Ulam-Hyers Stability of Caputo–Katugampola Generalized Hukuhara Type Partial Differential Symmetry Coupled Systems

Abstract

The purpose of this paper is to investigate a class of novel symmetric coupled fuzzy fractional partial differential equation system involving the Caputo–Katugampola (C-K) generalized Hukuhara (gH) derivative. Within the framework of C-K gH differentiability, two types of gH weak solutions are defined, and their existence is rigorously established through explicit constructions via employing Schauder fixed point theorem, overcoming the limitations of traditional Lipschitz conditions and thereby extending applicability to non-smooth and nonlinear systems commonly encountered in practice. A typical numerical example with potential applications is proposed to verify the existence results of the solutions for the symmetric coupled system. Furthermore, we introduce Ulam–Hyers stability (U-HS) theory into the analysis of such symmetric coupled systems and establish explicit stability criteria. U-HS ensures the existence of approximate solutions close to the exact solution under small perturbations, and thereby guarantees the reliability and robustness of the systems, while it prevents significant deviations in system dynamics caused by minor disturbances. We not only enrich the theoretical framework of fuzzy fractional calculus by extending the class of solvable systems and supplementing stability analysis, but also provide a practical mathematical tool for investigating complex interconnected systems characterized by uncertainty, memory effects, and spatial dynamics.

1. Introduction

Based on the Caputo–Katugampola fractional derivative approach due to Katugampola [1], which expands the theoretical framework of fractional differential via unifying Riemann–Liouville and Hadamard fractional derivatives, Singh et al. [2] investigated the following prey–predator fractional-order model of population carrying capacity and understanding their interactivity:

$$\left\{\begin{array}{c}{\displaystyle {}_a{}^{\mathrm{KC}}D_t^{\mu ,\rho }X(t)=X\left({\delta }_1-{\displaystyle \frac{{\delta }_{1}X}{P_1}}\right)-{\sigma }_{1}XY,}\hfill \\ {\displaystyle {}_a{}^{\mathrm{KC}}D_t^{\mu ,\rho }Y(t)=Y\left(-{\delta }_2+{\sigma }_{2}X\right),}\hfill \\ X(0)=X_0={\alpha }_1>0,\hfill \\ Y(0)=Y_0={\alpha }_2>0,\hfill \end{array}\right.$$

(1)

which contributes significantly to ecological community, where $X(t)$ and $Y(t)$ denote the population densities of the prey and predator, respectively, ${\delta }_1$ represents the growth rate of the prey, $P_1$ the carrying capacity, ${\sigma }_1$ and ${\sigma }_2$ the competitive interaction rates, and ${\delta }_2$ the growth rate of the predator; all these coefficients are positive constants. Building upon this model, this paper, by extending to the two-dimensional spatial dimension and incorporating mixed boundary conditions with unified C-K fractional derivatives, can most directly and accurately characterize the distribution, diffusion, and interaction of species under real-world spatial heterogeneity, and is more consistent with the actual scenarios of complex ecosystems and other systems.

We remark that the C-K fractional derivative has been widely adopted in recent years owing to its ability to capture local differential and integral characteristics, formulating a theoretical structure for fractional–exponent systems and laying a foundation for subsequent research [3,4]. With the in-depth investigation of complex systems, traditional integer-order derivative models exhibit limitations in describing processes with memory, heredity, or nonlocal characteristics, which has promoted the development of fractional partial differential equations (FPDEs) [5]. FPDEs serve as core modeling tools in fields such as ecology and engineering, capable of describing the dynamic evolution of natural phenomena [6] and modeling the spatiotemporal evolution of physical quantities in multivariable systems [7]. For FPDEs involving uncertainties, fuzzy theory can effectively characterize parameter perturbations, yet existing approaches are mostly limited to linear systems. In terms of numerical solutions, the finite element method and finite difference method are the mainstream techniques, while enriched elements (e.g., the fuzzy PDE discretization framework [8]) have been verified to excel at capturing solution singularities, providing support for the modeling of nonlinear coupled systems. Moreover, the successful applications of this mathematical tool in noise suppression [9], biological engineering [10], and physical science [11], etc., have further promoted the integration of FPDE theory with practical problems.

In contrast, biodiversity is an inherent property that underpins every ecosystem. Nevertheless, prior investigations predominantly centered on the survival and proliferation of individual species, thereby overlooking the competitive dynamics engendered by the coexistence of multiple species. Such mutual interrelations are defined as “coupling” [12] when two or more entities engage in reciprocal interaction and influence. As noted by Ding et al. [13], coupling mechanisms are capable of effectively characterizing the interaction dynamics between two competing species in ecological systems. A typical example of such coupled structures is the elliptic system below [14]:

$$\left\{\begin{array}{cc}\Delta u={\phi }_1(x,u,v)\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ \Delta v={\phi }_2(x,u,v)\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u=v=0\hfill & \mathrm{on}\mathsf{\partial }\mathrm{\Omega },\hfill \end{array}\right.$$

(2)

where ${\phi }_1,{\phi }_1\in C\left(\overline{\mathrm{\Omega }}\times {\mathbb{R}}^{+}\times {\mathbb{R}}^{+},{\mathbb{R}}^{+}\right)$, ${\mathbb{R}}^{+}=[0,+\infty )$ and $\mathrm{\Omega }\subset {\mathbb{R}}^n$$(n>3)$ is a smooth bounded domain; this system possesses a significant ability to capture the intrinsic dynamics of ecosystems. Furthermore, Zhang et al. [15] applied the fuzzy fractional coupled partial differential equation of the Caputo derivative to the initial value problem, and established an existence theory of solutions under the gH-type derivative framework. Muatjetjeja et al. [16] analyzed a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry using Noether symmetry analysis.

It is well known that practical engineering systems often have to face issues such as parameter uncertainties, measurement noise, or model inaccuracies. The deterministic framework of traditional fractional-order models is difficult to fully accommodate these challenges, while the introduction of fuzzy theory provides an effective approach to address such uncertainty problems [17,18]. Osman [19] introduced the fuzzy Adomian decomposition and modified Laplace decomposition methods, successfully applying them to the fuzzy fractional Navier–Stokes equations. Additionally, a fuzzy Elzaki transform was developed to address the linear–nonlinear Schrödinger equation. Pandey et al. [20] realized an efficient numerical approximation of variable-order fuzzy partial differential equations (PDEs) by using Bernstein spectral technique. Mazandarani [21] solved the numerical solution problem of initial value problems involving fuzzy fractional dynamics by improving the fractional Euler method. Through the use of Caputo’s fractional derivative definition and gH-difference sets, Singh et al. [22] examined fuzzy differential equations and presented a numerical approach for solving fuzzy fractional differential equations. This method leverages power series approximation in conjunction with Taylor’s theorem. It is worth mentioning that Pythagorean fuzzy fractional calculus provides a new paradigm for complex system analysis by virtue of its strong uncertainty modeling ability. For example, Akram et al. [23] successfully analyzed the fractional-order fuzzy wave equation by means of multivariate Pythagorean fuzzy Fourier transform. Baleanu et al. [24] analyzed the stability of differential equations under such derivatives by combining Adomian polynomials and fractional Taylor series. Hoa et al. [25] constructed analytical solutions for C-K fuzzy fractional differential equations by solving equivalent fuzzy integer-order differential equations, simultaneously establishing the solution’s existence and uniqueness under a generalized Lipschitz condition. These findings substantially advance the theoretical and applicative framework of fractional calculus.

Since then, researchers have explored symmetric coupled systems from multiple dimensions. In fuzzy fractional population dynamics models, stability conditions reveal the long-term evolution laws of species quantities under fractal habitats and fuzzy carrying capacities, providing guidance for the protection of endangered species. Fractional differential equations have become a research hotspot in stability analyses: Wang et al. [26] constructed a fractional predator–prey model, characterized the fractal characteristics of habitats using the Caputo derivative, and found that changes in fractional order significantly affect the stability of population equilibrium points, which is consistent with the conclusion of Alharbi [27], who studied neutral fractional equations with finite time delays, confirmed that fractional order and time delay jointly determine system stability, and applied the results to fractional nonlinear wave equations to verify robustness. Huang and Luo [28] developed a complete theory for a class of fuzzy fractional-order switched implicit systems with non-instantaneous impulses that incorporated explicit solutions, existence, and U-H stability by integrating switching laws and fractional-order memory effects. The stability of solutions varies across different equations: Awad and Alkhezi [29] extended their research to coupled systems with hybrid integral boundary conditions in Banach algebras, revealing the influence of algebraic structures on the stability of implicit solutions of symmetric coupled systems. Ali et al. [30] obtained the Ulam stability of solutions for fractal fractional symmetric coupled systems by means of the Banach contraction principle and Krasnoselski fixed-point theorem:

$$\left\{\begin{array}{cc}{}^{FD}D^{\eta }r(t)=\xi t^{\xi -1}f\left(t,r(t),p(t)\right),\hfill & t\in I=[0,T],\\ {}^{FD}D^{\eta }p(t)=\xi t^{\xi -1}g\left(t,r(t),p(t)\right),\hfill \\ r(0)=r_0+\varphi (r),\hfill \\ p(0)=p_0+\psi (p),\hfill \end{array}\right.$$

where ${}^{FD}D^{\eta }$ is the Caputo fractional derivative and $r_0,p_0\in \mathbb{R}$, $\eta ,\xi \in (0,1]$, $f,g:I\times {\mathbb{R}}^2\to \mathbb{R}$ and $\varphi ,\psi :I=[0,T]\times \mathbb{R}\to \mathbb{R}$ are continuous functions. This method has been enriched by multiple tools: Anderson [31] corrected errors in the stability analysis of linear equations using the Kamal integral transform and derived the optimal constant; Kahouli et al. [32] incorporated stochastic factors into the stability framework for fractional stochastic equations with proportional delays. Damag [33] establishes that symmetric structures enhance stability through the derivation of the spectral properties of fractional symmetric operators, providing a theoretical foundation for this paper’s analysis. András et al. [34] studied Ulam–Hyers stability (U-HS) of a class of elliptic PDEs:

$$\left\{\begin{array}{cc}-\Delta u=f,\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u=0,\hfill & \mathrm{on}\mathsf{\partial }\mathrm{\Omega },\hfill \end{array}\right.$$

which are defined on a bounded domain with Lipschitz boundary by using the direct technique and the abstract method of Picard operator.

Moreover, in the study of calculus theory, the Lipschitz condition has long been the core premise of classical results. However, in practical problems, due to the complex nonlinear and non-smooth characteristics of the systems, its applicability is limited [35]. Long et al. [36] introduced the Schauder-type nonlinear substitution technique to deal with fuzzy-valued continuous functions that do not satisfy Lipschitz condition. By circumventing the need for local smoothness and instead leveraging the compactness criterion from topological fixed point theory, a second existence result for two types of gH-weak solutions in special coupled systems was established. This finding broadens the theory’s applicability, offering a more adaptable tool for modeling complex systems. Building on this, Zhang et al. [15] employed the same methodology to demonstrate the existence of gH-weak solutions for coupled fractional equations, thereby advancing the theory of multi-scale symmetric coupled systems. The model adopted in this paper utilizes the C-K fractional derivative, which can unify multiple fractional derivative theories and more flexibly characterize the temporal memory and spatial nonlocal effects of the system. Meanwhile, a bidirectional symmetric coupling structure is constructed, which can more accurately reflect the bidirectional interaction relationships among variables in reality. The strict requirements of Lipschitz conditions on the smoothness of functions make it difficult to cover a large number of non-ideal situations in practice. Thus, we consider the related problems without Lipschitz conditions.

This study modifies system (2) to augment its ability to model intrinsic ecological characteristics, including slow diffusion and historical uncertainty. Drawing inspiration from [2,15,34], our improvements involve the incorporation of a C-K gH-type differentiable operator on the left-hand side, a more precise specification of the right-hand side, and the application of fuzzy initial conditions. We consider a fuzzy number space M, with $M_c$ being its subspace such that for any $\vartheta \in M$, the function $\varrho \to {\left[\vartheta \right]}^{\varrho }$ is continuous in the Hausdorff metric on [0, 1], where the $\varrho$-level set is defined as ${\left[\vartheta \right]}^{\varrho }=\{x\in \mathbb{R}|\vartheta (x)\ge \varrho \}$ and is a $\varrho$-level set of $\vartheta$. We therefore proceed to examine the following symmetric coupled system of fuzzy fractional partial differential equations (FFPDEs), governed by the concept of C-K gH-type differentiability:

$$\left\{\begin{array}{c}{}_{gh}{}^{CK}D_k^{\mathit{\alpha },p}{h}_1(x,y)=\widehat{f_1}(x,y,h_1(x,y),h_2(x,y)),\hfill \\ {}_{gh}{}^{CK}D_k^{\mathit{\beta },p}{h}_2(x,y)=\widehat{f_2}(x,y,h_1(x,y),h_2(x,y)),\hfill \\ h_1(x,0)={\zeta }_1(x),h_2(x,0)={\eta }_1(x),\hfill \\ h_1(0,y)={\zeta }_2(y),h_2(0,y)={\eta }_2(y),\hfill \end{array}\right.$$

(3)

for any $(x,y)\in \varpi =[0,a]\times [0,b]$, $h_1,h_2\in M$, and $k\in \{1,2\}$, the operator ${}_{gh}{}^{CK}D_k^{\gamma ,p}$ denotes a C-K gH-type derivative of fractional order $\gamma =({\gamma }_1,{\gamma }_2)\in (0,1]\times (0,1]$. The functions ${\zeta }_1,{\eta }_1:[0,a]\to M_c$ and ${\zeta }_2,{\eta }_2:[0,b]\to M_c$ are continuous, and $p>0$ is a real parameter. It is crucial to note that the operators ${}_{gh}{}^{CK}D_k^{\mathit{\alpha },p}$ and ${}_{gh}{}^{CK}D_k^{\mathit{\beta },p}$ in (3) presuppose the existence of both Hukuhara (H-) difference and gH-difference.

Remark 1.

The following points should be noted:

• The system (3) can flexibly characterize the temporal memory characteristics and spatial nonlocal effects of the system, expanding the application scope of fractional calculus in the modeling of complex systems.
• It is easy to see that the right-hand side of the second equation in (3) is a function of $h_1(x,y)$ and $h_2(x,y)$. A bidirectional symmetric coupled system is constructed instead of the “unidirectional coupling” in [15], which more accurately reflects the relationship of “interaction and symmetric restriction” among variables in reality (e.g., the bidirectional influence between prey and predator in an ecosystem).
• Notably, the C-K gH-type derivative operator in (3) is a generalization of the Caputo gH-type derivative operator in [15]. The C-K fractional derivative can unify multiple fractional derivative theories and flexibly characterize the temporal memory and spatial nonlocal effects of the system, and the constructed bidirectional symmetric coupling structure can more accurately reflect the bidirectional interaction among variables in reality, further enhancing the adaptability to complex scenarios.
• In addition, this paper also investigates the U-HS of this system, while stability analysis is not addressed in [15]. This extension fills the gap in robustness evaluation under perturbations for similar systems, can more comprehensively reveal the behavioral reliability of the system under external disturbances, and provides theoretical support for the anti-interference design and performance evaluation of systems in practical ecological fields.
  Notably, the C-K gH-type derivative operators in (3) generalize the Caputo gH-type derivative operators in [15].

Consequently, the system (3) is entirely novel and merits in-depth study.

The organization of this work begins with a presentation of necessary preliminaries in Section 2. The study then proceeds, in Section 3, to demonstrate the existence of two kinds of gH-weak solutions for Equation (3) via the Schauder fixed point theorem. Subsequently, a numerical illustration is presented in Section 4. The Ulam–Hyers stability (U-HS) of the symmetric coupled system (3) is investigated in Section 5, followed by a concluding section that summarizes the work and outlines prospects for future research.

2. Preliminaries

To establish a foundation for our analysis, this section outlines the fractional integral and C-K gH-derivative for fuzzy-valued multivariate functions, along with the theory of relative compactness in fuzzy number space. It is acknowledged that deeper treatments of certain concepts can be found in [2,13,15].

Definition 1

([15]). Let M represent the space of fuzzy numbers on $\mathbb{R}$, comprising all mappings $\vartheta :\mathbb{R}\to [0,1]$ that are normal, fuzzy convex, upper semi-continuous, and have compact support. For a fuzzy number ϑ, its ϱ-level set are defined by

$${\left[\vartheta \right]}^{\varrho }=\left\{\begin{array}{cc}\left\{x\in \mathbb{R}\mid \vartheta (x)\ge \varrho \right\},\hfill & if0<\varrho \le 1,\hfill \\ cl(supp\vartheta ),\hfill & if\varrho =0,\hfill \end{array}\right.$$

(4)

Here, cl denotes set closure, and supp $supp\vartheta$ is the support of ϑ.

The $\varrho$-level set of a fuzzy number $\vartheta$ forms a closed bounded interval $[{\vartheta }_{\varrho }^{-},{\vartheta }_{\varrho }^{+}]$, with endpoints ${\left[\vartheta \right]}^{\varrho }$ and ${\vartheta }_{\varrho }^{+}$. Its diameter is defined as $len{\left[\vartheta \right]}^{\varrho }={\vartheta }_{\varrho }^{+}-{\vartheta }_{\varrho }^{-}$. For any two distinct mappings $\vartheta ,\theta \in M$, the supremum metric $d_{\infty }$ on M is given by

$$\begin{array}{cc}{d}_{\infty }(\vartheta ,\theta )\hfill & =\underset{0\le \varrho \le 1}{sup}{d}_H({\left[\vartheta \right]}^{\varrho },{\left[\theta \right]}^{\varrho })\hfill \\ & =\underset{0\le \varrho \le 1}{sup}max\left\{|{\vartheta }_{\varrho }^{-}-{\theta }_{\varrho }^{-}|,|{\vartheta }_{\varrho }^{+}-{\theta }_{\varrho }^{+}|\right\},\forall \vartheta ,\theta \in M,\hfill \end{array}$$

(5)

where ${\left[\vartheta \right]}^{\varrho }=[{\vartheta }_{\varrho }^{-},{\vartheta }_{\varrho }^{+}]$. In $C(\varpi ,M)$, we consider the corresponding supremum metric:

$$\begin{array}{c}D(h_1,h_2)=\underset{(x,y)\in \varpi }{sup}{d}_{\infty }(h_1(x,y),h_2(x,y)),\hfill \end{array}$$

(6)

consequently, $(M,d_{\infty })$ and $(M,D)$ are complete metric spaces.

Reference [37] establishes the following level-wise arithmetic for all $\vartheta ,\theta \in M$ and $\varrho \in [0,1]$:

$$\left\{\begin{array}{c}(i){[\vartheta +\theta ]}^{\varrho }={\left[\vartheta \right]}^{\varrho }+{\left[\theta \right]}^{\varrho },\hfill \\ (ii){\left[k\vartheta \right]}^{\varrho }=k{\left[\vartheta \right]}^{\varrho },k\in \mathbb{R}\setminus \{0\},\hfill \\ (iii){[\vartheta \ominus \theta ]}^{\varrho }=[{\vartheta }_{\varrho }^{-}-{\theta }_{\varrho }^{-},{\vartheta }_{\varrho }^{+}-{\theta }_{\varrho }^{+}],\hfill \end{array}\right.$$

(7)

with ⊖ being the H-difference, which is assumed to be universally existent. The subspaces $M_c^i$$(i=1,2)$ consist of $H_m$-continuous fuzzy numbers. As complete metric semilinear spaces with cancellation property [15,38], they induce Banach semilinear spaces $C(\varpi ,M_c^i)$ of fuzzy-valued continuous functions. The corresponding Lebesgue integrable spaces are denoted as $\widehat{L}(\varpi ,M_c^i)$.

For any $r>0$, the closed ball $B(\widehat{0},r)$ in $\left(C\left(\varpi ,M_c^i\right),D\right)$ is defined by $D(\vartheta ,\widehat{0})\le r$, where $\widehat{0}=\vartheta \ominus \vartheta$ for any $\vartheta \in M_c^i$, characterized by $\widehat{0}(x)=1$ if $x=0$.

Lemma 1

([15]). For any $\mu ,\upsilon ,\omega ,\lambda \in M_c^i$$(i=1,2)$, the following properties are satisfied:

(i)

The metric satisfies $d_{\infty }(\mu +\upsilon ,\omega +\lambda )\le d_{\infty }(\mu +\omega )+d_{\infty }(\upsilon +\lambda )$.

(ii)

If $\mu \ominus \upsilon$ and $\omega \ominus \lambda$ hold, then $d_{\infty }(\mu \ominus \upsilon ,\omega \ominus \lambda )\le d_{\infty }(\mu ,\omega )+d_{\infty }(\upsilon ,\lambda )$.

(iii)

If $\mu \ominus (\upsilon +\omega )$ exist, then $\mu \ominus \upsilon \ominus \omega$ hold and $\mu \ominus (\upsilon +\omega )=\mu \ominus \upsilon \ominus \omega$.

(iv)

If $\mu \ominus \upsilon$ and $\upsilon \ominus \omega$ are defined, then $\mu \ominus (\upsilon \ominus \omega )$ is defined and satisfies $\mu \ominus (\upsilon \ominus \omega )=(\mu \ominus \upsilon )+\omega$.

(v)

The existence of $\mu \ominus \upsilon$ implies the existence of $(-1)\mu \ominus (-1)\upsilon$, with $(-1)(\mu \ominus \upsilon )=(-1)\mu \ominus (-1)\upsilon$.

Remark 2.

The applicability of properties $(\mathrm{ii})$–$(\mathrm{v})$ in Lemma 1 is contingent upon the existence of the H-difference. These properties are foundational to the derivation of this paper’s main results.

Definition 2

([15]). Let $\iota \in \mathbb{L}$, a fuzzy-valued mapping $f\in C(\varpi ,M_c^i)$$(i=1,2)$ is said to possess an ${\iota }^{\mathrm{th}}$ order gH-type derivative with respect to x at $(x_0,y_0)\in \varpi$ when

(i)

The function has gH-type derivatives of all orders from 1 to $\iota -1$ at the point;

(ii)

There exists an element ${\displaystyle \frac{{\mathsf{\partial }}^{\iota }f(x_0,y_0)}{\mathsf{\partial }{x}^{\iota }}}\in M_c^i$ such that for all sufficiently small $\hslash >0$ with $(x_0+\hslash ,y_0)\in \varpi$, the gH-difference $f^{\iota -1}(x_0+\hslash ,y_0){\ominus }_{gH}{f}^{\iota -1}(x_0,y_0)$ exists, and satisfies

$$\underset{\hslash \to 0}{lim}{\displaystyle \frac{f^{\iota -1}(x_0+\hslash ,y_0){\ominus }_{gH}{f}^{\iota -1}(x_0,y_0)}{\delta }}={\displaystyle \frac{{\mathsf{\partial }}^{\iota }f(x_0,y_0)}{\mathsf{\partial }{x}^{\iota }}}.$$

(8)

The gH-difference operation is characterized by

$$f_1{\ominus }_{gH}{f}_2=\xi ⟺\left\{\begin{array}{c}\left(†\right)f_1=f_2+\xi ,\hfill \\ \left(‡\right)f_2=f_1+(-1)\xi .\hfill \end{array}\right.$$

(9)

This limit element is denoted the ι-order gH-type derivative of f with respect to x at $(x_0,y_0)$.

Remark 3.

The framework established in Definition 2 for gH-type differentiability with respect to x is naturally extended to define higher-order partial derivatives with respect to y. In the first-order case $(\iota =1)$, the general expression (8) simplifies to

$$\underset{\hslash \to 0}{lim}{\displaystyle \frac{f(x_0+\hslash ,y_0){\ominus }_{gH}f(x_0,y_0)}{\delta }}={\displaystyle \frac{\mathsf{\partial }f(x_0,y_0)}{\mathsf{\partial }x}},$$

(10)

representing the first-order partial derivative at $(x_0,y_0)$.

Definition 3

([39]). Let $f(x,y)$ be a fuzzy-valued multivariate function belonging to $C(\varpi ,M_c^i)\cap \widehat{L}(\varpi ,M_c^i)$ for i = 1, 2. For fractional orders $\mathit{\alpha }=({\alpha }_1,{\alpha }_2)\in (0,1]\times (0,1]$ and parameter $p>0$, the mixed Riemann–Liouville fractional integral operator is defined as follows. At each point $(x,y)\in \varpi$ and for every ${\varrho }_1\in [0,1]$, with the level set $\left[f{(x,y)}^{{\varrho }_1}\right]=[f_{{\varrho }_1}^{-}(x,y),f_{{\varrho }_1}^{+}(x,y)]$, the operator is given by

$$\begin{array}{c}{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}f(x,y)={\displaystyle \frac{p^{2-{\alpha }_1-{\alpha }_2}}{\mathrm{\Gamma }({\alpha }_1)\mathrm{\Gamma }({\alpha }_2)}}\underset{0}{\overset{x}{\int }}\underset{0}{\overset{y}{\int }}{(x^p-s^p)}^{{\alpha }_1-1}{(y^p-t^p)}^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}f(s,t)dtds.\hfill \end{array}$$

(11)

Definition 4

([15]). The mapping $\widehat{f_i}$$(i=1,2)$: $\varpi \times C(\varpi ,M_c^1)\times C(\varpi ,M_c^2)\to M_c^i$ is said to be jointly continuous at the point $(x_0,y_0,\mathrm{\Psi },\mathrm{\Phi })\in \varpi \times C(\varpi ,M_c^1)\times C(\varpi ,M_c^2)$ if for every $\epsilon >0$, there exists $\widehat{\delta }>0$ such that whenever $|x-x_0|$$+|y-y_0|$$+d_{\infty }(h_1,\mathrm{\Psi })+d_{\infty }(h_2,\mathrm{\Phi })<\widehat{\delta }$, the following inequalities hold: $d_{\infty }(\widehat{f_i}(x,y,h_1,h_2),$$\widehat{f_i}(x_0,y_0,\mathrm{\Psi },\mathrm{\Phi }))<\epsilon$.

For all $(x,y)\in \varpi =[0,a]\times [0,b]$, define

$$\begin{array}{c}\mathrm{\Psi }(x,y)={\zeta }_2(y)+[{\zeta }_1(x)\ominus {\zeta }_1(0)],\hfill \\ \mathrm{\Phi }(x,y)={\eta }_2(y)+[{\eta }_1(x)\ominus {\eta }_1(0)],\hfill \end{array}$$

(12)

where ${\zeta }_1\in C([0,a],M_c^1)$, ${\eta }_1\in C([0,a],M_c^2)$, ${\zeta }_2\in C([0,b],M_c^1)$ and ${\eta }_2\in C([0,b],M_c^2)$ are given continuous functions with the specified domains and codomains, and the differences ${\zeta }_2(y)\ominus {\zeta }_1(0)$ and ${\eta }_2(y)\ominus {\eta }_1(0)$ are assumed to exist.

Then, define the function spaces

$$\begin{array}{cc}{\widehat{C}}_{\mathrm{\Psi }}^{\widehat{f_1}}(\varpi ,M_c^1)=\hfill & \{h_1\in C(\varpi ,M_c^1)|\mathrm{\Psi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\hfill \\ & \widehat{f_1}(x,y,h_1(x,y),h_2(x,y))exist,\forall (x,y)\in \varpi \},\hfill \end{array}$$

(13)

$$\begin{array}{cc}{\widehat{C}}_{\mathrm{\Phi }}^{\widehat{f_2}}(\varpi ,M_c^2)=\hfill & \{h_2\in C(\varpi ,M_c^2)|\mathrm{\Phi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}\hfill \\ & \widehat{f_2}(x,y,h_1(x,y),h_2(x,y))exist,\forall (x,y)\in \varpi \},\hfill \end{array}$$

(14)

where $\mathrm{\Psi }$ and $\mathrm{\Phi }$ are defined by (12).

Additionally, for nonnegative integers $m,n$ = 0,1,2 the space $C_{gH}^{m,n}(\varpi ,M_c^i)$ $(i=1,2)$ comprises all functions $\tau :\varpi \subset {\mathbb{R}}^2\to M_c^i$ that admit partial gH-type derivatives of orders up to m in x and up to n in y.

Definition 5

([39]). Let $f(x,y)$ be a twice partially gH-differentiable fuzzy-valued function in the space $C_{gH}^{2,2}(\varpi ,M_c^i)$ $(i=1,2)$. For $\mathit{\alpha }=({\alpha }_1,{\alpha }_2)\in (0,1]\times (0,1]$, the C-K gH-type derivative operator of order α with respect to x and y is defined by

$$\begin{array}{cc}{}_{gH}{}^{CK}\mathcal{D}_k^{\mathit{\alpha },p}h(x,y)\hfill & ={}_F{}^{RL}\mathcal{I}_{0^{+}}^{1-\mathit{\alpha },p}\left({\displaystyle \frac{{\mathsf{\partial }}^{2}h(x,y)}{\mathsf{\partial }x\mathsf{\partial }y}}\right)\hfill \\ \hfill & ={\displaystyle \frac{p^{{\alpha }_1+{\alpha }_2}}{\mathrm{\Gamma }(1-{\alpha }_1)\mathrm{\Gamma }(1-{\alpha }_2)}}{\int }_0^x{\int }_0^y{(x^p-s^p)}^{-{\alpha }_1}{(y^p-t^p)}^{-{\alpha }_2}{\displaystyle \frac{{\mathsf{\partial }}^{2}h(s,t)}{\mathsf{\partial }s\mathsf{\partial }t}}dtds,\hfill \end{array}$$

(15)

This definition requires the right-hand side to be well-defined and uses the parameter vector $1-\mathit{\alpha }=(1-{\alpha }_1,1-{\alpha }_2)\in [0,1)\times [0,1)$.

Particularly, we need to distinguish two cases corresponding to $(†)$ and $(‡)$ in (9) for any $h_1\in M_c^1$, as follows:

(i)

The function $h_1$ is considered C-K gH-type differentiability of order $\mathit{\alpha }$ with respect to both x and y if the mixed partial derivative ${\displaystyle \frac{{\mathsf{\partial }}^2{h}_1(·,·)}{\mathsf{\partial }x\mathsf{\partial }y}}$ is a gH-type derivative of type $(†)$ (i.e., with parameter k = 1 as defined in (3)) at the point $(x,y)$. This property is denoted by ${}_{gH}{}^{CK}\mathcal{D}_1^{\mathit{\alpha }}{h}_1(x,y)$.

(ii)

The function $h_1$ is classified as $(‡)$-C-K gH-type differentiable of order $\mathit{\alpha }$ in x and y when ${\displaystyle \frac{{\mathsf{\partial }}^2{h}_1(·,·)}{\mathsf{\partial }x\mathsf{\partial }y}}$ acts as a gH-type derivative of type $(‡)$ (i.e., where k = 2 in (3)) at $(x,y)$, For this, the notation ${}_{gH}{}^{CK}\mathcal{D}_2^{\mathit{\alpha }}{h}_1(x,y)$ is used.

Definition 6

([15]). A subset $S\subset M_c^i$$(i=1,2)$ is equicontinuous at $w_0\in M_c^i$ if for every $\epsilon >0$, there exists $\delta >0$ such that for all $w\in M_c^i$ with $d_{\infty }(w,w_0)<\delta$ and all $T\in S$, we have $d_{\infty }(T(w),T(w_0))<\epsilon$. If this holds at every ${\omega }_0\in M_c^i$, then S is equicontinuous.

In order to facilitate understanding, we present an example. Let $M_c^1$ be the space of fuzzy numbers (equipped with the supremum metric $d_{\infty }$), and consider the family of operators $S={\{T_k\}}_{k\in {\mathbb{N}}^{+}}$, where the $\tau$-level set of $T_k(w)$ is $\left[{\displaystyle \frac{a(\tau )}{k}},{\displaystyle \frac{b(\tau )}{k}}\right]$. Take $w_0$ with ${\left[w_0\right]}^{\tau }=[\tau ,1-\tau ]$. For any $\epsilon >0$, choose $\delta =\epsilon$. If $d_{\infty }(w,w_0)<\delta$, then for any $T_k\in S$, $d_H\left({\left[T_k(w)\right]}^{\tau },{\left[T_k(w_0)\right]}^{\tau }\right)={\displaystyle \frac{1}{k}}{d}_H\left({\left[w\right]}^{\tau },{\left[w_0\right]}^{\tau }\right)\le {\displaystyle \frac{1}{k}}\delta <\epsilon ,(\sin \mathrm{ce}k\ge 1)$; thus, S is equicontinuous at $w_0$.

Lemma 2

(Ascoli–Arzelá’s theorem [40]). Let $\mathbb{B}$ be a Banach space with norm $\parallel ·\parallel$, and let $C([a,b],\mathbb{B})$ the space of continuous functions from $[a,b]\subset \mathbb{R}$ to $\mathbb{B}$. Suppose $S\subseteq C([a,b],\mathbb{B})$ satisfies the following:

(i)

There exists $M>0$ such that $\parallel f(t)\parallel \le M$, for all $t\in [a,b]$, $f\in S$.

(ii)

For every $\epsilon >0$, there exists $\delta (\epsilon )>0$ such that $\parallel f(t)-f(s)\parallel \le \epsilon$, whenever $|t-s|\le \delta (\epsilon )$, for all $f\in S$.

(iii)

For each $t\in [a,b]$, the set $\{f(t):f\in S\}$ is relatively compact in $\mathbb{B}$.

Then, S is a relatively compact subset of $C([a,b],\mathbb{B})$.

Definition 7

([15]). A subset $S\subset M_c^i$$(i=1,2)$ is said to be compactly supported if there exists a compact set $K\subseteq \mathbb{R}$ such that for every fuzzy number $w\in S$, the support of w, denoted by ${\left[w\right]}^0$, is contained in K.

To elucidate the definition, we present an explanatory example. Fix a compact set $K=[0,1]$, and define the set $S=w\in M_c^i{|\left[w\right]}^o\subseteq [0,1]$. For any $w\in S$, its support ${\left[w\right]}^0$ is contained in the same compact set K, which satisfies the definition. Therefore, S is compactly supported.

Definition 8

([15]). A subset $S\subseteq M_c^i$$(i=1,2)$ is said to be level-equicontinuous at a point ${\tau }_0\in [0,1]$ if for every $\epsilon >0$, there exists $\delta >0$, such that for all $\tau \in [0,1]$ with $|\tau -{\tau }_0|<\delta$, the Hausdorff distance satisfies $d_H({\left[w\right]}^{\tau },{\left[w\right]}^{{\tau }_0})<\epsilon$ for each $w\in S$. If this condition holds for every ${\tau }_0\in [0,1]$, then S is termed level-equicontinuous on $[0,1]$.

An illustrative example is given below to aid comprehension. Let $M_c^1$ be the space of fuzzy numbers, and let w have the $\tau$-level set ${\left[w\right]}^{\tau }=[\tau ,3-2\tau ]$, with $S=\{w\}$. Take ${\tau }_0=0.3$; for any $\epsilon >0$, choose $\delta =\epsilon /2$. If $|\tau -0.3|<\delta$, $d_H\left({\left[w\right]}^{\tau },{\left[w\right]}^{0.3}\right)=max\left\{|\tau -0.3|,|0.6-2\tau |\right\}<\epsilon$; thus’ S is level-equicontinuous at ${\tau }_0=0.3$.

Definition 9

([15]). Let U and V be $H_m$-continuous fuzzy number spaces. A continuous mapping $\Lambda :U\to V$ is called a compact operator if for every bounded subset $S\subseteq U$, the image $\Lambda (S)$ is relatively compact in V, that is, the closure of $\Lambda (S)$ forms a compact subset of V.

For clarity, an example is presented to illustrate the definition. Consider the zero operator $\Lambda :U\to V$, defined by $\Lambda (w)=0$. It maps any bounded subset $S\subseteq U$ to the singleton set 0. In a metric space, a singleton set is compact. Thus, $\Lambda (S)$ is compact, and $\Lambda$ is a compact operator.

Lemma 3

([15]). A subset S in $M_c^i$$(i=1,2)$ is compactly supported if and only if it is relatively compact in the metric space $(M_c^i,d_{\infty })$ and level-equicontinuous on $[0,1]$.

Remark 4.

Lemma 3 establishes the equivalence between three properties for a subset $S\subseteq M_c^i$$(i=1,2)$, relative compactness in $(M_c^i,d_{\infty })$, compact support, and level-equicontinuity on $[0,1]$. Specifically, S is relatively compact if and only if it is both compactly supported and level-equicontinuous.

Lemma 4.

Let Ψ and Φ be defined as in (12), with $\widehat{f_1}$ and $\widehat{f_2}$ being jointly continuous as per Definition 4. For fuzzy-valued functions $h_1\in C_{gH}^{2,2}(\varpi ,M_c^1)$ and $h_2\in C_{gH}^{2,2}(\varpi ,M_c^2)$, system (3) is equivalent to the following nonlinear fractional Volterra integro-differential symmetry coupled systems:

• for case $k=1$:

  $$\left\{\begin{array}{cc}{h}_1(x,y)\hfill & =\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y)),\hfill \\ h_2(x,y)\hfill & =\mathrm{\Phi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y)),\hfill \end{array}\right.$$

  (16)
• for case $k=2$:

  $$\left\{\begin{array}{cc}{h}_1(x,y)\hfill & =\mathrm{\Psi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y)),\hfill \\ h_2(x,y)\hfill & =\mathrm{\Phi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y)).\hfill \end{array}\right.$$

  (17)

Proof. 

This equivalence can be derived using a similar methodology to that employed in the proof of Lemma 3 in [41]), and is thus omitted for conciseness. □

3. Main Result

In this section, we employ Schauder fixed point theorem to establish the existence of both $(†)$-weak and $(‡)$-weak solutions for system (3). It is important to note that our results are obtained without assuming the Lipschitz condition. We begin by recalling the main content of Schauder’s fixed point theorem.

Lemma 5

(Schauder fixed point theorem in semilinear spaces, [15]). Let S be a nonempty, bounded, closed and convex subset of a Banach semilinear space $C(\varpi ,M_c)$ endowed with the cancellation property. If h: $S\to S$ is a compact operator, then h admits at least one fixed point in S.

Remark 5.

To establish the compactness of an operator in the sense of Definition 9, one must verify its continuity and show that it maps bounded sets to relatively compact sets, thereby fulfilling the conditions of Lemma 5.

The first step consists of proving the continuity of the operator.

Lemma 6.

Let $\mathsf{\ell }>0$ be a constant such that the operators $\widehat{f_i}:\varpi \times B(\widehat{0},\mathsf{\ell })\times B(\widehat{0},\mathsf{\ell })\to M_c^i$ ($i=1,2$) are compact, with $\mathrm{\Psi },\mathrm{\Phi }\in C(\varpi ,B(\widehat{0},{\displaystyle \frac{\mathsf{\ell }}{2}}))$. Then there exist $\widehat{a}\in [0,a]$ and $\widehat{b}\in [0,b]$ such that the operator ${\tilde{\aleph }}_1:\overline{N}\to \overline{N}$ is continuous, where

$$\overline{N}:=\left\{(\vartheta ,\theta )\in C\left(\widehat{\varpi },M_c^1\right)\times C(\widehat{\varpi },M_c^2)|max\left\{D(\vartheta ,\widehat{0}),\phantom{D(\theta ,\widehat{0})}D(\theta ,\widehat{0})\phantom{D(\vartheta ,\widehat{0})}\right\}\le \mathsf{\ell }\right\},$$

with $\widehat{\varpi }:=[0,\widehat{a}]\times [0,\widehat{b}]$, and ${\tilde{\aleph }}_1$ is defined by

$${\tilde{\aleph }}_1(h_1,h_2)=\left(\begin{array}{c}{\tilde{W}}_1(h_1,h_2)\\ {\tilde{Q}}_1(h_1,h_2)\end{array}\right),\forall (h_1,h_2)\in \overline{N},$$

(18)

where $\tilde{W}:\overline{N}\to C(\widehat{\varpi },M_c^1)$, ${\tilde{Q}}_1:\overline{N}\to C(\widehat{\varpi },M_c^2)$ are respectively determined by

$$\begin{array}{cc}\hfill & {\tilde{W}}_1(h_1,h_2)=\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y)),\hfill \\ \hfill & {\tilde{Q}}_1(h_1,h_2)=\mathrm{\Phi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y)).\hfill \end{array}$$

(19)

Proof. 

For any two pairs of functions $\left(\begin{array}{c}{h}_{11}(x,y)\\ h_{21}(x,y)\end{array}\right),\left(\begin{array}{c}{h}_{12}(x,y)\\ h_{22}(x,y)\end{array}\right)\in C(\varpi ,M_c^1)\times C(\varpi ,M_c^2)$, we have

$$\begin{array}{cc}& D({\tilde{\aleph }}_1(h_{11},h_{21}),{\tilde{\aleph }}_1(h_{12},h_{22}))\hfill \\ & =max\left\{D({\tilde{W}}_1(h_{11},h_{21}),{\tilde{W}}_1(h_{12},h_{22})),D({\tilde{Q}}_1(h_{11},h_{21}),{\tilde{Q}}_1(h_{12},h_{22}))\right\}\hfill \\ & =max\left\{\underset{(x,y)\in \varpi }{sup}{d}_{\infty }\left({\tilde{W}}_1(h_{11}(x,y),h_{21}(x,y)),{\tilde{W}}_1(h_{12}(x,y),h_{22}(x,y))\right),\right.\hfill \\ & \left.\underset{(x,y)\in \varpi }{sup}{d}_{\infty }\left({\tilde{Q}}_1(h_{11}(x,y),h_{21}(x,y)),{\tilde{Q}}_1(h_{12}(x,y),h_{22}(x,y))\right)\right\}.\hfill \end{array}$$

(20)

The compactness of $\widehat{f_1}$ and $\widehat{f_2}$ implies that they are bounded. Define

$$L_i=\underset{((x,y),\vartheta ,\theta )\in \varpi \times B(\widehat{0},\mathsf{\ell })\times B(\widehat{0},\mathsf{\ell })}{sup}D({\widehat{f}}_i(x,y,\vartheta ,\theta ),\widehat{0}),\mathrm{for}i=1,2.$$

Then, for any $\mathit{\alpha }=({\alpha }_1,{\alpha }_2)$, $\mathit{\beta }=({\beta }_1,{\beta }_2)\in (0,1]\times (0,1]$, there exist $a_1,a_2,b_1,b_2>0$ such that ${\displaystyle \frac{p^{-({\alpha }_1+{\alpha }_2)}{a}_1^{{\alpha }_{1}p}{b}_1^{{\alpha }_{2}p}{L}_1}{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}}\le {\displaystyle \frac{\mathsf{\ell }}{2}}$ and ${\displaystyle \frac{p^{-({\beta }_1+{\beta }_2)}{a}_2^{{\beta }_{1}p}{b}_2^{{\beta }_{2}p}{L}_2}{\mathrm{\Gamma }({\beta }_1+1)\mathrm{\Gamma }({\beta }_2+1)}}\le {\displaystyle \frac{\mathsf{\ell }}{2}}$, (since $a^{ip}{b}^{jp}$ are positive power polynomials for any $a,b\in \mathbb{R}$ and $i,j\in (0,1]$). Taking $\widehat{a}=min\{a,a_1,a_2\}$ and $\widehat{b}=min\{b,b_1,b_2\}$ and denoting $\widehat{\varpi }=[0,\widehat{a}]\times [0,\widehat{b}]\subset \varpi$, we obtain

$${\displaystyle \frac{p^{-({\alpha }_1+{\alpha }_2)}{\widehat{a}}^{{\alpha }_{1}p}{\widehat{b}}^{{\alpha }_{2}p}{L}_1}{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}}\le {\displaystyle \frac{\mathsf{\ell }}{2}},{\displaystyle \frac{p^{-({\beta }_1+{\beta }_2)}{\widehat{a}}^{{\beta }_{1}p}{\widehat{b}}^{{\beta }_{2}p}{L}_2}{\mathrm{\Gamma }({\beta }_1+1)\mathrm{\Gamma }({\beta }_2+1)}}\le {\displaystyle \frac{\mathsf{\ell }}{2}}.$$

(21)

We first show that ${\tilde{\aleph }}_1(h_1,h_2)$ is a self-mapping on $\overline{N}$, i.e., ${\tilde{\aleph }}_1(\overline{N})\subset \overline{N}$. According to (20), for any $(h_1,h_2)\in \overline{N}$, we have

$$d_{\infty }\left({\tilde{\aleph }}_1(h_1,h_2),\widehat{0}\right)=max\left\{d_{\infty }\left({\tilde{W}}_1(h_1,h_2),\widehat{0}\right),d_{\infty }\left({\tilde{Q}}_1(h_1,h_2),\widehat{0}\right)\right\}.$$

(22)

Using Lemma 1 (i), one obtains

$$\begin{array}{c}{d}_{\infty }\left({\tilde{W}}_1(h_1(x,y),h_2(x,y),\widehat{0}\right)\hfill \\ \le d_{\infty }(\mathrm{\Psi }(x,y),\widehat{0})+{\displaystyle \frac{p^{2-({\alpha }_1+{\alpha }_2)}}{\mathrm{\Gamma }({\alpha }_1)\mathrm{\Gamma }({\alpha }_2)}}\underset{0}{\overset{x}{\int }}\underset{0}{\overset{y}{\int }}{(x^p-s^p)}^{{\alpha }_1-1}{(y^p-t^p)}^{{\alpha }_2-1}\hfill \\ ·s^{p-1}{t}^{p-1}{d}_{\infty }\left({\widehat{f}}_1(s,t,h_1(s,t),h_2(s,t)),\widehat{0}\right)dtds\hfill \\ \le D(\mathrm{\Psi },\widehat{0})+{\displaystyle \frac{p^{-({\alpha }_1+{\alpha }_2)}{\widehat{a}}^{{\alpha }_{1}p}{\widehat{b}}^{{\alpha }_{2}p}{L}_1}{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}}.\hfill \end{array}$$

(23)

From $\mathrm{\Psi }\in C(\varpi \times B(\widehat{0},{\displaystyle \frac{\mathsf{\ell }}{2}}))$, we obtain $D(\mathrm{\Psi },\widehat{0})\le {\displaystyle \frac{\mathsf{\ell }}{2}}$. Based on $D(\mathrm{\Psi },\widehat{0})\le {\displaystyle \frac{\mathsf{\ell }}{2}}$. Substituting into (21) and (23) yields

$$d_{\infty }\left({\tilde{W}}_1(h_1,h_2),\widehat{0}\right)\le {\displaystyle \frac{\mathsf{\ell }}{2}}+{\displaystyle \frac{\mathsf{\ell }}{2}}=\mathsf{\ell }.$$

(24)

Similarly,

$$d_{\infty }\left({\tilde{Q}}_1(h_1,h_2),\widehat{0}\right)\le \mathsf{\ell }.$$

(25)

Combining (24), (25) and (22), yields $H\left({\tilde{\aleph }}_1(h_1,h_2),\widehat{0}\right)\le \mathsf{\ell }$; hence, ${\tilde{\aleph }}_1(h_1,h_2)\in \overline{N}$.

We now prove the continuity of ${\tilde{\aleph }}_1$. Let $(h_{1n},h_{2n})$ tend to $(h_1,h_2)$ in $\overline{N}$. Using Lemma 1 (i), we have

$$\begin{array}{cc}{d}_{\infty }\hfill & \left({\tilde{W}}_1(h_{1n},h_{2n}),{\tilde{W}}_1(h_1,h_2)\right)\hfill \\ \le \hfill & 0+{\displaystyle \frac{p^{2-({\alpha }_1+{\alpha }_2)}}{\mathrm{\Gamma }({\alpha }_1)\mathrm{\Gamma }({\alpha }_2)}}\underset{0}{\overset{x}{\int }}\underset{0}{\overset{y}{\int }}{(x^p-s^p)}^{{\alpha }_1-1}{(y^p-t^p)}^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}\hfill \\ & \times d_{\infty }\left({\widehat{f}}_1(s,t,h_{1n}(s,t),h_{2n}(s,t)),{\widehat{f}}_1(s,t,h_1(s,t),h_2(s,t))\right)dtds\hfill \\ \le \hfill & {\displaystyle \frac{p^{-({\alpha }_1+{\alpha }_2)}{\widehat{a}}^{{\alpha }_{1}p}{\widehat{b}}^{{\alpha }_{2}p}}{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}}\hfill \\ & \times \underset{(s,t)\in \widehat{\varpi }}{sup}{d}_{\infty }\left({\widehat{f}}_1(s,t,h_{1n}(s,t),h_{2n}(s,t)),{\widehat{f}}_1(s,t,h_1(s,t),h_2(s,t))\right).\hfill \end{array}$$

(26)

The compactness and continuity of ${\widehat{f}}_1$ imply the continuity of ${\tilde{W}}_1$. Similarly,

$$\begin{array}{cc}{d}_{\infty }\hfill & \left({\tilde{Q}}_1(h_{1n},h_{2n}),{\tilde{Q}}_1(h_1,h_2)\right)\hfill \\ \le \hfill & {\displaystyle \frac{p^{-({\beta }_1+{\beta }_2)}{\widehat{a}}^{{\beta }_{1}p}{\widehat{b}}^{{\beta }_{2}p}}{\mathrm{\Gamma }({\beta }_1+1)\mathrm{\Gamma }({\beta }_2+1)}}\hfill \\ & \times \underset{(s,t)\in \widehat{\varpi }}{sup}{d}_{\infty }\left({\widehat{f}}_2(s,t,h_{1n}(s,t),h_{2n}(s,t)),{\widehat{f}}_2(s,t,h_1(s,t),h_2(s,t))\right),\hfill \end{array}$$

(27)

Hence, ${\tilde{Q}}_1$ is also continuous. Combining (26), (27) and (20) yields $d_{\infty }\left({\tilde{\aleph }}_1(h_{1n},h_{2n}),\right.\left.{\tilde{\aleph }}_1(h_1,h_2)\right)\to 0.$ Thus, the proof is concluded. □

The second step is to demonstrate that bounded sets are mapped by the operator into relatively compact sets.

Lemma 7.

Under the hypotheses of Lemma 6, if Ψ and Φ are compactly supported, Then the image ${\tilde{\aleph }}_1(\overline{N})$ is relatively compact in $C(\widehat{\varpi },M_c^1)\times C(\widehat{\varpi },M_c^2)$.

Proof. 

We present the argument in two stages.

Step 1: First, we establish the equicontinuity of ${\tilde{\aleph }}_1(\overline{N})$ in $C(\widehat{\varpi },M_c^1)\times C(\widehat{\varpi },M_c^2)$. For any $(x_1,y_1),(x_2,y_2)\in \widehat{\varpi }$ with $x_1<x_2$, $y_1<y_2$, and each $(h_1,h_2)\in \overline{N}$, let $s_1=x_1^p-s^p$, $s_2=x_2^p-s^p$, $t_1=y_1^p-t^p$, $t_2=y_2^p-t^p$; then, we have

$$\begin{array}{cc}& d_{\infty }({\int }_0^{x_1}{\int }_0^{y_1}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\hfill \\ & {\int }_0^{x_1}{\int }_0^{y_1}{s}_1^{{\alpha }_1-1}{t}_1^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds)\hfill \\ & =d_{\infty }({\int }_0^{x_1}{\int }_0^{y_1}(s_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}-s_1^{{\alpha }_1-1}{t}_1^{{\alpha }_2-1})s^{p-1}{t}^{p-1}·{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\hfill \\ & {\int }_0^{x_1}{\int }_0^{y_1}(s_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}-s_1^{{\alpha }_1-1}{t}_1^{{\alpha }_2-1})s^{p-1}{t}^{p-1}·\widehat{0}dtds)\hfill \\ & \le L_1·{\int }_0^{x_1}{\int }_0^{y_1}(s_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}-s_1^{{\alpha }_1-1}{t}_1^{{\alpha }_2-1})s^{p-1}{t}^{p-1}dtds,\hfill \end{array}$$

(28)

$$\begin{array}{cc}& d_{\infty }\left({\int }_{x_1}^{x_2}{\int }_0^{y_1}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\widehat{0}\right)\hfill \\ & \le L_1·{\int }_{x_1}^{x_2}{\int }_0^{y_1}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}dtds,\hfill \end{array}$$

(29)

$$\begin{array}{cc}& d_{\infty }\left({\int }_0^{x_1}{\int }_{y_1}^{y_2}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\widehat{0}\right)\hfill \\ & \le L_1·{\int }_0^{x_1}{\int }_{y_1}^{y_2}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}dtds,\hfill \end{array}$$

(30)

and

$$\begin{array}{cc}& d_{\infty }\left({\int }_{x_1}^{x_2}{\int }_{y_1}^{y_2}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\widehat{0}\right)\hfill \\ & \le L_1·{\int }_{x_1}^{x_2}{\int }_{y_1}^{y_2}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}dtds.\hfill \end{array}$$

(31)

Then, using (28)–(31), one obtains

$$\begin{array}{cc}{d}_{\infty }\hfill & ({\int }_0^{x_2}{\int }_0^{y_2}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\hfill \\ & {\int }_0^{x_1}{\int }_0^{y_1}{s}_1^{{\alpha }_1-1}{t}_1^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds)\hfill \\ \le \hfill & d_{\infty }({\int }_0^{x_1}{\int }_0^{y_1}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\hfill \\ & {\int }_0^{x_1}{\int }_0^{y_1}{s}_1^{{\alpha }_1-1}{t}_1^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds)\hfill \\ & +d_{\infty }\left({\int }_{x_1}^{x_2}{\int }_0^{y_1}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\widehat{0}\right)\hfill \\ & +d_{\infty }\left({\int }_0^{x_1}{\int }_{y_1}^{y_2}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\widehat{0}\right)\hfill \\ & +d_{\infty }\left({\int }_{x_1}^{x_2}{\int }_{y_1}^{y_2}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\widehat{0}\right)\hfill \\ \le \hfill & {\displaystyle \frac{L_1}{{\alpha }_1{\alpha }_2{p}^2}}\left(x_2^{{\alpha }_{1}p}{y}_2^{{\alpha }_{2}p}-x_1^{{\alpha }_{1}p}{y}_1^{{\alpha }_{2}p}\right),\hfill \end{array}$$

by virtue of Lemma 1 (i), we obtain

$$\begin{array}{c}{d}_{\infty }\left({\tilde{W}}_1(h_1(x_2,y_2),h_2(x_2,y_2)),{\tilde{W}}_1(h_1(x_1,y_1),h_2(x_1,y_1))\right)\hfill \\ \le d_{\infty }(\mathrm{\Psi }((x_2,y_2),\mathrm{\Psi })(x_1,y_1))+{\displaystyle \frac{p^{2-({\alpha }_1+{\alpha }_2)}}{\mathrm{\Gamma }({\alpha }_1)\mathrm{\Gamma }({\alpha }_2)}}\hfill \\ \times d_{\infty }({\int }_0^{x_2}{\int }_0^{y_2}{s}_2^{{\alpha }_1-1}{t}_2^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds,\hfill \\ {\int }_0^{x_1}{\int }_0^{y_1}{s}_1^{{\alpha }_1-1}{t}_1^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{\widehat{f}}_1\left(s,t,h_1(s,t),h_2(s,t)\right)dtds)\hfill \\ \le d_{\infty }\left(\mathrm{\Psi }((x_2,y_2),\mathrm{\Psi }(x_1,y_1))\right)+{\displaystyle \frac{p^{-({\alpha }_1+{\alpha }_2)}{L}_1}{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}}\left(x_2^{{\alpha }_{1}p}{y}_2^{{\alpha }_{2}p}-x_1^{{\alpha }_{1}p}{y}_1^{{\alpha }_{2}p}\right).\hfill \end{array}$$

The continuity of $\mathrm{\Psi }$ yields

$$\begin{array}{cc}\underset{\left(x_1,y_1\right)\to (x_2,y_2)}{lim}{d}_{\infty }\hfill & \left({\tilde{W}}_1(h_1(x_2,y_2),h_2(x_2,y_2)),\right.\left.{\tilde{W}}_1(h_1(x_1,y_1),h_2(x_1,y_1))\right)=0.\hfill \end{array}$$

(32)

Similarly, for $(x_1,y_1)\to (x_2,y_2)$,

$$\begin{array}{cc}& d_{\infty }\left({\tilde{Q}}_1(h_1(x_2,y_2),h_2(x_2,y_2)),{\tilde{Q}}_1(h_1(x_1,y_1),h_2(x_1,y_1))\right)\hfill \\ & \le d_{\infty }(\mathrm{\Phi }(x_2,y_2),\mathrm{\Phi }(x_1,y_1))+{\displaystyle \frac{p^{-({\beta }_1+{\beta }_2)}{L}_2}{\mathrm{\Gamma }({\beta }_1+1)\mathrm{\Gamma }({\beta }_2+1)}}\left(x_2^{{\beta }_{1}p}{y}_2^{{\beta }_{2}p}-x_1^{{\beta }_{1}p}{y}_1^{{\beta }_{2}p}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\underset{(x_1,y_1)\to (x_2,y_2)}{lim}\hfill & d_{\infty }(\mathrm{\Phi }(x_2,y_2),\mathrm{\Phi }(x_1,y_1))\hfill \\ & +{\displaystyle \frac{p^{-({\beta }_1+{\beta }_2)}{L}_2}{\mathrm{\Gamma }({\beta }_1+1)\mathrm{\Gamma }({\beta }_2+1)}}\left(x_2^{{\beta }_{1}p}{y}_2^{{\beta }_{2}p}-x_1^{{\beta }_{1}p}{y}_1^{{\beta }_{2}p}\right)=0,\hfill \end{array}$$

(33)

which follows from the continuity of $\mathrm{\Psi }$. Combining (6), (20), (32) and (33) yields $d_{\infty }({\tilde{\aleph }}_1(h_1(x_2,y_2),h_2(x_2,y_2))$, ${\tilde{\aleph }}_1(h_1(x_1,y_1),h_2(x_1,y_1)))\to 0$. Hence, ${\tilde{\aleph }}_1(\overline{N})$ is equicontinuous on $C(\widehat{\varpi },M_c^1)\times C(\widehat{\varpi },M_c^2)$.

Step 2: Next, we demonstrate the relative compactness of ${\tilde{\aleph }}_1(\overline{N})$ is relatively compact. Using Lemma 3, it is sufficient to verify that ${\tilde{\aleph }}_1(\overline{N})$ is both level-equicontinuous and compactly supported in $M_c^1\times M_c^2$.

First, verify ${\tilde{\aleph }}_1(\overline{N})$ is level-equicontinuous. For any fixed $(x,y)\in \widehat{\varpi }$, ${\tilde{\aleph }}_1(\overline{N})\in C(\widehat{\varpi },M_c^1)\times C(\widehat{\varpi },M_c^2)$. If ${({\hslash }_1,{\hslash }_2)}^T\in {\tilde{\aleph }}_1(\overline{N})$, then there exists ${(h_1,h_2)}^T\in \overline{N}$ such that

$$\begin{array}{cc}\hfill & {\hslash }_1(x,y)=\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y)),\hfill \\ \hfill & {\hslash }_2(x,y)=\mathrm{\Phi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y)).\hfill \end{array}$$

Let $\widehat{f_1}$, $\widehat{f_2}$ be compact operators (whose relevant conditions can be derived from Definitions 6–8) with $\widehat{f_1}$ relatively compact on $M_c^1$ and $\widehat{f_2}(\widehat{\varpi }\times \overline{N_{*}}$) relatively compact in $M_c^2$, where $\overline{N_{*}}:=\{ı\in C(\widehat{\varpi },M_c^2)|H(ı,\widehat{0})\le \mathsf{\ell }\}$. Using Lemma 3, $\widehat{f_1}(\widehat{\varpi }\times \overline{N})$ and $\widehat{f_2}(\widehat{\varpi }\times \overline{N_{*}})$ are level-equicontinuous. Consequently, for any $\epsilon >0$, there exists $\delta >0$ such that for all $(x,y)\in \widehat{\varpi }$ and $(h_1,h_2)\in \overline{N}$, when $|n_1-n_2|<\delta$,

$$\begin{array}{cc}& d_H\left({\left[\widehat{f_1}\left(x,y,h_1(x,y),h_2(x,y)\right)\right]}^{n_1},{\left[\widehat{f_1}\left(x,y,h_1(x,y),h_2(x,y)\right)\right]}^{n_2}\right)\hfill \\ & <{\displaystyle \frac{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}{p^{-({\alpha }_1+{\alpha }_2)}{a}^{{\alpha }_{1}p}{b}^{{\alpha }_{2}p}}}·{\displaystyle \frac{\epsilon }{2}},\hfill \\ & d_H\left({\left[\widehat{f_2}\left(x,y,h_1(x,y),h_2(x,y)\right)\right]}^{n_1},{\left[\widehat{f_2}\left(x,y,h_1(x,y),h_2(x,y)\right)\right]}^{n_2}\right)\hfill \\ & <{\displaystyle \frac{\mathrm{\Gamma }({\beta }_1+1)\mathrm{\Gamma }({\beta }_2+1)}{p^{-({\beta }_1+{\beta }_2)}{a}^{{\beta }_{1}p}{b}^{{\beta }_{2}p}}}·{\displaystyle \frac{\epsilon }{2}},\hfill \end{array}$$

and

$$\begin{array}{c}{d}_H\left({\left[\mathrm{\Psi }(x,y)\right]}^{n_1},{\left[\mathrm{\Psi }(x,y)\right]}^{n_2}\right)\le {\displaystyle \frac{\epsilon }{2}}{d}_H\left({\left[\mathrm{\Phi }(x,y)\right]}^{n_1},{\left[\mathrm{\Phi }(x,y)\right]}^{n_2}\right)\le {\displaystyle \frac{\epsilon }{2}}.\hfill \end{array}$$

Since

$$\begin{array}{cc}& d_H\left({\left[{\hslash }_1\right]}^{n_1},{\left[{\hslash }_2\right]}^{n_1}\right)=d_H\left({\left[{\tilde{W}}_1(h_1,h_2)\right]}^{n_1},{\left[{\tilde{W}}_1(h_1,h_2)\right]}^{n_2}\right)\hfill \\ & \le d_H\left({\left[\mathrm{\Psi }(x,y)\right]}^{n_1},{\left[\mathrm{\Psi }(x,y)\right]}^{n_2}\right)+{\displaystyle \frac{p^{2-({\alpha }_1+{\alpha }_2)}}{\mathrm{\Gamma }({\alpha }_1)\mathrm{\Gamma }({\alpha }_2)}}\hfill \\ & ·d_H\left(\left[{\int }_0^x{\int }_0^y{(x^p-s^p)}^{{\alpha }_1-1}{(y^p-t^p)}^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}\right.\right.\hfill \\ & {\left.·{\widehat{f}}_1(s,t,h_1(s,t),h_2(s,t))dtds\right]}^{n_1},\left[{\int }_0^x{\int }_0^y{(x^p-s^p)}^{{\alpha }_1-1}{(y^p-t^p)}^{{\alpha }_2-1}\right.\hfill \\ & \left.{\left.·s^{p-1}{t}^{p-1}{\widehat{f}}_1(s,t,h_1(s,t),h_2(s,t))dtds\right]}^{n_2}\right)\hfill \\ & \le {\displaystyle \frac{\epsilon }{2}}+{\displaystyle \frac{\epsilon }{2}}=\epsilon ,\hfill \end{array}$$

(34)

and

$$\begin{array}{c}{d}_H\left({\left[{\hslash }_2\right]}^{n_1},{\left[{\hslash }_2\right]}^{n_2}\right)=d_H\left({\left[{\tilde{Q}}_1(h_1,h_2)\right]}^{n_1},{\left[{\tilde{Q}}_1(h_1,h_2)\right]}^{n_2}\right)<\epsilon ,\hfill \end{array}$$

(35)

hold for $|n_1-n_2|<\delta$; it follows from (34) and (35) that

$$\begin{array}{cc}& d_H\left({\left[{\tilde{\aleph }}_1(h_1,h_2){]}^{n_1},[{\tilde{\aleph }}_1(h_1,h_2)\right]}^{n_2}\right)\hfill \\ & :=max\left\{d_H\left({\left[{\tilde{W}}_1(h_1,h_2)\right]}^{n_1},{\left[{\tilde{W}}_1(h_1,h_2)\right]}^{n_2}\right),\right.\left.d_H\left({\left[{\tilde{Q}}_1(h_1,h_2)\right]}^{n_1},{\left[{\tilde{Q}}_1(h_1,h_2)\right]}^{n_2}\right)\right\}\hfill \\ & <\epsilon .\hfill \end{array}$$

(36)

This implies ${\tilde{\aleph }}_1(\overline{N})$ is level-equicontinuous on $M_c^1\times M_c^2$.

Then, verify ${\tilde{\aleph }}_1(\overline{N})$ is compactly supported in $M_c^1\times M_c^2$. Given the relative compactness of $\widehat{f_1}(\widehat{\varpi }\times \overline{N})$ and $\widehat{f_2}(\widehat{\varpi }\times \overline{N_{*}})$, Lemma 3 implies that $\widehat{f_1}(\widehat{\varpi }\times \overline{N})$ and $\widehat{f_2}(\widehat{\varpi }\times \overline{N_{*}})$ possess compact supports and are level-equicontinuous on [0, 1]. According to Definition 7, there exist compact sets $K_{11},K_{21}\subseteq \mathbb{R}$ such that $(x,y,\mathrm{\Psi },\mathrm{\Phi })\in \widehat{\varpi }\times \overline{N}$, ${\left[\widehat{f_1}(x,y,h_1(x,y),h_2(x,y))\right]}^0\subseteq K_{11}$, and ${\left[\widehat{f_2}(x,y,h_1(x,y),h_2(x,y))\right]}^0\subseteq K_{21}$ for all $(x,y,\mathrm{\Psi },\mathrm{\Phi })\in \widehat{\varpi }\times \overline{N_{*}}$.

Furthermore, the compact supports of $\mathrm{\Psi }$ and $\mathrm{\Phi }$ guarantee the existence of compact sets $K_{12},K_{22}\subseteq \mathbb{R}$ satisfying ${\left[\mathrm{\Psi }(x,y)\right]}^0\in K_{12}$, ${\left[\mathrm{\Phi }(x,y)\right]}^0\in K_{22}$. We obtain the inclusion relation:

$$\begin{array}{cc}& {\left[{\tilde{W}}_1(h_1,h_2)\right]}^0\hfill \\ & ={\left[\mathrm{\Psi }(x,y)\right]}^0+{\displaystyle \frac{p^{2-({\alpha }_1+{\alpha }_2)}}{\mathrm{\Gamma }({\alpha }_1)\mathrm{\Gamma }({\alpha }_2)}}{\int }_0^x{\int }_0^y{(x^p-s^p)}^{{\alpha }_1-1}{(y^p-t^p)}^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}\hfill \\ & ·{\left[\widehat{f_1}(s,t,h_1(s,t),h_2(s,t))\right]}^{0}dtds\hfill \\ & \subseteq K_{12}+{\displaystyle \frac{p^{-({\alpha }_1+{\alpha }_2)}{x}^{{\alpha }_{1}p}{y}^{{\alpha }_{2}p}}{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}}{K}_{11}.\hfill \end{array}$$

(37)

Since $x^{{\alpha }_{1}p}{y}^{{\alpha }_{2}p}$ is bounded over the region $\widehat{\varpi }$, there exists a compact subset $K_1\in \mathbb{R}$ such that ${\left[{\tilde{W}}_1(h_1,h_2)\right]}^0\subseteq K_1$, establishing the compact support of ${\tilde{W}}_1(\overline{N})$. Similarly, ${\left[{\tilde{Q}}_1(h_1,h_2)\right]}^0\subseteq K_{22}+{\displaystyle \frac{p^{-({\beta }_1+{\beta }_2)}{x}^{{\beta }_{1}p}{y}^{{\beta }_{2}p}}{\mathrm{\Gamma }({\beta }_1+1)\mathrm{\Gamma }({\beta }_2+1)}}{K}_{21}$ for some compact $K_2\subseteq \mathbb{R}$, proving ${\tilde{Q}}_1(\overline{N})$ is compact supported. From (18), we obtain

$$\begin{array}{c}{\left[{\tilde{\aleph }}_1(h_1,h_2)\right]}^0\subseteq \left\{K_i\subseteq \mathbb{R}|K_i\subseteq K_j,i,j=1,2\right\},\hfill \end{array}$$

(38)

confirming ${\tilde{\aleph }}_1(h_1,h_2)$ is compactly supported.

Thus, ${\tilde{\aleph }}_1(\overline{N})$ is relatively compact on $M_c^1\times M_c^2$, and by Lemma 2, and also on $C(\widehat{\varpi },M_c^1)\times C(\widehat{\varpi },M_c^2)$. □

Combining Lemmas 6 and 7 yields the following theorem.

Theorem 1.

Assume there exists $\mathsf{\ell }>0$ such that $\widehat{f_i}:\varpi \times B(\widehat{0},\mathsf{\ell })\times B(\widehat{0},\mathsf{\ell })\to M_c^i$$(i=1,2)$ is a compact operator and $\mathrm{\Psi },\mathrm{\Phi }\in C(B(\widehat{0},{\displaystyle \frac{\mathsf{\ell }}{2}}))$ are compact-supported. Then there exist $\widehat{a}\in (0,a]$ and $\widehat{b}\in (0,b]$ such that the Equation (3) admits at least one $(†)$-weak solution on $C(\widehat{\varpi },M_c^1)\times C(\widehat{\varpi },M_c^2)$, where $\widehat{\varpi }=[0,\widehat{a}]\times [0,\widehat{b}]\subset \varpi$.

Proof. 

Define the operator ${\tilde{\aleph }}_1:\overline{N}\to C(\widehat{\varpi },M_c^1)\times C(\widehat{\varpi },M_c^2)$ as in (18), and define the operators ${\tilde{W}}_1:\overline{N}\to C(\widehat{\varpi },M_c^1)$ and ${\tilde{Q}}_1:\overline{N}\to C(\widehat{\varpi },M_c^2)$ according to (19). It is straightforward to check that ${\tilde{\aleph }}_1$ is well-defined. According to Lemma 6, ${\tilde{\aleph }}_1:\overline{N}\to \overline{N}$ is continuous. Moreover, combining Lemma 7 with Lemma 2 shows that ${\tilde{\aleph }}_1(\overline{N})$ is relatively compact. Therefore, according to Definition 9, ${\tilde{\aleph }}_1$ is a compact operator. Finally, by applying Lemma 5, we conclude that ${\tilde{\aleph }}_1$ has at least one fixed point in $\overline{N}$, which yields a $(†)$-weak solution of (3). □

Remark 6.

Our proof of Theorem 1 via Schauder fixed point theorem differs fundamentally from the methods in [2] and addresses a model that generalizes [2] into a more complex coupled fractional system. Moreover, system (3) extends the model in [15] by allowing the nonlinear term in its second equation to depend on both unknown functions. This approach establishes the existence of a broader class of systems and significantly expands the scope of the underlying methodology.

We now demonstrate the existence of a $(‡)$-weak solution to Equation (3) based on the subsequent hypotheses:

$({\mathbf{J}}_{\mathbf{1}})$${\widehat{C}}_{\mathrm{\Psi }}^{\widehat{f_1}}(\varpi ,M_c^1)\ne ⌀$, ${\widehat{C}}_{\mathrm{\Phi }}^{\widehat{f_2}}(\varpi ,M_c^2)\ne ⌀$.

$({\mathbf{J}}_{\mathbf{2}})$ If $h_1\in {\widehat{C}}_{\mathrm{\Psi }}^{\widehat{f_1}}(\varpi ,M_c^1)$, then for all $(x,y)\in \varpi =[0,a]\times [0,b]$ and each $h_2\in {\widehat{C}}_{\mathrm{\Phi }}^{\widehat{f_2}}(\varpi ,M_c^2)$, the following inclusions hold:

$$\begin{array}{cc}& H_1(x,y)=\mathrm{\Psi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y))\in {\widehat{C}}_{\mathrm{\Psi }}^{\widehat{f_1}}(\varpi ,M_c^1),\hfill \\ & H_2(x,y)=\mathrm{\Phi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0+}^{\mathit{\beta },p}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y))\in {\widehat{C}}_{\mathrm{\Phi }}^{\widehat{f_2}}(\varpi ,M_c^2).\hfill \end{array}$$

Lemma 8.

Assume hypotheses   $({\mathbf{J}}_{\mathbf{1}})$  and  $({\mathbf{J}}_{\mathbf{2}})$ hold, and there exists $\mathsf{\ell }>0$ such that

(i)

For $i=1,2$, $\widehat{f_i}:\varpi \times B(\widehat{0},\mathsf{\ell })\times B(\widehat{0},\mathsf{\ell })\to M_c^i$ is a compact operator.

(ii)

$\mathrm{\Psi },\mathrm{\Phi }\in C(B(\widehat{0},{\displaystyle \frac{\mathsf{\ell }}{2}}))$.

• Consequently, one can select parameters $\widehat{a}\in [0,a]$ and $\widehat{b}\in [0,b]$ such that the operator ${\tilde{\aleph }}_2(h_1,h_2)=\left(\begin{array}{c}{\tilde{W}}_2(h_1,h_2)\\ {\tilde{Q}}_2(h_1,h_2)\end{array}\right)$ constitutes a continuous operator from $\overline{N}$ to itself, where for $(x,y)\in \widehat{\varpi }=[0,\widehat{a}]\times [0,\widehat{b}]\subset \varpi$,

  $$\begin{array}{cc}\hfill & {\tilde{W}}_2(h_1,h_2):=\mathrm{\Psi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y)),\hfill \\ \hfill & {\tilde{Q}}_2(h_1,h_2):=\mathrm{\Phi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y)),\hfill \end{array}$$

  with $\overline{N}:=\left\{(h_1,h_2)\in {\widehat{C}}_{\mathrm{\Psi }}^{\widehat{f_1}}(\widehat{\varpi },M_c^1)\times {\widehat{C}}_{\mathrm{\Phi }}^{\widehat{f_2}}(\widehat{\varpi },M_c^2)\mid max\left\{H(h_1,\widehat{0}),H(h_2,\widehat{0})\right\}\right.$$\left.\le \mathsf{\ell }\right\}$.

Proof. 

Since hypotheses $({\mathbf{J}}_{\mathbf{1}})$ and $({\mathbf{J}}_{\mathbf{2}})$ hold for all $(x,y)\in \varpi$ and every $\left(\begin{array}{c}{h}_{11}(x,y)\\ h_{21}(x,y)\end{array}\right),\left(\begin{array}{c}{h}_{12}(x,y)\\ h_{22}(x,y)\end{array}\right)\in {\widehat{C}}_{\mathrm{\Psi }}^{\widehat{f_1}}(\varpi ,M_c^1)\times {\widehat{C}}_{\mathrm{\Phi }}^{\widehat{f_2}}(\varpi ,M_c^2)$, we obtain

$$\begin{array}{cc}& H({\tilde{\aleph }}_2(h_{11},h_{21}),{\tilde{\aleph }}_2(h_{12},h_{22}))\hfill \\ & =max\left\{\underset{(x,y)\in \varpi }{sup}{d}_{\infty }\left({\tilde{W}}_2(h_{11}(x,y),h_{21}(x,y)),{\tilde{W}}_2(h_{12}(x,y),h_{22}(x,y))\right),\right.\hfill \\ & \left.\underset{(x,y)\in \varpi }{sup}{d}_{\infty }\left({\tilde{Q}}_2(h_{11}(x,y),h_{21}(x,y)),{\tilde{Q}}_2(h_{12}(x,y),h_{22}(x,y))\right)\right\}.\hfill \end{array}$$

As the proof follows identical reasoning to Lemma 6, the detailed derivation is omitted here. □

By adapting the proof technique of Lemma 7, we arrive at the subsequent result.

Lemma 9

([15]). Suppose that the hypotheses of Lemma 8 are satisfied and that the functions are of compact support. Then, the set is relatively compact in ${\widehat{C}}_{\mathrm{\Psi }}^{\widehat{f_1}}(\varpi ,M_c^1)\times {\widehat{C}}_{\mathrm{\Phi }}^{\widehat{f_2}}(\varpi ,M_c^2)$.

Theorem 2.

If all conditions of Theorem 1 hold and $({\mathbf{J}}_{\mathbf{1}})$ and $({\mathbf{J}}_{\mathbf{2}})$ are satisfied, then there exist $\widehat{a}\in (0,a]$, $\widehat{b}\in (0,b]$ such that (3) admits at least one $(‡)$-weak solution on ${\widehat{C}}_{\mathrm{\Psi }}^{\widehat{f_1}}(\varpi ,M_c^1)\times {\widehat{C}}_{\mathrm{\Phi }}^{\widehat{f_2}}(\varpi ,M_c^2)$, where $\widehat{\varpi }$ is as defined in Lemma 8.

Proof. 

The proof follows identical reasoning to Theorem 1 and is therefore omitted. □

Remark 7.

Inspired by [15], this paper proves Theorem 2 without Lipschitz continuity using Lemmas 2 and 8. The proof differs from that in [14], as the stronger coupling in our $(‡)$-weak solutions introduces greater technical difficulty.

4. Numerical Example with Potential Applications

In the sequel, we present the following numerical example with potential applications to verify our main results: for each $(x,y)\in \varpi =[0,a]\times [0,a]$ and $k=1,2$,

$$\left\{\begin{array}{c}\begin{array}{c}{}_{gh}{}^{CK}D_k^{\frac{1}{2},2}{h}_1(x,y)={\displaystyle \frac{{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}{8{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}}{x}^{-1}{y}^{-1}{h}_1(x,y)+{\displaystyle \frac{P{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}{4{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}}-P\pi \hfill \end{array}\hfill \\ +{\displaystyle \frac{{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}{2{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}}{x}^{-\frac{5}{2}}{y}^{-\frac{5}{2}}{h}_2(x,y)+{\displaystyle \frac{5P{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}{2{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}}{x}^{-\frac{5}{2}}{y}^{-\frac{5}{2}},\hfill \\ {}_{gh}{}^{CK}D_k^{\frac{1}{2},2}{h}_2(x,y)={\displaystyle \frac{9{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}{2{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}}{h}_1(x,y)-18x^{-\frac{1}{2}}{y}^{-\frac{1}{2}}{h}_2(x,y)-90P{x}^{-\frac{1}{2}}{y}^{-\frac{1}{2}}\hfill \\ -{\displaystyle \frac{9P{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}{{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}},\hfill \\ h_1(x,0)=h_1(0,y)=h_1(0,0)=2P,\hfill \\ h_2(x,0)=h_2(0,y)=h_2(0,0)=-5P,\hfill \end{array}\right.$$

(39)

in which $h_1(x,y)$ and $h_2(x,y)$ are defined as fuzzy-valued functions, and P is a fuzzy number. Corresponding to the system (3), it is easily verified that the functions $\widehat{f_1}(x,y,h_1(x,y),h_2(x,y))$$:={\displaystyle \frac{{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}{8{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}}·x^{-1}{y}^{-1}{h}_1(x,y)$$+{\displaystyle \frac{{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}{2{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}}{x}^{-\frac{5}{2}}{y}^{-\frac{5}{2}}{h}_2(x,y)+{\displaystyle \frac{5P{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}{2{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}}{x}^{-\frac{5}{2}}{y}^{-\frac{5}{2}}+{\displaystyle \frac{P{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}{4{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}}-P\pi$ and $\widehat{f_2}(x,y,h_1(x,y),h_2(x,y)):={\displaystyle \frac{9{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}{2{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}}$$h_1(x,y)-18x^{-\frac{1}{2}}{y}^{-\frac{1}{2}}{h}_2(x,y)-90P{x}^{-\frac{1}{2}}{y}^{-\frac{1}{2}}-{\displaystyle \frac{9P{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}{{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}}$ are compact operators in (39). Furthermore, from (12), we immediately obtain $\mathrm{\Psi }(x,y)=2P$ and $\mathrm{\Phi }(x,y)=-5P$.

We note that in [2], stated that the prey–predator system (1) is connected to ecological models, a relationship attributed to shared features of memory and fractal behavior. Now, we extend model (1) to FFPDE (i.e., (39)), where the right-hand side of the equations is further generalized to represent a multi-species biological population model under uncertain environments and consider the symmetry coupled system (39) to verify the existence of solutions to (3).

Let $P=(3,4,5)$ be a triangular fuzzy number. From [16], its $\varrho$-level set is given by

$$\begin{array}{c}{\left[P\right]}^{{\varrho }_1}=[P_{{\varrho }_1}^{-},P_{{\varrho }_1}^{+}]=[{\varrho }_1+3,5-{\varrho }_1],{\left[P\right]}^{{\varrho }_2}=[P_{{\varrho }_2}^{-},P_{{\varrho }_2}^{+}]=[{\varrho }_2+3,5-{\varrho }_2].\hfill \end{array}$$

Consequently, we obtain

$$\begin{array}{cc}& {\left[\mathrm{\Psi }(x,y)\right]}^{{\varrho }_1}={\left[2P\right]}^{{\varrho }_1}=2[{\varrho }_1+3,5-{\varrho }_1]=[2{\varrho }_1+6,10-2{\varrho }_1],\hfill \\ & {\left[\mathrm{\Phi }(x,y)\right]}^{{\varrho }_2}={[-5P]}^{{\varrho }_2}=[-5{\varrho }_2-15,-25+5{\varrho }_2].\hfill \end{array}$$

According to Definition 7, one has

$$\begin{array}{c}{\left[\mathrm{\Psi }(x,y)\right]}^0=2[3,5]\subseteq [6,10],{\left[\mathrm{\Phi }(x,y)\right]}^0=(-5)[3,5]=[-25,-15].\hfill \end{array}$$

Let $K_1=[6,10]$ and $K_2=[-25,-15]$ be compact sets, implying that $\mathrm{\Psi }(x,y)$ and $\mathrm{\Phi }(x,y)$ possess compact supports. Define ${\mathsf{\ell }}_1=4D(P,\widehat{0}),{\mathsf{\ell }}_2=10D(P,\widehat{0})$, and take $\mathsf{\ell }=max\{{\mathsf{\ell }}_1,{\mathsf{\ell }}_2\}=10D(P,\widehat{0})$. According to the metric properties: $D(\mathrm{\Psi }(x,y),\widehat{0})=D(2P,\widehat{0})=2D(P,\widehat{0})\le {\displaystyle \frac{{\mathsf{\ell }}_1}{2}}$, $H(\mathrm{\Phi }(x,y),\widehat{0})=D(-5P,\widehat{0})=5D(P,\widehat{0})\le {\displaystyle \frac{{\mathsf{\ell }}_2}{2}}$. Hence, it is established that $\mathrm{\Psi }(x,y)$, $\mathrm{\Phi }(x,y)\in C\left(\varpi ,B\left(\widehat{0},\frac{\mathsf{\ell }}{2}\right)\right)$.

(Case I) For $k=1$, applying the Buckley–Feuring (BF) fuzzification strategy along with [Definition 4.1] [42], and combining Theorem 1 with the compact support and continuity results, the $(†)$-weak solution of system (39) is obtained as:

$$\left\{\begin{array}{c}{h}_1(x,y)=2P-2P{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}-2Pxy,\hfill \\ h_2(x,y)=-5P-{\displaystyle \frac{1}{2}}P{x}^{\frac{3}{2}}{y}^{\frac{3}{2}}.\hfill \end{array}\right.$$

(Case II) For k = 2, based on Lemma 1 (iii)–(v) and (17), while adopting a strategy analogous to (Case I), the BF solution of (39) is derived as

$$(h_1(x,y),h_2(x,y))=(2P\ominus 2P{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}\ominus 2Pxy,-5P\ominus {\displaystyle \frac{1}{2}}P{x}^{\frac{3}{2}}{y}^{\frac{3}{2}}).$$

According to the continuity of the extension principle, the level sets of the fuzzy solutions to (39) are

$$\begin{array}{cc}& {[\Delta (x,y,P)]}^{{\varrho }_1}=\left[2({\varrho }_1+3)-2({\varrho }_1+3)x^{\frac{1}{2}}{y}^{\frac{1}{2}}-2({\varrho }_1+3)xy,\right.\hfill \\ & \left.2(5-{\varrho }_1)-2(5-{\varrho }_1)x^{\frac{1}{2}}{y}^{\frac{1}{2}}-2(5-{\varrho }_1)xy\right],\hfill \end{array}$$

(40)

$$\begin{array}{cc}& {\left[\mathrm{\Theta }(x,y,P)\right]}^{{\varrho }_2}=\left[-5({\varrho }_2+3)-{\displaystyle \frac{1}{2}}({\varrho }_2+3)x^{\frac{3}{2}}{y}^{\frac{3}{2}},\right.\left.-5(5-{\varrho }_2)-{\displaystyle \frac{1}{2}}(5-{\varrho }_2)x^{\frac{3}{2}}{y}^{\frac{3}{2}}\right].\hfill \end{array}$$

(41)

Now, we present the numerical simulations of (40) and (41). First, we fix the variable x; from this, along with (40) and (41), we can derive the relevant data points. The command function “polt” of MATLAB R2024a (MathWorks, Natick, MA, USA) was used and all code was executed on a Windows 11 system equipped with an AMD Ryzen 7 8845H processor with Radeon 780M Graphics (Advanced Micro Devices, Santa Clara, CA, USA, base frequency 3.80 GHz); simulations were then performed based on these data points. Due to the vast amount of data, only a relatively critical subset of data points is provided here, as shown in

Table 1. Partial data of $\Delta (x,y,P)$ level set.

x	y	${\mathit{\varrho }}_1$	$\mathit{Upper}$	$\mathit{Lower}$
0.6	0.4	0.88	2.095991919	2.225640904
0.6	0.4	0.89	2.10139396	2.220238863
0.6	0.4	0.9	2.106796001	2.214836822
0.6	0.4	0.91	2.112198042	2.209434781
0.6	0.4	0.92	2.117600083	2.20403274
0.6	0.4	0.93	2.123002124	2.198630699
0.6	0.4	0.94	2.128404165	2.193228658
0.6	0.4	0.95	2.133806206	2.187826617
0.6	0.4	0.96	2.139208247	2.182424576
0.6	0.4	0.97	2.144610288	2.177022535
0.6	0.4	0.98	2.150012329	2.171620494
0.6	0.4	0.99	2.155414371	2.166218453
0.6	0.4	1	2.160816412	2.160816412
0.6	0.5	0	0.913664655	1.522774425
0.6	0.5	0.01	0.916710204	1.519728876
0.6	0.5	0.02	0.919755753	1.516683327
0.6	0.5	0.03	0.922801302	1.513637778
0.6	0.5	0.04	0.92584685	1.51059223
0.6	0.5	0.05	0.928892399	1.507546681
0.6	0.5	0.06	0.931937948	1.504501132

and

Table 2. Partial data of $\mathrm{\Theta }(x,y,P)$ level set.

x	y	${\mathit{\varrho }}_2$	$\mathit{Upper}$	$\mathit{Lower}$
0.6	0.4	0.88	−19.62809648	−20.84220555
0.6	0.4	0.89	−19.67868436	−20.79161767
0.6	0.4	0.9	−19.72927224	−20.74102979
0.6	0.4	0.91	−19.77986012	−20.69044191
0.6	0.4	0.92	−19.830448	−20.63985404
0.6	0.4	0.93	−19.88103587	−20.58926616
0.6	0.4	0.94	−19.93162375	−20.53867828
0.6	0.4	0.95	−19.98221163	−20.4880904
0.6	0.4	0.96	−20.03279951	−20.43750253
0.6	0.4	0.97	−20.08338738	−20.38691465
0.6	0.4	0.98	−20.13397526	−20.33632677
0.6	0.4	0.99	−20.18456314	−20.28573889
0.6	0.4	1	−20.23515102	−20.23515102
0.6	0.45	0	−15.21044417	−25.35074029
0.6	0.45	0.01	−15.26114565	−25.30003881
0.6	0.45	0.02	−15.31184713	−25.24933733
0.6	0.45	0.03	−15.36254861	−25.19863585
0.6	0.45	0.04	−15.4132501	−25.14793437
0.6	0.45	0.05	−15.46395158	−25.09723289
0.6	0.45	0.06	−15.51465306	−25.04653141

below:

Based on all data points, the three−dimensional (3D) visualization plots of $\mathrm{\Theta }(x,y,P)$ and $\Delta (x,y,P)$ were obtained as shown below:

Figure 1 presents the simulation results of the level sets for the fuzzy solutions in (40) and (41). The left and right subfigures show seven level sets of $\mathrm{\Theta }(x,y,P)$ and $\Delta (x,y,P)$ at seven fixed x values, respectively. Each surface group corresponds to a ${\varrho }_1$ or ${\varrho }_2$ level set, with inter-surface distances characterizing the fuzzy solutions. When seven y values are fixed, the variations in level sets with x and $\varrho$ are consistent with Figure 1 The curves in the $\varrho Oy$ planes represent contour lines of ${\left[\mathrm{\Theta }(x,y,P)\right]}^{{\varrho }_2}$ and ${[\Delta (x,y,P)]}^{{\varrho }_1}$, respectively.

To verify $({\mathbf{J}}_{\mathbf{1}})-({\mathbf{J}}_{\mathbf{1}})$, let $L_1={\displaystyle \frac{{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}{{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}}$, $L_2={\displaystyle \frac{{\left[\mathrm{\Gamma }(\frac{3}{4})\right]}^2}{{\left[\mathrm{\Gamma }(\frac{1}{4})\right]}^2}}$. Since

$$\begin{array}{cc}& {\left[h_1\right]}^{{\varrho }_2}={\left[2P\ominus 2P{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}\ominus 2Pxy\right]}^{{\varrho }_2}=\left(2-2x^{\frac{1}{2}}{y}^{\frac{1}{2}}-2xy\right)[{\varrho }_2+3,5-{\varrho }_2],\hfill \end{array}$$

(42)

$$\begin{array}{cc}& {\left[h_2\right]}^{{\varrho }_1}={\left[-5P\ominus \frac{1}{2}P{x}^{\frac{3}{2}}{y}^{\frac{3}{2}}\right]}^{{\varrho }_1}=\left(-5-{\displaystyle \frac{1}{2}}{x}^{\frac{3}{2}}{y}^{\frac{3}{2}}\right)[{\varrho }_1+3,5-{\varrho }_1],\hfill \end{array}$$

(43)

we obtain

$$\begin{array}{cc}& len{\left[h_1\right]}^{{\varrho }_2}=(2-2{\varrho }_2)\left(2-2x^{\frac{1}{2}}{y}^{\frac{1}{2}}-2xy\right),\hfill \\ & len{\left[h_2\right]}^{{\varrho }_1}=(2-2{\varrho }_1)\left(-5-{\displaystyle \frac{1}{2}}{x}^{\frac{3}{2}}{y}^{\frac{3}{2}}\right),\hfill \end{array}$$

and using (7), one has

$$\begin{array}{cc}& len{\left[(-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y))\right]}^{{\varrho }_2}\hfill \\ & =L_2(-72x^{\frac{5}{2}}{y}^{\frac{5}{2}}+{\displaystyle \frac{18}{\pi }}{x}^2{y}^2+{\displaystyle \frac{18}{\pi }}{x}^{\frac{3}{2}}{y}^{\frac{3}{2}})·len{\left[A\right]}^{{\varrho }_2}\hfill \\ & \ge -0.414(2-2{\varrho }_2),\hfill \\ & len{\left[\mathrm{\Phi }(x,y)\right]}^{{\varrho }_2}=-5(2-2{\varrho }_2).\hfill \end{array}$$

Hence,

$$len{\left[\mathrm{\Phi }(x,y)\right]}^{{\varrho }_2}\le len{\left[(-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y))\right]}^{{\varrho }_2}.$$

According to Proposition 21 (b) of [37], the H-difference $\mathrm{\Phi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}$$\widehat{f_2}(x,y,h_1(x,y),h_2(x,y))$ exists. Taking

$$H_1(x,y)=\mathrm{\Psi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y)),$$

its level set is

$$\begin{array}{cc}{\left[H_1(x,y)\right]}^{{\varrho }_2}=\hfill & \left(2-{\displaystyle \frac{\pi {(\mathrm{\Gamma }(\frac{1}{4}))}^2}{8{(\mathrm{\Gamma }(\frac{3}{4}))}^2}}{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}-{\displaystyle \frac{({\pi }^2-4){(\mathrm{\Gamma }(\frac{1}{4}))}^2+16\pi {(\mathrm{\Gamma }(\frac{3}{4}))}^2}{8\pi {(\mathrm{\Gamma }(\frac{3}{4}))}^2}}xy\right.\hfill \\ & \left.-{\displaystyle \frac{(1-\pi ){(\mathrm{\Gamma }(\frac{1}{4}))}^2}{8{(\mathrm{\Gamma }(\frac{3}{4}))}^2}}\right)·[{\varrho }_2+3,5-{\varrho }_2],\hfill \end{array}$$

with interval length

$$\begin{array}{cc}len{\left[H_1(x,y)\right]}^{{\varrho }_2}=\hfill & \left(2-{\displaystyle \frac{\pi {(\mathrm{\Gamma }(\frac{1}{4}))}^2}{8{(\mathrm{\Gamma }(\frac{3}{4}))}^2}}{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}-{\displaystyle \frac{({\pi }^2-4){(\mathrm{\Gamma }(\frac{1}{4}))}^2+16\pi {(\mathrm{\Gamma }(\frac{3}{4}))}^2}{8\pi {(\mathrm{\Gamma }(\frac{3}{4}))}^2}}xy\right.\hfill \\ & \left.-{\displaystyle \frac{(1-\pi ){(\mathrm{\Gamma }(\frac{1}{4}))}^2}{8{(\mathrm{\Gamma }(\frac{3}{4}))}^2}}\right)·(2-2{\varrho }_2).\hfill \end{array}$$

Using an analogous computational approach, we obtain

$${\left[(-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_2}(x,y,H_1(x,y),h_2(x,y))\right]}^{{\varrho }_2}=\frac{1}{2}{x}^{\frac{3}{2}}{y}^{\frac{3}{2}}·len{\left[P\right]}^{{\varrho }_2},$$

$$len{\left[(-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_2}(x,y,H_1(x,y),h_2(x,y))\right]}^{{\varrho }_2}\le \frac{1}{2}(2-2{\varrho }_2).$$

Hence, the H-difference $\mathrm{\Phi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_2}(x,y,H_1(x,y),h_2(x,y))$ exists.

Similarly, from (42) and (43),

$$\begin{array}{cc}& {\left[(-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y))\right]}^{{\varrho }_1}\hfill \\ & =\left({\displaystyle \frac{\pi {(\mathrm{\Gamma }(\frac{1}{4}))}^2}{8{(\mathrm{\Gamma }(\frac{3}{4}))}^2}}{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}+{\displaystyle \frac{({\pi }^2-4){(\mathrm{\Gamma }(\frac{1}{4}))}^2+16\pi {(\mathrm{\Gamma }(\frac{3}{4}))}^2}{8\pi {(\mathrm{\Gamma }(\frac{3}{4}))}^2}}xy\right.\left.+{\displaystyle \frac{(1-\pi ){(\mathrm{\Gamma }(\frac{1}{4}))}^2}{8{(\mathrm{\Gamma }(\frac{3}{4}))}^2}}\right)·{\left[P\right]}^{{\varrho }_1}.\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}& len{\left[\mathrm{\Psi }(x,y)\right]}^{{\varrho }_1}=2(2-2{\varrho }_1),\hfill \\ & len{\left[(-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y))\right]}^{{\varrho }_1}\le 1.175(2-2{\varrho }_1).\hfill \end{array}$$

This implies

$$len{\left[\mathrm{\Psi }(x,y)\right]}^{{\varrho }_1}\ge len{\left[(-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_1}(x,y,h_1(x,y),h_2(x,y))\right]}^{{\varrho }_1}.$$

According to Proposition 21(a) of [37], the H-difference ${\left[H_2(x,y)\right]}^{{\varrho }_1}=\mathrm{\Phi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y))$ exists. Using an identical methodology,

$$len{\left[H_2(x,y)\right]}^{{\varrho }_1}=\left(-5+72L_2{x}^{\frac{5}{2}}{y}^{\frac{5}{2}}-{\displaystyle \frac{18}{\pi }}{L}_2{x}^2{y}^2-{\displaystyle \frac{18}{\pi }}{L}_2{x}^{\frac{3}{2}}{y}^{\frac{3}{2}}\right)·(2-2{\varrho }_1),$$

$$len{\left[(-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_1}(x,y,h_1(x,y),H_2(x,y))\right]}^{{\varrho }_1}\le 1.175(2-2{\varrho }_1).$$

The existence of the H-difference $\mathrm{\Psi }(x,y)\ominus (-1){}_F{}^{RL}\mathcal{I}_{0^{+}}^{\frac{1}{2},2}\widehat{f_1}(x,y,h_1(x,y),H_2(x,y))$ is thereby established. This, in turn, fulfills the hypotheses $({\mathbf{J}}_{\mathbf{1}})$ and $({\mathbf{J}}_{\mathbf{2}})$ of Theorem 2. Furthermore, since $\mathrm{\Psi }(x,y)$ and $\mathrm{\Phi }(x,y)$ are compact-supported and lie in $C(\varpi ,B(\widehat{0},{\displaystyle \frac{\mathsf{\ell }}{2}}))$, Theorem 2 asserts the existence of a $(‡)$-weak solution to (39) on ${\widehat{C}}_{\mathrm{\Psi }}^{\widehat{f_1}}(\varpi ,M_c^1)\times {\widehat{C}}_{\mathrm{\Phi }}^{\widehat{f_2}}(\varpi ,M_c^2)$, which takes the following form:

$$\left\{\begin{array}{c}{h}_1(x,y)=2P\ominus 2P{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}\ominus 2Pxy,\hfill \\ h_2(x,y)=-5P\ominus \frac{1}{2}P{x}^{\frac{3}{2}}{y}^{\frac{3}{2}}.\hfill \end{array}\right.$$

5. U-HS Analysis

This section is dedicated to investigating the U-HS of solutions for system (3), with all subsequent results predicated on the assumption of their existence. The study of U-HS is not merely of theoretical interest but is critically important for practical applications in analysis. Its principal value is most apparent when addressing complex, often nonlinear, mathematical models for which deriving an exact solution is either profoundly challenging or outright impossible. In such scenarios, which are commonplace in applied mathematics, the U-HS framework provides a rigorous mathematical assurance: it guarantees the existence of an approximate solution that remains within a precisely defined bound of any hypothetical exact solution. Thus, rather than abandoning analytically intractable models, U-HS equips us with a robust methodology to obtain and justify the use of well-justified approximations, thereby bridging a crucial gap between theoretical modeling and practical computation.

Definition 10

([34]). The system (3) is said to be U-HS if there exists ϵ = $max\{{ϵ}_1,{ϵ}_2\}>0$, such that the inequality system

$$\left\{\begin{array}{c}\begin{array}{c}\left|{}_{gh}{}^{CK}D_k^{\mathit{\alpha },p}{h}_1(x,y)-\widehat{f_1}(x,y,h_1(x,y),h_2(x,y))\right|\le {ϵ}_1,\hfill \end{array}\hfill \\ \left|{}_{gh}{}^{CK}D_k^{\mathit{\beta },p}{h}_2(x,y)-\widehat{f_2}(x,y,h_1(x,y),h_2(x,y))\right|\le {ϵ}_2,\hfill \end{array}\right.$$

(44)

holds. For notational simplicity in subsequent proofs, denote $h_i(x,y)=h_i$ and ${\overline{h}}_i(x,y)={\overline{h}}_i$ for $i=1,2,3,4$. If there exists a constant $\overline{C}>0$ such that for every solution $(h_1,h_2)$ of (44), there exists a solution $({\overline{h}}_1,{\overline{h}}_2)$ of (3) satisfying

$$\left|(h_1,h_2)-({\overline{h}}_1,{\overline{h}}_2)\right|\le \overline{C}ϵ,$$

then system (3) is U-HS.

Remark 8.

$(h_1,h_2)$ is a solution of (44) if and only if for each $i=1,2$ and ${ϵ}_i$ as in Definition 8, there exists $g_i\in (\varpi ,M_c^i)$ (depending on $(h_1,h_2)$) satisfying $\left|g_i(x,y)\right|\le {ϵ}_i$ and the perturbed system

$$\left\{\begin{array}{c}\begin{array}{c}{}_{gh}{}^{CK}D_k^{\mathit{\alpha },p}{h}_1=\widehat{f_1}(x,y,h_1,h_2)+g_1(x,y),\hfill \end{array}\hfill \\ {}_{gh}{}^{CK}D_k^{\mathit{\beta },p}{h}_2=\widehat{f_2}(x,y,h_1,h_2)+g_2(x,y),\hfill \end{array}\right.$$

holds.

The following proof is provided only for type $(†)$ solutions, as the proof for type $(‡)$ solutions follows an identical procedure and is therefore omitted.

Lemma 10.

Let $(h_1,h_2)$ be a solution of (44). Then, the following inequality holds:

$$\left\{\begin{array}{c}\begin{array}{c}{h}_1-\left(\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1,h_2)\right)\le {\delta }_1{ϵ}_1,\hfill \end{array}\hfill \\ h_2-\left(\mathrm{\Phi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}\widehat{f_2}(x,y,h_1,h_2)\right)\le {\delta }_2{ϵ}_2.\hfill \end{array}\right.$$

Proof. 

Using Remark 8, the system can be written as

$$\left\{\begin{array}{c}\begin{array}{c}{}_{gh}{}^{CK}D_k^{\mathit{\alpha },p}{h}_1=\widehat{f_1}(x,y,h_1,h_2)+g_1(x,y),\hfill \end{array}\hfill \\ {}_{gh}{}^{CK}D_k^{\mathit{\beta },p}{h}_2=\widehat{f_2}(x,y,h_1,h_2)+g_2(x,y),\hfill \\ h_1(x,0)={\zeta }_1(x),h_2(x,0)={\eta }_1(x),\hfill \\ h_1(0,y)={\zeta }_2(y),h_2(0,y)={\eta }_2(y).\hfill \end{array}\right.$$

From Lemma 4, we obtain

$$\left\{\begin{array}{c}\begin{array}{c}{h}_1=\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1,h_2)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}{g}_1(x,y),\hfill \end{array}\hfill \\ h_2=\mathrm{\Phi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}\widehat{f_2}(x,y,h_1,h_2)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}{g}_2(x,y).\hfill \end{array}\right.$$

Thus,

$$\begin{array}{cc}& \left|h_1-\left(\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1,h_2)\right)\right|=\left|{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}{g}_1(x,y)\right|\hfill \\ & \le \left|{\displaystyle \frac{p^{2-{\alpha }_1-{\alpha }_2}}{\mathrm{\Gamma }({\alpha }_1)\mathrm{\Gamma }({\alpha }_2)}}\underset{0}{\overset{x}{\int }}\underset{0}{\overset{y}{\int }}{(x^p-s^p)}^{{\alpha }_1-1}{(y^p-t^p)}^{{\alpha }_2-1}{s}^{p-1}{t}^{p-1}{ϵ}_{1}dtds\right|\hfill \\ & \le {\delta }_1{ϵ}_1,\hfill \end{array}$$

where ${\delta }_1={\displaystyle \frac{p^{-({\alpha }_1+{\alpha }_2)}{x}^{{\alpha }_{1}p}{y}^{{\alpha }_{2}p}}{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}}$. Similarly,

$$\left|h_2(x,y)-\left(\mathrm{\Phi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\beta },p}\widehat{f_2}(x,y,h_1(x,y),h_2(x,y))\right)\right|\le {\delta }_2{ϵ}_2,$$

where ${\delta }_2={\displaystyle \frac{p^{-({\beta }_1+{\beta }_2)}{x}^{{\beta }_{1}p}{y}^{{\beta }_{2}p}}{\mathrm{\Gamma }({\beta }_1+1)\mathrm{\Gamma }({\beta }_2+1)}}$. □

The main result of this section relies on the following two hypotheses:

$({\mathbf{A}}_{\mathbf{1}})$ Letting $(h_1,h_2)$ and $(h_3,h_4)$ be any solutions of (3), there exist $L_{\widehat{f_1}}$ and $L_{\widehat{f_2}}$ such that

$$\begin{array}{cc}& \left|\widehat{f_1}(x,y,h_1,h_2)-\widehat{f_1}(x,y,h_3,h_4)\right|\le L_{\widehat{f_1}}(\left|h_1-h_3\right|+\left|h_2-h_4\right|),\hfill \\ & \left|\widehat{f_2}(x,y,h_1,h_2)-\widehat{f_2}(x,y,h_3,h_4)\right|\le L_{\widehat{f_2}}\left(\left|h_1-h_3\right|+\left|h_2-h_4\right|\right).\hfill \end{array}$$

$({\mathbf{A}}_{\mathbf{2}})$ There exist some constants $c_i,d_i>0$ (i = 1, 2) and $c_3,d_3\ge 0$ with

$$\begin{array}{cc}& \left|\widehat{f_1}(x,y,h_1,h_2)\right|\le c_1|h_1|+c_2|h_2|+c_3,\hfill \\ & \left|\widehat{f_2}(x,y,h_1,h_2)\right|\le d_1|h_1|+d_2|h_2|+d_3.\hfill \end{array}$$

Theorem 3.

Under the conditions in Theorems 1 and 2 and assumptions $({\mathbf{A}}_{\mathbf{1}})$ and $({\mathbf{A}}_{\mathbf{2}})$, the symmetry coupled system (3) is U-HS if $c_1{c}_2{L}_{\widehat{f_1}}{L}_{\widehat{f_2}}\ne 1$.

Proof. 

Since $(h_1,h_2)$ and $(h_3,h_4)$ are arbitrary solutions of (3), it follows from (16) that

$$\begin{array}{cc}\hfill & |h_1-h_3|\hfill \\ \hfill & =|\left(\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1,h_2)\right)-\left(\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_3,h_4)\right)|\hfill \\ \hfill & =\left(\mathrm{\Psi }(x,y)-\left(\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1,h_2)\right)\right.\hfill \\ \hfill & +\left(\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1,h_2)\right)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1,h_2)\hfill \\ \hfill & -\left(\mathrm{\Psi }(x,y)+{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_3,h_4)\right)\hfill \\ \hfill & \le {\delta }_1{ϵ}_1+\left|{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_1,h_2)-{}_F{}^{RL}\mathcal{I}_{0^{+}}^{\mathit{\alpha },p}\widehat{f_1}(x,y,h_3,h_4)\right|\hfill \\ \hfill & \le {\delta }_1{ϵ}_1+{\displaystyle \frac{p^{-({\alpha }_1+{\alpha }_2)}{x}^{{\alpha }_{1}p}{y}^{{\alpha }_{2}p}}{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}}\times \left|\widehat{f_1}(x,y,h_1,h_2)-\widehat{f_1}(x,y,h_3,h_4)\right|\hfill \\ \hfill & \le {\delta }_1{ϵ}_1+{\delta }_1{L}_{\widehat{f_1}}\left|h_2-h_4\right|+{\delta }_1{L}_{\widehat{f_1}}\left|h_1-h_3\right|.\hfill \end{array}$$

Thus, we have

$$|h_1-h_3|-{\displaystyle \frac{{\delta }_1{L}_{\widehat{f_1}}}{1-{\delta }_1{L}_{\widehat{f_1}}}}|h_2-h_4|\le {\displaystyle \frac{{\delta }_1{ϵ}_1}{1-{\delta }_1{L}_{\widehat{f_1}}}}.$$

(45)

Similarly, one can obtain

$$|h_2-h_4|-{\displaystyle \frac{{\delta }_2{L}_{\widehat{f_2}}}{1-{\delta }_2{L}_{\widehat{f_2}}}}|h_1-h_3|\le {\displaystyle \frac{{\delta }_2{ϵ}_2}{1-{\delta }_2{L}_{\widehat{f_2}}}}.$$

(46)

According to (45) and (46), its matrix can be expressed in the following form:

$$\left(\begin{array}{cc}1& -L_{\widehat{f_1}}{c}_1\\ -L_{\widehat{f_2}}{c}_2& 1\end{array}\right)\times \left(\begin{array}{c}|h_1-h_3|\\ |h_2-h_4|\end{array}\right)\le \left(\begin{array}{c}{c}_1{ϵ}_1\\ c_2{ϵ}_2\end{array}\right),$$

(47)

where $c_1={\displaystyle \frac{{\delta }_1}{1-{\delta }_1{L}_{\widehat{f_1}}}}$ and $c_2={\displaystyle \frac{{\delta }_2}{1-{\delta }_2{L}_{\widehat{f_2}}}}$. Letting $c_1{c}_2{L}_{\widehat{f_1}}{L}_{\widehat{f_2}}\ne 1$, then we solve the matrix inequality (47) and have

$$\begin{array}{c}|h_1-h_3|\le {\displaystyle \frac{c_1{ϵ}_1}{1-c_1{c}_2{L}_{\widehat{f_1}}{L}_{\widehat{f_2}}}}+{\displaystyle \frac{c_1{c}_2{L}_{\widehat{f_1}}{ϵ}_2}{1-c_1{c}_2{L}_{\widehat{f_1}}{L}_{\widehat{f_2}}}},\hfill \\ |h_2-h_4|\le {\displaystyle \frac{c_2{ϵ}_2}{1-c_1{c}_2{L}_{\widehat{f_1}}{L}_{\widehat{f_2}}}}+{\displaystyle \frac{c_1{c}_2{L}_{\widehat{f_2}}{ϵ}_1}{1-c_1{c}_2{L}_{\widehat{f_1}}{L}_{\widehat{f_2}}}},\hfill \end{array}$$

which means that

$$\begin{array}{c}|(h_1,h_2)-(h_3,h_4)|\le |h_1-h_3|+|h_2-h_4|\le \overline{C}ϵ,\hfill \end{array}$$

where $\overline{C}={\displaystyle \frac{c_1+c_2+c_1{c}_2(L_{\widehat{f_1}}+L_{\widehat{f_2}})}{1-c_1{c}_2{L}_{\widehat{f_1}}{L}_{\widehat{f_2}}}}$. Thus, the new fuzzy fractional partial differential symmetry coupled system (3) is U-HS. □

It follows that the stability constant $\overline{C}$ in Theorem 3 depends on the key parameters of the system. From the relationship between $\overline{C}$ and parameters such as $c_1,c_2,{\delta }_1,{\delta }_2,\alpha ,\beta ,a,b$, it can be concluded that $\overline{C}$ decreases as $\alpha ,\beta$ increase and increases as $a,b$ increase. Therefore, for System 3 to be U-HS, it is necessary that $c_1{c}_2{L}_{{\widehat{f}}_1}{L}_{{\widehat{f}}_2}<1$.

Remark 9.

Regarding the practical significance and methodological considerations of system (3), the following hold:

• In the predator–prey ecosystem, the model (3) is of great significance: the "memory effect" of fractional derivatives matches the cumulative influence of long-term environmental perturbations such as global warming, and this effect can render population oscillations more stable and smooth. Meanwhile, the U-HS of the model (3) guarantees that even if the initial populations or interspecific interactions suffer minor disturbances, the dynamics of predators and prey can still stay bounded and coordinated, which offers a reliable foundation for ecosystem sustainability and population management under climate change(see [43]).
• Compared with Mittag–Leffler stability, although Mittag–Leffler stability is unique to fractional-order systems, proving Mittag–Leffler stability usually requires constructing complex Lyapunov functions (see [44]), which is a huge challenge for our coupled fuzzy system. The method based on the fixed point theorem adopted in this paper is a classical and effective approach to proving U-HS. This method circumvents the difficulty of constructing Lyapunov functions and provides a feasible technical path for dealing with such complex systems.

6. Conclusions

The present analysis focuses on the existence and stability of solutions for a class of C-K FFPDE symmetric coupled systems with gH-differences. The system is formulated as follows:

$$\left\{\begin{array}{c}\begin{array}{c}{}_{gh}{}^{CK}D_k^{\mathit{\alpha },p}{h}_1(x,y)=\widehat{f_1}(x,y,h_1(x,y),h_2(x,y)),\hfill \end{array}\hfill \\ {}_{gh}{}^{CK}D_k^{\mathit{\beta },p}{h}_2(x,y)=\widehat{f_2}(x,y,h_1(x,y),h_2(x,y)),\hfill \\ h_1(x,0)={\zeta }_1(x),h_2(x,0)={\eta }_1(x),\hfill \\ h_1(0,y)={\zeta }_2(y),h_2(0,y)={\eta }_2(y),\hfill \end{array}\right.$$

(48)

for any $(x,y)\in \varpi =[0,a]\times [0,a]$ and $k=1,2$. By employing Schauder fixed point theorem (i.e., Lemma 5), the relevant results were obtained under more general situations without Lipschitz conditions, which is common and very important.

In the work of this paper, the points of innovation are as follows:

• A novel class of fuzzy fractional partial differential coupled systems is introduced, which unifies and extends existing fuzzy fractional models, offering broader applicability.
• Within the C-K gH-differentiability framework, two types of gH-weak solutions are defined, and their existence is demonstrated through the construction of concrete examples.
• The U-HS theory is introduced to analyze such fuzzy fractional partial differential coupled systems.

While this study establishes a theoretical framework for fuzzy fractional partial differential coupled systems, its analysis is limited to deterministic models, excluding real-world uncertainties like stochastic perturbations and time-delay effects. Further, we fully recognize the importance of numerical simulations and visual presentations illustrating the impact of fractional order $\gamma$ on system characteristics, as these elements are crucial for verifying theoretical results and enhancing the persuasiveness of the research. Future work should address these limitations by extending the framework to fuzzy stochastic systems, developing machine learning-based stability analysis methods, and comparing U-HS with other stability theories. Additionally, to advance numerical investigations, some numerical figures or simulations are worthy of study to show the effect of fractional order $\gamma$ on the system (48), and enriched Crouzeix–Raviart finite elements [45] can be introduced; these elements adapt to discontinuities in fuzzy solutions, enhance accuracy, optimize fractional-order discretization, support framework construction, expand application scenarios, and thus form a theory–numerics–application closed loop, with results that can be further applied to practical problems in ecology, epidemiology, and engineering.