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Weighted pseudo almost periodicity on neutral type CNNs involving multi-proportional delays and D operator

  • Received: 23 September 2020 Accepted: 16 November 2020 Published: 02 December 2020
  • MSC : 34K13, 34C25

  • In this paper, we aim to explore the dynamic behaviors of cellular neural networks with D operator and multi-proportional delays. By using the fixed point theorem and some differential inequality techniques, we derive some new sufficient criteria on the existence and exponential stability of weighted pseudo almost periodic solutions for the proposed neural networks, which generalize and improve all known consequences in Yang et al. [22] and Yu [Neural Process Lett., 45 (2017), 141-151]. Finally, the effectiveness of the obtained results is illustrated by a numerical simulation example.

    Citation: Hedi Yang. Weighted pseudo almost periodicity on neutral type CNNs involving multi-proportional delays and D operator[J]. AIMS Mathematics, 2021, 6(2): 1865-1879. doi: 10.3934/math.2021113

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  • In this paper, we aim to explore the dynamic behaviors of cellular neural networks with D operator and multi-proportional delays. By using the fixed point theorem and some differential inequality techniques, we derive some new sufficient criteria on the existence and exponential stability of weighted pseudo almost periodic solutions for the proposed neural networks, which generalize and improve all known consequences in Yang et al. [22] and Yu [Neural Process Lett., 45 (2017), 141-151]. Finally, the effectiveness of the obtained results is illustrated by a numerical simulation example.


    The fundamental concept of fractional calculus involves replacing natural numbers with rational numbers in the order of differentiation operators. Although this concept may initially appear undeveloped and straightforward, it carries profound implications and yields significant results that effectively describe phenomena in various fields, including bioengineering, dynamics, modeling, control theory, and medicine. For more detailed information on this topic, we refer the reader to the following references [1,2,3,4]. Fractional differential equations (FDEs) play a critical role in modeling the growth of systems, particularly through nonlocal Cauchy-type problems that describe nonnegative quantities like species concentrations or temperature distributions. Before applying these models to real-world phenomena, it is crucial to establish the existence of solutions, which can be addressed using fixed point theory. Furthermore, it is essential to develop stable algorithms and reliable numerical procedures to obtain accurate results and ensure the practicality and validity of the models in real-life applications. The articles [5,6,7,8] and their references explore a wide range of interesting and innovative findings related to the existence and stability of coupled systems in various types of FDEs. These studies delve into different aspects of coupled systems and contribute to our understanding of their dynamics and properties. They provide valuable insights into the behavior and solutions of coupled FDEs, opening up avenues for further research and applications in this field. New strategies and methodologies have been developed to describe the dynamics of real problems. In particular, Caputo and Fabrizio [9] introduced a new class of fractional derivatives within the exponential kernel, which Losada and Nieto [10] explored further, uncovering intriguing characteristics. Atangana and Baleanu [11] investigated a novel type of fractional derivatives using Mittag-Leffler (ML) kernels. Abdeljawad [12] extended the ABC fractional derivative to higher arbitrary orders and included their associated integral operators. The reader may find theoretical works on the ABC fractional operator in references [13,14,15,16].

    It is evident that the challenges posed by operators of variable order are more complex compared to operators of constant order. Recently, several authors have explored the applications of variable order derivatives in various scientific fields, including multi-fractional Gaussian noises, mechanical applications, and anomalous diffusion models. Among these, numerous studies have focused on developing numerical techniques for a specific class of variable-order FDEs, as exemplified by [17,18,19,20]. In this regard, Lorenzo et al. [21], utilizing fractional operators with variable order, addressed the problem of fractional diffusion. Subsequently, fractional variable order has found applications in various areas of science and engineering (see [22,23,24]). Sun et al. [25] conducted a comparative study between constant and variable order models to describe systems with memory. Various techniques are available for approximating solutions to fractional boundary value problems with variable orders. For instance, the authors in [26] developed a nonlinear model of alcoholism within the framework of FDEs with variable order and examined its solutions numerically and analytically. Recently, Li et al [27] investigated the following fractional problem using a novel numerical strategy

    {MLDω(ι)0+y(ι)+a(ι)y(ι)=H(ι,y(ι)),ι[0,1],B(y)=0, (1.1)

    where, MLDω(ι)0+ is the Mittag-Leffler fractional derivative of order ω(ι). Kaabar et al. [28] examined the qualitative characteristics of the following implicit FDEs with variable order

    {CDω(ι)0+y(ι)=H(ι,y(ι),CDω(ι)0+y(ι)),  ι[0,b],y(0)=0,y(b)=0, (1.2)

    where, CDω(ι)0+ is the Caputo fractional differential (FD) of variable order ω(ι). Jeelani et al. [29] employed generalized intervals, Krasnoselskii and Banach's fixed point theorems, and piecewise constant functions to conduct qualitative analyses on a nonlinear fractional integrodifferential equation.

    Based on the provided justifications, we are motivated to evaluate and investigate the necessary conditions for a coupled system within the context of the ABC-fractional derivative with a variable order as the following

    {ABCDω(ι)0+y1(ι)=H1(ι,y2(ι),ABCDω(ι)0+y1(ι)),ιJ:=(0,b],ABCDω(ι)0+y2(ι)=H2(ι,y1(ι),ABCDω(ι)0+y2(ι)),ιJ:=(0,b],y1(0)=0,y1(b)=nk=1y1(τk), τk(0,b),y2(0)=0,y2(b)=nk=1y2(γk), γk(0,b), (1.3)

    where

    i)ABCDω(ι)0+ is the ABC-FD of variable order ω(ι)(1,2].

    ii) τk,γkR, k=1,2,...,n, are prefixed points satisfying 0<κ1κ2...κn<b and 0<γ1γ2...γn<b.

    iii) H1,H2:J×R2R are continuous functions that comply with a few later-described presumptions.

    To the best of our knowledge, no studies in the aforementioned body of research address an implicit coupled system of fractional variable order. The proposed implicit coupled system (1.3) is a more general form compared to the previous problems (1.1) and (1.2) in [26,28,29]. The proposed implicit coupled system (1.3) contains nonlocal conditions and variable order ω(ι)(1,2]. The complexity of calculations and partitioning the principal time interval has limited the number of studies specifically addressing the stability and existence of fractional integrodifferential boundary value problems with variable order. To bridge this gap, we conduct a comprehensive qualitative analysis of a new implicit coupled system (1.3) with a variable order within the framework of ABC fractional operators. By employing extended intervals and piecewise constant functions, we accurately transform the coupled system into an equivalent standard system with constant order in the context of the ABC-fractional derivative. The existence and uniqueness of solutions to the coupled system (1.3) are then demonstrated using fixed point techniques such as Krasnoselskii and Banach theorems. The Ulam-Hyers stability result of the suggested problem is also discussed. Additionally, we establish suitable conditions for the existence of effective solutions to the coupled system (1.3). The contributions of this work lie in providing a framework to address the challenges associated with variable-order FDEs involving the ABC fractional operator. The study offers transformation techniques, investigation of qualitative properties, existence and uniqueness analysis, stability analysis, and illustrative examples to support its findings. The results obtained in this study will contribute to understanding the stability properties of the considered implicit coupled system of fractional variable order. This knowledge is valuable for designing and controlling systems modeled using such equations, ensuring their robustness, and predicting their behavior accurately.

    The paper's structure is as follows: Section 2 offers an overview of fractional calculus notations, definitions, and relevant lemmas, including a crucial lemma that converts the coupled system of ABC-fractional derivative into an equivalent integral equation. Section 3 focuses on establishing the existence and uniqueness of solutions for the coupled system, while Section 4 investigates the Ulam-Hyers stability of the system. To illustrate the obtained results, Section 5 presents numerical examples. Finally, the paper concludes with a summary of the findings, concluding remarks, and suggestions for potential future research directions.

    Let C(J,R) be a Banach space consisting of all continuous functions y:JR, where J:=(0,b] with the following supremum norm

    y=sup{|y(ι)|:ιJ}.

    Define the product space Υ=C(J,R)×C(J,R). Thus, Υ\ is Banach space with the following norm

    (y1,y2)Υ=y1+y2,

    for each (y1,y2)Υ.

    Derivatives and integrals that have orders as functions of specific variables belong to the category of operators with variable order, which is a more complex classification. Variable order fractional integrals and derivatives can be defined in various ways.

    Definition 2.1. [17] Let ω(ι)(n1,n], yH1(J), then, the left ABC fractional derivative of order ω(ι) for function y(ι) with the lower limit zero are defined as

    ABCDω(ι)0+y(ι)=Λ(ω(ι))1ω(ι)ι0Eω(ι)(ω(ι)ω(ι)1(ιθ)ω)y(θ)dθ, ι>0. (2.1)

    The normalization function Λ(ω(ι)) satisfies Λ(0)=Λ(1)=1, where Eω(ι) is the ML function defined by

    Eω(ι)(y)=k=0ykΓ(kω(ι)+1), Re(ω)>0,yC.

    The left Atangana-Baleanu (AB) fractional integral of order ω(ι) for function y(ι) are defined as

    ABIω(ι)0+y(ι)=1ω(ι)Λ(ω(ι))y(ι)+ω(ι)Λ(ω(ι))Γ(ω(ι))ι0(ιs)ω(ι)1y(s)ds.

    Lemma 2.2. [12] Let y(ι) be a function defined on the interval [0,b] and n<ωn+1, nN0. Then, we have

    (ABIωABC0+Dω0+y)(ι)=y(ι)nk=0y(k)(0)k!ιk.

    Theorem 2.3. [30] Let X be a Banach space and S\ X be a nonempty, closed, convex and bounded subset. If there exist two operators Φ1,Φ2 satisfying the following conditions:

    (1) Φ1u+Φ2vX, for all u,vX.

    (2) Φ1 is completely continuous and Φ2 is a contraction mapping.

    Then, there exists a function zS such that z=Φ1z+Φ2z.

    Remark 2.4. Let u(ι),v(ι)C(J,R) be two continuous functions. Then, we have

    ABIu(ι)0+ ABIv(ι)0+y(ι) ABIu(ι)+v(ι)0+y(ι).

    Lemma 2.5. [12] Let ω(1,2] and C([a,b],R). Then, the following integral equation

    y(ι)=c1+c2(ιa)+ABIωa+(ι),

    is an equivalent to the following linear problem

    {ABCDωa+y(ι)=(ι),y(a)=c1,y(a)=c2,

    where

    ABIωa+(ι)=2ωΛ(ω1)ιa(s)ds+ω1Λ(ω1)Γ(ω)ιa(ιs)ω1(s)ds. (2.2)

    For the sake of simplicity in our analysis, we will use the following notations:

    Θ1=(bibi1)nk=1(τkbi1)0,Θ2=(bibi1)nk=1(γkbi1)0,
    P1=2ωiΛ(ωi1),P2=ωi1Λ(ωi1),P3=2ωiΛ(ωi1),P4=ωi1Λ(ωi1)Γ(ωi),
    ZΠ=(P3(bibi1)+P4Γ(ωi+1)(bibi1)ωi),
    GΠ1=P1(bibi1)Θ1(nk=1(τkbi1)+(bibi1))+P2(bibi1)Θ1Γ(ωi+1)(nk=1(τkbi1)ωi+(bibi1)ωi),
    GΠ2=P1(bibi1)Θ2(nk=1(γkbi1)+(bibi1))+P2(bibi1)Θ2Γ(ωi+1)(nk=1(γkbi1)ωi+(bibi1)ωi),
    RΠ1=GΠ1+ZΠ and RΠ2=GΠ2+ZΠ,

    and

    Ω1=RΠ1NH11NH1 and Ω2=RΠ2NH21NH2. (2.3)

    In this part, we convert the coupled system (1.3) with variable-order ω(ι) into an equivalent integral equations. To begin, we transform the coupled system with variable order into a coupled system with constant order by using generalized intervals and piecewise constant functions. Let B be a partition of the interval J defined as

    B={J1=[b0,b1],J2=(b1,b2],J3=(b2,b3],.....,Jn=(bn1,bn]},

    such that b0=0,bn=b and nN. Let ω(ι):J(1,2] be a piecewise constant function concerning B. This means that the function ω(ι) takes values in the interval (1,2] and is constant over different intervals or "pieces" defined by the set B. In other words, for each interval in B, the function ω(ι) remains constant within that interval. The specific values of ω(ι) within each interval can vary, but they will always be within the range of (1,2]. The set B defines the partition or division of the domain J into different intervals, and within each interval, the function ω(ι) takes on a constant value. So, the function ω(ι) can be a representation as

    ω(ι)=ni=1ωiUi(ι)={ω1,    if ιJ1,ω2,    if ιJ2,...ωn,    if ιJn,

    where, ωi(1,2] are constants numbers, and Ui is the indicator of the interval Ji:=(bi1,bi], i=1,2,...,n such that

    Ui(ι)={1,   if ιJi,0,   for elsewhere.

    Let C(Ji,R), i{1,2,...,n} be a Banach space of all continuous functions defined as yi:JiR with the following norm

    yi=sup{|y(ι)|:ιJi}.

    By (2.1), the coupled system (1.3) can be expressed as

    {Λ(ω(ι))1ω(ι)ι0Eω(ι)(ω(ι)ω(ι)1(ιθ)ω)y1(θ)dθ=H1(ι,y2(ι),ABCDω(ι)0+y1(ι)),Λ(ω(ι))1ω(ι)ι0Eω(ι)(ω(ι)ω(ι)1(ιθ)ω)y2(θ)dθ=H2(ι,y1(ι),ABCDω(ι)0+y2(ι)). (2.4)

    Then, for ιJi,i=1,2,...,n the coupled system (2.4) on Ji can express as a sum of left ABC-coupled systems of constant-orders ω1,ω2,....ωn as follows

    {Λ(ω1)1ω1b10Eω1(ω1(ιθ)ω1ω11)y1(θ)dθ+...+Λ(ωi)1ωiιbi1Eωi(ωi(ιθ)ωiωi1)y1(θ)dθ=H1(ι,y2(ι),ABCDωi0+y1(ι)),Λ(ω1)1ω1b10Eω1(ω1(ιθ)ω1ω11)y2(θ)dθ+...+Λ(ωi)1ωiιbi1Eωi(ωi(ιθ)ωiωi1)y2(θ)dθ=H2(ι,y1(ι),ABCDωi0+y2(ι)). (2.5)

    Let the function (y1,y2) be such that y1(ι)y2(ι)0 on ι[0,bi1],i=1,2,....,n, and such that it solves the system (2.5). Hence, the coupled system (1.3) with variable-order ω(ι) is reduced to the following system

    {ABCDωib+i1y1(ι)=H1(ι,y2(ι),ABCDωib+i1y1(ι)),ιJi,ABCDωib+i1y2(ι)=H2(ι,y1(ι),ABCDωib+i1y2(ι)),ιJi,y1(bi1)=0, y1(bi)=nk=1y1(τk), τk(0,b),y2(bi1)=0, y2(bi)=nk=1y2(γk), γk(0,b). (2.6)

    Now, to analyze the properties of solutions using fixed point theorems, we transform the coupled system (2.6) into an equivalent integral equation. For that, we present the following theorem.

    Theorem 2.6. For i=1,2,...,n, let ωi(1,2], and H:Ji×R2R be a continuous function and Θ1,Θ20. If (y1,y2)Υ is a solution of coupled system (2.6), then, (y1,y2) satisfies the following fractional integral equations

    y1(ι)=(ιbi1)Θ1[nk=1ABIωib+i1Hy1(τk)ABIωib+i1Hy1(bi)]+ABIωib+i1Hy1(ι), (2.7)

    and

    y2(ι)=(ιbi1)Θ2[nk=1ABIωib+i1Hy2(γk)ABIωib+i1Hy2(bi)]+ABIωib+i1Hy2(ι), (2.8)

    where

    Hy1(ι)=H1(ι,y2(ι),ABCDωib+i1y1(ι)),Hy2(ι)=H2(ι,y1(ι),ABCDωib+i1y2(ι)).

    Proof. We assume that (y1,y2)Υ is a solution of coupled system (2.6). Apply the operator ABIωib+i1 to both sides of (2.6) and using Lemma 2.5, we get

    y1(ι)=c1+c2(ιbi1)+ABIωib+i1Hy1(ι), (2.9)

    and

    y2(ι)=c3+c4(ιbi1)+ABIωib+i1Hy2(ι). (2.10)

    By conditions y1(bi1)=0 and y2(bi1)=0, we get c1=c3=0. Hence. Eqs (2.9) and (2.10) become as

    y1(ι)=c2(ιbi1)+ABIωib+i1Hy1(ι),

    and

    y2(ι)=c4(ιbi1)+ABIωib+i1Hy2(ι).

    By second conditions y1(bi)=nk=1y1(τk) and y2(bi)=nk=1y2(γk), we get

    c2=1Θ1[nk=1 ABIωib+i1Hy1(τk)ABIωib+i1Hy1(bi)],

    and

    c4=1Θ2[nk=1 ABIωib+i1Hy2(γk)ABIωib+i1Hy2(bi)].

    Substitute the values of c1,c2,c3,c4 into (2.9) and (2.10), and we get (2.7) and (2.8). Conversely, assume that (y1,y2)Υ satisfies (2.7) and (2.8). Then, by applying ABCDωib+i1 on both sides of (2.7) and (2.8), with using the fact ABCDωib+i1(ιbi1)=0, we get (2.6). Next, take ιτk in (2.7), and we get

    nk=1y1(τk)=nk=1(τkbi1)Θ1[nk=1 ABIωib+i1Hy1(τk)ABIωib+i1Hy1(bi)]+nk=1 ABIωib+i1Hy1(τk)=(bibi1)Θ1Θ1[nk=1 ABIωib+i1Hy1(τk)ABIωib+i1Hy1(bi)]+nk=1 ABIωib+i1Hy1(τk)=y1(bi).

    In the same manner, we get y2(bi)=nk=1y2(γk). Thus, the nonlocal conditions are satisfied.

    For further analysis, we simplify the solutions (2.7) and (2.8) as

    y1(ι)=P1(ιbi1)Θ1(nk=1τkbi1Hy1(s)dsbibi1Hy1(s)ds)+P2(ιbi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1Hy1(s)dsbibi1(bis)ωi1Hy1(s)ds)+P3ιbi1Hy1(s)ds+P4Γ(ωi)ιbi1(ιs)ωi1Hy1(s)ds,

    and

    y2(ι)=P1(ιbi1)Θ2(nk=1γkbi1Hy2(s)dsbibi1Hy2(s)ds)+P2(ιbi1)Θ2Γ(ωi)(nk=1γkbi1(γks)ωi1Hy2(s)dsbibi1(bis)ωi1Hy2(s)ds)+P3ιbi1Hy2(s)ds+P4Γ(ωi)ιbi1(ιs)ωi1Hy2(s)ds.

    In this part, we will examine the necessary conditions for the existence and uniqueness for the coupled system (2.6) by using some fixed point theorems.

    Theorem 3.1. Suppose that

    (H1) The functions H1,H2:Ji×R2R are continuous and there exist constant numbers NH1,NH2>0 such that

    |H1(ι,x,y)H1(ι,¯x,¯y)|NH1(|x¯x|+|y¯y|),NH1>0,|H2(ι,x,y)H2(ι,¯x,¯y)|NH2(|x¯x|+|y¯y|),NH2>0,

    for all x,y,¯x,¯yC(Ji,R). Then, the coupled system (2.6) has a unique solution provided that max{Ω1,Ω2}<1, where Ω1 and Ω2 are defined by (2.3).

    Proof. On the light of Theorem 2.6, we define the operator Π:ΥΥ as

    Π(y1,y2)(ι)=(Π1(y1,y2)(ι),Π2(y1,y2)(ι)),

    where

    Π1(y1,y2)(ι)=P1(ιbi1)Θ1(nk=1τkbi1Hy1(s)dsbibi1Hy1(s)ds)+P2(ιbi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1Hy1(s)dsbibi1(bis)ωi1Hy1(s)ds)+P3ιbi1|Hy1(s)|ds+P4Γ(ωi)ιbi1(ιs)ωi1Hy1(s)ds,

    and

    Π2(y1,y2)(ι)=P1(ιbi1)Θ2(nk=1γkbi1Hy2(s)dsbibi1Hy2(s)ds)+P2(ιbi1)Θ2Γ(ωi)(nk=1γkbi1(γks)ωi1Hy2(s)dsbibi1(bis)ωi1Hy2(s)ds)+P3ιbi1|Hy2(s)|ds+P4Γ(ωi)ιbi1(ιs)ωi1Hy2(s)ds.

    Let βηi be a closed ball defined as

    βηi={(y1,y2)Υ:(y1,y2)Υηi},

    with radius ηi(Φ1+Φ2)1max{Ω1,Ω2}, and

    Φi=RΠiLHi,i=1,2,

    where LHi=maxιJi|Hi(ι,0,0)|. Now, we show that Πβηiβηi. For the operator Π1, let (y1,y1)βηi and ιJi. Then, we have

    |Π1(y1,y2)(ι)|=P1(ιbi1)Θ1(nk=1τkbi1|Hy1(s)|dsbibi1|Hy1(s)|ds)+P2(ιbi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1|Hy1(s)|dsbibi1(bis)ωi1|Hy1(s)|ds)+P3ιbi1|Hy1(s)|ds+P4Γ(ωi)ιbi1(ιs)ωi1|Hy1(s)|ds.

    By (H1), we have

    |Hy1(ι)|=|H1(ι)|=|H1(ι,y2(ι),ABCDωib+i1y1(ι))H1(ι,0,0)|+|H1(ι,0,0)|NH11NH1|y2(ι)|+LH1. (3.1)

    Hence

    Π1(y1,y2)P1(ιbi1)Θ1(nk=1τkbi1|Hy1(s)|dsbibi1|Hy1(s)|ds)+P2(ιbi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1|Hy1(s)|dsbibi1(bis)ωi1|Hy1(s)|ds)+P3ιbi1|Hy1(s)|ds+P4Γ(ωi)ιbi1(ιs)ωi1|Hy1(s)|dsΩ1y2+Φ1.

    In the same manner, for the operator Π2, we get

    Π2(y1,y2)Ω2y1+Φ2.

    Hence, by the fact Π(y1,y2)Υ=Π1(y1,y2)+Π2(y1,y1), we get

    Π(y1,y2)ΥΩ1y1+Ω2y2+(Φ1+Φ2)max{Ω1,Ω2}(y1+y2)+(Φ1+Φ2)max{Ω1,Ω2}(y1,y2)Υ+(Φ1+Φ2)ηi.

    Thus Π(y1,y2)βηi. Now, we shall prove that Π(y1,y2) is a contraction. Let (y1,y2),(ˆy1,ˆy2)βηi and ιJi. Then, for the operator Π1, we have

    |Π1(y1,y2)(ι)Π1(ˆy1,ˆy2)(ι)|P1(ιbi1)Θ1(nk=1τkbi1|Hy1(s)Hˆy1(s)|ds+bibi1|Hy1(s)Hˆy1(s)|ds)+P2(ιbi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1|Hy1(s)Hˆy1(s)|ds+bibi1(bis)ωi1|Hy1(s)Hˆy1(s)|ds)+P3ιbi1|Hy1(s)Hˆy1(s)|ds+P4Γ(ωi)ιbi1(ιs)ωi1|Hy1(s)Hˆy1(s)|ds.

    By (H1), we obtain

    |Hy1(s)Hˆy1(s)|=|H1(s,y2(s),ABCDωib+i1y1(s))H1(s,ˆy2(s),ABCDωib+i1ˆy1(s))|Nf(|y2(s)ˆy2(s)|+|Hy1(s)Hˆy1(s)|)Nf1Nf(y2ˆy2). (3.2)

    Hence

    Π1(y1,y2)Π1(ˆy1,ˆy2)Ω1(y2ˆy2).

    In the same manner, for the operator Π1, we get

    Π2(y1,y2)Π2(ˆy1,ˆy2)Ω2(y1ˆy1).

    Thus, we have

    Π(y1,y2)Π(ˆy1,ˆy2)Υ=Π1(y1,y1)Π1(ˆy1,ˆy1)+Π2(y1,y1)Π2(ˆy1,ˆy1)Ω1y2ˆy2+Ω2y1ˆy1max{Ω1,Ω2}(y1,y2)(ˆy1,ˆy2)Υ.

    Due to max{Ω1,Ω2}<1, we deduce that the operator Π is a contraction mapping. Thus, Π has a unique fixed point. By Banach contraction principle, we deduce that the coupled system (2.6) has a unique solution.

    Theorem 3.2. Under the hypothesis (H1) mentioned in Theorem 3.1. If ZΠmax(NH11NH1,NH21NH2)<1, then, the coupled system (2.6) has at least one solution

    Proof. Define the operators F1,G1,F2,G2:ΥΥ as

    F1(y1,y2)(ι)=P3ιbi1|Hy1(s)|ds+P4Γ(ωi)ιbi1(ιs)ωi1Hy1(s)ds,F2(y1,y2)(ι)=P3ιbi1|Hy2(s)|ds+P4Γ(ωi)ιbi1(ιs)ωi1Hy2(s)ds,G1(y1,y2)(ι)=P1(ιbi1)Θ1(nk=1τkbi1Hy1(s)dsbibi1Hy1(s)ds)+P2(ιbi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1Hy1(s)dsbibi1(bis)ωi1Hy1(s)ds),G2(y1,y2)(ι)=P1(ιbi1)Θ2(nk=1γkbi1Hy2(s)dsbibi1Hy2(s)ds)+P2(ιbi1)Θ2Γ(ωi)(nk=1γkbi1(γks)ωi1Hy2(s)dsbibi1(bis)ωi1Hy2(s)ds).

    From the above mentioned operators, we can write

    Π1(y1,y2)(ι)=F1(y1,y2)(ι)+G1(y1,y2)(ι),

    and

    Π2(y1,y2)(ι)=F2(y1,y2)(ι)+G2(y1,y2)(ι).

    Thus, the operator Π can be expressed as

    Π(y1,y2)(ι)=(Π1(y1,y2)(ι),Π2(y1,y2)(ι)).

    We split the proof into the following steps:

    Step 1:In this step, we shall prove that Π is continuous. The continuity of Hy1 and Hy2 implies the continuity of F1,G1,F2 and G2. Consequently, Π is continuous.

    Step 2: In this step, we shall prove that G1 and G2 are bounded. By (3.1), we have

    G1(y1,y2)P1(ιbi1)Θ1(nk=1τkbi1|Hy1(s)|dsbibi1|Hy1(s)|ds)+P2(ιbi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1|Hy1(s)|dsbibi1(bis)ωi1|Hy1(s)|ds)P1(bibi1)Θ1(nk=1(τkbi1)+(bibi1))(NH11NH1y1+LH1)+P2(bibi1)Θ1Γ(ωi+1)(nk=1(bibi1)ωi+(bibi1)ωi)(NH11NH1y1+LH1)GΠ1NH11NH1y1+GΠ1LH1=Q1.

    In the same manner, we get

    G2(y1,y2)GΠ2NH21NH2y2+GΠ2LH2=Q2,

    which implies

    G(y1,y2)Υmax(Q1,Q2),

    where

    Qi=GΠi[NHi1NHiyi+LHi].

    This proves G1,G2 are uniformly bounded.

    Step 3:In this step, we will prove that G1 and G2 are equicontinuous. For this purpose, let (y1,y2)βηl and ι1,ι2J such that 0ι2<ι1b. Then, we have

    |G1(y1,y2)(ι2)G1(y1,y2)(ι1)|P1[(ι2bi1)(ι1bi1)]Θ1(nk=1τkbi1|Hy1(s)|dsbibi1|Hy1(s)|ds)+P2(ι2bi1)(ι1bi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1|Hy1(s)|dsbibi1(bis)ωi1|Hy1(s)|ds).

    Thus

    G1(y1,y2)(ι2)G1(y1,y2)(ι1)0 as ι2ι1.

    In the same manner, we get

    G2(y1,y2)(ι2)G2(y1,y2)(ι1)0 as ι2ι1.

    We conclude that the operator G1 and G2 are relatively compact. Based on the previous steps and using the Arzela-Ascoli theorem, we deduce that G is completely continuous.

    Step 4:In this step, we will prove that F1 and F2 are contractions. Let (y1,y2),(ˆy1,ˆy2)βηi and ιJi. Then, for the operator F1, we have

    F1(y1,y2)(ι)F1(ˆy1,ˆy2)(ι)P3ιbi1|Hy1(s)Hˆy1(s)|ds+P4Γ(ωi)ιbi1(ιs)ωi1|Hy1(s)Hˆy1(s)|ds(P3(bibi1)+P4Γ(ωi+1)(bibi1)ωi)NH11NH1.

    Thus, we have

    F1(y1,y2)F1(ˆy1,ˆy2)ZΠNH11NH1(y1ˆy1).

    In the same manner, we get

    F2(y1,y2)F2(ˆy1,ˆy2)ZΠNH21NH2(y2ˆy2).

    Consequently,

    F(y1,y2)F(ˆy1,ˆy2)ΥZΠmax(NH11NH1,NH21NH2)(y1,y2)(ˆy1,ˆy2)Υ.

    Due to ZΠmax(NH11NH1,NH21NH2)<1, we deduce that the operator F is contraction map. Hence, by Theorem 2.3, we see that Π at least one fixed point, which is the corresponding solution (y1,y2)i of the coupled system (2.6). For i{1,2,...,n}, we define the functions

    (y1,y2)i={0, ι[0,bi1],(˜y1,˜y2)i, ιJi.     (3.3)

    As a result of this, it's well-known that, (y1,y2)iC(Ji,R)×C(Ji,R) given by (3.3) satisfies the following problem

    Λ(ω(ι))1ω(ι)b10Eω(ι)(ω(ι)(ι1θ)ω(ι)ω(ι)1)y1i(θ)dθ+...+Λ(ω(ι))1ω(ι)ιbi1Eω(ι)(ω(ι)(bθ)ω(ι)ω(ι)1)y1i(θ)dθ=H1(ω,y1i(ω),ABCDω(ι)0+y2i(ω)),

    and

    Λ(ω(ι))1ω(ι)b10Eω(ι)(ω(ι)(ι1θ)ω(ι)ω(ι)1)y2i(θ)dθ+...+Λ(ω(ι))1ω(ι)ιbi1Eω(ι)(ω(ι)(bθ)ω(ι)ω(ι)1)y2i(θ)dθ=H2(ω,y2i(ω),ABCDω(ι)0+y1i(ω)),

    where (y1,y2)i is a solution to system (2.6) equipped with (y1,y2)i(0)(y1,y2)i(bi)(˜y1,˜y2)i(bi)0. Then

    (y1,y2)(ι)={(y1,y2)1(ι),    ιJ1,                 (y1,y2)2(ι)={0,     ιJ1,(˜y1,˜y2)2,   ιJ2,...(y1,y2)i(ι)={0,    ι[0,bi1],(˜y1,˜y2)n,        ιJi,

    is the solution for coupled system (2.6).

    In this section, we discuss Ulam-Hyers (UH) stability results for coupled system (2.6). For more information about the following definition see [31,32,33]. Let ε1,ε2>0, and y be a function such that y(ι)C(Ji,R) satisfies the following inequalities:

    {|ABCDω(ι)0y1(ι)Hy1(ι)|ε1, ιJ,|ABCDω(ι)0y2(ι)Hy2(ι)|ε2, ιJ. (4.1)

    Now, we defined the functions (y1,y2)i(ι),(ˆy1,ˆy2)i(ι), (k1,k2)i(ι), ιJi as

    (y1,y2)i(ι)={0,   if ι[0,bi1],(y1,y2)(ι)          if ιJi, (4.2)
    (ˆy1,ˆy2)i(ι)={0,   if ι[0,bi1],(ˆy1,ˆy2)(ι)          if ιJi, (4.3)

    and

    (k1,k2)i(ι)={0,   if ι[0,bi1],(k1,k2)(ι)          if ιJi.

    In the forthcoming analyses, we will use the above notations (y1,y2)(ι),(ˆy1,ˆy2)(ι) and (k1,k2)(ι).

    Definition 4.1. The coupled system (2.6) is UH stable if there is a real number CH=max{CH1,CH2}>0 such that for each ε=max{ε1,ε2}>0, and each solution (ˆy1,ˆy2)Υ of coupled system (2.6) satisfying the inequalities (4.1), there exists a solution (y1,y2)Υ of coupled system (2.6) such that

    (ˆy1,ˆy2)(y1,y2)ΥCHε,

    where (y1,y2) and (ˆy1,ˆy2) defined in (4.2) and (4.3), respectively.

    Remark 4.2. Let (ˆy1,ˆy2)Υ be solutions of the inequalities (4.1) if and only if there is a function (k1,k2)(ι)Υ (which depends on (y1,y2))\ satisfying the following conditions,

    i) {|k1(ι)|ε1 for all ιJ,|k2(ι)|ε2 for all ιJ,

    ii) {ABCDωib+i1ˆy1(ι)=Hˆy1(ι)+k1(ι),ιJ,ABCDωib+i1ˆy2(ι)=Hˆy2(ι)+k2(ι),ιJ.

    Lemma 4.3. Let (ˆy1,ˆy2)Υ be satisfying (4.1), then (ˆy1,ˆy2) satisfy the following inequalities

    {|ˆy1(ι)Ψˆy1P3ιbi1Hˆy1(s)dsP4Γ(ωi)ιbi1(ιs)ωi1Hˆy1(s)ds|ε1RΠ1,|ˆy2(ι)Ψˆy2P3ιbi1Hˆy2(s)dsP4Γ(ωi)ιbi1(ιs)ωi1Hˆy2(s)ds|ε2RΠ2,

    where

    Ψˆy1=P1(ιbi1)Θ1(nk=1τkbi1Hˆy1(s)dsbibi1Hˆy1(s)ds)+P2(ιbi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1Hˆy1(s)dsbibi1(bis)ωi1Hˆy1(s)ds),

    and

    Ψˆy2=P1(ιbi1)Θ2(nk=1γkbi1Hˆy2(s)dsbibi1Hˆy2(s)ds)+P2(ιbi1)Θ2Γ(ωi)(nk=1γkbi1(γks)ωi1Hˆy2(s)dsbibi1(bis)ωi1Hˆy2(s)ds).

    Proof. By Remark 4.2, we have

    {ABCDωib+i1ˆy1(ι)=Hˆy1(ι)+k1(ι),ιJ,ABCDωib+i1ˆy2(ι)=Hˆy2(ι)+k2(ι),ιJ. (4.4)

    Then, by Theorem 2.6, the solutions of the system (4.4) are given as

    ˆy1(ι)=Ψˆy1+P3ιbi1Hy1(s)ds+P4Γ(ωi)ιbi1(ιs)ωi1Hy1(s)ds+P1(ιbi1)Θ1(nk=1τkbi1k1(s)dsbibi1k1(s)ds)+P2(ιbi1)Θ1Γ(ωi)(nk=1τkbi1(τks)ωi1k1(s)dsbibi1(bis)ωi1k1(s)ds)+P3ιbi1k1(s)ds+P4Γ(ωi)ιbi1(ιs)ωi1k1(s)ds,

    and

    ˆy2(ι)=Ψˆy2+P3ιbi1Hy2(s)ds+P4Γ(ωi)ιbi1(ιs)ωi1Hy2(s)ds+P1(ιbi1)Θ2(nk=1γkbi1k2(s)dsbibi1k2(s)ds)+P2(ιbi1)Θ2Γ(ωi)(nk=1γkbi1(γks)ωi1k2(s)dsbibi1(bis)ωi1k2(s)ds)+P3ιbi1k2(s)ds+P4Γ(ωi)ιbi1(ιs)ωi1k2(s)ds.

    By some calculations, we get

    \begin{equation*} \left\{ \begin{array}{c} \left\vert \widehat{y}_{1}(\iota )-\Psi _{\widehat{y}_{1}}-P_{3} \int_{b_{i-1}}^{\iota }\mathcal{H}_{\widehat{y}_{1}}\left( s\right) ds \mathfrak{-}\frac{P_{4}}{\Gamma (\omega _{i})}\int_{b_{i-1}}^{\iota }\left( \iota -s\right) ^{\omega _{i}-1}\mathcal{H}_{\widehat{y}_{1}}\left( s\right) ds\right\vert \leq \varepsilon _{1}\mathcal{R}_{\Pi _{1}}, \\ \left\vert \widehat{y}_{2}(\iota )-\Psi _{\widehat{y}_{2}}-P_{3} \int_{b_{i-1}}^{\iota }\mathcal{H}_{\widehat{y}_{2}}\left( s\right) ds \mathfrak{-}\frac{P_{4}}{\Gamma (\omega _{i})}\int_{b_{i-1}}^{\iota }\left( \iota -s\right) ^{\omega _{i}-1}\mathcal{H}_{\widehat{y}_{2}}\left( s\right) ds\right\vert \leq \varepsilon _{2}\mathcal{R}_{\Pi _{2}}. \end{array} \right. \end{equation*}

    Theorem 4.4. Suppose that \left(H_{1}\right) holds. If

    \begin{equation*} \left( P_{3}\left( b_{i}-b_{i-1}\right) +\frac{P_{4}\left( b_{i}-b_{i-1}\right) ^{\omega _{i}}}{\Gamma (\omega _{i}+1)}\right) \frac{ \mathfrak{N}_{f}(1+\mathcal{M})}{1-\mathfrak{N}_{f}} < 1, \end{equation*}

    then the coupled system (2.6) is UH stable.

    Proof. Let \varepsilon = \max \{\varepsilon _{1}, \varepsilon _{2}\} > 0 and \left(\widehat{y}_{1}, \widehat{y}_{2}\right) \in \Upsilon be a solution of coupled system (2.6) satisfying (4.1), and let \left(y_{1}, y_{2}\right) \in \Upsilon be the unique solution of the coupled system (2.6). Then, by Theorem 2.6, the solution of coupled system (2.6) is given by

    \begin{eqnarray*} y_{1}(\iota ) & = &\Psi _{\widehat{y}_{1}}+P_{3}\int_{b_{i-1}}^{\iota } \mathcal{H}_{y_{1}}\left( s\right) ds+\frac{P_{4}}{\Gamma (\omega _{i})} \int_{b_{i-1}}^{\iota }\left( \iota -s\right) ^{\omega _{i}-1}\mathcal{H} _{y_{1}}\left( s\right) ds, \\ y_{2}(\iota ) & = &\Psi _{\widehat{y}_{2}}+P_{3}\int_{b_{i-1}}^{\iota } \mathcal{H}_{y_{2}}\left( s\right) ds+\frac{P_{4}}{\Gamma (\omega _{i})} \int_{b_{i-1}}^{\iota }\left( \iota -s\right) ^{\omega _{i}-1}\mathcal{H} _{y_{2}}\left( s\right) ds. \end{eqnarray*}

    Hence by Lemma 4.3 and \left(H_{1}\right), we have

    \begin{eqnarray*} \left\vert \widehat{y}_{1}(\iota )-y_{2}(\iota )\right\vert &\leq &\left\vert \widehat{y}_{1}(\iota )-\Psi _{\widehat{y}_{1}}-P_{3} \int_{b_{i-1}}^{\iota }\mathcal{H}_{\widehat{y}_{1}}\left( s\right) ds \mathfrak{-}\frac{P_{4}}{\Gamma (\omega _{i})}\int_{b_{i-1}}^{\iota }\left( \iota -s\right) ^{\omega _{i}-1}\mathcal{H}_{\widehat{y}_{1}}\left( s\right) ds\right\vert \\ &&+P_{3}\int_{b_{i-1}}^{\iota }\left\vert \mathcal{H}_{\widehat{y}_{1}}(s)- \mathcal{H}_{y_{1}}\left( s\right) \right\vert ds+\frac{P_{4}}{\Gamma (\omega _{i})}\int_{b_{i-1}}^{\iota }\left( \iota -s\right) ^{\omega _{i}-1}\left\vert \mathcal{H}_{\widehat{y}_{1}}\left( s\right) -\mathcal{H} _{y_{1}}\left( s\right) \right\vert ds \\ &\leq &\varepsilon _{1}\mathcal{R}_{\Pi _{1}}+\left( P_{3}\left( b_{i}-b_{i-1}\right) +\frac{P_{4}}{\Gamma (\omega _{i}+1)}\left( b_{i}-b_{i-1}\right) ^{\omega _{i}}\right) \frac{\mathfrak{N}_{f}}{1- \mathfrak{N}_{f}}\left\Vert \widehat{y}_{1}-y_{1}\right\Vert _{\Upsilon }. \end{eqnarray*}

    Thus

    \begin{equation*} \left\Vert \widehat{y}_{1}-y_{1}\right\Vert \leq C_{\mathcal{H} _{1}}\varepsilon _{1}, \end{equation*}

    where

    \begin{equation*} C_{\mathcal{H}_{1}} = \frac{\mathcal{R}_{\Pi _{1}}}{1-\left( P_{3}\left( b_{i}-b_{i-1}\right) +\frac{P_{4}}{\Gamma (\omega _{i}+1)}\left( b_{i}-b_{i-1}\right) ^{\omega _{i}}\right) \frac{\mathfrak{N}_{f}}{1- \mathfrak{N}_{f}}} > 0. \end{equation*}

    In the same manner, we have

    \begin{equation*} \left\Vert \widehat{y}_{2}-y_{2}\right\Vert _{\Upsilon }\leq C_{\mathcal{H} _{2}}\varepsilon _{2}, \end{equation*}

    where

    \begin{equation*} C_{\mathcal{H}_{2}} = \frac{\mathcal{R}_{\Pi _{1}}}{1-\left( P_{3}\left( b_{i}-b_{i-1}\right) +\frac{P_{4}}{\Gamma (\omega _{i}+1)}\left( b_{i}-b_{i-1}\right) ^{\omega _{i}}\right) \frac{\mathfrak{N}_{f}}{1- \mathfrak{N}_{f}}} > 0. \end{equation*}

    Thus, we have

    \begin{eqnarray*} \left\Vert \left( \widehat{y}_{1}, \widehat{y}_{2}\right) -\left( y_{1}, y_{2}\right) \right\Vert _{\Upsilon } & = &\left\Vert \widehat{y} _{1}-y_{1}\right\Vert +\left\Vert \widehat{y}_{2}-y_{2}\right\Vert \\ &\leq &C_{\mathcal{H}_{1}}\varepsilon _{1}+C_{\mathcal{H}_{2}}\varepsilon _{2} \\ &\leq &2C_{\mathcal{H}}\varepsilon . \end{eqnarray*}

    Hence, coupled system (2.6) is UH stability. Now, by choosing C_{\mathcal{H }}(\varepsilon) = C_{\mathcal{H}}\varepsilon such that C_{\mathcal{H} }(0) = 0 , we conclude that coupled system (2.6) has Generalized UH stability.

    To illustrate our key results, this section presents numerical examples addressing the coupled system (2.6) in the context of ABC-fractional derivative with variable order \omega (\iota) . These specific examples have been chosen based on the conditions outlined in the theorems used, the specific conditions mentioned in our proposed conclusions, and the values of parameters and fractional-order derivatives. These examples serve as an illustration of the various dynamic applications that can be created by combining the ABC-fractional derivative with variable order \omega (\iota) .

    Example 5.1. Let us consider the following coupled system of ABC fractional problem with variable order \omega (\iota)

    \begin{equation*} \left\{ \begin{matrix} {^{ABC}}\mathbf{D}_{0^{+}}^{\omega (\iota )}y_{1}(\iota ) = \frac{\iota ^{2}}{ 5e^{\iota }}\left( e^{-\iota }+\frac{\left\vert y_{2}(\iota )\right\vert }{ 1+\left\vert y_{2}(\iota )\right\vert }+\frac{^{ABC}\mathbf{D} _{0^{+}}^{\omega (\iota )}y_{1}(\iota )}{1+^{ABC}D_{0^{+}}^{\omega (\iota )}y_{1}(\iota )}\right) , \quad \;\, \iota \in \mathcal{J}: = \left( 0, 2\right] , \\ {^{ABC}}\mathbf{D}_{0^{+}}^{\omega (\iota )}y_{2}(\iota ) = \frac{\iota ^{3}}{ 2\left( e^{\iota }-1\right) }\left( \frac{1}{e^{\iota }}+\frac{\left\vert y_{1}(\iota )\right\vert }{1+\left\vert y_{1}(\iota )\right\vert }+\frac{ ^{ABC}\mathbf{D}_{0^{+}}^{\omega (\iota )}y_{2}(\iota )}{1+^{ABC}D_{0^{+}}^{ \omega (\iota )}y_{2}(\iota )}\right) , \quad \;\, \iota \in \mathcal{J} : = \left( 0, 2\right] , \\ y_{1}(0) = 0, y_{2}(0) = 0, y_{1}(2) = \frac{1}{2}, y_{2}(2) = \frac{1}{3}. \end{matrix} \right. \end{equation*}

    Here, a = 0, b = 2 , and

    \begin{eqnarray*} \mathcal{H}_{y_{1}}(\iota ) & = &\frac{\iota ^{2}}{5e^{\iota }}\left( e^{-\iota }+\frac{\left\vert y_{2}(\iota )\right\vert }{1+\left\vert y_{2}(\iota )\right\vert }+\frac{^{ABC}\mathbf{D}_{0^{+}}^{\omega (\iota )}y_{1}(\iota )}{1+^{ABC}D_{0^{+}}^{\omega (\iota )}y_{1}(\iota )}\right) , \\ && \\ \mathcal{H}_{y_{2}}(\iota ) & = &\frac{\iota ^{3}}{2\left( e^{\iota }-1\right) }\left( \frac{1}{e}+\frac{\left\vert y_{1}(\iota )\right\vert }{1+\left\vert y_{1}(\iota )\right\vert }+\frac{^{ABC}\mathbf{D}_{0^{+}}^{\omega (\iota )}y_{2}(\iota )}{1+^{ABC}D_{0^{+}}^{\omega (\iota )}y_{2}(\iota )}\right) . \end{eqnarray*}

    Let \iota \in \left[0, 1\right], y_{1}, \overline{y}_{1}\in C\left(\mathcal{ J}_{i}, \mathcal{ \mathbb{R} }\right). Then

    \begin{eqnarray*} &&\left\vert \mathcal{H}_{y_{1}}(\iota )-\mathcal{H}_{\overline{y} _{1}}(\iota )\right\vert \\ &\leq &\frac{1}{5}\left( \left\vert y_{2}(\iota )-\overline{y}_{2}(\iota )\right\vert +\left\vert {^{ABC}}\mathbf{D}_{0^{+}}^{\omega (\iota )}y_{1}(\iota )-{^{ABC}}\mathbf{D}_{0^{+}}^{\omega (\iota )}\overline{y} _{1}(\iota )\right\vert \right) , \end{eqnarray*}

    and

    \begin{eqnarray*} &&\left\vert \mathcal{H}_{y_{2}}(\iota )-\mathcal{H}_{\overline{y} _{2}}(\iota )\right\vert \\ &\leq &\frac{1}{2}\left( \left\vert y_{1}(\iota )-\overline{y}_{1}(\iota )\right\vert +\left\vert {^{ABC}}\mathbf{D}_{0^{+}}^{\omega (\iota )}y_{2}(\iota )-{^{ABC}}\mathbf{D}_{0^{+}}^{\omega (\iota )}\overline{y} _{2}(\iota )\right\vert \right) . \end{eqnarray*}

    Therefore, hypothesis (H _{1} ) holds with \mathfrak{N}_{\mathcal{H}_{1}} = \frac{1}{5} and \mathfrak{N}_{\mathcal{H}_{2}} = \frac{1}{2}. Let

    \begin{equation*} \omega (\iota ) = \left\{ \begin{array}{c} \frac{3}{2}, { \ if }\iota \in \left( 0, 1\right] , \\ \frac{5}{2}{ \ \ \ \ if }\iota \in \left( 1, 2\right] . \end{array} \right. \end{equation*}

    Then, for i = 1, by some calculation, we get \Omega _{1} = 0.65 and \Omega _{2} = 0.52 and hence \Omega = \max \left\{ \Omega _{1}, \Omega _{2}\right\} = 0.65 < 1. Thus, by Theorem 3.1, the coupled system of ABC (2.6) has a unique solution. For every \varepsilon = \max \{\varepsilon _{1}, \varepsilon _{2}\} > 0 and each \left(\widehat{y}_{1}, \widehat{y} _{2}\right) \in \Upsilon satisfying

    \begin{equation*} \left\{ \begin{array}{c} \left\vert {^{ABC}}\mathbf{D}_{0}^{\omega (\iota )}\widehat{y}_{1}(\iota )- \mathcal{H}_{\widehat{y}_{1}}(\iota )\right\vert \leq \varepsilon _{1}, \\ \left\vert {^{ABC}}\mathbf{D}_{0}^{\omega (\iota )}\widehat{y}_{2}(\iota )- \mathcal{H}_{\widehat{y}_{2}}(\iota )\right\vert \leq \varepsilon _{2}, \end{array} \right. \end{equation*}

    there exists a solution \left(y_{1}, y_{2}\right) \in \Upsilon of the coupled system of ABC (2.6) with

    \begin{equation*} \left\{ \begin{array}{c} \left\Vert \widehat{y}_{1}-y_{1}\right\Vert \leq C_{\mathcal{H} _{1}}\varepsilon _{1}, \\ \left\Vert \widehat{y}_{2}-y_{2}\right\Vert \leq C_{\mathcal{H} _{2}}\varepsilon _{2}, \end{array} \right. \end{equation*}

    where

    \begin{equation*} C_{\mathcal{H}_{1}} = \frac{\mathcal{R}_{\Pi _{1}}}{1-\left( P_{3}\left( b_{i}-b_{i-1}\right) +\frac{P_{4}}{\Gamma (\omega _{i}+1)}\left( b_{i}-b_{i-1}\right) ^{\omega _{i}}\right) \frac{\mathfrak{N}_{f}}{1- \mathfrak{N}_{f}}} > 0, \end{equation*}

    and

    \begin{equation*} C_{\mathcal{H}_{2}} = \frac{\mathcal{R}_{\Pi _{1}}}{1-\left( P_{3}\left( b_{i}-b_{i-1}\right) +\frac{P_{4}}{\Gamma (\omega _{i}+1)}\left( b_{i}-b_{i-1}\right) ^{\omega _{i}}\right) \frac{\mathfrak{N}_{f}}{1- \mathfrak{N}_{f}}} > 0. \end{equation*}

    Thus,

    \begin{eqnarray*} \left\Vert \left( \widehat{y}_{1}, \widehat{y}_{2}\right) -\left( y_{1}, y_{2}\right) \right\Vert _{\Upsilon } &\leq &\left\Vert \widehat{y} _{1}-y_{1}\right\Vert _{\Upsilon }+\left\Vert \widehat{y}_{2}-y_{2}\right \Vert _{\Upsilon } \\ &\leq &C_{\mathcal{H}_{1}}\varepsilon _{1}+C_{\mathcal{H}_{2}}\varepsilon _{2} \\ &\leq &2C_{\mathcal{H}}\varepsilon . \end{eqnarray*}

    Therefore, by Theorem 4.4, the coupled system (2.6) is UH stable. Now, let C_{\mathcal{H}}(\varepsilon) = C_{\mathcal{H}}\varepsilon such that C_{\mathcal{H}}(0) = 0 , then the coupled system of ABC-fractional derivative (2.6) is GUH stability.

    For i = 2, with some calculation, we have \Omega _{1} = 0.55 and \Omega _{2} = 0.43 , and hence, \Omega = \max \left\{ \Omega _{1}, \Omega _{2}\right\} = 0.55 < 1. Then, by Theorem 3.1, the coupled system (2.6) has a unique solution. For every \varepsilon = \max \{\varepsilon _{1}, \varepsilon _{2}\} > 0 and each \left(\widehat{y}_{1}, \widehat{y}_{2}\right) \in \Upsilon satisfying

    \begin{equation*} \left\{ \begin{array}{c} \left\vert {^{ABC}}\mathbf{D}_{0}^{\omega (\iota )}\widehat{y}_{1}(\iota )- \mathcal{H}_{\widehat{y}_{1}}(\iota )\right\vert \leq \varepsilon _{1}, \\ \left\vert {^{ABC}}\mathbf{D}_{0}^{\omega (\iota )}\widehat{y}_{2}(\iota )- \mathcal{H}_{\widehat{y}_{2}}(\iota )\right\vert \leq \varepsilon _{2}, \end{array} \right. \end{equation*}

    there exists a solution \left(y_{1}, y_{2}\right) \in \Upsilon of the coupled system of ABC-fractional derivative (2.6) with

    \begin{eqnarray*} \left\Vert \left( \widehat{y}_{1}, \widehat{y}_{2}\right) -\left( y_{1}, y_{2}\right) \right\Vert _{\Upsilon } &\leq &\left\Vert \widehat{y} _{1}-y_{1}\right\Vert _{\Upsilon }+\left\Vert \widehat{y}_{2}-y_{2}\right \Vert _{\Upsilon } \\ &\leq &2C_{\mathcal{H}}\varepsilon , \end{eqnarray*}

    where C_{\mathcal{H}} = \max \left\{ C_{\mathcal{H}_{1}}, C_{\mathcal{H} _{2}}\right\} > 0 . Therefore, by Theorem 4.4, the coupled system (2.6) is UH stable. Now, by choosing C_{ \mathcal{H}}(\varepsilon) = C_{\mathcal{H}}\varepsilon such that C_{ \mathcal{H}}(0) = 0 , the coupled system (2.6) is a Generalized UH stability.

    Recent research has emphasized the Atangana-Baleanu fractional derivative, leading to the study and development of various qualitative features of solutions to FDEs incorporating these operators. This research paper presents a novel investigation into an implicit coupled system of fractional variable order, filling a gap in the existing literature. In this context, we have established and examined the necessary conditions for the existence and uniqueness of solutions to the coupled system of the ABC fractional problem with variable order, without relying on the semigroup property. We utilized extended intervals, piecewise constant functions, and fixed point theorems of the Banach-type and Krasnoselskii-type to reduce the proposed ABC system into a fractional integral equation. Furthermore, we employed mathematical analysis tools to investigate the stability results in the UH sense. An illustrative example in two cases was provided to support the findings of the study. By extending the existing conclusions examined for the ABC operator, this research contributes to advancing the understanding of variable-order FDEs. The paper provides a solid theoretical foundation that can be further explored through analysis, numerical simulations, and practical applications. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research in the field. Overall, this research paper expands the knowledge and understanding of variable-order FDEs, offering valuable insights and paving the way for further advancements in the field. We believe that the results obtained in this study will be significant for future research in the field of fractional calculus theory, given the recent extensive explorations and applications of the Mittag-Leffler power law.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    It is noted that the authors declare no conflicts of interest in the research study.

    This research was funded by the Zhejiang Normal University Research Fund under Grant YS304223919.



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