# Impulsive Coupled System of Fractional Differential Equations with Caputo–Katugampola Fuzzy Fractional Derivative

Authors: Leila Sajedi; Nasrin Eghbali; Hassen Aydi
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we investigate the existence, uniqueness, and different kinds of Ulam–Hyers stability of solutions of an impulsive coupled system of fractional differential equations by using the Caputo–Katugampola fuzzy fractional derivative. We applied the Perov-type fixed point theorem to prove the existence and uniqueness of the proposed system. Furthermore, the Ulam–Hyers–Rassias stability and Ulam–Hyers–Rassias–Mittag-Leffler’s stability results for the given system are discussed.


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## 1. Introduction

Fractional differential equations have become powerful tools for describing the mathematical modeling of systems. They have been widely used in modeling real-world phenomena and various fields such as economics, biology, mechanics, bioengineering, physics, and other applied sciences. Many systems can be described precisely by them. Kilbas et al. [1] introduced the properties of fractional integrals and derivatives. Agrawal et al. in [2] provided the generalized calculation of multiparametric fractional derivatives and in [3] introduced some generalized fractional integral and differential operators and their properties. Almeida in [4] suggested a Caputo fractional derivative of one function to another function. The generalization of the Riemann–Liouville and Caputo derivatives has presented the effect of using it in mathematical physics equations. Later, Katugampola [5] introduced a new fractional derivative, which generalizes both derivatives in one.Recently, fuzzy fractional and differential calculus has been developed in wide domains. Bede et al. [6–8] presented a great number of applications of this calculus.This theory, as well as the benefits of fractional calculus, has attracted numerous mathematicians so that the research studies about the existence, uniqueness, and stability have also developed; for example, Allahviranloo et al. [9] applied the generalized Hukuhara Caputo differentiability to investigate the existence and uniqueness of the solution for fuzzy fractional differential equations. Mazandarani and Najariyan [10] studied fundamental theories for numerical investigation. Ahmadian et al. [11] studied fuzzy solutions for fractional differential systems. Besides, Hoa [12] investigated some results to fuzzy fractional differential equations with delay. Authors in [13] presented a few basic theories of different IVP of implicit fuzzy fractional differential equations involving different fuzzy fractional derivatives. For more recent developments about this concept, [12–19] and the references therein are some papers that include basic outcomes in the theory of fuzzy fractional differential equations and the method for investigating numerical solutions of them.After the definition of Caputo–Katugampola fractional derivative by Katugampola [5], authors in [19] suggested Caputo–Katugampola fuzzy fractional derivative and investigated the existence and uniqueness of the solutions to the IVP of fractional fuzzy differential equations with this new derivative.Ulam–Hyers stability and its various types are another interesting topic for mathematicians. In 1940, the first controversial question in the theory of functional equations was originated by Ulam [20]. In 1941, Hyers [21] gave the first answer to the Ulam’s question. In 1987, Rassias [22] generalized it for linear mappings. Research into stability theory is now very widespread, and numerous papers have been published. Abbas [23] and Adjabi [24] studied different topics and used the Caputo–Hadamard derivative in Cauchy problems, respectively. For more details, references [25–37] are some of the studies extended by researchers on the Ulam stability problem to functional and fractional-order differential equations.Impulsive differential equations are a suitable mathematical model to study the evolutionary processes which suddenly or abruptly change their state at certain moments. These equations have a wide variety of applications in dynamic systems and many phenomena in natural sciences, economics, biology, space sciences, and control of the spacecraft (see [38–42]). Also, the theory of impulsive fractional differential equations and their applications have made considerable progress (see [43–48]). In 2001, Lakshmikantham and McRae [49] studied the fuzzy impulsive differential equations. In 2003, Vatsala investigated impulsive hybrid fuzzy differential equations [50]. There are not too many papers on impulsive fuzzy differential equations; however, some of the main findings on impulsive fuzzy differential equations can be found in [49–53].Motivated by the aforementioned works, in this paper, we study the existence, uniqueness, and Ulam-type stability of the following coupled system of fractional differential equations of the form:(1)Da+σ,ρCζz=ℱ1z,ζ,℘,z∈a,T,z≠zℓ,ℓ∈1,…,r,Da+ϱ,ρCζz=ℱ2z,ζ,℘,z∈a,T,z≠zℓ,ℓ∈1,…,r,Δζzℓ=ℐℓζzℓ−,℘zℓ−,ℓ∈1,…,r,Δζzℓ=ℐℓ¯ζzℓ−,℘zℓ−,ℓ∈1,…,r,ζa=ζ0,℘a=℘0,where Da+σ,ρC and Da+ϱ,ρC are the Caputo–Katugampola fractional derivatives of order σ,ϱ, 0<σ,ϱ<1,ρ>0. ℱ1,ℱ2:ℐ×Ω×Ω⟶Ω are continuous functions, where Ω is the space of fuzzy sets, and ℐℓ,ℐℓ¯∈CΩ×Ω,Ω for all ℓ∈1,…,r, and zℓ satisfy a=z0<z1<⋯<zr<zr+1=T, Δζzℓ=ζzℓ+−ζzℓ−, and ζzℓ+=limℏ⟶0ζzℓ+ℏ, ζzℓ−=limℏ⟶0ζzℓ−ℏ indicate the right and left limits of ζzℓ at z=zℓ, respectively.This paper is organized as follows. Section2 is related to some definitions, theorems, lemmas, and remarks which will be needed in the later sections. In Section 3, appropriate conditions are proposed for the existence and uniqueness of solutions and Ulam–Hyers–Rassias stability for the given problem. Section 4 is devoted to the study of the Ulam–Hyers–Rassias–Mittag-Leffler stability for system (1). The final section provides a conclusion.


## 2. Preliminaries

Here, we provide some definitions, theorems, and properties that will be useful throughout this work. LetΩ be a family of fuzzy numbers, i.e., normal, convex, upper semicontinuous, and compactly supported fuzzy subsets of the real figures. For s∈0,1, denote ξ˜s=z∈ℝ|ξ˜z≥s and ξ˜0=z∈ℝ|ξ˜z>0¯. So, the s-level set of ξ˜,ξ˜s=ξ˜¯s,ξ˜¯s is a bounded closed interval for any s∈0,1.Definition 1.
(see [18]). Let ξ˜1,ξ˜2∈Ω and δ∈ℝ. Based on Zadeh’s extension principle, ξ˜1+ξ˜2 and δ⋅ξ˜1 are in Ω and defined as follows:(2)ξ˜1+ξ˜2s=ξ˜1s+ξ˜2s,δ⋅ξ˜1s=δξ˜1s,∀s∈0,1,where ξ˜1s+ξ˜2s indicates that two intervals of ℝ have been added and δξ˜1s is the product from a scalar to a real interval number. The diameter of the s-level set of ξ˜ in Ω is presented by dξ˜s=ξ˜¯s−ξ˜¯s.Definition 2.
(see [18]). The distance D0ξ˜1,ξ˜2 between two fuzzy numbers is presented as follows:(3)D0ξ˜1,ξ˜2=sup0≤s≤1Hξ˜1s,ξ˜2s,∀ξ˜1,ξ˜2∈Ω,where Hξ˜1s,ξ˜2s=maxξ˜1¯s−ξ˜2¯s,ξ˜1¯s−ξ˜2¯s is the Hausdorff distance between ξ˜1s and ξ˜2s.Definition 3.
(see [8]). The generalized Hukuhara difference of two fuzzy numbers ξ˜1,ξ˜2∈Ω (gH-difference) is given by(4)ξ˜1⊖gHξ˜2=ξ˜3⇔aξ˜1=ξ˜2+ξ˜3,orbξ˜2=ξ˜1+−1ξ˜3.
In thes-levels, we have that for all s∈0,1,(5)ξ˜1⊖gHξ˜2s=minξ˜1¯s−ξ˜2¯s,ξ˜1¯s−ξ˜2¯s,maxξ˜1¯s−ξ˜2¯s,ξ˜1¯s−ξ˜2¯s.
Also, the condition for the existence ofξ˜1⊖gHξ˜2 in (a) is dξ˜1s≥dξ˜2s, and the condition for the existence of ξ˜1⊖gHξ˜2 in (b) is dξ˜2s≥dξ˜1s.Definition 4.
(see [18]). Let ϰ:a,b⟶Ω be a fuzzy function. Then, for each s∈0,1, the function z⟼dϰzs is nondecreasing (nonincreasing) on a,b, we say that ϰ is ξ˜-increasing (ξ˜-decreasing) on a,b. If ϰ is ξ˜-increasing or ξ˜-decreasing on a,b, then ϰ is called ξ˜-monotone on a,b.Definition 5.
(see [8]). Let ϰ:a,b⟶Ω and z belong to a,b. We say that the fuzzy function ϰ is generalized Hukuhara differentiable at z if there exists an element ϰ′z∈Ω such that the following statement is true:(6)ϰ′z=limℏ⟶0ϰz+ℏ⊖gHϰzℏ.
Letℐ=a,T,Cℐ,Ω is formed by all continuous fuzzy functions with value in Ω on the interval a,T;ACℐ,Ω denotes all absolutely continuous fuzzy functions with value in Ω on a,T;Lℐ,Ω is the set of all fuzzy functions ϰ:a,T⟶Ω such that the functions z⟼D0ϰz,0 belong to La,T;ForT>a>0, let ℐℓ=zℓ,zℓ+1, ℓ∈1,…,r. Denote by PCℐ,Ω the space of piecewise continuous fuzzy functions ℘:a,T⟶Ω and ℘ℓ∈Cℐℓ,Ω for ℓ∈1,…,r+1 such that ℘zℓ+and℘zℓ− exist for ℓ∈1,…,r with respect to the uniform norm as follows:(7)℘PC=supD0℘z,0,z∈ℐ.Definition 6.
(see [19]). Consider ϰ∈Lℐ,Ω as a fuzzy function, and σ∈0,1, then the Riemann–Liouville generalized fractional integral (RLGFI) is given by(8)ϰz≔ℑa+σ,ρϰz=ρ1−σΓσ∫azωρ−1zρ−ωρσ−1ϰωdω,z>a,where Γσ is the Gamma function.Definition 7.
(see [19]). Let σ∈0,1, then the fuzzy Riemann–Liouville–Katugampola generalized Hukuhara fractional derivative for a function ϰ∈Lℐ,Ω is expressed as(9)Da+σ,ρRLϰz≔z1−ρℑa+1−σ,ρϰ′z,z∈a,T,if there gH-derivative ϰ1−σ,ρ′z exists for z∈a,T, where(10)ϰ1−σ,ρz≔ℑa+1−σ,ρϰz=ρσΓ1−σ∫azωρ−1zρ−ωρ−σϰωdω,z∈a,T.Definition 8.
(see [19]). Consider ϰ∈ACℐ,Ω as a ξ˜-monotone fuzzy function, then the Caputo–Katugampola generalized fractional derivative is given by(11)Da+σ,ρCϰz≔ρΓ1−σ∫azzρ−ωρ−σϰ′ωdω,z∈a,T.Theorem 1 (see [19]).
Letσ∈0,1 and ϰ∈ACℐ,Ω be a ξ˜-monotone fuzzy function, then(12)ℑa+σ,ρDa+σ,ρCϰz=ϰz⊖gHϰa,z∈a,T,Da+σ,ρCℑa+σ,ρϰz=ϰz,z∈a,T.Lemma 1 (see [19]).
Assume thatℱ:ℐ×Ω⟶Ω is a continuous function. Then, a ξ˜-monotone fuzzy function ζ∈PCℐ,Ω is a solution of the following IVP:(13)Da+σ,ρCζz=ℱz,ζz,z∈a,T,ζa=ζ0,if and only if ζ∈PCℐ,Ω satisfies the integral equation provided as follows:(14)ζz⊖gHζ0=ρ1−σΓσ∫azωρ−1zρ−ωρσ−1ℱω,ζωdω,z∈a,T,and the fuzzy function t⟶ℑa+σ,ρℱz,ζ is ξ˜-increasing on a,T.Lemma 2.
Letσ∈0,1 and ℱ:ℐ×Ω×Ω⟶Ω. A ξ˜-monotone continuous function ζ is a solution of the following fractional integral equation:(15)ζz⊖gHζ0=ρ1−σΓσ∫azωρ−1zρ−ωρσ−1ℱω,ζω,℘ωdω,ifz∈a,z1,ζz⊖gHζ0=ρ1−σΓσ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1ℱω,ζω,℘ωdω+ρ1−σΓσ∫zℓzωρ−1zρ−ωρσ−1ℱω,ζω,℘ωdω+∑i=1ℓℐiζzi−,℘zi−,ifz∈zℓ,zℓ+1,if and only if ζ is a ξ˜-monotone solution of the impulsive fractional IVP:(16)Da+σ,ρCζz=ℱz,ζ,℘,z∈ℐ,Δζzℓ=ℐℓζzℓ−,℘zℓ−,ℓ∈1,…,r,ζa=ζ0.Proof.
Assume thatζ satisfies (16). By using condition (16) and Lemma 1,if z∈a,z1, then(17)ζz⊖gHζ0=ρ1−σΓσ∫azωρ−1zρ−ωρσ−1ℱω,ζω,℘ωdω.
Ifz∈z1,z2, we have(18)ζz⊖gHζz1+=ρ1−σΓσ∫z1zωρ−1zρ−ωρσ−1ℱω,ζω,℘ωdω,then(19)ζz⊖gHζ0=ℐ1ζz1−,℘z1−+ρ1−σΓσ∫az1ωρ−1zρ−ωρσ−1ℱω,ζω,℘ωdω+ρ1−σΓσ∫z1zωρ−1zρ−ωρσ−1ℱω,ζω,℘ωds.
Ifz∈z2,z3, we have(20)ζz⊖gHζz2+=ρ1−αΓσ∫z2zωρ−1zρ−ωρσ−1ℱω,ζω,℘ωdω,then(21)ζz⊖gHζ0=ℐ1ζz1−,℘z1−+ℐ2ζz2−,℘z2−+ρ1−σΓσ∫az1ωρ−1z1ρ−ωρσ−1ℱω,ζω,℘ωdω+ρ1−σΓσ∫z1z2ωρ−1z2ρ−ωρσ−1ℱω,ζω,℘ωdω+ρ1−σΓσ∫z2zωρ−1zρ−ωρσ−1ℱω,ζω,℘ωdω.
Repeating the same process forz∈zℓ,zℓ+1, we obtain(22)ζz⊖gHζ0=ρ1−σΓσ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1ℱω,ζω,℘ωdω+ρ1−σΓσ∫zℓzωρ−1zρ−ωρσ−1ℱω,ζω,℘ωdω+∑i=1ℓℐiζzi−,℘zi−.
Conversely, suppose thatζ satisfies equation (15). If z∈a,z1, then ζa=ζ0, and applying the fact that Da+σ,ρC is the left inverse of ℑa+σ,ρC, we obtain(23)Da+σ,ρC=ℱz,ζ,℘for allz∈a,z1∪zℓ,zℓ+1,ℓ∈1,…,r.
And we can easily prove thatΔ|z=zℓ=ℐℓζzℓ−,℘zℓ− for ℓ∈1,…,r.Remark 1.
In the usual metric space and the generalized metric space according to Perov, the notions of convergence, Cauchy sequence, completeness, and open and closed subsets are the same.Definition 9.
(see [54] ). A square matrix ℬ of real numbers is convergent to zero if and only if ℬn⟶0 as n approaches to ∞.Lemma 3 (see [24]).
Letℬ∈ℳn×nℝ+. Then, the following statements are equivalent:The matrixℬ is convergent to 0The eigenvalues ofℬ lie in the open unit disc, i.e., λ<1, for each λ∈ℂ with detℬ−λI=0The matrixI−ℬ is nonsingular and I−ℬ−1=I+ℬ+⋯+ℬn+⋯I−ℬ−1 has nonnegative elementsFor everyq∈ℝm, ℬnq⟶0 and qℬn⟶0 as n approaches ∞Theorem 2 (Perov’s fixed point theorem [48]).
LetX,d be a complete generalized metric space. Let G:X⟶X be a mapping having the property that there exists a matrix ℬ that converges to 0 such that dGϰ,Gj≤ℬdϰ,j for all ϰ,j∈X. Then, G has a unique fixed point ϰ∗, and we have(24)dGℓϰ0,ϰ∗≤ℬℓI−ℬ−1dϰ0,Gϰ0,∀ℓ∈G.Theorem 3 (the Gronwall inequality [55]).
Supposeϑandζ are two nonnegative integrable functions and h˜ is a nonnegative and nondecreasing continuous function on a,b such that(25)ϑz≤ζz+h˜zρ1−σ∫azωρ−1zρ−ωρσ−1ϑωdω,∀z∈a,b,then(26)ϑz≤ζz+∫az∑ℓ=1∞ρ1−ℓσh˜zΓσℓΓℓσωρ−1zρ−ωρℓσ−1ζωdω,∀z∈a,b.
Furthermore, ifζz is nondecreasing on [a, b], then(27)ϑ≤ζzEσh˜zΓσzρ−aρρσ,∀z∈a,b.


## 3. Ulam–Hyers–Rassias (UHR) Stability of System (1)

Our goal in this part is to prove that an impulsive coupled system of fractional differential equations (1) is Ulam–Hyers–Rassias stable.Letε=εσ,εϱ>0, η=ησ,ηϱ>0, and φ=φσ,φϱ∈PCℐ,ℝ+. Consider the following inequalities:(28)D0Da+σ,ρCϰz,ℱ1z,ϰ,j≤εσ,z∈ℐ,D0Δϰzℓ,ℐℓϰzℓ,jzℓ≤εσ,ℓ∈1,…,r,D0Da+ϱ,ρCϰz,ℱ2z,ϰ,j≤εϱ,z∈ℐ,D0Δϰzℓ,ℐℓ¯ϰzℓ,jzℓ≤εϱ,k∈1,…,r,(29)D0Da+σ,ρCϰz,ℱ1z,ϰ,j≤φσz,z∈ℐ,D0Δϰzℓ,ℐℓϰzℓ,jzℓ≤ησ,ℓ∈1,…,r,D0Da+ϱ,ρCϰz,ℱ2z,ϰ,v≤φϱz,z∈ℐ,D0Δϰzℓ,ℐℓ¯ϰzℓ,jzℓ≤ηϱ,ℓ∈1,…,r,(30)D0Da+σ,ρCϰz,ℱ1z,ϰ,j≤εφσz,z∈ℐ,D0Δϰzℓ,ℐℓϰzℓ,jzℓ≤εησ,ℓ∈1,…,r,D0Da+ϱ,ρCϰz,ℱ2z,ϰ,j≤εφϱz,z∈ℐ,D0Δϰzℓ,ℐℓ¯ϰzℓ,jzℓ≤εηϱ,ℓ∈1,…,r.Remark 2.
(a)
A fuzzy functionϰ,j∈PCℐ,Ω×PCℐ,Ω is a solution of (28) if and only if there exist δ1,δ2∈PCℐ,Ω and a sequence of fuzzy functions λ1ℓ,λ2ℓ, ℓ∈1,…,r such that(i)
D0δ1z,0≤εσ, D0δ2z,0≤εϱ, D0λ1ℓz,0≤ησ, and D0λ2ℓz,0≤ηϱ for all z∈I.(31)Da+α,ρCϰz=ℱ1z,ϰ,j+δ1z,z∈ℐ,Da+β,ρCjz=ℱ2z,ϰ,j+δ2z,z∈ℐ,Δϰzℓ=ℐℓϰzℓ−,jzℓ−+λ1ℓz,ℓ∈1,…,r,Δjzℓ=ℐℓ¯ϰzℓ−,jzℓ−+λ2ℓz,ℓ∈1,…,r.(b)
A fuzzy functionϰ,j∈PCℐ,Ω×PCℐ,Ω is a solution of (30) if and only if there exist δ3,δ4∈PCℐ,Ω and a sequence of fuzzy functions λ3ℓ,λ4ℓ, ℓ∈1,…,r such that(i)
D0δ3z,0≤εσφσz, D0δ4z,0≤εϱφϱz, D0λ3ℓz,0≤εησ, and D0λ4ℓz,0≤εηϱ for all z∈ℐ.(32)Da+α,ρCϰz=ℱ1z,ϰ,j+δ3z,z∈ℐ,Da+β,ρCjz=ℱ2z,ϰ,j+δ4z,z∈ℐ,Δϰzℓ=ℐℓϰzℓ−,jzℓ−+λ3ℓz,ℓ∈1,…,r,Δjzℓ=ℐℓ¯ϰzℓ−,jzℓ−+λ4ℓz,ℓ∈1,…,r.Definition 10.
System (1) is∗ Ulam–Hyers (UH) stable if ∃λσ,ϱ=λσ,λϱ>0 such that for each ε=εσ,εϱ>0 and for each solution ϰ,j∈PCℐ,Ω×PCℐ,Ω of inequality (28) ∃ζ,℘∈PCℐ,Ω×PCℐ,Ω satisfying (1) with(33)D0ϰ,j,ζ,℘≤λσ,ϱε,z∈ℐ.∗ Generalized Ulam–Hyers (GUH) stable if ∃φσ,ϱ=φσ,φϱ∈Cℐ,ℝ+ with φσ,ϱ0=0 such that for each solution ϰ,j∈PCℐ,Ω×PCℐ,Ω of inequality (28) ∃ζ,℘∈PCℐ,Ω×PCℐ,Ω satisfying (1) with(34)D0ϰ,j,ζ,℘≤φσ,ϱε,z∈ℐ.∗ Ulam–Hyers–Rassias (UHR) stable with respect to φσ,ϱ,ησ,ϱ if ∃λσ,ϱ,φ>0 such that for each ε>0 and for each solution ϰ,j∈PCℐ,Ω×PCℐ,Ω of (30) ∃ζ,℘∈PCℐ,Ω×PCℐ,Ω satisfying (1) with(35)D0ϰ,j,ζ,℘≤λσ,ϱ,φεφz+η,z∈ℐ.∗ Generalized Ulam–Hyers–Rassias (GUHR) stable with respect to φσ,ϱ,ησ,ϱ if ∃λσ,ϱ,φ>0 such that for each solution ϰ,j∈PCℐ,Ω×PCℐ,Ω of (29) ∃ζ,℘∈PCℐ,Ω×PCℐ,Ω of (1) with(36)D0ϰ,j,ζ,℘≤λσ,ϱ,φφz+η,z∈ℐ.
To prove the existence, uniqueness, and the (UH) type stability in Definition10, the following hypotheses are needed in the sequel:1∃Li>0i=1,…,4 such that for all ϰ,ϰ¯,j,j¯∈Ω, we have(37)D0ℱ1z,ζ,℘,ℱ1z,ζ¯,℘¯≤L1D0ζ,ζ¯+L2D0℘,℘¯,D0ℱ2z,ζ,℘,ℱ2z,ζ¯,℘¯≤L3D0ζ,ζ¯+L4D0℘,℘¯.2∃a1ℓ,a2ℓ,b1ℓ,b2ℓ≥0 , ℓ∈1,…,r such that for all ζ,ζ¯,℘,℘¯∈Ω, we have(38)D0ℐℓζ,℘,ℐℓζ¯,℘¯≤a1ℓD0ζ,ζ¯+a2ℓD0℘,℘¯,D0ℐ¯ℓζ,℘,ℐ¯ℓζ¯,℘¯≤b1ℓD0ζ,ζ¯+b2ℓD0℘,℘¯.3 Let φ=φσ,φϱ∈CI,ℝ+ be a nondecreasing function, then ∃λφ>0 such that(39)ρ1−σΓσ∫azωρ−1zρ−ωρσ−1φωdω≤λφφz.Theorem 4.
Letð1−ð3 be satisfied. We suppose that Aσ=2/Γσ+1Tρ−aρ/ρσ, Aϱ=2/Γϱ+1Tρ−aρ/ρϱ, ℜ=1−ℜ2ℜ1∗/1−ℜ11−ℜ2∗≠0, ℜ1<1, ℜ2∗<1, A=AσL1+∑i=1ℓa1iAσL2+∑i=1ℓa2iAϱL3+∑i=1ℓb1iAϱL4+∑i=1ℓb2i converges to zero, and ℱ1.,0,0=ℱ2.,0,0=ℐℓ0,0=ℐℓ¯0,0. Then, for every ε>0, if ϰ,j∈PCℐ,Ω×PCℐ,Ω satisfies inequality (30), there exists a unique solution ζ,℘∈PCℐ,Ω×PCℐ,Ω of system (1). Moreover, system (1) has UHR stability and GUHR stability.Proof.
Consider the operatorϒ:PC×PC⟶PC×PC defined by(40)ϒζ,℘=ϒ1ζ,℘,ϒ2ζ,℘,where(41)ϒ1ζ,℘z⊖gHζ0=ρ1−σΓσ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1ℱ1ω,ζω,℘ωdω+ρ1−σΓσ∫zℓzωρ−1zρ−ωρσ−1ℱ1ω,ζω,℘ωdω+∑i=1ℓℐiζzi,℘zi,ϒ2ζ,℘z⊖gH℘0=ρ1−ϱΓϱ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρϱ−1ℱ2ω,ζω,℘ωdω+ρ1−ϱΓϱ∫zℓzωρ−1zρ−ωρϱ−1ℱ2ω,ζω,℘ωdω+∑i=1ℓℐi¯ζzi,℘zi.
Obviously,ϒ is a well-defined operator, and fixed points of ϒ are solutions to system (1). We prove that the operator ϒ is the contractive operator. Suppose that ζ,℘,ζ¯,℘¯∈PC×PC, then(42)D0ϒ1ζ,℘,ϒ1ζ¯,℘¯≤ρ1−σΓσ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1D0ℱ1ω,ζω,℘ω,ℱ1ω,ζ¯ω,℘¯ωdω+ρ1−σΓσ∫zℓzωρ−1zρ−ωρσ−1D0ℱ1ω,ζω,℘ω,ℱ1ω,ζ¯ω,℘¯ωdω+∑i=1ℓD0ℐiζzi,℘zi,ℐiζ¯zi,℘¯zi≤ρ1−σΓσL1D0ζ,ζ¯+L2D0℘,℘¯∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1dω+ρ1−σΓσL1D0ζ,ζ¯+L2D0℘,℘¯∫zℓzωρ−1zρ−ωρσ−1dω+∑i=1ℓa1iD0ζ,ζ¯+a2iD0℘,℘¯≤2Γσ+1Tρ−aρρσL1D0ζ,ζ¯+L2D0℘,℘¯+∑i=1ℓa1iD0ζ,ζ¯+a2iD0℘,℘¯≤AσL1+∑i=1ℓa1iD0ζ,ζ¯+AσL2+∑i=1ℓa2iD0℘,℘¯.
Similarly, we have(43)D0ϒ2ζ,℘,ϒ2ζ¯,℘¯≤AϱL3+∑i=1ℓb1iD0ζ,ζ¯+AϱL4+∑i=1ℓb2iD0℘,℘¯.
Therefore,(44)D0ϒζ,℘,ϒζ¯,℘¯≤AD0ζ,ζ¯D0℘,℘¯,for all ζ,℘,ζ¯,℘¯∈PC×PC.
By Theorem2, the operator ϒ has a unique fixed point, which is a solution of system (1).
In the sequel, we shall show that system (1) is UHR stable. We observe that if ϰ,j∈PCℐ,Ω×PCℐ,Ω is a solution of inequality (30) for all z∈ℐ, then by Remark 2(b) and Lemma 2, we notice that(45)ϰz⊖gHϰ0=ρ1−σΓσ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1ℱ1ω,ϰω,jω+δ3ωdω+ρ1−σΓσ∫zℓzωρ−1zρ−ωρσ−1ℱ1ω,ϰω,jω+δ3ωdω+∑i=1ℓℐiϰzi,jzi+λ3i,jz⊖gHj0=ρ1−ϱΓϱ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρϱ−1ℱ2ω,ϰω,jω+δ4ωdω+ρ1−ϱΓϱ∫zℓzωρ−1zρ−ωρϱ−1ℱ2ω,ϰω,jω+δ4ωdω+∑i=1ℓℐi¯ϰzi,jzi+λ4i,D0ϰz⊖gHϰ0,ρ1−σΓσ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1ℱ1ω,ϰω,jωdω+ρ1−αΓσ∫zℓzωρ−1zρ−ωρσ−1ℱ1ω,ϰω,jωdω+∑i=1ℓℐiϰzi,jzi≤ρ1−σΓσ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1D0δ3ω,0dω+ρ1−σΓσ∫zℓzωρ−1zρ−ωρσ−1D0δ3ω,0dω+∑i=1ℓD0λ3i,0≤2εσφσzΓσ+1Tρ−aρρσ+ℓεσησ≤εσ2Γσ+1Tρ−aρρσφσz+ℓησ≤εσ2Γσ+1Tρ−aρρσ+ℓφσz+ησ≤εσλσφσz+ησ,where λσ=2/Γσ+1Tρ−aρ/ρσ+ℓ.
If we repeat the same procedure for the second equation, we obtain(46)D0jz⊖gHj0,ρ1−ϱΓϱ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρϱ−1ℱ2ω,ϱω,jωdω+ρ1−ϱΓϱ∫zℓzωρ−1zρ−ωρϱ−1ℱ2ω,ϰω,jωdω+∑i=1ℓℐi¯ϰzi,jzid≤εϱλϱφϱz+ηϱ,where λϱ=2/Γϱ+1Tρ−aρ/ρϱ+ℓ.
Now, letζ,℘ be a unique solution of system (1). By using Remark 2(b), the solution of system (1) is provided by(47)ζz⊖gHϰ0=ρ1−σΓσ∑i=1ℓ∫zi−1zisρ−1ziρ−ωρσ−1ℱ1ω,ζω,℘ωdω+ρ1−σΓσ∫zℓzωρ−1zρ−ωρσ−1ℱ1ω,ζω,℘ωdω+∑i=1ℓℐiζzi,℘zi,℘z⊖gHj0=ρ1−ϱΓϱ∑i=1ℓ∫zi−1zisρ−1ziρ−ωρϱ−1ℱ2ω,ζω,℘ωdω+ρ1−ϱΓϱ∫zℓzωρ−1zρ−ωρϱ−1ℱ2ω,ζω,℘ωdω+∑i=1ℓℐi¯ζzi,℘zi.
It follows that(48)D0ϰz,ζz≤D0ϰz⊖gHϰ0,ρ1−σΓσ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1ℱ1ω,ζω,jωdω+ρ1−σΓσ∫zℓzωρ−1zρ−ωρσ−1ℱ1ω,ζω,℘ωdω+∑i=1ℓℐiϰzi,℘zi+ρ1−σΓσ∑i=1ℓ∫zi−1ziωρ−1ziρ−ωρσ−1D0ℱ1ω,ϰω,jω,ℱ1ω,ζω,℘ωdω+ρ1−σΓσ∫zℓzωρ−1zρ−ωρσ−1D0ℱ1ω,ϰω,jω,ℱ1ω,ζω,℘ωdω+∑i=1ℓD0ℐiϰzi,jzi,ℐiζzi,℘zi≤ϵσλσφσz+ησ+AσL1D0ϰ,ζ+L2D0j,℘+∑i=1ℓa1iD0ϰ,ζ+a2iD0j,℘≤ϵσλσφσz+ησ+AσL1+∑i=1ℓa1iD0ϰ,ζ+AσL2+∑i=1ℓa2iD0j,℘≤ϵσλσφσz+ησ+ℜ1D0ϰ,ζ+ℜ2D0j,℘,where ℜ1=AσL1+∑i=1ℓa1i and ℜ2=AσL2+∑i=1ℓa2i.
We obtain(49)D0ϰz,ζz≤λσεσφσz+ησ1−ℜ1+ℜ21−ℜ1D0j,℘.
Similarly, it follows(50)D0jz,℘z≤λϱεϱφϱz+ηϱ1−ℜ2∗+ℜ1∗1−ℜ2∗D0ϰ,ζ,where ℜ1∗=AϱL3+∑i=1ℓb1i and ℜ2∗=AϱL4+∑i=1ℓb2i.
From equations (49) and (50), we have(51)1−ℜ21−ℜ1−ℜ1∗1−ℜ2∗1D0ϰ,ζD0j,℘≤λσεσφσz+ησ1−ℜ1λϱεϱφϱz+ηϱ1−ℜ2∗.
If we solve the above inequality, we obtain(52)D0ϰ,ζD0j,℘≤1ℜℜ2ℜ1−ℜ1−ℜ1∗ℜ1−ℜ2∗1ℜλσεσφσz+ησ1−ℜ1λϱεϱφϱz+ηϱ1−ℜ2∗,where ℜ=1−ℜ2ℜ1∗/1−ℜ11−ℜ2∗.
So, we get(53)D0ϰ,ζ≤λσεσφσz+ησℜ1−ℜ1+ℜ2λϱεϱφϱz+ηϱℜ1−ℜ11−ℜ2∗,D0j,℘≤λϱεϱφϱz+ηϱℜ1−ℜ2∗+ℜ1∗λσεσφσz+ησℜ1−ℜ11−ℜ2∗.
We have(54)D0ϰ,ζ+D0j,℘≤λσεσφσz+ησℜ1−ℜ1+λϱεϱφϱz+ηϱℜ1−ℜ2∗+ℜ2λϱεϱφϱz+ηϱℜ1−ℜ11−ℜ2∗+ℜ1∗λσεσφσz+ησℜ1−ℜ11−ℜ2∗.
Letε=maxεσ,εϱ, η=maxησ,ηϱ, and φ=maxφσ,φϱ, then from (54), we have(55)D0ϰ,j,ζ,℘≤λσ,ϱεφz+η,where(56)λσ,ϱ=λσℜ1−ℜ+λϱℜ1−ℜ2∗+ℜ2λϱℜ1−ℜ2∗1−ℜ1+ℜ1∗λσℜ1−ℜ11−ℜ2∗.
Therefore, we verify that system (1) is UHR stable concerning φ,η. If we take ε=1, it implies that system (1) is GUHR stable.


## 4. Ulam–Hyers–Rassias–Mittag-Leffler (UHRML) Stability

We reconsider inequalities (28)–(30). In the fuzzy concept, we have the following definitions.Definition 11.
(see [19]). Let βf≥0. System (1) is∗ Ulam–Hyers–Mittag-Leffler (UHML) stable if ∃λσ,ϱ=λσ,λϱ>0 such that for each ε=εσ,εϱ>0 and for each solution ϰ,j∈PCℐ,Ω×PCℐ,Ω of inequality (28) ∃ζ,℘∈PCℐ,Ω×PCℐ,Ω satisfying (1) with(57)D0ϰ,j,ζ,℘≤λσ,ϱεEσ,1βfzρ−aρρσ,z∈ℐ.∗ Generalized Ulam–Hyers–Mittag-Leffler (GUHML) stable if ∃φσ,ϱ=φσ,φϱ∈Cℐ,ℝ+ with φσ,ϱ0=0 such that for each ε=εσ,εϱ>0 and for each solution ϰ,j∈PCℐ,Ω×PCℐ,Ω of (28), ∃ζ,℘∈PCI,Ω×PCℐ,Ω satisfying (1) with(58)D0ϰ,j,ζ,℘≤φσ,ϱεEσ,1βfzρ−aρρσ,z∈ℐ.∗ Ulam–Hyers–Rassias–Mittag-Leffler (UHRML) stable with respect to φσ,ϱ,ησ,ϱ, ∃λσ,ϱ,φ>0 such that for each ε>0 and for each solution ϰ,j∈PCℐ,Ω×PCℐ,Ω of (30) ∃ζ,℘∈PCℐ,Ω×PCℐ,Ω satisfying (1) with(59)D0ϰ,j,ζ,℘≤λσ,ϱ,φεEσ,1βfzρ−aρρσφz+η,z∈ℐ.∗ Generalized Ulam–Hyers–Rassias–Mittag-Leffler (GUHRML) stable with respect to φσ,ϱ,ησ,β if ∃λσ,ϱ,φ>0 such that for each ε and for each solution ϰ,j∈PCℐ,Ω×PCI,Ω of (29) ∃ζ,℘∈PCℐ,Ω×PCℐ,Ω satisfying (1) with(60)D0ϰ,j,ζ,℘≤λσ,ϱ,φEσ,1βfzρ−aρρσφz+η,z∈ℐ.Theorem 5.
Letφ=φσ,φϱ∈Cℐ,ℝ be a nondecreasing function and hypothesis ð1−ð3 be satisfied, then for every ε>0, if a continuous function ϰ,j∈PCℐ,Ω×PCℐ,Ω satisfied in inequality (30) for all z∈ℐ, then there exists a unique solution ζ,℘∈PCℐ,Ω×PCℐ,Ω of system (1), and system (1) has UHRML stability. In addition, system (1) is GUHRML stable.Proof.
It is similar to the proof of Theorem4. In view of Theorem 4, we obtained(61)D0ϰ,j,ζ,℘≤λσ,ϱεφz+η.
Now, we take the RLGFI ona,T of the order 0<σ<1 on both sides of (61) and by ð3,(62)ρ1−σΓσ∫azωρ−1zρ−ωρσ−1D0ϰω,jω,ζω,℘ωdω≤ελσ,ϱρ1−σΓσ∫azωρ−1zρ−ωρσ−1φω+ηdω≤ελσ,ϱ1Γσ+1zρ−ωρρσ+λφη+φz.
It implies that(63)D0ϰz,jz,ζz,℘z+ρ1−σΓσ∫azωρ−1zρ−ωρσ−1D0ϰω,jω,ζω,℘ωdω≤ελσ,ϱ1Γσ+1zρ−ωρρσ+λφ+1η+φz.
Therefore, we get(64)D0ϰz,jz,ζz,℘z≤Pz+1Γσ∫azωρ−1zρ−ωρρσ−1D0ϰω,jω,ζω,℘ωdω,where Pz=ελσ,ϱ1/Γσ+1zρ−ωρ/ρσ+λφ+1η+φz. Set ℛ=1/Γσ and Sz=D0ϰω,jω,ζω,℘ω. Then, in (64), we have(65)Sz≤Pz+ℛz∫azωρ−1zρ−ωρρσ−1Sωdω.
Clearly,P is nondecreasing on ℐ. Then, by Theorem 3, we get(66)Sz≤PzEσ,1ℛzΓσzρ−ωρρσ.
Then, we have the following estimate:(67)D0ϰz,jz,ζz,℘z≤ελσ,ϱ1Γσ+1zρ−ωρρσ+λφ+1η+φzEσ,1×ℛzΓσzρ−ωρρσ,∀z∈ℐ.
Hence, system (1) has UHRML stability, where(68)λσ,ϱ,φ≔λσ,ϱ1Γσ+1zρ−ωρρσ+λφ+1,and βf:=1.
Furthermore, assumingε=1, we can conclude that system (1) is GUHRML stable.


## 5. Conclusion

In this work, a coupled system of fractional differential equations is modified by the Caputo–Katugampola fuzzy fractional derivative. Firstly, certain definitions are set out to use in the sequel. Then, we impose conditions for the existence and uniqueness of solutions of the proposed system by implementing Perove’s fixed point theorem. Finally, we have shown that the system has UHR and UHRML stabilities.



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