EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5963 ISSN 1307-5543 – ejpam.com Published by New York Business Global Investigation of Existence and Ulam’s type Stability for Coupled Fractal Fractional Differential Equations Amjad Ali1, Farha Bibi1, Zeeshan Ali2,∗, Kamal Shah3, Bahaaeldin Abdalla3, Thabet Abdeljawad3 1 Department of Mathematics and Statistics, University of Swat, KPK, Pakistan 2 Department of Information Management, National Yunlin University of Science and Technology Douliu Taiwan, R. O.C 2 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia Abstract. In this study, we investigate the coupled system of fractal fractional differential equa- tions (FFDEs) from an existence and stability point of view. The area related to coupled FFDEs is significant because its permitting us to analyze and predict the relationships between several variables throughout the real-world phenomena. Coupled systems have numerous applications in different fields, such as modeling of brain activities and spreads of disease in biology and medicine, modeling of mechanical systems, electrical circuits in engineering, and modeling of population dy- namics, predator-prey models in ecology and financial mathematical in economics. Furthermore, the afore mentioned area is significant for environmental science in pollution control, climate change modeling, artificial intelligence, and control systems in technology. We analysis a coupled system and make more informed choices in a variety of fields through coupled system of FFDEs. Keeping these informative aspects of coupled systems in mind, this article aims to explore the qualitative analysis such as existence, uniqueness, and stability for the solution of underlying coupled systems of FFDEs. The tools of functional analysis (FA) and fixed point theory (FPT) have been applied to deduce our required results. We have used the Banach contraction principle (BCP) and the Krasnoselskii fixed point theorem (KFPT) to demonstrate the conditions for existence and unique- ness of a solutions (EUS). Additionally, the results related to stability have demonstrated by using Ulam’s concept. Subsequently, a suitable example is given to illustrates the validity of this work. 2020 Mathematics Subject Classifications: 26A33, 34A08, 35A07 Key Words and Phrases: Coupled System, Fractal fractional Operator, Existence, Stability, Fixed Point Theorems ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5963 Email addresses: amjadalijc18@gmail.com (A. Ali), farhabibi987@gmail.com (F.Bibi), zeeshan@yuntech.edu.tw (Z. Ali),kshah@psu.edu.sa (K. Shah), babdallah@psu.edu.sa (B.Abdalla), tabdeljawad@psu.edu.sa (T. Abdeljawad) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 2 of 19 1. Introduction The concept of fractional calculus (FC) is an emerging area of recent research due to its significant implementations in different field of life sciences, physical sciences, and social sciences [1]. The idea of fractional order derivative (FODs) was initiated from the well-known correspondence between Leibniz’s and L’Hospital in the seventeenth hundred century [2]. The FODs have some advantages over the classical derivatives, these are more flexible, non-local in nature, having memory effects, great degree of freedom and more reliable [3]. FC is the capability to describe the complex dynamics of all those real-world scenarios, which can not be formulated by conventional calculus [4]. Furthermore, FC is substantial and has a multifaceted of applications throughout numerous fields, such as finance, bio-engineering, ecology, physics and graphics. Of course one of the significant aspects of FC is that it can describe and investigate the real-world situation in the form of fractional differential equations (FDEs), or system of FDEs [5]. Researchers are interested to investigate FDEs to provides a powerful framework for formulating those dynamical phenomena which is not describe by classical order DEs accurately [1]. Due to aforesaid applications, mathematicians have been taken keen interest to investigate FDEs. Conse- quently, researchers have well explored the theory of FDEs from different aspects, such as stability analysis, numerical approximation, and existence of solution, and published plenty of books, articles, and manographs [6]. Meanwhile, the researchers introduced different types of fractional operators (FOs), such as the conformable fractional deriva- tives (CFDs), the Riemann-Liouville fractional derivatives (RLFDs), the Caputo fractional derivative (CFD), the Grunwald-Letnikov fractional derivative (GLFD) [7], non singu- lar fractional derivatives (NSFDs) like Atangana-Baleanu fractional derivative (ABFD), Caputo-Fabrizio fractional derivative (CFFD), etc [8]. The theory of fractals derivative is old as the FC. In 2017, researchers have extended the concepts of fractional derivatives to fractal and introduced fractals fractional derivatives (FFDs) [9]. The concept of this new type of FFDs was introduced by Atangana [10]. This men- tioned class of FFDs consist of two types of orders, the first is usual fractional order, while the second part is known as fractal dimension. An important aspect of the consider derivatives is that they can investigate both FODs and FFDs at same stage simultane- ously in the same problem [11]. The FFDs have advanced over the traditional FODs to understanding their geometrical meaning as well as relation with them [12]. The FFDs have very useful consequences in situation, where discontinuity, power-law, and crossover occur, or decaying memory fail to capture the complex behaviors [13]. The concerned FFDs has a accomplished form of FODs which combined both fractal and fractional to more compact form is known as fractal fractional calculus [14]. This newly emerging area of research has attracted the attention of researchers to investigate various features [15]. The coupled system of FDEs have been studied by researchers from various aspects and furnish theory up-to large extend in the sense of ordinary FODs, such as existence, numerical solution and stability analysis [16]. To best to our knowledge the consider type of coupled system of FFDEs are very rarely investigated by researchers. In this context, the aforementioned research directions needs further attention of researchers to explore A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 3 of 19 this theory. It is worthy to highlight the key features of underlying coupled system of FFDEs, that in some circumstances, the actual dynamics behaviors of phenomena cannot be expressed with ordinary tools of FDEs [17]. While, the proposed tools of coupled system of FFDEs are remarkable achievement for investigating such problems. Some well-known examples of coupled systems are host-parasite, plant- pollinator of biological process, grass land-fire, Lorenz model [18], atmosphere-ocean current and tides-coastal erosion are ex- amples of physical systems [19]. According to our study the coupled system of FDEs in sense of fractal derivatives are very rarely studied by researchers and need more attention. Therefore, we investigate the coupled system of FDEs in sense of fractal operator. Researchers from different fields are interested to investigate numerous aspects as this theory has the potential to address a variety of issues and is a fast growing field of re- search [20]. There are many methods to develop the EUS of FFDEs, such as FPT and topological degree theory [21]. Another subsequent aspect of fractal fractional calculus is investigating stability analysis of initial value problems of FFDEs. Stability analysis is a technique, used to find the ability of the system to get back to its equilibrium position af- ter introducing some changes. There are different kinds of stability analysis in the present literature of fractal FC [22]. Ulam, raised a question that under what condition does there exists an additive mapping near an approximately additive mapping? In reply to this question Hyres answered Ulam’s question for the “additive mapping in complete norm spaces” [23]. After it was considered a type of stability and called it Ulam -Hyers (UH) stability. The said concept was further extended to generalized (GUH), and Ulam-Hyers- Rassias (UHR) and generalized (GUHR) analysis. This concept has many applications in different mathematical fields, including DEs, operator theory, physics, engineering, and computer science. The connection between stability analysis and Hyers-Ulam stability are interconnected concepts of understanding the strength of systems and equations by apply- ing stability analysis to functional equations [24]. This integrated approach can lead to a deeper understanding of complex dynamics and their behavior. The proposed stability has the main concern of researchers, due to extensive applications in various fields. Some the major contributions in proposed field we discuss here. For instance, Xiadjie et al., [25] studied the following non-linear problem in sense of the Caputo’s derivative with order ξ as: Dξϕ(t) = θ1(t, ϕ(t)), t ∈ Ω = [0, T ], ϕ(0) + ψ(ϕ) = ϕ0, where f : Ω × R → R is continuous function. The investigation of coupled system for non-linear FDEs has the main focuss of researchers from existence point of view. In this regards, the author’s [26] studied the uniqueness of solution for coupled system of FDEs with integral boundary conditions. In the same way, Su [27] established the conditions for EUS for the following coupled system with two-point boundary conditions (BCs) in sense A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 4 of 19 of singular kernel operator Dηϕ(t) = θ1(t, ψ(t), D pψ(t)), t ∈ Ω = [0, 1], Dξϕ(t) = θ2(t, ϕ(t), D qϕ(t)), ϕ(0) = ϕ(1) = ψ(0) = ψ(1) = 0, where η, ξ ∈ (1, 2], such that p − η and q − ξ > 0, θ1, θ2 ∈ C([0, 1] × R2, R). On the same fashion, Wang et al., [28] have been presented sufficient conditions for the EUS to the following non-linear CS with three point BCs Dηϕ(t) = θ1(t, ψ(t)), t ∈ Ω = [0, 1], Dξϕ(t) = θ2(t, ϕ(t)), ϕ(0) = ψ(0) = 0, ϕ(1) = qϕ(p), ψ1bψ(p), where η, ξ ∈ (1, 2] and 0 ≤ a, b ≤ 1, 0 < p ≤ 1, θ1, θ2 ∈ C([0, 1]×R,R). After the comprehensive study of present literature, it is clear that FDEs and their coupled systems are well investigated by researchers in term of singular kernel fractional operators. We felt that coupled system of FFDEs are very rarely investigated by re- searchers. For the better understanding of consider coupled system FFDEs, we carry out investigations and analysis for model via the tools of mathematical analysis. Therefore, main focuses of our research work is to establish the conditions of EUS and stability anal- ysis for underlying coupled system of FFDEs. We use the results of the functional analysis along with FPT [29, 30] to obtain required results. The considered problem is described as follows:  FDDηr(t) = ξtξ−1f (t, r(t), p(t)) , t ∈ I = [0, T ], FDDηp(t) = ξtξ−1g(t, r(t), p(t)), r(0) = r0 + ϕ(r), p(0) = p0 + ψ(p), (1) where FDDη is the Caputo fractional derivative and r0, p0 ∈ R,η, ξ ∈ (0, 1], f, g : I × R2 → R, ϕ, ψ : I = [0, T ] × R → R are continuous functions. The qualitative analysis of the proposed coupled system of FFDEs offer deeper understanding into the complex dynamics systems exhibiting self-similarity and non-locality. The consider class have many applications in various discipline that can model complex phenomena. The coupled system of FFDEs can accurately describe the behavior of complex system in physics, biology, finance and predictive capabilities and understanding the qualitative behavior of solution. The result of this research will also provide new insight into the behavior of coupled system of FFDEs and will have the ability to inspire new applications and research directions. 2. Background Information In this work, we will use the following notation: the norm define by ∥u∥ = sup{|u(t)| : t ∈ [0, T ]} for the Banach space denoted by U,V = C([0, T ], R), where S represents a A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 5 of 19 family of all bounded subsets of B(U×V), and the product U×V is a Banach space with norm ∥(u, v)∥ = ∥u∥+ ∥v∥. Definition 1. [2] The Riemann-Liouville derivative is defined by FDDηy(t) =  1 Γ(m− η) ( d dt )m ∫ t 0 (t− δ)m−η−1y(δ)dδ, m− 1 < η < m, 1 Γ(1− η) d dt ∫ t 0 (t− δ)−ηy(δ)dδ, 0 < η < 1, dmy dtm , η = m. In the same way, the Caputo fractional derivative is defined by FDDηy(t) =  1 Γ(m− η) ∫ t 0 (t− δ)m−η−1 ( d dδ )m y(δ)dδ, m− 1 < η < m, 1 Γ(1− η) ∫ t 0 (t− δ)−η d dδ y(δ)dδ, 0 < η < 1, dmy dtm , η = m. The corresponding fractional integral is defined by FIIηy(t) = 1 Γ(η) ∫ t 0 (t− δ)η−1y(δ)dδ, provided that integral on right exists. Definition 2. [10] The FFD in sense of Riemann-Liouville operator for function y which is differentiable both in fractional and in fractals sense over [0, T ] given by RFFDη,ξy(t) =  1 Γ(m− η) ( d dtξ )m ∫ t 0 (t− δ)m−η−1y(δ)dδ, m− 1 < η < m, 1 Γ(1− η) d dtξ ∫ t 0 (t− δ)−ηy(δ)dδ, 0 < η, ξ < 1, 1 Γ(m− η) ( d dt )m ∫ t 0 (t− δ)m−η−1y(δ)dδ, ξ = 1,m− 1 < η < m, dmy dtm , ξ = 1, η = m. Definition 3. [10] Let y is continuous function, then the fractals fractional integral is defined as follows: RFF Iη,ξf(t) = ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ)dδ, where ξ is fractal dimension and η is fractional order. A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 6 of 19 Lemma 1. [2] The solution of FDEs FDDηr(t)− ξtξ−1e(t) = 0, r(0) = r0, is given by r(t) = r0 + FI Iηξtξ−1e(t). Definition 4. (Fixed Point:)[29] Let U × V be a norm linear space and mapping H : U×V → U×V, then (r, p) ∈ U×V is be fixed point, if H(r, p) = (r, p) Definition 5. [29] (Equi-continuous:) Let a bounded subset B of U × V and (rn, pn) have a sequence in B for several (rn, pn) in B and for all possible ϵ > 0 |ζ(rn, pn)(t)− ζ(rn, pn)(τ)| ≤ ϵ Theorem 1. [29] Let U × V, be a Banach space and mapping H : U × V → U × V, is contraction, with H(r, p) = (r, p), then H has a unique fixed point in U×V. Theorem 2. (Krasnoselskii Theorem:)[30] Let B be a Banach space and H and G be two operators, such that • H is contraction. • G is compact operator. • Then p = (H +G)p has a fixed point. 3. Existence of solution This section of research work is devoted to EUS for the proposed coupled system of FFDEs (1). Lemma 2. Let t ∈ I and η, ξ ∈ (0, 1], then the solution of coupled system of FFDEs FDDηr(t) = ξtξ−1f (t, r(t), p(t)) , FDDηp(t) = ξtξ−1g(t, r(t), p(t)), r(0) = r0 + ϕ(r), p(0) = p0 + ψ(p), (2) is provided by  r(t) = r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ, p(t) = p0 + ψ(p) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1g(δ, r(δ), p(δ))dδ. (3) A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 7 of 19 Proof. Let us consider the coupled system of FFDEs (2) in the form of linear equations as:  FDDηr(t) = ξtξ−1f (t, r(t), p(t)) = θ1(t), FDDηp(t) = ξtξ−1g(t, r(t), p(t)) = θ2(1), r(0) = r0 + ϕ(r), p(0) = p0 + ψ(p), (4) where θ1(t), θ2(t) : I → R are continuous functions. Applying FIIη on coupled system of FFDEs (4) and in view of Lemma 1, we obtain{ r(t) = h0 + FI Ipξtξ−1θ1(t), p(t) = h∗0 + FI Iηξtξ−1θ2(t). (5) Using the condition r(0) = r0 + ϕ(r) and p(0) = p0 + ψ(t, e(t)), we obtain equation (5) in the form of  r(t) = r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1θ1(δ)dδ. p(t) = p0 + ψ(p) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1θ2(δ)dδ. (6) In-sight of system of equations (4), the system of equations (6) is equivalent to the follow- ing: r(t) = r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ. p(t) = p0 + ψ(p) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1g(δ, r(δ), p(δ))dδ. (7) The above system (7), represents the integral representation for our proposed problem (2). Let us define the following operators, H : U×V → U×V, such that H(r, p)(t) = (H1r(t), H2p(t)), (8) . where H1r(t) = r0 + ϕ(r), H2p(t) = p0 + ψ(p). Also G∗∗ : U×V → U×V, such that G∗∗(r, p)(t) = (G∗ 1(r, p)(t), G ∗ 2(r, p)(t)) , (9) where G∗ 1(r, p)(t) = ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ, G∗ 2(r, p)(t) = ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1g(δ, r(δ), p(δ))dδ. A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 8 of 19 Furthermore, let us define K = H + G∗∗, then the operator equation corresponding to system of equation (6) as: (r, p) = K(r, p)(t) = H(r, p)(t) +G∗∗(r, p)(t), which is a solution of the system in operator form. Now we present some hypothesis which are helpful in building our main existence result. (M1) There exist some constant Lϕ and Lψ ∈ (0, 1] such that |ϕ(r)− ϕ(r)| ≤ Lϕ|r − r|, |ψ(p)− ψ(p)| ≤ Lψ|p− p|. (M2) There exist some constant c1, c2, d1, d2 > 0 and c3, d3 ⩾ 0, with |f(t, r(t), p(t))| ≤ c1|r|+ c2|p|+ c3, |g(t, r(t), p(t)| ≤ d1|r|+ d2|p|+ d3. (M3) There exist Lf and Lg such that ||f(t, r, p)(t)− f(t, r(t), r(t)|| ≤ Lf (||r − r||+ ||p− p||) , ||g(t, r, p)(t)− g(t, r(t), p(t)|| ≤ Lg (||r − r||+ ||p− p||) . Theorem 3. Under the assumption (M1) and (M2) the coupled system of FFDE (2) has at lest one solution. Proof. To obtained results, we needs the following: Step1: We need to show that H is contraction mapping, so for any (r, p), (r, p). ||H(r, p)(t)−H(r, p)(t)|| ≤ ||ϕ(r)− ϕ(r)||+ ||ψ(p)− ψ(p|| ≤ Lϕ|r − r|+ Lψ|p− p|, which implies that ||H(r, p)(t)−H(r, p)(t)|| ≤ L||(r, p)− (r, p)||, (10) where L = max(Lϕ, Lψ). Hence H is contraction. Step2: Next, we will to prove G∗∗ is bounded , for this let S = {||(r, p)|| ≤ r; (r, p)} be closed and bounded set, then ∥G∗∗(r, p)(t)∥ = ∥∥∥∥ ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1g(δ, r(δ), p(δ))dδ ∥∥∥∥ ≤ ∥∥∥∥ ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ ∥∥∥∥+ ∥∥∥∥ ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1g(δ, r(δ), p(δ))dδ ∥∥∥∥ ≤ T η+ξ−1B(η, ξ)(c1|r|+ c2|p|+ c3) + T η+ξ−1B(η, ξ)(d1|r|+ d2|p|+ d3), = T η+ξ−1B(η, ξ) [(c1 + d1)|r|+ (c2 + d2)|p|+ (c3 + d3)] ≤ ∞. A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 9 of 19 Thus, G∗∗ is bounded. For the equi-continuity of G, let τ < t ∈ I, and ∥G∗ 1(r, p)(t)−G∗ 1(r, p)(τ)∥ = ∥∥∥∥ ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ − ξ Γ(η) ∫ τ 0 (τ − δ)η−1δξ−1f(δ, r(δ), p(δ))dδ ∥∥∥∥ ≤ ∥f(δ, r(δ), p(δ))∥ (∫ t 0 (t− δ)η−1δξ−1dδ − ∫ τ 0 (τ − δ)η−1δξ−1dδ ) ≤ ∥f(δ, r(δ), p(δ))∥B(η, ξ)(tη+ξ−1 − τη+ξ−1) → 0 as t→ τ. It follow that ∥G∗ 1(r, p)(t)−G∗ 1(r, p)(τ)∥ → 0 as t→ τ. Similarly, one can obtained the following result for: ∥G∗ 2(r, p)(t)−G∗ 2(r, p)(τ)∥ → 0 as t→ τ. Therefore G∗ 1 and G∗ 1 are continuous and hence G∗∗ is continuous and also bounded. Thanks to Krasnoselskii’s FPT the system (2) has at lest one solution. Theorem 4. Under the assumptions (M1) and (M3) and such Υ = L+WT η+ξ−1B(η, ξ) ≤ 1 holds, then the consider system FFDE have unique solution. Proof. For this let (r, p), (r, p) ∈ U×V, then from equation (10), one have ||H(r, p)(t)−H(r, p)(t)|| ≤ L||(r, p)− (r, p)||, (11) and ∥G∗ 1(r, p)(t)−G∗ 1(r, p)(t)∥ = ∥∥∥∥ ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ − ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ ∥∥∥∥, = ξ Γ(η) ∫ T 0 (t− δ)η−1δξ−1∥f(δ, r(δ), p(δ))− f(δ, r(δ), p(δ))∥dδ, ≤ ξ Γ(η) T η+ξ−1B(η, ξ)Lf (||r − r||+ ||p− p||, which implies that ∥G∗ 1(r, p)(t)−G∗ 1(r, p)(t)∥ ≤ T η+ξ−1B(η, ξ)Lf∥(r, p)(t)− (r, p)(t)∥, (12) and ∥G∗ 2(r, p)(t)−G∗ 2(r, p)(t)∥ ≤ T η+ξ−1B(η, ξ)Lg∥(r, p)(t)− (r, p)(t)∥. (13) A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 10 of 19 Hence from (12) and (13), we have ∥G∗∗(r, p)(t)−G∗∗(r, p)(t)∥ ≤ T η+ξ−1B(η, ξ)Lf∥(r, p)(t)− (r, p)(t)∥ + T η+ξ−1B(η, ξ)Lg∥(r, p)(t)− (r, p)(t)∥, ≤ T η+ξ−1B(η, ξ)(Lf + Lg)|(r, p)(t)− (r, p)(t)|. which implies that ∥G∗∗(r, p)(t)−G∗∗(r, p)(t)∥ ≤ T η+ξ−1B(η, ξ)W∥(r, p)(t)− (r, p)(t)∥, (14) where W = Lf + Lg. Therefore, from (11) and (14), we have ∥K(r, p)(t)−K(r, p)(t)∥ ≤ L∥(r, p)(t)− (r, p)(t)∥+WT η+ξ−1B(η, ξ)∥(r, p)(t)− (r, p)(t)∥ ≤ Υ∥(r, p)(t)− (r, p)(t)∥, where Υ = L +WT η+ξ−1B(η, ξ). Therefore, K is contraction, thus the system (2) has a unique solution. 4. Stability analysis This section, is devoted to establishing stability results for the considered coupled system of FFDEs (2). We contend that UH stability idea is important for practical issues in analysis. The interesting feature of stability is that to search a UH stability of a system, who exact solution does not exist, which is typically challenging or time consuming. According to UH stability, there exist a close approximate solution of system to exact solution. As we know that mostly a mathematical models are non-linear in nature and some time its exact solution does not exist or difficult to be obtained. Therefore, we need to find best approximate solution for such problems. Definition 6. To obtained the UH stability of the system (2), suppose that exist ϵ = max (ϵ1, ϵ2) > 0, and the system of inequality given by{ |FDDηr(t)− ξtξ−1f(t, r(t), p(t))| ≤ ϵ1, |FDDηp(t)− ξtξ−1g(t, r(t), p(t))| ≤ ϵ2. (15) Then our system (2) is UH stable, ∃C(1,2) > 0, with every solution (r, p) of inequality equation (15) there exist unique solution (r, p) with |(r, p)(t)− (r, p)(t)| ≤ ϵC(1,2). (16) Remark : We say that (r, p) is a solution of the system (15), if there exist ω1, ω2 ∈ C(I,R) which defend upon (r, p), such that • |ω1(t)| ≤ ϵ1, A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 11 of 19 • |ω2(t)| ≤ ϵ2. and the perturbed system of coupled system of FFDEs as:{ FDDηr(t) = ξtξ−1f(t, r(t), p(t)) + ξtξ−1ω1(t), FDDηp(t) = ξtξ−1g(t, r(t), p(t)) + ξtξ−1ω2(t). (17) Lemma 3. Suppose that (r, p) be the solution of the inequality (15), then the following system of inequality hold.∣∣∣∣r(t)− ( (r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, ri(δ), pi(δ))dδ) )∣∣∣∣ ≤ δϵ1,∣∣∣∣p(t)− ( (p0 + ψ(p) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1g(δ, xi(δ), pi(δ))dδ) )∣∣∣∣ ≤ δϵ2. Proof. In-view of Remark(4), we can express the consider system as: FDDηr(t) = ξtξ−1f(t, r(t), p(t)) + ξtξ−1ω1(t), FDDηp(t) = ξtξ−1g(t, r(t), p(t)) + ξtξ−1ω2(t), r(0) = (r0 + ϕ(r)), p(0) = (p0 + ψ(p)). In light of Lemma 2, we can obtained the following results: r(t) = (r0 + ϕ(r)) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1ω1(δ)dδ, p(t) = (p0 + ψ(p)) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1g(δ, r(δ), p(δ))dδ) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1ω2(δ)dδ. Now∣∣∣∣r(t)− ( r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ) )∣∣∣∣ = ∣∣∣∣ ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1ω1(δ)dδ ∣∣∣∣ ≤ ξ Γ(η) ∫ T 0 (t− δ)η−1δξ−1ϵ1dδ ≤ B(η, ξ)T η+ξ−1ϵ1 = δϵ1, where δ = B(η, ξ)T η+ξ−1. Similarly, one can obtain the following result for second com- partment of proposed model:∣∣∣∣p(t)− ( p0 + ψ(p) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ )∣∣∣∣ ≤ δϵ2, which completes the proof A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 12 of 19 Theorem 5. Under the assumptions (M1)-(M3), the solution of coupled system (2) is UH and GUH stable, if c1c2LfLg ̸= 1. Proof. Let (r, p) be the arbitrary solution and (τ, κ) be the unique solution of (18): FDDη,ξr(t) = ξtξ−1f(t, r(t), p(t)), FDDη,ξp(t) = ξtξ−1g(t, r(t), p(t)), r(0) = r0 + ϕ(r), p(0) = p0 + ψ(p). (18) The solution of system (18) is given by: τ(t) = (r0 + ϕ(τ)) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, τ(δ), κ(δ))dδ), κ(t) = (p0 + ψ(κ)) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1g(δ, τ(δ), κ(δ))dδ). Now |r(t)− τ(t)| = ∣∣∣∣(r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ)− (r0 + ϕ(τ) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, τ(δ), κ(δ))dδ) ∣∣∣∣, = ∣∣∣∣r(t)− ( r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ ) + (r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ)| − ( r0 + ϕ(τ) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, τ(δ), κ(δ)dδ ) ∣∣∣∣, ≤ δϵ1 + |ϕ(r)− ϕ(τ)|+ β(η, ξ)T η+ξ−1|f(δ, t(δ), p(δ))dδ)− f(δ, τ(δ), κ(δ)dδ|, ≤ δϵ1 + (Lϕ + δLf )|r(t)− τ(t)|+ δLf |p(t)− κ(t)|, Hence, we have |r(t)− τ(t)| − δLf 1− (Lϕ + δLf ) |p(t)− κ(t)| ≤ δϵ1 1− (Lϕ + δLf ) . (19) Similarly, one can obtain the following result |p(t)− κ(t)| − δLg 1− (Lψ + δLg) |r(t)− τ(t)| ≤ δϵ2 1− (Lψ + δLg) . (20) The matrix representation the above inequalities as: A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 13 of 19( 1 −Lfc1 −Lgc2 1 ) × ( |r − τ | |p− κ| ) ≤ ( c1ϵ1 c2ϵ2, ) , where c1 = δ 1−(Lϕ+δLf ) and c2 = δ 1−(Lψ+δLg) , then solving the above matrix, we have |r − τ | ≤ c1ϵ1 1− (c1c2LfLg) + c1c2Lf ϵ2 1− (c1c2LfLg) , (21) and |p− κ| ≤ c2ϵ2 1− (c1c2LfLg) + c1c2Lgϵ1 1− (c1c2LfLg) (22) Adding (21) and (22), we gets |r − τ |+ |p− κ| ≤ c1ϵ1 1− (c1c2LfLg) + c1c2Lf ϵ2 1− (c1c2LfLg) + c2ϵ2 1− (c1c2LfLg) + c1c2Lgϵ1 1− (c1c2LfLg) , ≤ (c1 + c2 + c1c2(Lf + Lg))ϵ 1− (c1c2LfLg) , which implies that |(r, p)− (τ, κ)| ≤ C(1,2)ϵ, (23) where C(1,2) = (c1 + c2 + c1c2(Lf + Lg)) 1− (c1c2LfLg) , Thus, the solution of (2) is UH stable. Further, the solution of coupled system of FFDEs (2) is GUH stable,if ψ : (0, 1) → (0,∞) such that ψ(ϵ) = ϵ,be a non decreasing function,then we have |(r, p)− (τ, κ)| ≤ C(1,2)ψ(ϵ), where ψ(0) = 0. (24) Thus CS is (2) GUH stable For any function b,we assume the following inequalities holds Iηbi(t) ≤ λinbi(t),where bi = {1, 2} Lemma 4. The solution (r, p) for the selected coupled system of FFDEs{ FDDηr(t) = ξtξ−1f(t, r(t), p(t)) + ξtξ−1b1(t), FDDηp(t) = ξtξ−1g(t, r(t), p(t)) + ξtξ−1b2(t), will obeys the following:∣∣∣∣r(t)− ( (r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, ri(δ), pi(δ))dδ) )∣∣∣∣ ≤ λ1nb1(t)ϵ1,∣∣∣∣p(t)− ( (p0 + ψ(p) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1g(δ, ri(δ), pi(δ))dδ) )∣∣∣∣ ≤ λ2nb2(t)ϵ2. (25) A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 14 of 19 Proof. Using remark 4, its proof is similar to (3) Theorem 6. Under the assumptions (M1)-(M3) and consider Lemma 4 with the condition c1c2LfLg ̸= 1, then the selected system is both UHR and GUHR stable. Proof. Let (r, p) be the arbitrary and (τ, κ) unique solution of the coupled system of FFDEs, then one have |r(t)− τ(t)| = ∣∣∣∣(r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ)− (r0 + ϕ(τ) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, τ(δ), κ(δ))dδ) ∣∣∣∣, = ∣∣∣∣r(t)− ( r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ ) + (r0 + ϕ(r) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, r(δ), p(δ))dδ)| − ( r0 + ϕ(τ) + ξ Γ(η) ∫ t 0 (t− δ)η−1δξ−1f(δ, τ(δ), κ(δ)ds ) ∣∣∣∣, ≤ λinb(t)ϵ1 + |(ϕ(r − ϕ(t, τ(v))|+ β(η, ξ)T η+ξ−1|f(δ, r(δ), p(δ)))− f(δ, τ(δ), κ(δ)|, ≤ λinb1(t)ϵ1 + Lϕ|r(t)− τ(t)|+ δLf (||p(t)− τ(t)|+ |p(t)− κ(t)|). From the above, we gets |r(t)− τ(t)| − δLf 1− (Lϕ + δLf ) |p(t)− κ(t)| ≤ λinb1(t)ϵ1 1− (Lϕ + δLf ) . (26) Similarly, one can obtain the following result |p(t)− κ(t)| − δLg 1− (Lψ + δLg) |r(t)− τ(t)| ≤ λinb2(t)ϵ2 1− (Lψ + δLg) . (27) In matrix form, the inequalities (26) and (27) can be expressed as( 1 −Lfc1 −Lgc2 1 ) × ( |r − τ | |p− κ| ) ≤ ( c1λinb1(t)ϵ1 c2λinb2(t)ϵ2 ) , Let max{λinb1(t), λinb2(t)} = λb(t), max{ϵ1, ϵ2} = ϵ and solving the above matrix, we get |(r, p)− (τ, κ)| ≤ D(1,2)ϵλb(t), (28) where D(1,2) = (c1 + c2 + c1c2(Lf + Lg))λ 1− (c1c2LfLg) , Hence, the solution of coupled system of FFDEs (2) is UHR stable. Now, by using above inequality with D(1,2,ϵ) = (c1 + c2 + c1c2(Lf + Lg))λϵ 1− (c1c2LfLg) , A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 15 of 19 we have |(r, p)− (τ, κ)| ≤ D(1,2,ϵ)λb(t). (29) So the coupled system of FFDEs (2) is GUHR stable. 5. Numerical Application This section of our work, is committed to numerical application of our finding. We present a numerical example to elaborate the main finding of this work. Example 1. Consider the coupled system of FFDEs with t ∈ [0, T ] as follows FDDηr(t) = ξtξ−1 [ e−πt 42 + t2 + |r(t)| et(50 + t3) + t2|sinp(t)| 51 ] , FDDηp(t) = ξtξ−1 [ t3 75 + e−tcos|r(t)| 39 + t5 + |p(t)| 39 + |cosp(t)| ] , r(0) = 3 73 + sin(r) 76 , p(0) = 17 49 + √ p 45 (30) From (30), we have the following f(t) = ξtξ−1 [ e−πt 42 + t2 + |r(t)| et(500 + t3) + t2|sinp(t)| 500 ] , g(t) = ξtξ−1 [ t3 75 + e−tcos|(t)| 400 + t5 + |p(t)| 400 + |cosp(t)| ] , ϕ(0) = sin(r) 76 , ψ(0) = √ p 45 . From coupled system (30), we can easily obtained, Lϕ = 1 76 , Lψ = 1 45 , c1 = c2 = 1 500 , c3 = 1 42d1 = d2 = 1 400 , d3 = 1 75 and Lf = 1 500 , Lg = 1 400. Also, L = max{Lϕ, Lψ} = 1 45 < 1. Let particularly use T = 1, 2, we have Hence, the given coupled system of FFDE (30) has at least one solution. Now further we have L+WT η+ξ−1β(η, ξ) = 1 45 + T η+ξ−1 ( 1 500 + 1 400 ) B (η, ξ) < 1. (31) Here, we demonstrate graphically to show that the quantity Υ = L+WT η+ξ−1B (η, ξ) < 1 for η, ξ ∈ (0, 1] in figure 1 A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 16 of 19 Figure 1: Graphical illustration of quantity Υ in η, ξ space by using T = 1, 2. From figure 1, we see that as the values of ξ, η increase over (0, 1], the value of Υ does not exceed 1, which authenticate the result (31) for all values η, ξ ∈ (0, 1]. Hence, the coupled system (30) has a unique solution. The problem (30) is UH and GUH stable, since c1c2LfLg ̸= 1 for c1, c2. Similarly, by the same procedure the coupled system (30) is UHR and GUHR stable with w(t) = s(t) = t for t ∈ (0, 1). 6. Conclusion In this research work, we have investigated a coupled system of FFDEs for existence theory and stability analysis. We have deduced sufficient conditions for the existence, uniqueness of solution to the mentioned problem by using FPT. In addition, appropriate results devoted to UH stability has also been developed. Different kinds of UH stability were deduced for the considered problem. Here, it is remarkable that coupled systems have very rarely investigated for the existence theory of solution and stability analysis. The mentioned type systems have numerous applications in variety of disciplines of science and technology. Also, in the future, the coupled systems of drug therapy, coupled system of mechanical oscillators and coupled system of planetary system can be studied by following our established analysis. Acknowledgements Authors are thankful to Prince Sultan University for APC and support through TAS research lab. A. Ali et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5963 17 of 19 Competing interest Does not exist. References [1] HongGuang Sun, Yong Zhang, Dumitru Baleanu, Wen Chen, and YangQuan Chen. A new collection of real world applications of fractional calculus in science and engineer- ing. 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