

(26)

where ${\delta }_{rj}$ is the Kronecker delta, with $r=1,\cdots ,\nu$ and $j=1,\cdots ,\nu .$

Remark 6.

When the orders are commensurate (i.e., ${\alpha }_j=\alpha$ for all j), this function reduces to the classical Mittag–Leffler function: $E_{\infty ,\overrightarrow{\alpha }}(A,t)=E_{\alpha }\left(A{t}^q\right).$

Theorem 7.

Let $A\in {\mathbb{R}}^{\nu \times \nu },$ and $\overrightarrow{\alpha }\in {(0,\infty )}^{\nu }.$ Then,

$$\overrightarrow{x}\left(t\right)=E_{\infty ,\overrightarrow{\alpha }}(A,t-a){\overrightarrow{x}}_a$$

(27)

is a continuous mild solution to System (25) with $w_i\left(t\right)=t$.

Proof. 

First, we translate the time scale by $a.$ For System (25), substituting $t\mapsto t+a$ gives

$${}_a\overrightarrow{\mathfrak{D}}_{t+a}^{\overrightarrow{\alpha }}\overrightarrow{x}\left(t\right)(t+a)=A\overrightarrow{x}(t+a)\overrightarrow{x}\left(a\right)={\overrightarrow{x}}_a,t\in [0,b-a].$$

(28)

Define a new state function $\overrightarrow{y}\left(t\right):=\overrightarrow{x}(t+a)$ for $t\in [0,b-a]$. Using the time-shift property of the fractional derivative:

$${}_a\overrightarrow{\mathfrak{D}}_{t+a}^{\overrightarrow{\alpha }}\overrightarrow{x}\left(t\right)(t+a)={}_0\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha }}\overrightarrow{y}\left(t\right)\left(t\right),t\in [0,b-a].$$

(29)

Thus, System (28) transforms to

$${}_0\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha }}\overrightarrow{y}\left(t\right)\left(t\right)=A\overrightarrow{y}\left(t\right),\overrightarrow{y}\left(0\right)={\overrightarrow{x}}_a,t\in [0,b-a].$$

(30)

By Theorem 2.4 in [32], the mild solution to (30) is

$$\overrightarrow{y}\left(t\right)=E_{\infty ,\overrightarrow{\alpha }}(A,t){\overrightarrow{x}}_a,t\in [0,b-a].$$

(31)

Substituting back $\overrightarrow{x}\left(t\right)=\overrightarrow{y}(t-a)$, we obtain

$$\overrightarrow{x}\left(t\right)=E_{\infty ,\overrightarrow{\alpha }}(A,t-a){\overrightarrow{x}}_a,t\in [a,b],$$

which is a continuous mild solution of Equation (25) with $w\left(t\right)=t$. □

Theorem 8.

Suppose $w_i=w$ for all $i=1,\dots ,\nu$ (commensurate weight function), where $w\in C^1([a,b],\mathbb{R})$ satisfies Conditions 1–4 and Condition 8. Then,

$$\overrightarrow{x}\left(t\right)=E_{\infty ,\overrightarrow{\alpha }}\left(A,w\left(t\right)-w\left(a\right)\right){\overrightarrow{x}}_a$$

(32)

is the mild solution of the IFDS (25).

Proof. 

According to Theorem 6, the mild solution of (25) with a commensurate weight is given by $\overrightarrow{x}\left(t\right)=\overrightarrow{y}\left(w\left(t\right)\right)$, where $\overrightarrow{y}$ satisfies

$${}_{w\left(a\right)}\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha }}\overrightarrow{y}\left(t\right)=A\overrightarrow{y}\left(t\right),t\in [w\left(a\right),w\left(b\right)]$$

(33)

From Theorem 7, we have $\overrightarrow{y}\left(t\right)=E_{\infty ,\overrightarrow{\alpha }}\left(A,t-w\left(a\right)\right){\overrightarrow{x}}_a$. Substituting $t\mapsto w\left(t\right)$ into this expression yields $\overrightarrow{x}\left(t\right)=\overrightarrow{y}\left(w\left(t\right)\right)=E_{\infty ,\overrightarrow{\alpha }}\left(A,w\left(t\right)-w\left(a\right)\right){\overrightarrow{x}}_a$, which completes the proof. □

5. Existence of a Unique Mild Solution for IFDS with a Commensurate Weight Function

The existence of a mild solution for generalized incommensurate-order FDEs with a commensurate weight function can be established based on the equivalence established in Theorem 6 and previous works [24,33].

Theorem 9

([24]). Let $f_i:[a,b]\times {\mathbb{R}}^{\nu }\to \mathbb{R}$ be continuous with respect to $t\in [a,b]$ and Lipschitz continuous with respect to $\overrightarrow{x}\in {\mathbb{R}}^{\nu }$; that is, there exist real constants $L_i>0$ such that

$$|f_i(t,\overrightarrow{x})-f_i(t,\overrightarrow{y})|\le L_i·\underset{i=1,\dots ,\nu }{\max }|x_i-y_i|,\forall \overrightarrow{x},\overrightarrow{y}\in {\mathbb{R}}^{\nu },t\in [a,b].$$

Then, for $\overrightarrow{\alpha }\in {(0,1)}^{\nu }$, System (3) with the initial condition $\overrightarrow{x}\left(a\right)={\overrightarrow{x}}_a$ and commensurate weight functions $w_i\left(t\right)=t$ (corresponding to classical non-generalized IFDEs) admits a unique mild solution in $C[a,b]$.

Proof. 

The proof again relies on translating the equations from the interval $[a,b]$ to $[0,b-a]$. Substituting $t\mapsto t+a$ into System (3) yields the following:

$${}_a\overrightarrow{\mathfrak{D}}_{t+a}^{\overrightarrow{\alpha },t}\overrightarrow{x}(t+a)=\overrightarrow{f}(t+a,\overrightarrow{x}(t+a)),t\in [0,b-a].$$

(34)

Define $\overrightarrow{y}\left(t\right):=\overrightarrow{x}(t+a)$ for $t\in [0,b-a]$. From (34), we derive:

$${}_0\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha }}\overrightarrow{y}\left(t\right)=\overrightarrow{f}(t+a,\overrightarrow{y}\left(t\right)),\overrightarrow{y}\left(0\right)={\overrightarrow{x}}_a.$$

(35)

By Theorem 12 in [24], Equation (35) has a unique mild solution $\overrightarrow{y}$ on $[0,b-a]$. Since (35) is equivalent to (3), the function $\overrightarrow{x}\left(t\right)=\overrightarrow{y}(t-a)$ is the unique mild solution of System (3) on $[a,b]$. □

Based on Theorem 6, we establish the existence and uniqueness of solutions for generalized IFDSs with a commensurate weight function.

Theorem 10.

Let $w_i=w$ for all $i=1,\dots ,\nu$ where $w\in C^1([a,b],\mathbb{R})$ satisfying Conds. 1–4 and 8. Suppose $f_i:[a,b]\times {\mathbb{R}}^{\nu }\to \mathbb{R}$ are continuous in $t\in [a,b]$ and Lipschitz continuous in $\overrightarrow{x}\in {\mathbb{R}}^{\nu }$. Then, System (3) with initial condition $\overrightarrow{x}\left(a\right)={\overrightarrow{x}}_a$ admits a unique mild solution in $C[a,b]$.

Proof. 

By Theorem 6, the generalized IFDS (3) is equivalent to non-generalized IFDS (19). Theorem 9 implies (19) has a unique mild solution $\overrightarrow{y}\in C[w\left(a\right),\left(b\right)].$ Thus, $\overrightarrow{x}\left(t\right)=\overrightarrow{y}\left(w\left(t\right)\right)$ is the unique mild solution of (3) on $[a,b].$ □

6. HU Stability

Recently, the HU stability of systems of equations, especially systems of differential equations and fractional differential equations, has received significant attention among researchers [38,39,40]. The authors of [33] noticed that HU stability may play an important role in the convergence analysis of numerical methods for IFDSs. We recall that the first concept of HU stability was introduced in another area of mathematics, specifically for the homomorphism problem for original groups [41,42]. However, this concept has been developed in many areas of mathematics, including systems of differential and integral equations. Ulam, Hyers, and Rassias have made substantial contributions to this topic [41,42,43]. This concept, also referred to as Hyers-Ulam-Rassias stability, takes its name from these foundational researchers.

Definition 5

(HU stability).We say the mild solution of the system (3) is HU stable if, for a given $ϵ>0$ and any solution $\overrightarrow{y}={[y_1,\dots ,y_{\nu }]}^T\in {\left(C[a,b]\right)}^{\nu }$ to the perturbed system $t\in [a,b]$

$$\underset{t\in [a,b]}{\sup }|y_i\left(t\right)-x_i\left(a\right)-{}_a\mathfrak{I}_t^{{\alpha }_i,w_i\left(t\right)}\left(f_i(t,y_1\left(t\right),\dots ,y_{\nu }\left(t\right))\right)\left(t\right)|\le ϵ$$

for $i=1,\dots ,\nu$, there exists a function $h:[0,\infty )\to \mathbb{R}$ (with $h\left(ϵ\right)\to 0$ as $ϵ\to 0$) such that

$$\underset{i=1,\dots ,\nu }{\max }\left(\underset{t\in [a,b]}{\sup }|x_i\left(t\right)-y_i\left(t\right)|\right)\le h\left(ϵ\right).$$

(36)

for any solution $\overrightarrow{x}\in {\left(C[a,b]\right)}^{\nu }$ of System (20).

The HU stability of non-homogeneous linear IFDSs (3)

$$\overrightarrow{f}(t,\overrightarrow{x})=A\overrightarrow{x}\left(t\right)+\overrightarrow{q}\left(t\right)$$

(37)

where $\overrightarrow{q}={[q_{,}\dots ,q_{\nu }]}^T\in {\left(C[a,b]\right)}^{\nu }$ ($a=0$) is studied in [33]; by a similar argument of the previous section, it can be extended to an arbitrary interval.

Theorem 11.

Let $\overrightarrow{q}\in {\left(C[a,b]\right)}^{\nu }$, $a,b\in \mathbb{R}$ with $a<b$, and $A=\left(a_{ij}\right)\in {\mathbb{R}}^{\nu \times \nu }$. Then the mild solution of the non-homogeneous linear IFDS

$${}_a\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha },t}\overrightarrow{x}\left(t\right)\left(t\right)=A\overrightarrow{x}\left(t\right)+\overrightarrow{q}\left(t\right)$$

(38)

is HU stable on ${\left(C[a,b]\right)}^{\nu }$.

Proof. 

Suppose $\overrightarrow{y}\in {\left(C[a,b]\right)}^{\nu }$ satisfies

$$\underset{t\in [a,b]}{\sup }\left|e_i\left(t\right)\right|\le ϵ,$$

where $e_i\left(t\right)$ is defined as

$$e_i\left(t\right):=y_i\left(t\right)-x_i\left(a\right)-{}_a\mathfrak{I}_t^{{\alpha }_i,t}\left({\left(A\overrightarrow{y}\right)}_i\left(t\right)\right)\left(t\right)+{}_a\mathfrak{I}_t^{{\alpha }_i,t}\left(q_i\left(t\right)\right)\left(t\right),\mathrm{for}t\in [a,b].$$

Substituting the variable transformation $t\mapsto t+a$ yields the following:

$$e_i(t+a):=y_i(t+a)-x_i\left(a\right)-{}_a\mathfrak{I}_{t+a}^{{\alpha }_i,t}\left({\left(A\overrightarrow{y}\right)}_i\left(t\right)\right)(t+a)+{}_a\mathfrak{I}_{t+a}^{{\alpha }_i,t}\left(q_i\left(t\right)\right)(t+a)$$

for $t\in [0,b-a]$. Define $\overrightarrow{z}\left(t\right):=\overrightarrow{y}(t+a)$ for $t\in [0,b-a]$; substituting this into the expression for $e_i(t+a)$ gives the following:

$$e_i(t+a):=z_i\left(t\right)-x_i\left(a\right)-{}_0\mathfrak{I}_t^{{\alpha }_i,t}\left({\left(A\overrightarrow{z}\right)}_i\left(t\right)\right)\left(t\right)+{}_0\mathfrak{I}_t^{{\alpha }_i,t}\left(q_i(t+a)\right)\left(t\right)$$

for $t\in [0,b-a]$. It follows that

$$\underset{t\in [0,b-a]}{\sup }\left|z_i\left(t\right)-x_i\left(a\right)-{}_0\mathfrak{I}_t^{{\alpha }_i,t}\left({\left(A\overrightarrow{z}\right)}_i\left(t\right)\right)\left(t\right)+{}_0\mathfrak{I}_t^{{\alpha }_i,t}\left(q_i(t+a)\right)\left(t\right)\right|\le ϵ.$$

(39)

Since $\overrightarrow{z}\in {\left(C[0,b-a]\right)}^{\nu }$, by Corollary 6.4 in [33], there exist $\overrightarrow{g}\in {\left(C[0,b-a]\right)}^{\nu }$ and a function $h\left(ϵ\right)$ (with $h\left(ϵ\right)\to 0$ as $ϵ\to 0$) such that

$$g_i\left(t\right)-x_i\left(a\right)-{}_0\mathfrak{I}_t^{{\alpha }_i,t}\left({\left(A\overrightarrow{g}\right)}_i\left(t\right)\right)\left(t\right)+{}_0\mathfrak{I}_t^{{\alpha }_i,t}\left(q_i(t+a)\right)\left(t\right)=0$$

(40)

for $i=1,2,\dots ,\nu$ and $t\in [0,b-a]$, and

$$\underset{i=1,\dots ,\nu }{\max }\left(\underset{t\in [0,b-a]}{\sup }\left|g_i\left(t\right)-z_i\left(t\right)\right|\right)\le h\left(ϵ\right).$$

(41)

Applying the variable transformation $t\mapsto t-a$ to Equation (40) gives the following:

$$g_i(t-a)-x_i\left(a\right)-{}_0\mathfrak{I}_{t-a}^{{\alpha }_i,t}\left({\left(A\overrightarrow{g}\right)}_i\left(t\right)\right)(t-a)+{}_0\mathfrak{I}_{t-a}^{{\alpha }_i,t}\left(q_i\left(t\right)\right)(t-a)=0$$

(42)

for $t\in [a,b]$. Define $\overrightarrow{x}\left(t\right):=\overrightarrow{g}(t-a)$ for $t\in [a,b]$; substituting this into Equation (42) yields:

$$x_i\left(t\right)-x_i\left(a\right)-{}_a\mathfrak{I}_t^{{\alpha }_i,t}\left({\left(A\overrightarrow{x}\right)}_i\left(t\right)\right)\left(t\right)+{}_a\mathfrak{I}_t^{{\alpha }_i,t}\left(q_i\left(t\right)\right)\left(t\right)=0$$

(43)

for $t\in [a,b]$. Thus, $\overrightarrow{x}$ is a mild solution of the IFDS (38).

From Equation (41) and the definition of $\overrightarrow{x}\left(t\right)$ (with $\overrightarrow{z}\left(t\right)=\overrightarrow{y}(t+a)$), we have the following:

$$\underset{i=1,\dots ,\nu }{\max }\left(\underset{t\in [a,b]}{\sup }\left|x_i\left(t\right)-y_i\left(t\right)\right|\right)\le h\left(ϵ\right).$$

(44)

Therefore, the mild solution of the IFDS (38) is HU stable. □

Theorem 12.

Let $w_i=w$ for all $i=1,\dots ,\nu$ where $w\in C^1([a,b],\mathbb{R})$ satisfies Conds. 1–4 and Cond. 8. Let $\overrightarrow{q}\in {\left(C[a,b]\right)}^{\nu }$, $a,b\in \mathbb{R}$ with $a<b$, and $A=\left(a_{ij}\right)\in {\mathbb{R}}^{\nu \times \nu }$. Then, the mild solution of the non-homogeneous linear generalized IFDS

$${}_a\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha },\overrightarrow{w}}\overrightarrow{x}\left(t\right)\left(t\right)=A\overrightarrow{x}\left(t\right)+\overrightarrow{q}\left(t\right)$$

(45)

is HU stable on ${\left(C[a,b]\right)}^{\nu }$.

Proof. 

Suppose $\overrightarrow{y}\in {\left(C[a,b]\right)}^{\nu }$ satisfies

$$\underset{t\in [a,b]}{\sup }\left|e_i\left(t\right)\right|\le ϵ,\mathrm{for}\mathrm{all}i=1,\cdots ,\nu ,$$

where $e_i\left(t\right)$ is defined as

$$e_i\left(t\right):=y_i\left(t\right)-x_i\left(a\right)-{}_a\mathfrak{I}_t^{{\alpha }_i,w}\left({\left(A\overrightarrow{y}\right)}_i\left(t\right)\right)\left(t\right)+{}_a\mathfrak{I}_t^{{\alpha }_i,w}\left(q_i\left(t\right)\right)\left(t\right),\mathrm{for}t\in [a,b].$$

Substituting $t=w^{-1}\left(s\right)$ for $s\in \left[w\right(a),w(b\left)\right]$, we rewrite $e_i\left(t\right)$ as follows:

$$e_i\left(w^{-1}\left(s\right)\right):=y_i\left(w^{-1}\left(s\right)\right)-x_i\left(a\right)-{}_a\mathfrak{I}_t^{{\alpha }_i,w}\left({\left(A\overrightarrow{y}\right)}_i\left(t\right)\right)\left(w^{-1}\left(s\right)\right)+{}_a\mathfrak{I}_t^{{\alpha }_i,w}\left(q_i\left(t\right)\right)\left(w^{-1}\left(s\right)\right),$$

Recall the definition of generalized RL integral:

$$\begin{array}{cc}\hfill {}_a\mathfrak{I}_t^{{\alpha }_i,w}\left(q_i\left(t\right)\right)\left(w^{-1}\left(s\right)\right)=& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_a^{w^{-1}\left(s\right)}{(w\left(w^{-1}\left(s\right)\right)-w\left(\tau \right))}^{\alpha -1}{w}^{\prime }\left(\tau \right)q_i\left(\tau \right)d\tau \hfill \\ \hfill =& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{w\left(a\right)}^s{(s-z)}^{\alpha -1}{q}_i\left(w^{-1}\left(z\right)\right)dz\hfill \\ \hfill =& {}_{w\left(a\right)}\mathfrak{I}_t^{{\alpha }_i,t}\left(q_i\left(w^{-1}\left(t\right)\right)\right)\left(s\right).\hfill \end{array}$$

(46)

Thus,

$$e_i\left(w^{-1}\left(s\right)\right):=y_i\left(w^{-1}\left(s\right)\right)-x_i\left(a\right)-{}_{w\left(a\right)}\mathfrak{I}_t^{{\alpha }_i,t}\left({\left(A\overrightarrow{y}\right)}_i\left(w^{-1}\left(t\right)\right)\right)\left(s\right)+{}_{w\left(a\right)}\mathfrak{I}_t^{{\alpha }_i,t}\left(q_i\left(w^{-1}\left(t\right)\right)\right)\left(s\right).$$

Introducing $z_i\left(t\right)=y_i\left(w^{-1}\left(t\right)\right)$ for $s\in \left[w\right(a),w(b\left)\right]$, we obtain the following:

$$e_i\left(w^{-1}\left(s\right)\right):=z_i\left(s\right)-x_i\left(a\right)-{}_{w\left(a\right)}\mathfrak{I}_t^{{\alpha }_i,t}\left({\left(A\overrightarrow{z}\right)}_i\left(t\right)\right)\left(s\right)+{}_{w\left(a\right)}\mathfrak{I}_t^{{\alpha }_i,t}\left(q_i\left(w^{-1}\left(t\right)\right)\right)\left(s\right).$$

By Theorem 11, there exists $h:[0,\infty )\to \mathbb{R}$ with $h\left(ϵ\right)\to 0$ as $ϵ\to 0$ such that for any solution $\overrightarrow{g}={[g_1,\dots ,g_{\nu }]}^T\in {\left(C[w\left(a\right),w\left(b\right)]\right)}^{\nu }$ of the system

$$g_i\left(s\right)-x_i\left(a\right)-{}_{w\left(a\right)}\mathfrak{I}_s^{{\alpha }_i,t}\left({\left(A\overrightarrow{g}\right)}_i\left(t\right)\right)\left(s\right)+{}_{w\left(a\right)}\mathfrak{I}_s^{{\alpha }_i,t}\left(q_i\left(w^{-1}\left(t\right)\right)\right)\left(s\right)=0,$$

(47)

for $i=1,\dots ,\mu ,s\in \left[w\right(a),w(b\left)\right],$ (which is unique) we have the following:

$$\underset{i=1,\dots ,\nu }{\max }\left(\underset{s\in \left[w\right(a),w(b\left)\right]}{\sup }\left|g_i\left(s\right)-z_i\left(s\right)\right|\right)\le h\left(ϵ\right).$$

(48)

Substituting $s=w\left(t\right)$ gives

$$\underset{i=1,\dots ,\nu }{\max }\left(\underset{t\in [a,b]}{\sup }\left|g_i\left(w\left(t\right)\right)-z_i\left(w\left(t\right)\right)\right|\right)\le h\left(ϵ\right).$$

(49)

Define $\overrightarrow{x}\left(t\right)=\overrightarrow{g}\left(w\left(t\right)\right),$ for $t\in [a,b]$. Since $z_i\left(w\left(t\right)\right)=y_i\left(t\right)$, this simplifies to the following:

$$\underset{i=1,\dots ,\nu }{\max }\left(\underset{t\in [a,b]}{\sup }\left|x_i\left(t\right)-y_i\left(t\right)\right|\right)\le h\left(ϵ\right).$$

(50)

It remains to verify $\overrightarrow{z}$ is a mild solution of generalized IFDS (45) (which is unique). Substituting $s=w\left(t\right)$ into (47), we get

$$g_i\left(w\left(t\right)\right)-x_i\left(a\right)-{}_{w\left(a\right)}\mathfrak{I}_t^{{\alpha }_i,t}\left({\left(A\overrightarrow{g}\right)}_i\left(t\right)\right)\left(w\left(t\right)\right)+{}_{w\left(a\right)}\mathfrak{I}_t^{{\alpha }_i,t}\left(q_i\left(w^{-1}\left(t\right)\right)\right)\left(w\left(t\right)\right)=0.$$

(51)

By property (1) of Theorem (2):

$$g_i\left(w\left(t\right)\right)-x_i\left(a\right)-{}_a\mathfrak{I}_t^{{\alpha }_i,w}\left({\left(A\overrightarrow{g}\right)}_i\left(w\left(t\right)\right)\right)\left(t\right)+{}_a\mathfrak{I}_t^{{\alpha }_i,w}\left(q_i\left(t\right)\right)\left(t\right)=0.$$

(52)

Substituting back $\overrightarrow{z}\left(t\right)=\overrightarrow{g}\left(w\left(t\right)\right)$, gives the following:

$$z_i\left(t\right)-x_i\left(a\right)-{}_a\mathfrak{I}_t^{{\alpha }_i,w}\left({\left(A\overrightarrow{z}\right)}_i\left(t\right)\right)\left(t\right)+{}_a\mathfrak{I}_t^{{\alpha }_i,w}\left(q_i\left(t\right)\right)\left(t\right)=0,$$

(53)

confirming $\overrightarrow{z}$ is a mild solution of generalized IFDS (45). It completes the proof. □

7. Further Improvement of Existence Analysis for Incommensurate Weight Functions

For the general case involving incommensurate weight functions, a distinct analytical approach is required. Assume that each $w_i$ satisfies Conditions 1–4 and 8. Using Property 5 of Theorem 2, Equation (3) can be rewritten as

$$\overrightarrow{x}\left(t\right)-\overrightarrow{x}\left(a\right)={}_a\mathfrak{J}_t^{\overrightarrow{\alpha },\overrightarrow{w}\left(t\right)}\overrightarrow{f}\left(t,\overrightarrow{x}\left(t\right)\right).$$

(54)

We impose the following assumptions on $f_i$ for $i=1,\dots ,\nu$:

H(a) 

$f_i$ is continuous with respect to all its arguments.

H(b) 

$f_i$ satisfies a local Lipschitz condition with uniform Lipschitz constant L, i.e.,

$$\left|f_i\left(t,\overrightarrow{x}\left(t\right)\right)-f_i\left(t,\overrightarrow{y}\left(t\right)\right)\right|\le L\left|x_i\left(t\right)-y_i\left(t\right)\right|,t\in [a,b].$$

(55)

Assumptions H(a) and H(b) are necessary and mild: continuity well-defines the fractional integral of $f_i$, allowing limit interchange inside $f_i$, with local Lipschitz preventing unbounded growth and ensuring Picard convergence [23,24].

The following lemma is needed for the proof of Theorem 13 and 14.

Lemma 1.

Let ${\alpha }_i\in (0,\infty )$, $\beta >0$, and ${\alpha }_m=\min \{{\alpha }_i:i=1,\dots ,\nu \}$. Define

$$M=\underset{t\in [a,b],i=1,\dots ,\nu }{\sup }\left|w_i^{\prime }\left(t\right)\right|$$

and let $t_1=a+{\displaystyle \frac{1}{M}}$. Then for all $\tau \in [0,b-a-t_1]$ and all $t\in [a,t_1]$,

$$\left|{}_a\mathfrak{J}_t^{{\alpha }_i,w_i(t+\tau )}\left({(w_i(t+\tau )-w_i(a+\tau ))}^{\beta }\right)\left(t\right)\right|\le C_i{\left(w_i(t+\tau )-w_i(a+\tau )\right)}^{{\alpha }_m+\beta },$$

(56)

where

$$C_i={\displaystyle \frac{\mathrm{\Gamma }\left({\alpha }_m\right)\mathrm{\Gamma }(\beta +1)}{\mathrm{\Gamma }\left({\alpha }_i\right)\mathrm{\Gamma }(\beta +{\alpha }_m+1)}}.$$

Proof. 

Since $w_i\in C^1[a,b]$ and is strictly increasing, the mean value theorem gives

$$w_i(t+\tau )-w_i(a+\tau )=w_i^{\prime }\left({\eta }_i\right)(t-a),{\eta }_i\in (a+\tau ,t+\tau ),t\ge a.$$

By the assumptions of the lemma, for all $t\in [a,t_1]$ and all $i=1,\dots ,\nu$,

$$0\le w_i(t+\tau )-w_i(a+\tau )\le {\displaystyle \frac{|w_i^{\prime }\left({\eta }_i\right)|}{M}}\le 1.$$

The remainder of the proof follows directly from Case (3) of Theorem 2, in a similar manner to Lemma 3.3 in [33]. □

7.1. Existence Analysis

We are now in a position to establish an existence result on the interval $[a,t_1]$. For generality and to support subsequent analysis, we prove the result for the following extended version of (54):

$$\overrightarrow{x}\left(t\right)=\overrightarrow{g}\left(t\right)+{}_a\mathfrak{J}_t^{\overrightarrow{\alpha },\overrightarrow{w}\left(t\right)}\overrightarrow{f}\left(t,\overrightarrow{x}\left(t\right)\right),$$

(57)

where $\overrightarrow{g}:[a,b]\to {\mathbb{R}}^{\nu }$ is a continuous vector-valued function. Equation (54) is recovered from (57) by setting $\overrightarrow{g}\left(t\right)=\overrightarrow{x}\left(a\right)$.

Theorem 13.

Let $g_i\in C[a,b]$, and let $f_i:[a,b]\times {\mathbb{R}}^{\nu }\to \mathbb{R}$ satisfy assumptions H(a) and H(b). Let $w_i:\mathbb{R}\to \mathbb{R}$ be incommensurate weight functions satisfying Conditions 1–4, with ${\alpha }_i>0$ for $i=1,\dots ,\nu$. Assume that $t_1$ is defined as in Lemma 1. Then system (57) (and hence also system (54)) admits a solution in ${\left(C[a,t_1]\right)}^{\nu }$.

Proof. 

From (57), we define the Picard iteration

$$x_i^{\left[k\right]}\left(t\right)=g_i\left(t\right)+{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}{f}_i\left(t,{\overrightarrow{x}}^{[k-1]}\left(t\right)\right),k\in \mathbb{N},$$

(58)

starting from the initial function

$$x_i^{\left[0\right]}\left(t\right)=g_i\left(t\right).$$

For the first step, we obtain from Lemma 1

$$\begin{array}{cc}\hfill \left|x_i^{\left[1\right]}\left(t\right)-x_i^{\left[0\right]}\left(t\right)\right|& =\left|{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}\left(|f_i(t,{\overrightarrow{x}}^{\left[0\right]}\left(t\right))|\right)\left(t\right)\right|\hfill \\ \hfill & \le M{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}\left({(w_i\left(t\right)-w_i\left(a\right))}^0\right)\left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{M\mathrm{\Gamma }\left({\alpha }_m\right)\mathrm{\Gamma }\left(1\right)}{\mathrm{\Gamma }\left({\alpha }_i\right)\mathrm{\Gamma }({\alpha }_m+1)}}{\left(w_i\left(t\right)-w_i\left(a\right)\right)}^{{\alpha }_m},t\in [a,t_1],\hfill \end{array}$$

(59)

where $M>0$ is a uniform bound for $|f_i(t,\overrightarrow{g}\left(t\right))|$, which exists by Assumption H(a) and continuity of $\overrightarrow{g}$.

Now using (59) and Assumption H(b), we get

$$\begin{array}{cc}\hfill |x_i^{\left[2\right]}\left(t\right)-x_i^{\left[1\right]}\left(t\right)|& =\left|{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}(|f_i(t,{\overrightarrow{x}}^{\left[1\right]}\left(t\right))-f_i(t,{\overrightarrow{x}}^{\left[0\right]}\left(t\right))|)\left(t\right)\right|\hfill \\ \hfill & \le L{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}|x_i^{\left[1\right]}\left(t\right)-x_i^{\left[0\right]}\left(t\right)|\left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{M\mathrm{\Gamma }\left({\alpha }_m\right)L}{\mathrm{\Gamma }\left({\alpha }_i\right)\mathrm{\Gamma }(2{\alpha }_m+1)}}{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}\left({(w_i\left(t\right)-w_i\left(a\right))}^{{\alpha }_m}\right)\left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{M\mathrm{\Gamma }{\left({\alpha }_m\right)}^{2}L}{\mathrm{\Gamma }{\left({\alpha }_i\right)}^2\mathrm{\Gamma }(2{\alpha }_m+1)}}{\left(w_i\left(t\right)-w_i\left(a\right)\right)}^{2{\alpha }_m}.\hfill \end{array}$$

(60)

By induction, we arrive at

$$\underset{t\in [a,t_1]}{\sup }\left|x_i^{\left[k\right]}\left(t\right)-x_i^{[k-1]}\left(t\right)\right|\le \underset{t\in [a,t_1]}{\sup }{\displaystyle \frac{M\mathrm{\Gamma }{\left({\alpha }_m\right)}^k{L}^{k-1}}{\mathrm{\Gamma }{\left({\alpha }_i\right)}^k\mathrm{\Gamma }(k{\alpha }_m+1)}}{\left(w_i\left(t\right)-w_i\left(a\right)\right)}^{k{\alpha }_m}.$$

(61)

Hence, for any $m>n$,

$$\begin{array}{cc}\hfill \underset{t\in [a,t_1]}{\sup }\left|x_i^{\left[m\right]}\left(t\right)-x_i^{\left[n\right]}\left(t\right)\right|& \le \underset{t\in [a,t_1]}{\sup }\sum _{k=n+1}^m\left|x_i^{\left[k\right]}\left(t\right)-x_i^{[k-1]}\left(t\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{M}{L}}\sum _{k=n+1}^m{\displaystyle \frac{{\mathsf{\Omega }}_i^k}{\mathrm{\Gamma }(k{\alpha }_m+1)}},\hfill \end{array}$$

(62)

where

$${\mathsf{\Omega }}_i=\underset{t\in [a,t_1]}{\sup }{\displaystyle \frac{L\mathrm{\Gamma }\left({\alpha }_m\right){\left(w_i\left(t\right)-w_i\left(a\right)\right)}^{{\alpha }_m}}{\mathrm{\Gamma }\left({\alpha }_i\right)}}.$$

The right-hand side of (62) is a tail of the Mittag–Leffler function $E_{{\alpha }_m}\left({\mathsf{\Omega }}_i\right)$, which is an entire function [44]. It is known that the Mittag–Leffler function converges uniformly on the entire complex plane for fractional orders with positive real part. Therefore, each sequence ${\left\{x_i^{\left[m\right]}\right\}}_{m=1}^{\infty }$ is a Cauchy sequence in $C[a,t_1]$ with respect to the supremum norm. Compactness of the continuous function space $C[a,t_1]$ Guarantees its uniform convergence to continuous limit function $y_i$, i.e.,

$$\underset{m\to \infty }{\lim }{x}_i^{\left[m\right]}=y_i\mathrm{uniformly}\mathrm{on}[a,t_1].$$

Thus the assumptions of Theorem 4 are satisfied, so we may interchange the limit and the generalized fractional integral operator:

$$\begin{array}{cc}\hfill \underset{k\to \infty }{\lim }{x}_i^{\left[k\right]}\left(t\right)& =\underset{k\to \infty }{\lim }{g}_i\left(t\right)+\underset{k\to \infty }{\lim }{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}{f}_i(t,{\overrightarrow{x}}^{[k-1]}\left(t\right))\hfill \\ & =g_i\left(t\right)+{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}\underset{k\to \infty }{\lim }{f}_i(t,{\overrightarrow{x}}^{[k-1]}\left(t\right)).\hfill \end{array}$$

By continuity of $f_i$, the limit also commutes with $f_i$, so

$$\begin{array}{cc}\hfill y_i\left(t\right)& =g_i\left(t\right)+{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}{f}_i(t,\underset{k\to \infty }{\lim }{\overrightarrow{x}}^{[k-1]}\left(t\right))\hfill \\ & =g_i\left(t\right)+{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}{f}_i(t,\overrightarrow{y}\left(t\right)).\hfill \end{array}$$

In vector form,

$$\overrightarrow{y}\left(t\right)=\overrightarrow{g}\left(t\right)+{}_a\mathfrak{J}_t^{\overrightarrow{\alpha },\overrightarrow{w}\left(t\right)}\overrightarrow{f}(t,\overrightarrow{y}\left(t\right)),t\in [a,t_1],$$

which shows that $\overrightarrow{y}$ is a solution of system (57). □

Remark 7.

The local existence interval $[a,t_1]$ with $t_1=a+1/M$ is explicitly dependent on the incommensurate weight functions $w_i\left(t\right)$, with M governing the interval length and tying the result to Conds. 1–4 of $w_i\left(t\right)$. This weight-dependent scaling of $t_1$ is absent in the global existence result: in Theorem (14), we prove that Conds. 1–4 alone for $w_i\left(t\right)$ suffice to extend the local solution to the full interval $[a,b]$ and guarantee a continuous mild solution on the entire domain.

Theorem 14.

Let $f_i:[a,b]\times {\mathbb{R}}^{\nu }\to \mathbb{R}$ satisfy assumptions H(a) and H(b), let $w_i:\mathbb{R}\to \mathbb{R}$ be incommensurate weight functions satisfying Conditions 1–4, and let ${\alpha }_i>0$ for $i=1,\dots ,\nu$. Then system (54) has a solution in ${\left(C[a,b]\right)}^{\nu }$.

Proof. 

We have already shown that there exists $t_1>a$, depending only on $w_i$, such that system (57) (and hence also system (54)) admits a continuous solution on $[a,t_1]$. We now extend this result step-by-step to the whole interval $[a,b]$ by showing that the same method applies to successive subintervals $[t_1,t_1+T]$, $[t_1+T,t_1+2T]$, etc., where $T=t_1-a$. We only present the extension to $[t_1,t_1+T]$; the remaining steps are analogous.

Let $\overrightarrow{y}\in {\left(C[a,t_1]\right)}^{\nu }$ be a solution of system (54) on $[a,t_1]$. From (54), for $t\in [t_1,t_1+T]$ we write

$$\begin{array}{cc}\hfill x_i\left(t\right)& =x_i\left(a\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\int }_a^{t_1}{\left(w_i\left(t\right)-w_i\left(\tau \right)\right)}^{{\alpha }_i-1}{f}_i\left(\tau ,\overrightarrow{y}\left(\tau \right)\right)w_i^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\int }_{t_1}^t{\left(w_i\left(t\right)-w_i\left(\tau \right)\right)}^{{\alpha }_i-1}{f}_i\left(\tau ,\overrightarrow{x}\left(\tau \right)\right)w_i^{\prime }\left(\tau \right)d\tau .\hfill \end{array}$$

(63)

Define

$$h_i\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\int }_a^{t_1}{\left(w_i\left(t\right)-w_i\left(\tau \right)\right)}^{{\alpha }_i-1}{f}_i\left(\tau ,\overrightarrow{y}\left(\tau \right)\right)w_i^{\prime }\left(\tau \right)d\tau .$$

By our assumptions, $h_i\in C[t_1,t_1+T]$ and is independent of the unknown $\overrightarrow{x}$. Substituting $t\mapsto t+T$, we obtain

$$\begin{array}{cc}\hfill x_i(t+T)& =h_i(t+T)\hfill \\ & +{\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\int }_{t_1}^{t+T}{\left(w_i(t+T)-w_i\left(\tau \right)\right)}^{{\alpha }_i-1}{f}_i(\tau ,\overrightarrow{x}\left(\tau \right))w_i^{\prime }\left(\tau \right)d\tau ,t\in [a,t_1].\hfill \end{array}$$

Now introduce the change of variable $\tau =z+T$, so that $d\tau =dz$. Then

$$\begin{array}{cc}\hfill x_i(t+T)& =h_i(t+T)\hfill \\ & +{\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\int }_a^t{\left(w_i(t+T)-w_i(z+T)\right)}^{{\alpha }_i-1}{f}_i(z+T,\overrightarrow{x}(z+T))w_i^{\prime }(z+T)dz.\hfill \end{array}$$

Set

$${\tilde{x}}_i\left(t\right)=x_i(t+T),{\tilde{g}}_i\left(t\right)=h_i(t+T),{\tilde{w}}_i\left(t\right)=w_i(t+T),{\tilde{f}}_i(z,\overrightarrow{\tilde{x}})=f_i(z+T,\overrightarrow{\tilde{x}}).$$

Then the equation becomes

$${\tilde{x}}_i\left(t\right)={\tilde{g}}_i\left(t\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\int }_a^t{\left({\tilde{w}}_i\left(t\right)-{\tilde{w}}_i\left(z\right)\right)}^{{\alpha }_i-1}{\tilde{f}}_i(z,\overrightarrow{\tilde{x}}\left(z\right)){\tilde{w}}_i^{\prime }\left(z\right)dz,t\in [a,t_1].$$

(64)

This system is exactly of the same form as the extended system (57). Moreover, ${\tilde{w}}_i$ satisfies the same assumptions used in Lemma 1 with $\tau =T$, so the same length $t_1$ is valid. Repeating the argument from Theorem 13 shows that system (64) has a solution ${\tilde{x}}_i\in C[a,t_1]$.

Reversing the change of variables, we conclude that $x_i\left(t\right)={\tilde{x}}_i(t-T)$ solves system (63) on $[t_1,t_1+T]$. By continuing this extension process, we obtain a solution defined on the full interval $[a,b]$, which completes the proof. □

7.2. Uniqueness Analysis

Theorem 15.

Assume that the hypotheses of Theorem 14 are satisfied. Then system (54) has a unique solution in ${\left(C[a,b]\right)}^{\nu }$.

Proof. 

Let $\overrightarrow{x}$ and $\overrightarrow{y}$ be two solutions of system (54) with the same initial condition. Then

$$\begin{array}{cc}\hfill \left|x_i\left(t\right)-y_i\left(t\right)\right|& =\left|{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}\left(|f_i(t,\overrightarrow{x}\left(t\right))-f_i(t,\overrightarrow{y}\left(t\right))|\right)\left(t\right)\right|\hfill \\ & \le L{}_a\mathfrak{J}_t^{{\alpha }_i,w_i\left(t\right)}\left|x_i\left(t\right)-y_i\left(t\right)\right|\left(t\right)\hfill \\ & ={\displaystyle \frac{L}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\int }_a^t{\left(w_i\left(t\right)-w_i\left(\tau \right)\right)}^{{\alpha }_i-1}\left|x_i\left(\tau \right)-y_i\left(\tau \right)\right|w_i^{\prime }\left(\tau \right)d\tau .\hfill \end{array}$$

We now use a direct combined substitution:

$$s=w_i\left(t\right)-w_i\left(a\right),z=w_i\left(\tau \right)-w_i\left(a\right),$$

so that

$$ds=w_i^{\prime }\left(t\right)dt,dz=w_i^{\prime }\left(\tau \right)d\tau ,t=w_i^{-1}(s+w_i\left(a\right)),\tau =w_i^{-1}(z+w_i\left(a\right)).$$

Define

$${\tilde{x}}_i\left(s\right)=x_i\left(w_i^{-1}(s+w_i\left(a\right))\right),{\tilde{y}}_i\left(s\right)=y_i\left(w_i^{-1}(s+w_i\left(a\right))\right).$$

Then the inequality becomes

$$\begin{array}{c}\hfill \left|{\tilde{x}}_i\left(s\right)-{\tilde{y}}_i\left(s\right)\right|\le {\displaystyle \frac{L}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\int }_0^s{(s-z)}^{{\alpha }_i-1}\left|{\tilde{x}}_i\left(z\right)-{\tilde{y}}_i\left(z\right)\right|dz,s\in [0,w_i\left(b\right)-w_i\left(a\right)].\end{array}$$

This is a weakly singular Gronwall-type inequality for incommensurate orders, for which Theorem 5 gives

$$\left|{\tilde{x}}_i\left(s\right)-{\tilde{y}}_i\left(s\right)\right|=0.$$

Reverting the substitution gives

$$x_i\left(t\right)=y_i\left(t\right)\mathrm{for}\mathrm{all}t\in [a,b],$$

which establishes uniqueness. □

7.3. HU Stability Analysis

To extend the HU stability analysis for Equation (45) with incommensurate $w_i$, we follow the method in [33] with the aid of the Laplace transform.

Theorem 16.

Let $w_i$, $i=1,\dots ,\nu$ satisfy Conditions 1–4 and Condition 8. Let $\overrightarrow{q}\in {\left(C[a,b]\right)}^{\nu }$, $a,b\in \mathbb{R}$ with $a<b$, and $A=\left(a_{ij}\right)\in {\mathbb{R}}^{\nu \times \nu }$. Then, the mild solution of the non-homogeneous linear generalized IFDS with incommensurate weights (45) is HU stable on ${\left(C[a,b]\right)}^{\nu }$.

Proof. 

Suppose $\overrightarrow{x}$ is a mild solution of (45) and $\overrightarrow{y}\in {\left(C[a,b]\right)}^{\nu }$ is a mild solution of the perturbed system satisfying

$$e_i\left(t\right)=y_i\left(t\right)-x_i\left(a\right)-{}_a\mathfrak{I}_t^{{\alpha }_i,w_i}\left({(A\overrightarrow{y})}_i\left(t\right)\right)\left(t\right)+{}_a\mathfrak{I}_t^{{\alpha }_i,w_i}\left(q_i\left(t\right)\right)\left(t\right)=0,\mathrm{for}t\in [a,b].$$

(65)

such that

$$\underset{i=1,\dots ,\nu }{\max }\underset{t\in [a,b]}{\sup }\left|e_i\left(t\right)\right|\le ϵ,$$

and let $\overrightarrow{z}=\overrightarrow{y}-\overrightarrow{x}$. Then $\overrightarrow{z}$ satisfies

$$e_i\left(t\right)=z_i\left(t\right)-{}_a\mathfrak{I}_t^{{\alpha }_i,w_i}\left({(A\overrightarrow{z})}_i\left(t\right)\right)\left(t\right),\mathrm{for}t\in [a,b].$$

(66)

Taking the weighted Laplace transform [45,46]

$${\mathcal{L}}_{w_i}\left\{x\left(t\right)\right\}\left(s\right)={\int }_a^{\infty }{e}^{-s\left(w_i\left(t\right)-w_i\left(a\right)\right)}x\left(t\right)w_i^{\prime }\left(t\right)dt$$

of (66), we obtain

$$\begin{array}{cc}\hfill {\mathcal{L}}_{w_i}\left\{e_i\left(t\right)\right\}\left(s\right)& ={\mathcal{L}}_{w_i}\left\{z_i\left(t\right)\right\}\left(s\right)-{\displaystyle \frac{1}{s^{{\alpha }_i}}}{\mathcal{L}}_{w_i}\left\{\left({(A\overrightarrow{z})}_i\left(t\right)\right)\right\}\left(s\right)\hfill \\ \hfill & ={\mathcal{L}}_{w_i}\left\{z_i\left(t\right)\right\}\left(s\right)-{\displaystyle \frac{1}{s^{{\alpha }_i}}}\left({(A{\mathcal{L}}_{w_i}\{\overrightarrow{z}\})}_i\left(t\right)\right)\left(s\right).\hfill \end{array}$$

(67)

Therefore,

$$\left(I-\mathrm{Diag}[s^{-{\alpha }_1},\dots ,s^{-{\alpha }_{\nu }}]A\right){\mathcal{L}}_{\overrightarrow{w}}\{\overrightarrow{z}\left(t\right)\}\left(s\right)={\mathcal{L}}_{\overrightarrow{w}}\{\overrightarrow{e}\left(t\right)\}\left(s\right),$$

(68)

where ${\mathcal{L}}_{\overrightarrow{w}}\{\overrightarrow{z}\left(t\right)\}\left(s\right)={[{\mathcal{L}}_{w_1}\{z_1\left(t\right)\}\left(s\right),\dots ,{\mathcal{L}}_{w_{\nu }}\left\{z_{\nu }\left(t\right)\right\}\left(s\right)]}^T$ and I denotes the identity matrix. Since

$${\left(I-\mathrm{Diag}[s^{-{\alpha }_1},\dots ,s^{-{\alpha }_{\nu }}]A\right)}^{-1}=I+\sum _{k=0}^{\infty }{\left(\mathrm{Diag}[s^{-{\alpha }_1},\dots ,s^{-{\alpha }_{\nu }}]A\right)}^{k+1},$$

it follows that

$${\mathcal{L}}_{\overrightarrow{w}}\{\overrightarrow{z}\left(t\right)\}\left(s\right)={\mathcal{L}}_{\overrightarrow{w}}\{\overrightarrow{e}\left(t\right)\}\left(s\right)+\sum _{k=0}^{\infty }{\left(\mathrm{Diag}[s^{-{\alpha }_1},\dots ,s^{-{\alpha }_{\nu }}]A\right)}^{k+1}{\mathcal{L}}_{\overrightarrow{w}}\{\overrightarrow{e}\left(t\right)\}\left(s\right).$$

(69)

Now observe that

$$\begin{array}{cc}\hfill {\left(\mathrm{Diag}[s^{-{\alpha }_1},\dots ,s^{-{\alpha }_{\nu }}]A\right)}_{ij}^1& =s^{-{\alpha }_i}{a}_{ij},\hfill \\ \hfill {\left(\mathrm{Diag}[s^{-{\alpha }_1},\dots ,s^{-{\alpha }_{\nu }}]A\right)}_{ij}^2& =\sum _{p_1=1}^{\nu }{s}^{-{\alpha }_i-{\alpha }_{p_1}}{a}_{i{p}_1}{a}_{p_{1}j},\hfill \\ \hfill ⋮& \\ \hfill {\left(\mathrm{Diag}[s^{-{\alpha }_1},\dots ,s^{-{\alpha }_{\nu }}]A\right)}_{ij}^{k+1}& =\sum _{p_k=1}^{\nu }\cdots \sum _{p_1=1}^{\nu }{s}^{-{\alpha }_i-{\alpha }_{p_1}-\cdots -{\alpha }_{p_k}}{a}_{i{p}_k}{a}_{p_k{p}_{k-1}}\cdots a_{p_{1}j}.\hfill \end{array}$$

(70)

Applying the inverse weighted Laplace transform ${\mathcal{L}}_{w_i}^{-1}$ to the i-th component of (69), we get

$$z_i\left(t\right)=e_i\left(t\right)+\sum _{k=0}^{\infty }\sum _{j=1}^{\nu }\sum _{p_k=1}^{\nu }\cdots \sum _{p_1=1}^{\nu }{a}_{i{p}_k}{a}_{p_k{p}_{k-1}}\cdots a_{p_{1}j}{}_a\mathfrak{I}_t^{{\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k},w_i}\{e_j\left(t\right)\}\left(t\right).$$

(71)

Note that

$$\begin{array}{cc}\hfill \left|{}_a\mathfrak{I}_t^{{\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k},w_i}\left\{e_j\left(t\right)\right\}\left(t\right)\right|\le & ϵ{}_a\mathfrak{I}_t^{{\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k},w_i}\left\{{(w_i\left(t\right)-w_i\left(a\right))}^0\right\}\left(t\right)\hfill \\ \hfill =& {\displaystyle \frac{ϵ{(w_i\left(t\right)-w_i\left(a\right))}^{{\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k}}}{\mathrm{\Gamma }({\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k}+1)}}.\hfill \end{array}$$

(72)

From (71), it follows that

$$|z_i\left(t\right)|\le ϵ+\sum _{k=0}^{\infty }\sum _{j=1}^{\nu }\sum _{p_k=1}^{\nu }\cdots \sum _{p_1=1}^{\nu }{a}_{i{p}_k}{a}_{p_k{p}_{k-1}}\cdots a_{p_{1}j}{\displaystyle \frac{ϵ{(w_i\left(t\right)-w_i\left(a\right))}^{{\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k}}}{\mathrm{\Gamma }({\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k}+1)}}.$$

(73)

Now define

$${\alpha }_M=\underset{i=1,\dots ,\nu }{\max }{\alpha }_i,{\alpha }_m=\underset{i=1,\dots ,\nu }{\min }{\alpha }_i,a_M=\underset{i,j=1,\dots ,\nu }{\max }\left|a_{ij}\right|,$$

$$W_M=\underset{i=1,\dots ,\nu }{\max }\underset{t\in [a,b]}{\sup }(w_i\left(t\right)-w_i\left(a\right)),W_M^{*}=\max \{1,W_M\}.$$

Inequality (73) can be simplified as

$$|z_i\left(t\right)|\le ϵ+\sum _{k=0}^{\infty }\sum _{j=1}^{\nu }\sum _{p_k=1}^{\nu }\cdots \sum _{p_1=1}^{\nu }{a}_M^{k+1}{\displaystyle \frac{ϵW_M^{(k+1){\alpha }_M}}{\mathrm{\Gamma }({\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k}+1)}}.$$

(74)

Furthermore, there exists an integer N such that for all $k>N$,

$$\mathrm{\Gamma }({\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k}+1)\ge \mathrm{\Gamma }((k+1){\alpha }_m+1),$$

since $\mathrm{\Gamma }\left(z\right)$ is increasing for $z>2$. We may then further simplify to obtain

$$\begin{array}{cc}\hfill |z_i\left(t\right)|& \le ϵM_1+\sum _{k=N+1}^{\infty }ϵ{\displaystyle \frac{a_M^{k+1}{\nu }^{k+1}{W}_M^{(k+1){\alpha }_M}}{\mathrm{\Gamma }((k+1){\alpha }_m+1)}}\hfill \\ \hfill & \le ϵM_1+\sum _{k=0}^{\infty }ϵ{\displaystyle \frac{{\left(a_M\nu W_M^{{\alpha }_M}\right)}^k}{\mathrm{\Gamma }(k{\alpha }_m+1)}}\hfill \\ \hfill & \le ϵ\left(M_1+E_{{\alpha }_m}\left(a_M\nu W_M^{{\alpha }_M}\right)\right),\hfill \end{array}$$

(75)

where

$$M_1=1+\sum _{k=0}^N\sum _{j=1}^{\nu }\sum _{p_k=1}^{\nu }\cdots \sum _{p_1=1}^{\nu }{a}_M^{k+1}{\displaystyle \frac{W_M^{(k+1){\alpha }_M}}{\mathrm{\Gamma }({\alpha }_i+{\alpha }_{p_1}+\cdots +{\alpha }_{p_k}+1)}}.$$

Equation (75) implies

$$|x_i\left(t\right)-y_i\left(t\right)|\le ϵ\left(M_1+E_{{\alpha }_m}\left(a_M\nu W_M^{{\alpha }_M}\right)\right),$$

and hence system (45) is HU stable. □

8. Application in Neural Networks

Activation functions such as the hyperbolic tangent function $\tanh (·)$ and the sigmoid function are widely used in neural networks [2,47]. These functions are bounded, possess bounded derivatives, and satisfy the assumptions required in Theorem 14. In particular, this property holds for HNN.

Let A denote a weight matrix, $\overrightarrow{b}$ a bias vector, and $\overrightarrow{N}$ the output of the neural network. The network output is defined by

$$\overrightarrow{N}=A\overrightarrow{x}+\overrightarrow{b}.$$

A one-layer Hopfield Neural Network governed by an IFDS can be expressed as

$${}_a\overrightarrow{\mathfrak{D}}_t^{\overrightarrow{\alpha },\overrightarrow{w}\left(t\right)}\overrightarrow{x}\left(t\right)=\overrightarrow{\sigma }(\overrightarrow{N}).$$

If the activation functions ${\sigma }_i$ are chosen as $\tanh (·)$ or sigmoid functions, the above system admits a unique mild solution by Theorems 14 and 15.

9. Conclusions

This paper systematically investigates GIFDSs with commensurate weight functions, addressing critical gaps in the analysis of generalized fractional frameworks. By introducing a state transformation $y_i\left(t\right)=x_i\left(w^{-1}\left(t\right)\right)$, we establish a fundamental equivalence between GIFDSs and classical IFDSs, enabling the extension of well-developed IFDS theories to the generalized case. For linear homogeneous GIFDSs, we derive an explicit mild solution using the incommensurate Mittag–Leffler function, providing a direct tool for modeling relaxation phenomena in fields like material science. For nonlinear GIFDSs, we prove the existence and uniqueness of mild solutions under continuity and Lipschitz conditions, leveraging the transformational equivalence and existing results for non-generalized IFDSs. Notably, these results do not depend on the Lipschitz constants, whereas most existence results derived via fixed-point theorems impose restrictions on the Lipschitz constants. Additionally, we verify the HU stability of non-homogeneous linear GIFDSs, which is pivotal for numerical convergence analysis in practical applications.

Future research can focus on sensitivity analysis of existence and uniqueness results with respect to Lipschitz constants in high-dimensional stiff systems, including fractional versions of the Van der Pol equation and Robertson chemical reaction model, the development of efficient numerical methods for incommensurate Mittag–Leffler function, and extensions to stochastic fractional systems.