EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 4071-4092 ISSN 1307-5543 – ejpam.com Published by New York Business Global On a Hybrid Class of p-Laplacian Initial Value Problems with Modified Mittag-Leffler Kernel Sowmiya Ramasamy1, Kavitha Velusamy1, Dumitru Baleanu2, Mallika Arjunan Mani3,∗ 1 Department of Mathematics, School of Sciences, Arts Media & Management, Karunya Institute of Technology and Sciences, Karunya Nagar, Coimbatore-641114, Tamil Nadu, India 2 Department of Computer Science and Mathematics, Labanese American University, Beirut, Lebanon 3 Department of Mathematics, School of Arts, Sciences, Humanities and Education, SASTRA Deemed to be University, Thanjavur-613401, Tamil Nadu, India Abstract. In this work, we establish key results on the existence theory for a category of initial value problems (IVPs) involving hybrid fractional integro-differential equations (HFIDEs) with a p-Laplacian operator, utilizing the modified Mittag-Leffler kernel. By employing Krasnoselskii and Banach fixed point theorems (FPTs), we determine the conditions required for the existence of solutions. Additionally, we examine the Hyers-Ulam (H-U) stability of the problem. Lastly, we present an example to confirm our theoretical results. 2020 Mathematics Subject Classifications: 26A33, 34A08, 34D20 Key Words and Phrases: Fractional-order, mABC fractional derivative, Existence and uniqueness, Stability, FPTs 1. Introduction This paper addresses the existence of solutions for a hybrid class of mABC-HFIDEs with a p-Laplacian operator given by the abstract form: CDβ Ψp mABCDρ 0+  w(7)− h(7, w(7)) Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ    = g(7, w(7)), w(0) = h(0, w(0)) +Q(0)ABIρ0+Θ, (1) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5455 Email addresses: sowmiyar@karunya.edu.in (R. Sowmiya), kavi velubagyam@yahoo.co.in (V. Kavitha), dumitru.baleanu@lau.edu.lb (D. Baleanu), arjunphd07@yahoo.co.in (M. M. Arjunan) https://www.ejpam.com 4071 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4072 where ρ, β, γ ∈ (0, 1) and mABCDρ 0+ represents the mABC derivative, ABIρ0+ is the Atangana-Baleanu fractional integral, CDβ signifies the Caputo fractional derivative, Ψp, p > 1 is a p-Laplacian operator, Θ ∈ R, Q : Ω → R, where Ω = [0, Z], f, g, h ∈ C(Ω× R,R) with Q(7) + Iγ0+f(7, w(7)) ̸= 0. Fractional differential equations extend traditional differential equations by incorporating derivatives of non-integer orders. This extension allows for the modeling of processes that involve complex dynamics, such as systems with memory and hereditary characteristics. Unlike standard derivatives, which are local operators, fractional derivatives consider the entire history of the function, making them ideal for modeling phenomena where past states influence the present and future behavior. This non-local nature of fractional derivatives has made them increasingly popular in various scientific and engineering disciplines, where they offer a more nuanced understanding of systems exhibiting non-traditional dynamics [6, 9, 15, 22, 26]. The use of fractional differential equations has become widespread across different fields due to their ability to model processes more accurately than traditional differential equations. In physics, they are used to describe anomalous diffusion processes, where the movement of particles does not follow the standard pattern seen in classical diffusion [24]. In biology, fractional differential equations help model complex biological processes, such as the diffusion of substances across cellular membranes and the dynamics of cell potentials [19–21]. In engineering, these equations are crucial for modeling materials with viscoelastic properties, where the relationship between stress and strain is not instantaneous but depends on the material’s history [27]. In finance, fractional models are employed to capture memory effects in stock prices and to model the dynamics of financial instruments over time. These applications highlight the versatility and effectiveness of fractional differential equations in providing deeper insights into various complex systems. The realm of fractional calculus has expanded remarkably with the introduction of various fractional derivative definitions, each bringing its own advantages and specific uses. Among the pioneering contributions is the Caputo derivative, introduced by Michele Caputo [7] in 1967, which has gained widespread recognition for its practical utility. Despite its widespread adoption, the Caputo derivative’s reliance on a single kernel presents certain constraints, particularly when modeling diverse phenomena. To overcome these limitations, Caputo and Fabrizio [8] proposed a new approach by introducing a non-singular derivative based on the exponential function. This innovation effectively addresses the issue of singularity, although it encounters difficulties when applied to systems that do not naturally follow exponential patterns. Seeking to further expand the modeling potential of fractional derivatives, Atangana and Baleanu [4] introduced a derivative based on the extended Mittag-Leffler function. This derivative allows for a more flexible description of non-local and non-singular kernels, thereby extending the range of phenomena that can be accurately represented. Building on these significant developments, Refai and Baleanu recently introduced the mABC-derivative, a novel operator that merges the strengths of both the Caputo and Atangana-Baleanu derivatives [2]. This new tool offers a robust solution for tackling complex problems that were previously challenging to address with existing methodologies, marking a substantial M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4073 advancement in the field of fractional calculus. Initial value problems (IVPs) play a pivotal role in the mathematical modeling of real-world systems, where the state of a system at a given initial time dictates its future behavior. In the context of p-Laplacian equations, IVPs involve determining the evolution of a system governed by a nonlinear differential operator. The analysis of IVPs for p- Laplacian equations is particularly challenging due to the nonlinearity of the operator, which can lead to complex dynamics, including the existence of multiple solutions, bifurcations, and sensitivity to initial conditions. Understanding these aspects is crucial for accurately predicting the behavior of the modeled systems, whether they pertain to physical processes, biological systems, or engineering applications. In recent times, a novel category known as hybrid boundary value problems has gained prominence, integrating aspects from both linear and nonlinear theories. This hybrid approach facilitates a more thorough comprehension of intricate systems, where conventional methods might be inadequate. A notable contribution to this field is the research conducted by Dhage [10, 13], which highlights the significance of hybrid differential equations (HDEs) in the analysis of dynamical systems. Dhage meticulously categorized HDEs based on different types of perturbations, emphasizing their importance in refining perturbation techniques within the expansive domain of differential and integral equations. His work underscores the potential of HDEs to provide deeper insights and more robust solutions to complex mathematical problems. Following Dhage’s pioneering contributions, numerous researchers in mathematics and related disciplines have focused on exploring various hybrid differential equations (HDEs). A key discovery from this extensive research is that fractional-order hybrid differential equations (FHDEs) offer a more detailed representation of hereditary and memory effects, particularly in fields such as biology, chemistry, and physics. This enhanced capability allows FHDEs to outperform traditional integer-order HDEs, capturing the interest of many scholars and prompting deeper investigations into their properties. Building on the foundational work of Dhage et al. [11, 12], who explored the conditions for the existence and uniqueness of solutions in FHDEs, Baleanu et al. [5] integrated Caputo fractional derivatives within a hybrid framework. Their study of a thermostat model demonstrated the effectiveness of this approach in revealing complex dynamical behaviors. The exploration of FHDEs has since led to a wealth of contributions, with researchers examining various derivatives such as the Hadamard derivative [1], the Riemann derivative [32], the Hilfer derivative [30], and the ABC derivative [3, 18, 20, 28, 29, 31]. Additionally, innovative formulations like the mABC derivative have been proposed [19, 21], further broadening the scope of fractional calculus in the context of HDEs. Despite the growing interest in fractional calculus, the application of the mABC fractional derivative in FHDEs involving the p-Laplacian operator with IVPs remains largely unexplored in the current literature. This intriguing gap presents a unique opportunity for further investigation and serves as the primary motivation for this work. This work makes the following key contributions to the field. (i) This research represents the first known attempt to address mABC-HFIDEs in conjunction with the p-Laplacian operator for the system described in (1). It thoroughly examines the existence, uniqueness, and stability of the proposed system. M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4074 (ii) By utilizing the properties of the mABC derivative, we have derived the solution for the system outlined in (1), as detailed in Lemma 2. (iv) Expanding upon the seminal contributions of previous research [17, 20, 21, 31], this study offers novel insights and significantly extends the applicability of prior findings. This paper is organized in a carefully structured way. Section 2 establishes the necessary foundation by providing the reader with essential background information. This includes the definition of the mABC fractional derivative, relevant results, and the fundamental concept of fixed-point theory that underpins our analysis. Section 3 addresses the core issues of existence and uniqueness of solutions for system (1). We effectively utilize both the Banach and Krasnoselskii FPTs to accomplish these objectives. To deepen our understanding, Section 4 thoroughly examines the stability properties of the system, highlighting its behavior in response to perturbations. Finally, Section 5 offers a numerical example to demonstrate the application and importance of our main findings. 2. Preliminaries and Hypotheses In this section, we provide a comprehensive overview of the Caputo-type Mittag-Leffler fractional derivative (CMLFD) and integral operator and several fundamental properties. Definition 1. [4] Let 0 < ρ < 1 and y ∈ H1(0, Z), where Z > 0, the CMLFD of order ρ of y is defined as(ABCDρ 0+y ) (7) = B(ρ) 1− ρ ∫ 7 0 Eρ (−µρ(7 − σ)ρ) y′(σ)dσ, 0 < 7 < Z (2) where µρ = ρ 1− ρ ,B(ρ) = 1 − ρ + ρ Γ(ρ) is a normalization function satisfying B(0) = B(1) = 1. Let Eρ denote the ML function, defined by Eρ(z) = ∞∑ k=0 zk Γ(ρk + 1) , ρ > 0, z ∈ C H1(0, Z) = { v ∈ L2(0, Z) | v′ ∈ L2(0, Z) } . Additionally, Lm denotes the space of functions for which the m-th power of their absolute value is Lebesgue integrable. Definition 2. [4] Let 0 < ρ < 1 and y ∈ L1(0, Z), where Z > 0, the fractional integral associated with the above CMLFD of order ρ of y is described as:(ABIρ0+y ) (7) = 1− ρ B(ρ) y(7) + ρ B(ρ) ( RLIρ0+y ) (7), 0 < 7 < Z (3) where RLIρ0+ denotes the Riemann-Liouville fractional integral operator, given by: RLIρ0+y(7) = 1 Γ(ρ) ∫ 7 0 (7 − σ)ρ−1y(σ)dσ, 7 > 0. (4) M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4075 Remark 1. In (2), the kernel Eρ (−µρ(7 − σ)ρ) is non-singular. Since Eρ (−µρ(7 − σ)ρ) = Eρ ( − ρ 1− ρ (7 − σ)ρ ) . As 7 approaches σ, the term (7 − σ)ρ approaches 0. Thus, we have Eρ (−µρ(7 − σ)ρ) = Eρ(0) = 1. From this, we observe that (ABCDρ 0+y ) (0) = 0, if y ∈ H1(0, Z). The aforementioned operators serve as fundamental tools in establishing various theoretical underpinnings. The efficacy of these operators in practical applications is significantly influenced by the underlying function spaces, as elucidated by Al-Refai et al. in [2]. For instance, we fix y(7) ∈ H1(0, Z) then the system (ABCDρ 0+y ) (7) − ωy(7) = 0, ω ∈ R, has only the trivial solution y(7) = 0. However, in this context, the space is restrictive for the Caputo derivative. If we fix the space χ(y) = {y : y′ ∈ L1[0, 1]}, then the following system (ABCDρ 0+y ) (7) = { ωy(7), 7 ∈ (0, Z); y0, 7 = 0 with 0 < ρ < 1, we have the solution y(7) = y0Eρ,1(−ω7ρ), where Eρ,1(z) = ∞∑ k=0 zk Γ(ρk + 1) . This demonstrates the significant impact of space. To address this challenge, Al-Refai et al. [2] published a research work in which a larger space is selected to eliminate the need for additional conditions. Following [2], we presents a mABC fractional derivative operator, which is applicable in a broader functional space to address the initialization problem effectively. Definition 3. [2, 25] Let y ∈ L1(0, Z), Z > 0 and ρ ∈ (0, 1), the mABC derivative is described by ( mABCDρ 0+y ) (7) = B(ρ) 1− ρ [y(7)− Eρ (−µρ7ρ) y(0) −µρ ∫ 7 0 (7 − σ)ρ−1Eρ,ρ (−µρ(7 − σ)ρ) y(σ)dσ ] , 0 < 7 < Z (5) where µρ = ρ 1− ρ , the normalized function B(ρ) satisfies the property B(0) = B(1) = 1 and Eρ,ρ is the two-parameters M-L function. M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4076 We see that Definitions 1 and 3 are the same in the space H1(0, Z) ⊆ L1(0, Z). However, if y ∈ L1(0, Z), it is not guaranteed that ( mABCDρ 0+y ) (0) = 0. To support this result, we have the following example: Example 1. Consider u(7) = { 7− 3 4 , 7 ̸= 0 B, 7 = 0 where B ∈ R and u ∈ L1(0, Z). For B(ρ) = 1 − ρ + ρ Γ(ρ) and µρ = ρ 1− ρ , the modified Atangana-Baleanu derivative in Caputo sense is: ( mABCDρ 0+u ) (7) = B(ρ) 1− ρ [ u(7)− Eρ (−µρ7ρ)u(0) − µρ ∫ 7 0 (7 − σ)ρ−1Eρ,ρ (−µρ(7 − σ)ρ)u(σ) dσ ] . If ρ = 3 4 , then above expression becomes( mABCD 3 4 0+u ) (7) = 3.25 [ u(7)− E 3 4 ( −37 3 4 ) u(0) − 3 ∫ 7 0 (7 − σ)− 1 4E 3 4 , 3 4 ( −3(7 − σ) 3 4 ) u(σ) dσ ] . (6) For 7 ̸= 0: u(7) = 7− 3 4 . For 7 = 0: u(0) = B. Since ∫ 7 0 (7 − σ)− 1 4E 3 4 , 3 4 ( −3(7 − σ) 3 4 ) σ− 3 4dσ = Γ ( 1 4 ) E 3 4 ( −37 3 4 ) using the fact that∫ 7 0 (7 − σ)ρ−1Eρ,ρ (−µρ(7 − σ)ρ)σ−ρdσ = Γ(1− ρ)Eρ (−µρ7ρ) . Thus (6) becomes( mABCD 3 4 0+u ) (7) = 3.25 [ 7− 3 4 − E 3 4 ( −37 3 4 ) B − 3Γ ( 1 4 ) E 3 4 ( −37 3 4 )] . Consequently mABCD 3 4 0+u(0) = −9.75Γ ( 1 4 ) ̸= 0. M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4077 Remark 2. In the preceding illustration, it is observed that the fractional derivative mABCD 3 4 0+u(0) ̸= 0 at the point 7 = 0. Finally, we recall the well-known related to Caputo fractional derivative and the properties of the p-Laplacian operator. Definition 4. [6, 26] For any ρ > 0 and u ∈ C(0, T ) ∩ L(0, Z), we have Iρ0+D ρ 0+u(7) = u(7) + c0 + c17 + . . .+ cn−17n−1, fore some ci ∈ R, i = 1, 2, . . . , n − 1. where n = [ρ] + 1. In particular, when ρ ∈ (0, 1), Iρ0+D ρ 0+u(7) = u(7) + c0. Definition 5. [23] The p-Laplacian operator is given by Ψp(ū) = |ū|p−2ū = ūp−1, ū ≥ 0, p > 1, (7) at which Ψ−1 p = Ψq where 1 p + 1 q = 1. Lemma 1. [23] Assume that Ψp(ū), p ≥ 2, be p-Laplacian operator and |ū|, |v̄| ≤ M, then |Ψp(ū)−Ψp(v̄)| ≤ (p− 1)Mp−2|ū− v̄|. (8) Definition 6. A function w ∈ AC(Ω,R) is called a solution of the system (1) if function g ∈ L1(Ω,R), and w fulfills (1). Lemma 2. Let 0 < ρ < 1 and g ∈ L1(0, Z). Then, the solution w ∈ AC(Ω,R) of the system (1) iff it is a solution to the subsequent integral equation: w(7) = ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) (×) [ ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ] + h(7, w(7)). (9) Proof. Since CDβ ( Ψp ( mABCDρ 0+ ( w(7)− h(7, w(7)) Q(7) + Iγ0+f(7, w(7)) ))) = g(7, w(7)). Taking Iβ0+ on both sides of the above equation, we have Ψp ( mABCDρ 0+ ( w(7)− h(7, w(7)) Q(7) + Iγ0+f(7, w(7)) )) = Iβ0+g(7, w(7)) + c0, M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4078 for some c0 ∈ R. At 7 = 0, we have Ψp ( mABCDρ 0+ ( w(0)− h(0, w(0)) Q(0) )) = c0. That is Ψp ( mABCDρ 0+ ( w(7)− h(7, w(7)) Q(7) + Iγ0+f(7, w(7)) )) −Ψp ( mABCDρ 0+ ( w(0)− h(0, w(0)) Q(0) )) = Iβ0+g(7, w(7)). By the properties of p-Laplacian operator, we have mABCDρ 0+ ( w(7)− h(7, w(7)) Q(7) + Iγ0+f(7, w(7)) ) −mABC Dρ 0+ ( w(0)− h(0, w(0)) Q(0) ) = Ψq ( Iβ0+g(7, w(7)) ) . Taking ABIρ0+ on both sides of the above equation, we have w(7)− h(7, w(7)) Q(7) + Iγ0+f(7, w(7)) − w(0)− h(0, w(0)) Q(0) = ABIρ0+ ( Ψq ( Iβ0+g(7, w(7)) )) by using the fact that (ABIρ0+ mABCDρ 0+w ) (7) = w(7)− w(0). Thus w(7) = ( Q(7) + Iγ0+f(7, w(7)) ) (ABIρ0+(Θ) + ABIρ0+ ( Ψq ( Iβ0+g(7, w(7)) ))) + h(7, w(7)) or equivalently w(7) = ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) (×) [ ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ] + h(7, w(7)). (10) Conversely, we have w(7)− h(7, w(7)) Q(7) + Iγ0+f(7, w(7)) = ABIρ0+(Θ) + ABIρ0+ ( Ψq ( Iβ0+(g(7, w(7))) )) . Taking mABCDρ 0+ derivative on both sides of the above equation, we have mABCDρ 0+ ( w(7)− h(7, w(7)) Q(7) + Iγ0+f(7, w(7)) ) M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4079 =mABC Dρ 0+ ( ABIρ0+(Θ) + ABIρ0+ ( Ψq ( Iβ0+g(7, w(7)) ))) = Θ+ ( Ψq ( Iβ0+g(7, w(7)) )) by utilizing the fact that ( mABCDρ 0+ (ABIρ0+w )) (7) = w(7). Taking p-Laplacian operator on both sides, we have Ψp ( mABCDρ 0+ ( w(7)− h(7, w(7)) Q(7) + Iγ0+f(7, w(7)) )) = Ψp(Θ) + Iβ0+g(7, w(7)). Taking CDβ on both sides, we get CDβ ( Ψp ( mABCDρ 0+ ( w(7)− h(7, w(7)) Q(7) + Iγ0+f(7, w(7)) ))) = g(7, w(7)) by utilizing the fact that CDβIβ0+w(7) = w(7) and further w(0) = h(0, w(0))+Q(0)ABIρ0+Θ. Describe the operator Φ : AC(Ω,R) → AC(Ω,R) by (Φw)(7) = ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) (×) [ ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ] + h(7, w(7)). (11) According to equation (11), each fixed point of the operator Φ is associated with the desired solution of the system (1). Note 2.1. For our convenience, we split the operator (11) as: (Φw)(7) = (Aw)(7) + (Bw)(7), 7 ∈ Ω, where (Aw)(7) = ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) (×) [ ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ] , (12) M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4080 and (Bw)(7) = h(7, w(7)), 7 ∈ Ω. (13) Now |(Aw)(7)− (Aw)(7)| = ∣∣∣∣∣ ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) (×) [ ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ] − ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) (×) [ ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ]∣∣∣∣∣ We denote Aw = ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) Bw = ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ; Aw = ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) Bw = ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ. The right side of the above expression can be written as |AwBw − AwBw|. Using the property of absolute values for products, we can express this as: |AwBw −AwBw| = |AwBw −AwBw +AwBw −AwBw| ≤ |Aw(Bw −Bw) +Bw(Aw −Aw)| M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4081 ≤ |Aw||Bw −Bw|+ |Bw||Aw −Aw|. (14) Now |Bw −Bw| ≤ ∣∣∣∣∣ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ − ABIρ0+(Θ)− 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) − ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ∣∣∣∣∣ ≤ 1− ρ B(ρ) {∣∣∣∣∣Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) −Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) ∣∣∣∣∣ } + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ∣∣∣∣∣Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ ) −Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ ) ∣∣∣∣∣dσ, (15) and |Aw −Aw| ≤ ∣∣∣∣∣ ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) − ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) ∣∣∣∣∣ ≤ ∣∣∣∣ 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1[f(σ,w(σ))− f(s, w(s))]dσ ∣∣∣∣ . (16) We will now outline the following assumptions: (A1) For positive constants Lf , Lg, Lh > 0, it holds that for any elements w,w1, w, w1 ∈ AC(Ω) |f(7, w(7))− f(7, w(7))| ≤ Lf |w(7)− w(7)|, |g(7, w(7))− g(7, w(7))| ≤ Lg|w(7)− w(7)|, and |h(7, w(7))− h(7, w(7))| ≤ Lh|w(7)− w(7)|. M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4082 (A2) There exist functions F,G ∈ L1(Ω,R+) such that |f(7, w(7))| ≤ F (7) and |g(7, w(7))| ≤ G(7), 7 ∈ Ω. (A3) For any constant LQ > 0, it follows that |Q(72)−Q(71)| ≤ LQ|72 − 71|, 71, 72 ∈ Ω. 3. Existence Results This section initiates a thorough analysis aimed at proving the existence of solutions for the system (1). To accomplish this, we effectively utilize two fundamental methodologies: the Banach contraction principle and Krasnoselskii FPTs [14, 16]. Theorem 1. Given the assumptions (A1) − (A3), the system (1) possesses a unique solution when ∆ = [( LQZ + |Q(0)|+ Zγ∥F∥L1 Γ(γ + 1) )( (q − 1)Mq−2Zβ B(ρ)Γ(β + 1) { 1− ρ+ Zρ Γ(ρ) } Lg ) + ( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) })( LfZ γ Γ(γ + 1) ) + Lh ] < 1. (17) Proof. Let w,w ∈ AC(Ω). Then from Note 2.1, we have ∥Φw − Φw∥ = max 7∈Ω |((A+B)w)(7)− ((A+B)w)(7)| ≤ max 7∈Ω |(Aw)(7)− (Aw)(7)|+max 7∈Ω |(Bw)(7)− (Bw)(7)|. (18) From (14)-(16), we have ∥Aw −Aw∥ ≤ max 7∈Ω {|Aw||Bw −Bw|+ |Bw||Aw −Aw|} . (19) We can now evaluate the expression mentioned earlier in the following manner: max 7∈Ω |Aw| = max 7∈Ω ∣∣∣∣Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ∣∣∣∣ ≤ max 7∈Ω |Q(7)−Q(0)|+ |Q(0)|+ 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1max 7∈Ω |f(σ,w(σ))|dσ ≤ LQZ + |Q(0)|+ 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1max 7∈Ω F (σ)dσ ≤ LQZ + |Q(0)|+ Zγ∥F∥L1 Γ(γ + 1) ; M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4083 max 7∈Ω |Bw −Bw| ≤ (q − 1)Mq−2Zβ B(ρ)Γ(β + 1) { 1− ρ+ Zρ Γ(ρ) } Lg∥w − w∥; since 1− ρ B(ρ) { max 7∈Ω ∣∣∣∣∣Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) −Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) ∣∣∣∣∣ } ≤ 1− ρ B(ρ) (q − 1)Mq−2 ∣∣∣∣∣ 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ − 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ∣∣∣∣∣ ≤ 1− ρ B(ρ) (q − 1)Mq−2 1 Γ(β) ∫ 7 0 (7 − σ)β−1|g(σ,w(σ))− g(σ,w(σ))|dσ ≤ 1− ρ B(ρ) (q − 1)Mq−2Zβ Γ(β + 1) Lg∥w − w∥, and max 7∈Ω ∣∣∣∣∣ ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ − ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ∣∣∣∣∣ ≤ ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1max 7∈Ω ∣∣∣∣∣Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ ) −Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ ) ∣∣∣∣∣dσ ≤ (q − 1)Mq−2Zβ+ρ Γ(β + 1)B(ρ)Γ(ρ) Lg∥w − w∥. max 7∈Ω |Bw| = max 7∈Ω ∣∣∣∣∣ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ∣∣∣∣∣. Since from (7), we have ∣∣∣∣Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ )∣∣∣∣ M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4084 ≤ Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1G(s)dσ ) ≤ Ψq ( Zβ∥G∥L1 Γ(β + 1) ) ∣∣ ∵ Ψq(ū) = ūq−1 = ( Zβ∥G∥L1 Γ(β + 1) )q−1 and ∣∣∣∣ ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ∣∣∣∣ ≤ ( Zβ∥G∥L1 Γ(β + 1) )q−1 ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1dσ ≤ ( Zβ∥G∥L1 Γ(β + 1) )q−1 · Zρ B(ρ)Γ(ρ) . Thus, we have max 7∈Ω |Bw| ≤ ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) } . Finally max 7∈Ω |Aw −Aw| = max 7∈Ω ∣∣∣∣∣ ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) − ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) ∣∣∣∣∣ ≤ 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1max 7∈Ω |f(σ,w(σ))− f(s, w(s))|dσ ≤ LfZ γ Γ(γ + 1) ∥w − w∥. Then (19) becomes ∥Aw −Aw∥ ≤ [( LQZ + |Q(0)|+ Zγ∥F∥L1 Γ(γ + 1) )( (q − 1)Mq−2Zβ B(ρ)Γ(β + 1) { 1− ρ+ Zρ Γ(ρ) } Lg ) + ( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) })( LfZ γ Γ(γ + 1) )] ∥w − w∥. Consequently (18) becomes ∥Φw − Φw∥ M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4085 ≤ [( LQZ + |Q(0)|+ Zγ∥F∥L1 Γ(γ + 1) )( (q − 1)Mq−2Zβ B(ρ)Γ(β + 1) { 1− ρ+ Zρ Γ(ρ) } Lg ) + ( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) })( LfZ γ Γ(γ + 1) ) + Lh ] ∥w − w∥ ≤ ∆∥w − w∥. From equation (17), we have the condition ∆ < 1, which guarantees that the operator Φ is a contraction. By applying Banach’s FPT, it follows that the hybrid system ofmABC- HFIDEs described in (1) possesses a unique solution, which corresponds to the fixed points of the operator Φ. Subsequently, utilizing Krasnoselskii FPT [14, 16], we establish the existence of solutions for the system (1). Theorem 2. Under the conditions set by hypotheses (A1)−(A3), the mABC-HFIDEs (1) is guaranteed to have at least one solution if Lh < 1. (20) Proof. Fix B = AC(Ω,R) and define a subset S of B by S = {w ∈ B : ∥w∥ ≤ Λ}, where Λ = ( LQZ + |Q(0)|+ Zγ∥F∥L1 Γ(γ + 1) )( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) }) + Lh∥w∥+H0 with H0 = max 7∈Ω |h(7, 0)|. It is evident that S is a closed, convex, and bounded subset of the Banach space B. We will now examine two operators A and B that map from S to B, which are defined as specified in equations (12) and (13), respectively. At this point, the expression in (9) can be rewritten as the operator equation w(7) = (Aw)(7) + (Bw)(7), 7 ∈ Ω. Step 1: Let w,w ∈ S. Then from (13) and (A1), we have ∥Bw −Bw∥ = max 7∈Ω |h(7, w(7))− h(7, w(7))| ≤ Lh∥w − w∥. Therefore, according to (20), the operator B acts as a contraction on S with a constant Lh < 1. M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4086 Step 2: We will demonstrate that A is a compact operator mapping from S to B. It suffices to show that A(S) forms a uniformly bounded and equi-continuous subset within B. First, consider an arbitrary element w ∈ S. Then, from (12) and (A2), we have ∥Aw∥ ≤ max 7∈Ω ( |Q(7)|+ 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1|f(σ,w(σ))|dσ ) (×) [ ABIρ0+ |Θ|+ 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1|g(σ,w(σ))|dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1|g(τ, w(τ))|dτ )) dσ ] ≤ ( LQZ + |Q(0)|+ Zγ∥F∥L1 Γ(γ + 1) )( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) }) . This indicates that A(S) is uniformly bounded within B. Conversely, let 71, 72 ∈ Ω be chosen arbitrarily such that 71 < 72. Then, for any w ∈ S, we obtain |(Aw)(72)− (Aw)(71)| ≤ ( |Q(72)|+ Zγ∥F∥L1 Γ(γ + 1) )( (q − 1)Mq−2Zβ B(ρ)Γ(β + 1) { 1− ρ+ Zρ Γ(ρ) } ∣∣∣∣∫ 72 71 G(σ)dσ ∣∣∣∣) + ( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) })( LQ|72 − 71| + Zγ Γ(γ + 1) ∣∣∣∣∫ 72 71 F (σ)dσ ∣∣∣∣ ) = ( |Q(72)|+ Zγ∥F∥L1 Γ(γ + 1) )( (q − 1)Mq−2Zβ B(ρ)Γ(β + 1) { 1− ρ+ Zρ Γ(ρ) } |ξ(72)− ξ(71)| ) + ( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) })( LQ|72 − 71| + Zγ Γ(γ + 1) |ζ(72)− ζ(71)| ) , where ξ(7) = ∫ 7 0 G(σ)dσ and ζ(7) = ∫ 7 0 F (σ)dσ. Given that the functions ξ and ζ are continuous on the compact interval Ω, they are also uniformly continuous. Therefore, for any ε > 0, ∃ a δ > 0 such that for all 71, 72 ∈ Ω and w ∈ S, the following holds: |72 − 71| < δ =⇒ |(Aw)(72)− (Aw)(71)| < ε. This establishes that A(S) is an equi-continuous subset of B. Since A(S) is both uniformly bounded and equi-continuous in B, it follows from the Arzelà-Ascoli theorem that A(S) is relatively compact. Consequently, we conclude that A is a compact operator on S. M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4087 Step 3: To demonstrate that A is a continuous operator from S to B, consider a sequence {wn} in S that converges to a point w ∈ S. By applying the Lebesgue dominated convergence theorem, we can derive the following result: lim n→∞ (Awn)(7) = lim n→∞ {( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,wn(σ))dσ ) (×) [ ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,wn(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, wn(τ))dτ )) dσ ]} = ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1 lim n→∞ f(σ,wn(σ))dσ ) (×) [ ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1 lim n→∞ g(σ,wn(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1 lim n→∞ g(τ, wn(τ))dτ )) dσ ] = ( Q(7) + 1 Γ(γ) ∫ 7 0 (7 − σ)γ−1f(σ,w(σ))dσ ) (×) [ ABIρ0+(Θ) + 1− ρ B(ρ) Ψq ( 1 Γ(β) ∫ 7 0 (7 − σ)β−1g(σ,w(σ))dσ ) + ρ B(ρ)Γ(ρ) ∫ 7 0 (7 − σ)ρ−1 ( Ψq ( 1 Γ(β) ∫ σ 0 (σ − τ)β−1g(τ, w(τ))dτ )) dσ ] = (Aw)(7) for all 7 ∈ Ω. This establishes that the sequence {Awn} converges point-wise to Aw on the interval Ω. Furthermore, by employing a similar argument as in Step 2, we can demonstrate that the sequence {Awn} is equi-continuous. Consequently, it follows that {Awn} converges uniformly to Aw, thereby confirming that A is a continuous operator on S. Step 4: We demonstrate that Aw+Bw ∈ S for all w,w ∈ S. For any w,w ∈ S and 7 ∈ Ω, it follows that |(Aw)(7) + (Bw)(7)| ≤ ( LQZ + |Q(0)|+ Zγ∥F∥L1 Γ(γ + 1) )( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) }) + Lf∥w∥+ F0 M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4088 ≤ Λ, it follows that Aw + Bw ∈ S for all w,w ∈ S. This confirms that all the conditions specified in [17, Theorem 2.6] are satisfied, leading to the conclusion that the operator equation Aw+Bw = w has a solution in the set S. Consequently, the mABC-HFIDEs (1) has a solution that is defined on the interval Ω. 4. Stability Analysis This section is dedicated to the study of U-H-stability of the mABC-HFIDEs (1). To proceed, we will first recall the following definition: Definition 7. The integral equation (11) is Ulam-Hyers stable, if for some λ1 > 0, we have ϑ > 0 with w satisfying ∥w − Φw∥ < ϑ (21) with w(7) of (11) with w(7) = Φw(7) (22) and ∥w − w∥ < ϑλ1. Theorem 3. Under the conditions of Theorem 1, the system (11) demonstrates U-H stability, which implies the U-H stability of the hybrid system of mABC-FDEs (1). Proof. For any w,w∗ ∈ AC(Ω,R), we have ∥Φw − Φw∗∥ = max 7∈Ω |((A+B)w)(7)− ((A+B)w)(7)| ≤ max 7∈Ω |(Aw)(7)− (Aw)(7)|+max 7∈Ω |(Bw)(7)− (Bw)(7)|. In view of Theorem 1, we have ∥Φw − Φw∗∥ ≤ [( LQZ + |Q(0)|+ Zγ∥F∥L1 Γ(γ + 1) )( (q − 1)Mq−2Zβ B(ρ)Γ(β + 1) { 1− ρ+ Zρ Γ(ρ) } Lg ) + ( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) })( LfZ γ Γ(γ + 1) ) + Lh ] ∥w − w∗∥ = ∆∥w − w∗∥. (23) For ∆ < 1, by (21)-(23), consider the following norm ∥w − w∗∥ = ∥w − Φw +Φw − w∗∥ M. M. Arjunan / Eur. J. Pure Appl. Math, 17 (4) (2024), 4071-4092 4089 ≤ ∥w − Φw∥+ ∥Φw − Φw∗∥ ≤ ϑ+∆∥w − w∗∥. Hence ∥w − w∗∥ ≤ ϑ 1−∆ with λ = 1 1−∆ . Hence (11) is stable. This implies the stability of the addressing system represented by (1). 5. Example In this section, we will provide a justification for our findings by presenting an illustrative example. Consider the given mABC-HFIDEs CDβ Ψp mABCDρ 0+  w(7)− 1 16 sinw(7) π+sin 7+ 1 Γ ( 1 3 ) ∫ 7 0 (7 − σ)− 2 3 1 σ + 25 sinw(σ)dσ    = 1 7 + 36 cosw(7), 7 ∈ [0, 1], w(0) = 1 16 sinw(0) + π. (24) Set β = 1 4 , ρ = 1 2 , γ = 1 3 , Z = 1, LQ = 1,Θ = 0.735,ABIρ0+Θ = 1, p = q = 2,M = 1,B(ρ) = 1− ρ+ ρ Γ(ρ) , h(7, w(7)) = 1 16 sinw(7), Q(7) = π + sin 7, Q(0) = π, f(7, w(7)) = 1 7 + 25 sinw(7), g(7, w(7)) = 1 7 + 36 cosw(7). Let w,w ∈ AC([0, 1]). Then, we have |f(7, w(7))− f(7, w(7))| ≤ 1 26 |w(7)− w(7)| , |g(7, w(7))− g(7, w(7))| ≤ 1 37 |w(7)− w(7)| , and |h(7, w(7))− h(7, w(7))| ≤ 1 16 |w(7)− w(7)| . Then the assumptions (A1)-(A3) holds with Lf = 1 26 , Lg = 1 37 , Lh = 1 16 , LQ = 1, ∥F∥L1 = ln ( 26 25 ) = 0.03922, ∥G∥L1 = ln ( 37 36 ) = 0.02731. REFERENCES 4090 At this point, we will examine the conditions outlined in the theorems to ensure they are satisfied. By carefully analyzing these conditions, we can derive the necessary conclusions and results that follow from them. This thorough verification process will allow us to confirm the validity of our findings. Now Λ = [( LQZ + |Q(0)|+ Zγ∥F∥L1 Γ(γ + 1) )( (q − 1)Mq−2Zβ B(ρ)Γ(β + 1) { 1− ρ+ Zρ Γ(ρ) } Lg ) + ( ABIρ0+ |Θ|+ 1 B(ρ) ( Zβ∥G∥L1 Γ(β + 1) )q−1{ 1− ρ+ Zρ Γ(ρ) })( LfZ γ Γ(γ + 1) ) + Lh ] = 0.2782 < 1. Consequently, we have established that the conditions specified in Theorem 1 are indeed met. As a result of this verification, we can confidently conclude that problem (24) possesses a unique solution. Next, Lh = 0.0625 < 1. Therefore, we can confirm that the criteria outlined in Theorem 2 are also fulfilled. This affirmation leads us to conclude that the problem presented in equation (24) has at least one solution. Also 1− 0.2782 = 0.7218 ̸= 0. Thus (24) is U-H stable. 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