EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6411 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Study of a Coupled System of Fractional Differential Equations with Two Points Integral Boundary Conditions Shaher Momani1,2, Hamzeh Zureigat3, Shrideh Al-Omari4,∗, Mona Mohammad Khandaqji5, Mohammed Al-Smadi2,6,7 1 Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan 2 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 20550, UAE 3 Department of Mathematics, Faculty of Science and Technology, Jadara University, 21110 Irbid, Jordan 4 Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 11134, Jordan 5 Department of Basic Science, Faculty of Arts and Science, Applied Science Private University, Amman 11931, Jordan 6 College of Commerce and Business, Lusail University, Lusail, Qatar 7 Department of Applied Science, Ajloun College, Al Balqa Applied University, Ajloun, 26816, Jordan Abstract. In this paper, a certain system of fractional differential equations of integral boundary conditions BCs at two points is discussed. The presented coupled fractional system are useful for describing real-world phenomena, such as in physics, biology, and engineering. By utilizing the contraction mapping principle, we demonstrate uniqueness of certain solutions of the given system. Next, we utilize the contraction mapping principle to prove uniqueness of each solution. Further, we address the Hyers-Ulam stability and provide its conditions to show that small changes in the input lead to small changes in the result. Moreover, we provide numerical examples to support and demonstrate our theoretical results. 2020 Mathematics Subject Classifications: 34A08, 34A12, 26A33, 47H10 Key Words and Phrases: Coupled system, fractional differential equation, fractional derivative, Hyers-Ulam stability ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6411 Email addresses: shahermm@yahoo.com (S. Momani), hamzeh.zu@jadara.edu.jo (H. Zureigat), shrideh@bau.edu.jo (S. Al-Omari), m_khandakji@asu.edu.jo (M. M. Khandaqji), malsmadi@lu.edu.qa (M. Al-Smadi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 2 of 16 1. Introduction Due to their numerous applications in the natural sciences and engineering [1, 2], fractional differential equations (FDEs) have garnered more attention recently [3, 4]. In the last decade, the study of FDEs has garnered significant attention, resulting in a fundamental of articles in many applications across various fields in mathematical modelling of complex systems [5, 6]. Such type of equations have been successfully applied to address real-life problems that often involve integral BCs of blood flow dynamics [7], underground water flow [8], and population dynamics [9, 10]. A coupled system of FDEs with boundary conditions BCs is a complex mathematical model used to describe biological [11], different physical [12, 13], and engineering processes [14, 15]. Unlike regular differential equations, the FDEs use derivatives of non-integer orders to provide a better capture the memory and history of a system. When these equations are coupled, it means they are linked together and must be solved at the same time because the variables affect each other. To ensure that the solutions to real-life problems are realistic, mathematicians used the BCs to show how the solution should behave at the edges of the problem domain [16, 17]. These models are important in fields like materials science since [18] they explain how materials behave over time and in control systems where they manage complex systems. Solving these equations is challenging and requires advanced mathematical methods. In the last decade, many mathematicians studied a coupled system of FDEs of certain integral BCs. Ntouyas and Obaid [19] discussed a system of FDEs of coupled equations with nonlocal integral boundaries. The Banach’s fixed-point theorem and Schauder’s alternative are discussed and implemented to investigate both uniqueness and existence of solutions for the proposed coupled FDEs under Riemann-Liouville integral BCs. Ahmed and Ntouyas [20] utilized Schauder’s fixed-point and Banach’s fixed-point theorems to demonstrate the existence of solutions for coupled fractional DEs. These equations have different cases including those with coupled integral boundary conditions. The Banach fixed-point theorem and the Leray-Schauder alternative considered impor- tant proof techniques of coupled systems of nonlinear FDEs due to their effectiveness in investigating existence and uniqueness of solutions for coupled systems of nonlinear FDEs in those coupled systems that involve complex BCs. The Banach’s fixed-point theorem is valuable for proving uniqueness since it ensures a unique fixed point for contraction mappings on a complete metric space. In contrast, the Leray-Schauder is used to prove existence by dealing with compact and continuous mappings that may not be strict con- tractions. This led to extending the applicability of the fixed-point theory. These theorems provide strong and dependable techniques that have been demonstrated in the literature, making them ideal for nonlinear situations without the need for linearisation. Using fixed- point theorems of Banach and the Leray-Schauder alternative together takes advantage of their strengths which offers a complete approach to handling the complexity of a coupled system of fractional DEs. Such flexibility and dependability make them perfect for the theoretical framework needed to manage the detailed behavior of the coupled system of fractional DEs. Studying coupled systems of FDEs with integral BCs is crucial for theoretical mathe- S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 3 of 16 matics and practical applications. These systems arise in various fields such as population dynamics and heat conduction [21, 22]. By considering integral BCs, we capture more re- alistic behavior and improve accuracy in modeling real-world phenomena [23]. Solutions to these systems provide insights into physical processes and optimal control strategies [24]. This field provides our understanding of complex dynamics and contributes to scientific advancements. Thus, it is crucial to concentrate on studying the coupled systems of FDEs with integral BCs in order to define complete and realistic models, fill in research gaps, and improve our comprehension of stability and control in these systems. In this paper, the coupled system of nonlinear FDEs with two points integral coupled BCs is defined as follows: { cDα (t) = u (τ, υ (τ) , ω (τ)) , τ ∈ [0,H] , 1 < α ≤ 2, cDβ (t) = h (t, υ (τ) , ω (t)) , τ ∈ [0,H] , 1 < β ≤ 2, (1) where H > 0 , enhanced with integral BCs in the following form:{ ∫ H 0 υ ′(t)dt = ζω ′(µ), ∫ H 0 ω′ (t) dt = ηυ′ (ρ) , µ, ρ ∈ [0,H] υ (0) = 0, ω (0) = 0, (2) where cDi represents the Caputo fractional derivatives of order i, i=α, β, and u, h∈C([0,H]×R2,R) are continuous functions, and ζ, η are real constants. The model in equations: E. (1) and E. (2) can describe real-life systems where the current state depends on past behavior, like stretchy materials, spreading substances, or controlling machines. It’s useful for accurately capturing how these systems behave over time. 2. Preliminaries In this section, we go over the meanings of fractional derivatives and integrals from [25, 26]. Definition 2.1: The Riemann-Liouville fractional integral of order p for a continuous function h is defined as Iph (t) = 1 Γ (p) ∫ t 0 h (s) (t− s)1−pds, p > 0. Definition 2.2: The Caputo fractional derivatives of order p for a continuous function h : [0,∞)→R is given as follows cDph (t) = 1 Γ (k − p) ∫ t 0 (t− s)k−p−1h(k) (s) ds, k − 1 < p < k, k = [p] + 1. We prove the following auxiliary lemma in order to specify the solution for the problem in E. (1) and E. (2). Lemma 2.3: Let x, y∈C([0,H],R then the unique solution for the problem for H > 0 cDαυ (τ) = x (τ) , τ ∈ [0,H] , 1 < α ≤ 2, cDβω (τ) = y (τ) , τ ∈ [0,H] , 1 < β ≤ 2,∫ H 0 υ′ (t) dt = ζω′ (µ) , ∫ H 0 ω′ (t) dt = ηυ′ (ρ) , υ (0) = 0, ω (0) = 0 , µ, ρ ∈ [0,H] (3) S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 4 of 16 is υ(τ) = τ Λ ( ζH ∫ µ 0 (µ− s)β−2 Γ(β − 1) y(s) ds−H ∫ H 0 ∫ s 0 (s− t)α−2 Γ(α− 1) x(t) dt ds + ζη ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) x(s) ds− ζ ∫ H 0 ∫ s 0 (s− t)β−2 Γ(β − 1) y(t) dt ds ) + ∫ τ 0 (τ − s)α−1 Γ(α) x(s) ds. (4) and ω(τ) = τ Λ ( ζη ∫ µ 0 (µ− s)β−2 Γ(β − 1) y(s) ds− η ∫ H 0 ∫ s 0 (s− t)α−2 Γ(α− 1) x(t) dt ds +Hη ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) x(s) ds−H ∫ H 0 ∫ s 0 (s− t)β−2 Γ(β − 1) y(t) dt ds ) + ∫ τ 0 (τ − s)β−1 Γ(β) y(s) ds. (5) where Λ=H2−ζη ̸=0. Proof: The general solution of the coupled system in E. (3) are referred to as follows [26] υ (τ) = a0τ + a1 + 1 Γ (α) ∫ τ 0 (τ − s)α−1x (s) ds, (6) ω (τ) = b0τ + b1 + 1 Γ (β) ∫ τ 0 (τ − s)β−1y (s) ds, (7) where a0, a1, b0, b1 are arbitrary constants. By applying the conditions υ (0)= 0 and ω (0)= 0, we obtain a1=b1= 0. Here, we have υ′ (τ) = a0 + 1 Γ (α− 1) ∫ τ 0 (τ − s)α−2x (s) ds, ω′ (τ) = b0 + 1 Γ (β − 1) ∫ τ 0 (τ − s)β−2y (s) ds. Therefore, in view of the conditions∫ H 0 υ′(s)ds = ζω′ (µ) , ∫ H 0 ω′ (s) ds = ηυ′ (ρ) , we get a0H+ ∫ H 0 ∫ s 0 (s− t)α−2 Γ (α− 1) x (t)dtds = ζb0 + ζ ∫ µ 0 (µ− s)β−2 Γ (β − 1) y (s) ds, and b0H+ ∫ H 0 ∫ s 0 (s− t)β−2 Γ (β − 1) y (t)dtds = a0η + η ∫ ρ 0 (ρ− s)α−2 Γ (α− 1) x (s) ds, S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 5 of 16 provided that a0 = 1 H ( ζb0 + ζ ∫ µ 0 (µ− s)β−2 Γ (β − 1) y (s) ds− ∫ H 0 ∫ s 0 (s− t)α−2 Γ (α− 1) x (t)dtds ) , and b0 = 1 H ( a0η + η ∫ ρ 0 (ρ− s)α−2 Γ (α− 1) x (s) ds− ∫ H 0 ∫ s 0 (s− t)β−2 Γ (β − 1) y (t)dtds ) . After substituting the value of a0 into b0, we derive the final result for such constants in the form b0 = 1 Λ ( ζη ∫ µ 0 (µ− s)β−2 Γ(β − 1) y(s) ds− η ∫ H 0 ∫ s 0 (s− t)α−2 Γ(α− 1) x(t) dt ds +Hη ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) x(s) ds−H ∫ T 0 ∫ s 0 (s− t)β−2 Γ(β − 1) y(t) dt ds ) . (8) and a0 = 1 Λ ( ζH ∫ µ 0 (µ− s)β−2 Γ(β − 1) y(s) ds−H ∫ T 0 ∫ s 0 (s− t)α−2 Γ(α− 1) x(t) dt ds + ζη ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) x(s) ds− ζ ∫ T 0 ∫ s 0 (s− t)β−2 Γ(β − 1) y(t) dt ds ) . (9) Hence, substituting the values of a0, a1, b0, b1 in E. (6) and E. (7), gives E. (4) and E. (5). The converse can be established after a straightforward computation. Thus, the proof is therefore completed. ■ In summary, the unique solution for the coupled fractional system defined in E. (3) is obtained by proving Lemma 2.3, where the solution is presented in E. (4) and E. (5). 3. Existence and Uniqueness This section covers uniqueness and existence of the connected fractional system defined by equations E. (1) and (2). Let’s define the space G = {υ (τ) , υ (τ) ∈ C([0,H]} , Z = {ω (τ) , ω (τ) ∈ C([0,H]} , with the norm ∥υ∥= sup 0≤τ≤H |υ (τ)| and∥ω∥= sup 0≤τ≤H |ω (τ)| , respectively. It is obvious that both (G, ∥ .∥) and (Z, ∥ .∥) are considered as Banach spaces. Therefore, the product space (G×Z, ∥(υ, ω)∥) is also a Banach space as well as ∥(υ, ω)∥= ∥υ∥+ ∥ω∥. But by considering Lemma (??), we establish the operator Q:G×Z→G×Z S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 6 of 16 Q (υ, ω) (τ) = (Q1 (υ, ω) (τ) , Q2 (υ, ω) (τ)) , where Q1(υ, ω)(τ) = τ Λ ( ζH ∫ µ 0 (µ− s)β−2 Γ(β − 1) h(s, υ(s), ω(s)) ds −H ∫ H 0 ∫ s 0 (s− t)α−2 Γ(α− 1) u(t, υ(t), ω(t)) dt ds + ζη ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) u(s, υ(s), ω(s)) ds − ζ ∫ H 0 ∫ s 0 (s− t)β−2 Γ(β − 1) h(t, υ(t), ω(t)) dt ds ) + ∫ τ 0 (τ − s)α−1 Γ(α) u(s, υ(s), ω(s)) ds. (10) and Q2(υ, ω)(τ) = τ Λ ( ζη ∫ µ 0 (µ− s)β−2 Γ(β − 1) h(s, υ(s), ω(s)) ds − η ∫ H 0 ∫ s 0 (s− t)α−2 Γ(α− 1) u(t, υ(t), ω(t)) dt ds +Hη ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) u(s, υ(s), ω(s)) ds −H ∫ H 0 ∫ s 0 (s− t)β−2 Γ(β − 1) h(t, υ(t), ω(t)) dt ds ) + ∫ τ 0 (τ − s)β−1 Γ(β) h(s, υ(s), ω(s)) ds. (11) Indeed, properties of existence and uniqueness of the solutions of E. (1) and E. (2) may be established by applying the Banach’s contraction mapping theory. Theorem 3.1: Let u, h: [0,H]×R2→R be jointly continuous functions. Assume that (i) there exist constants Θ,ϖ∈R such that ∀ z1, z2, k1, k2∈R,∈ [0,H], we have |u (,z1, z2)−u (,k1, k2)| ≤Θ(|z2 − z1|+ |k2 − k1|) , and |h (τ, z1, z2)− h (τ, k1, k2)| ≤ ϖ (|z2 − z1|+ |k2 − k1|) . (i) Θ(N1 +N3) +ϖ (N2 +N4) < 1. Then the coupled fractional system in the E. (1) and E. (2) has a unique solution on [0,H], where N1 = H |Λ| ( Hα+1 Γ (α+ 1) + |ζη| ρα−1 Γ (α) ) + Hα Γ (α+ 1) , S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 7 of 16 N2 = H |Λ| ( |ζ|Hµβ−1 Γ (β) + |ζ|Hβ Γ (β + 1) ) , N3 = H |Λ| ( |η|Hα Γ (α+ 1) + H |η| ρα−1 Γ (α) ) , N4 = H |Λ| ( |ζη|µβ−1 Γ (β) + Hβ+1 Γ (β + 1) ) + Hβ Γ (β + 1) . Proof: Let us define sup0≤τ≤Hu (τ, 0, 0)=u0<∞, sup0≤τ≤Hh (τ, 0, 0)=h0<∞ and Ψr= {(υ, ω)∈G×Z: ∥(υ, ω)∥≤r}, and r> 0, such that r ≥ (N1 +N3)u0 + (N2 +N4)h0 1− [Θ (N1 +N3) +ϖ (N2 +N4)] . Hence, we first show that QΨr⊆Ψr. By our assumption, (υ, ω)∈Ψr, τ∈ [0,H] , we have |u (τ, υ (τ) , ω (τ))| ≤ |u (τ, υ (τ) , ω (τ))− u (τ, 0, 0)|+ |u (τ, 0, 0)| , ≤ Θ(|υ (τ)|+ |ω (τ)|) + u0 ≤ Θ(∥υ∥+ ∥ω∥) + u ≤ Θr + u0, and |h (τ, υ (τ) , ω (τ))| ≤ ϖ (|υ (τ)|+ |ω (τ)|) + h0 ≤ ϖ (∥υ∥+ ∥ω∥) + h0 ≤ ϖr + h0, which lead to |Q1(υ, ω)(τ)| ≤ H |Λ| ( |ζ|H ∫ µ 0 (µ− s)β−2 Γ(β − 1) ds ( ϖ(∥υ∥+ ∥ω∥) + h0 ) +H ∫ H 0 ∫ s 0 (s− t)α−2 Γ(α− 1) dt ds ( Θ(∥υ∥+ ∥ω∥) + u0 ) + |ζη| ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) ds ( Θ(∥υ∥+ ∥ω∥) + u0 ) + |ζ| ∫ H 0 ∫ s 0 (s− t)β−2 Γ(β − 1) dt ds ( ϖ(∥υ∥+ ∥ω∥) + h0 )) + sup 0≤τ≤H ∫ τ 0 (τ − s)α−1 Γ(α) ds ( Θ(∥υ∥+ ∥ω∥) + u0 ) ≤ ( Θ(∥υ∥+ ∥ω∥) + u0 ) [H Λ ( Hα+1 Γ(α+ 1) + |ζη|ρα−1 Γ(α) ) + Hα Γ(α+ 1) ] + ( ϖ(∥υ∥+ ∥ω∥) + h0 ) [H Λ ( |ζ|Hµβ−1 Γ(β) + |ζ|Hβ Γ(β + 1) )] ≤ ( Θ(∥υ∥+ ∥ω∥) + u0 ) N1 + ( ϖ(∥υ∥+ ∥ω∥) + h0 ) N2 ≤ (Θr + u0)N1 + (ϖr + h0)N2. (12) In alike manner S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 8 of 16 |Q2 (υ, ω) (τ)| ≤ (Θ (∥υ∥+ ∥ω∥) + u0)N3+(ϖ (∥υ∥+ ∥ω∥) + h0)N4 ≤ (Θr + u0)N3+(ϖr + h0)N4. Hence, we derive ∥Q1 (υ, ω)∥ ≤ (Θr + u0)N1 + (ϖr + h0)N2, and ∥Q2 (υ, ω)∥ ≤ (Θr + u0)N3 + (ϖr + h0)N4. Consequently, we get ∥Q (υ, ω)∥ ≤ (Θr + u0) (N1 +N3) + (ϖr + h0) (N2 +N4) ≤ r . Therefore, it follows ∥Q (υ, ω)∥≤r, i.e., QΨr⊆Ψr. Now, let (υ1, ω1) , (υ2, ω2)∈G×Z,∀τ∈ [0,H], υ, ω ϵR. Then, we get |Q1(υ1, ω1)(τ)−Q1(υ2, ω2)(τ)| ≤ H |Λ| ( |ζ|H ∫ µ 0 (µ− s)β−2 Γ(β − 1) ds ϖ ( ∥υ2 − υ1∥+ ∥ω2 − ω1∥ ) +H ∫ H 0 ∫ s 0 (s− t)α−2 Γ(α− 1) dt ds Θ ( ∥υ2 − υ1∥+ ∥ω2 − ω1∥ ) + |ζη| ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) ds Θ ( ∥υ2 − υ1∥+ ∥ω2 − ω1∥ ) + |ζ| ∫ H 0 ∫ s 0 (s− t)β−2 Γ(β − 1) dt ds ϖ ( ∥υ2 − υ1∥+ ∥ω2 − ω1∥ )) + sup 0≤τ≤H ∫ τ 0 (τ − s)α−1 Γ(α) ds Θ ( ∥υ2 − υ1∥+ ∥ω2 − ω1∥ ) . (13) That is, ∥Q1 (υ1, ω1)−Q1 (υ2, ω2)∥ ≤ N1Θ(∥υ2 − υ1∥+ ∥ω2 − ω1∥)+N2ϖ (∥υ2 − υ1∥+ ∥ω2 − ω1∥) . (14) Similarly, it follows ∥Q2 (υ1, ω1)−Q2 (υ2, ω2)∥ ≤ N3Θ(∥υ2 − υ1∥+ ∥ω2 − ω1∥)+N4ϖ (∥υ2 − υ1∥+ ∥ω2 − ω1∥) . (15) For υ, ω ϵR we, from (14) and (15), deduced that ∥Q (υ1, ω1)−Q (υ2, ω2)∥ ≤ (Θ (N1 +N3) +ϖ (N2 +N4)) (∥υ2 − υ1∥+ ∥ω2 − ω1∥) , as Θ(N1 +N3)+ϖ (N2 +N4)< 1. Further, as the operator Q is a contraction operator, the operator Q possesses a unique fixed point according to the Banach’s fixed-point theorem. This, indeed, equates to the unique solution of Eqs. (1) and (2). The proof is now complete. ■ The following outcome relies on the Leray-Schauder alternative. Lemma 3.2: (Leray-Schauder alternative) [27]: Let F : E → E be a completely con- tinuous operator and assume E (F )= {x ∈ E : x = λF (x) for some 0 < λ < 1}. Then, either the set E (F ) is unbounded or F has at least one fixed point. S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 9 of 16 Theorem 3.3: Let u, h : [0,H]×R2 → R be a continuous function. Assume the following hold true (i) There exist 1, 2,ϕ1, ϕ2≥0 where 1, 2,ϕ1, ϕ2 are real constants and 0, ϕ0> 0 such that ∀i, i∈R, (i = 1, 2) , |u (τ, υ1, υ2)| ≤ ψ0 + ψ1 |υ1|+ ψ2 |υ2| , and |h (τ, υ1, υ2)| ≤ ϕ0 + ϕ1 |υ1|+ ϕ2 |υ2| . (i) (N1 +N3)ψ1 + (N2 +N4)ϕ1 < 1, and (N1 +N3)ψ2 + (N2 +N4)ϕ2 < 1, where Ni, i= 1, 2, 3, 4 are defined in Theorem 3.1. Then, there exists at least one solution for the coupled fractional system in E. (1) and E. (2). Proof: The proof of this theorem is divided into two parts: First Step: Show that Q:G×Z→G×Z is completely continuous, where the continuity of the operator Q holds by the continuity of the functions u, h. Let R⊆G×Z be bounded. Then, there exist positive constants λ1, λ2 such that |u (τ, υ (τ) , ω (τ))| ≤ λ1, |h (τ, υ (τ) , ω (τ))| ≤ λ2, ∀τ ∈ [0,H] . Therefore, ∀ (υ, ω)∈R, we have |Q1 (υ, ω) (τ)| ≤ N1λ1 +N2λ2. This implies that ∥Q1 (υ, ω)∥ ≤ N1λ1 +N2λ2. Similarly, we get ∥Q2 (υ, ω)∥ ≤ N3λ1 +N4λ2. Therefore, based on the inequalities stated earlier, it can be concluded that the operator Q is uniformly bounded because∥Q (υ, ω)∥ ≤ (N1 +N3)λ1 + (N2 +N4)λ2. Furthermore, we prove that the operator Q is equicontinuous. For, assume k1, k2∈ [0,H] with k1 0, we consider the following inequality ∥M1 (υ, ω)∥≤r1 and ∥M2 (υ, ω)∥≤r2. (20) S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 12 of 16 Definition 4.1: [27, 28] The coupled fractional system in E.(1) and E.(2) is said to be Hyers-Ulam stable, if there exist EQ1 , EQ2> 0, such that for every solution (υ∗, ω∗)∈C ([0,H] ,R)×C ([0,H] ,R) as given in Eq.(12) , there exists a unique solution (υ, ω)∈C([0,H],R)×C([0,H],R) of the system presented in Eq.(1) and Eq.(2) such that ∥(υ, ω)− (υ∗, ω∗)∥≤EQ1r1+EQ1r2. Theorem 4.2: Let the assumptions of Theorem 3.1 hold. Then, the coupled system in E.(1) and E.(2) is Hyers-Ulam stable. Proof. Let (υ, ω)∈C([0,H],R)×C([0,H],R) be the solution of the problems in E.(1) and E.(2) satisfying Eq.(8) and Eq.(9) and (υ∗, ω∗) be any solution satisfying E.(20): cDαυ∗ (τ)=u (τ, υ∗ (τ) , ω∗ (τ))+M1 (υ ∗, ω∗) (τ) , τ∈ [0,H] , cDβω∗ (τ)=h (τ, υ∗ (τ) , ω∗ (τ))+M2 (υ ∗, ω∗) (τ) , τ∈ [0,H] . Then, we have υ∗(τ) = Q1(υ ∗, ω∗)(τ) + τ Λ [ ζH ∫ µ 0 (µ− s)β−2 Γ(β − 1) M2(υ ∗, ω∗)(s) ds −H ∫ H 0 ∫ s 0 (s− t)α−2 Γ(α− 1) M1(υ ∗, ω∗)(t) dt ds + ζη ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) M1(υ ∗, ω∗)(s) ds − ζ ∫ H 0 ∫ s 0 (s− t)β−2 Γ(β − 1) M2(υ ∗, ω∗)(t) dt ds ] + ∫ τ 0 (τ − s)α−1 Γ(α) M1(υ ∗, ω∗)(s) ds. (21) It hence follows that |Q1(υ ∗, ω∗)(τ)− υ∗(τ)| ≤ ∣∣∣∣ τΛ ( ζη ∫ µ 0 (µ− s)β−2 Γ(β − 1) r2(s) ds− η ∫ H 0 ∫ s 0 (s− t)α−2 Γ(α− 1) r1(t) dt ds +Hη ∫ ρ 0 (ρ− s)α−2 Γ(α− 1) r1(s) ds−H ∫ H 0 ∫ s 0 (s− t)β−2 Γ(β − 1) r2(t) dt ds ) + ∫ τ 0 (τ − s)β−1 Γ(β) r2(s) ds ∣∣∣∣ ≤ ( H Λ ( Hα+1 Γ(α+ 1) + |ζη|ρα−1 Γ(α) ) + Hα Γ(α+ 1) ) r1 + ( H Λ ( |ζ|Hµβ−1 Γ(β) + |ζ|Hβ Γ(β + 1) )) r2 ≤ N1r1 +N2r2. S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 13 of 16 Similarly, |Q2 (υ ∗, ω∗) (τ)−ω∗ (τ)| ≤N3r1+N4r2. We therefore arrive to the following conclusion based on using the operator Q’s fixed-point property, given by Eqs. (10) and (11), as follows |υ(τ)− υ∗ (τ)| = |υ (τ)−Q1 (υ ∗, ω∗) (τ) +Q1 (υ ∗, ω∗) (τ)− υ∗ (τ)| ≤ |Q1 (υ, ω) (τ)−Q1 (υ ∗, ω∗) (τ)|+ |M1 (υ ∗, ω∗) (τ)−υ∗ (τ)| ≤ (N1Θ+N2ϖ) ∥(υ, ω)− (υ∗, ω∗)∥+N1r1+N2r2. (22) Similarly |υ(τ)− υ∗ (τ)| = |υ (τ)−Q2 (υ ∗, ω∗) (τ) +Q2 (υ ∗, ω∗) (τ)− υ∗ (τ)| ≤ |Q2 (υ, ω) (τ)−Q2 (υ ∗, ω∗) (τ)|+ |M2 (υ ∗, ω∗) (τ)−υ∗ (τ)| ≤ (N3Θ+N4ϖ) ∥(υ, ω)− (υ∗, ω∗)∥+N3r1+N4r2,. (??) From E.(22) and E.(??) it follows that ∥(υ, ω)− (υ∗, ω∗)∥≤ (N1Θ+N2ϖ+N3Θ+N4ϖ) ∥(υ, ω)− (υ∗, ω∗)∥+(N1+N3) r1+(N2+N4) r2, ∥(υ, ω)− (υ∗, ω∗)∥≤ (N1+N3) r1+(N2+N4) r2 1− ((N1+N3)Θ+ (N2+N4)ϖ) , where EQ1= (N1+N3) 1−((N1+N3)Θ+(N2+N4)ϖ) , and EQ2= (N2+N4) 1−((N1+N3)Θ+(N2+N4)ϖ) . Thus we obtain the Hyers-Ulam stability condition. In summary, the Hyers-Ulam stability condition for the coupled fractional system defined in E.(1) is obtained by proving Lemma 4.2 5. Numerical Examples Example 1: Examine the following coupled system of FDEs given as cD3/2υ (τ) = 1 6π √ 81+τ2 ( |υ(τ)| 3+|υ(τ)| + |ω(τ)| 5+|υ(τ)| ) , cD7/4ω (τ) = 1 12π √ 64+τ2 (sin (υ (τ)) + sin (ω (τ))) ,∫ 1 0 υ ′ (s) ds = 2ω′ (1) , ∫ 1 0 ω ′(s)ds = −υ′ (1/2) , υ (0) = 0, ω (0) = 0, (23) where α = 3 2 , β = 7 4 ,H = 1, ζ = 2, η = −1, ρ = 1 2 , µ = 1. Using the given data, we find that Λ= 3,N1= 1.269,N2= 1.1398,N3= 0.5167,N4= 1.554,Θ= 1 54π , ϖ= 1 48π . It’s clear that u, h are jointly continuous functions and Θ(N1+N3)+ϖ (N2+N4)< 1, such that 1 54π (1.269 + 0.5167)+ 1 48π (1.1398 + 1.554)= 0.0283 < 1. S. Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6411 14 of 16 Therefore, all conditions of Theorem 3.1 are met, and as a result of E.(23) it has a unique solution is in the interval [0,1]. Example 2: Consider the following coupled system of fractional DEs cD5/3υ (τ) = 1 80+τ4 + |υ(τ)| 120(1+ω2(τ)) + 1 4 √ 2500+τ2 e−3τcos (ω (τ)) , τ ∈ [0, 1] cD 6 5ω (τ) = 1√ 16+τ2 cosτ + 1 150e −3τsin (ω (τ)) + 1 180υ (τ) , τ ∈ [0, 1]∫ 1 0 υ ′(s)ds = −3ω′ (1/3) , ∫ 1 0 ω ′(s)ds = υ′ (1) , υ (0) = 0, ω (0) = 0, (24) Where α = 5 3 , β = 6 5 ,H = 1, ζ = −3, η = 1, ρ = 1, µ = 1/3. Using the given data, we find that Λ= 3,N1= 1.269,N2= 1.1398,N3= 0.5167,N4= 1.554,Θ= 1 54π , ϖ= 1 48π . It is clear that |u (τ, υ1, υ2)| ≤ 1 80 + 1 120 ∥υ∥+ 1 200 ∥ω∥ , and |h (τ, υ1, υ2)| ≤ 1 4 + 1 180 ∥υ∥+ 1 150 ∥ω∥ . Thus, ψ0 = 1 80 , ψ1 = 1 120 , ψ2 = 1 200 , ϕ0 = 1 4 , ϕ1 = 1 180 , ϕ2 = 1 150 . We find (N1+N3)ψ1+(N2+N4)ϕ1= 0.0298 < 1 and (N1+N3)ψ2+(N2+N4)ϕ2= 0.0269 < 1. Therefore, according to Theorem 3.3, E.(24) has at least one solution in the interval [0, 1]. 6. Conclusion This article discussed existence, uniqueness, and stability of a coupled system of FDEs with BCs that are two point integral coupled. The fixed-point principle is used to examine the uniqueness of the solutions of the given problem. Then, the existence of solutions for the given coupled system is also verified by using the Leray-Schauder’s alternative. Moreover, the Hyers-Ulam stability has been employed to discuss the proposed fractional coupled system’s stability. Furthermore, two numerical examples are provided to demonstrate our results. Such an alternative stability theories and availability of solutions for a nonlinear coupled system of three FDEs with nonlocal coupled BCs and non-separated BCs will be our further research. Author Contributions: Conceptualization, S. Momani; methodology, S. Al-Omari.; software, M. Khandaqji, validation, S. Al-Omari, M. Al-Smadi; formal analysis, H. Zureigat; investigation, H. Zureigat; writing-original draft preparation, M. 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