# Existence Results of Fuzzy Delay Impulsive Fractional Differential Equation by Fixed Point Theory Approach

Authors: Aziz Khan; Ramsha Shafqat; Azmat Ullah Khan Niazi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The main aim of this article is to study controllability and existence of solution of fuzzy delay impulsive fractional nonlocal integro-differential equation in the sense of Caputo operator. The existence and uniqueness of the solution have been carried out with the help of the Banach fixed point theorem. Moreover, for fuzzy fractional differential equations (FFDEs) driven by the Liu process, this present work introduced a concept of stability in credibility space. Finally, efficient examples are presented to demonstrate the main theoretical findings.


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## Body

## 1. Introduction

Fractional-order dynamical equations can be used to model a huge spectrum of physical processes in modern-world observations [1]. Due to its wide range application in various areas of sciences such as physics, chemistry, biology, electronics, thermal systems, electrical engineering, mechanics, signal processing, weapon systems, electrohydraulics, population modeling, robotics, and control, the concept of fuzzy sets continues to catch the attention of researchers [2]. As a result, in recent years, scholars have been increasingly interested in it. As a concept of describing a set with uncertain boundary, the fuzzy set was developed by Zadeh et al. [3]. The concept of possibility measure was studied by Zadeh [4] in 1978. Fuzzy set theory is a very useful technique for simulating uncertain problems. In fuzzy calculus, therefore, the concept of the fractional derivative is essential. Although the possibility measure provides the theoretical basis for the measurement of fuzzy events, it does not satisfy self-duality. Liu B. and Liu Y. [5] studied the concept of credibility measure in 2002, and a sufficient and necessary condition for credibility measure was derived by Li and Liu [6] in 2006. Fractional differential equations (FDEs) are differential equations with fractional derivatives. It is known from the research on fractional derivatives that they originate uniformly from major mathematical reasons. Different types of derivatives exist, such as Caputo and RL [7]. In 1965, Zadeh used the membership function to propose the concept of fuzzy sets for the first time. The FFDE is the most fascinating field. They are useful for understanding phenomena that have an underlying effect. Kwun et al. [8] and Lee et al. [9] investigated the solution of uniqueness-existence for FDEs. Controlled processes have been explored by several researchers. In the case of the fuzzy system, Kwun et al. [10] for the impulsive semilinear FDEs, controllability in n-dimension fuzzy vector space was demonstrated. Park et al. [11] controllability of semilinear fuzzy integro-differential equations with nonlocal conditions was investigated. Park et al. [12] established controllability of impulsive semilinear fuzzy integro-differential equations. Phu and Dung [13] studied stability analysis and controllability of fuzzy control set differential equations. According to Lee et al. [14], in the n-dimensional fuzzy space ENn of a nonlinear fuzzy control system, controllability with nonlocal initial conditions was examined.Balasubramaniam and Dauer [15] examined the controllability of stochastic systems in Hilbert space of quasilinear stochastic evolution equations, while Feng [16] explored the controllability of stochastic with control systems associated with time-variant coefficients. Arapostathis et al. [17] analyzed the controllability of stochastic differential systems of equations with linear-controlled diffusion affected by Lipschitz nonlinearity that is limited, smooth, and uniform. Stochastic differential equations given by Brownian motion are a well-known and well-studied area of modern mathematics. A new type of FDE was created using the Liu technique [18], which was described as follows:
(1)dXν=fXν,νdν+gXν,νdCν,whereCν denotes Liu operation and f and g are functions that have been assigned to it. This class of equations is solved using a fuzzy technique. For homogeneous FDEs, Chen and Qin [19] studied solutions of existence-uniqueness of few special FDEs. Liu [20] investigated an approximate method for solving unknown differential equations. Abbas et al. [21, 22] worked on a partial differential equation. Niazi et al. [23, 24], Iqbal et al. [25], Shafqat et al. [26], Abuasbeh et al. [27], and Alnahdi [28] existence-uniqueness of the FFEE were investigated. Arjunan et al. [29–32] worked on the fractional differential inclusions.Using conclusions of Liu [20], Jeong et al. [33] focused on exact controllability in credibility space for FDEs. Abstract FDEs’ complete controllability in credibility space is as follows:
(2)dxν,ϖ=Axν,ϖdν+fν,xν,ϖdCν+Buν,ν∈0,I,x0=x0.We used the Caputo derivative to prove controllability for the fuzzy delay impulsive fractional integro-evolution equation in credibility space with nonlocal condition; as a result of the above research,(3)D0Cνβuν,ζ=giν,uν+Auν,ζ+∫0νfν,uν,ζ,∫0sks,uν,ζdCν+BxνCxνdν,ν∈0,νi,i=1,2,⋯,N,u0=u0+hν1,ν2,⋯,νi,u.,whereU⊂EN and V⊂EN are two bounded spaces. EN is denoted for the set of numbers; all upper semicontinuously convex fuzzy on Rm, and Θ1,Pm,Cr, is the credibility space.The fuzzy coefficient is defined by the state functionu:0,I×Θ1,Pm,Cr⟶U. f:0,I×U⟶U is a fuzzy process. x:0,I×Θ1,Pm,Cr⟶V is regular fuzzy function, x:0,I×Θ1,Pm,Cr⟶V is control function, and B is linear bounded operator on V to U. The initial value is u0∈EN, and Cν denotes the Liu process.The goal of this work is to investigate the existence and stability of results to FDEs and the exact controllability driven by the Liu process, in order to deal with a fuzzy process. Some scholars discovered FDE results in the literature, although the vast majority of them were differential equations of the first order. We discovered the results for Caputo derivatives of order0,1 in our research. Stability, as a part of differential equation theory, is vital in both theory and application. As a result, stability is a key subject of study for researchers, and research papers on stability for FDE have been published in the last two decades, for example, essential conditions for solution stability and asymptotic stability of FDEs. We use fuzzy delay impulsive fractional integro-evolution equations with the nonlocal condition. The theory of fuzzy sets continues to gain scholars’ attention because of its huge range of applications in different fields of sciences such as engineering, robotics, mechanics, control, thermal systems, electrical, and signal processing.In Section2, we go over some basic notions relating to Liu’s processes and fuzzy sets. Section 3 demonstrates the existence of solutions of FDE and shows that FDE is precisely controllable. The concept of credibility stability for FDEs driven by the Liu process was developed in Section 4. Finally, in Section 5, several theorems for FDEs driven by the Liu process that is stable in credibility space were demonstrated.


## 2. Preliminary

IfMkRm be the family of all nonempty compact convex subsets of Rm, then addition and scalar multiplication are commonly defined as MkRm. Consider two nonempty bounded subsets of Rm, A1 and B1. The distance between A1 and B1 is measured using the Hausdorff metric as
(4)dAi,Bi=maxsupai∈Aiinfbi∈Biai−bi,supbi∈Biinfai∈Aiai−bi,where· indicates the usual Euclidean norm in Rm. It follows that MkRm,d is a separable and complete metric space [20]. Satisfy the below condition:
(5)Em=j:Rm⟶0,1jsatisfiesa−bbelow, where
(a)
j is normal; there exists an j0∈Rm such that jj0=1.(b)
j is fuzzy convex, such that is jλν+1−λs≥1.(c)
j is upper semicontinuous function on Rm, that is, jν0≥limk⟶∞jνk¯ for any νk∈Rmk=0,1,2,⋯,νk⟶ν0.(d)
j0=clu∈Rmjν>0 is compact.InRm [34], for 0<β<1, denote jβ=ν∈Rmuν≥β and u0 are nonempty compact convex sets. Then from (a) to (b), it concludes that β-level set jβν∈MkRm for all 0<β<1. Using Zadeh’s extension principle, we can have scalar multiplication and addition in fuzzy number space Em as follows:
(6)j⊕℘β=jβ⊕℘β,kjβ=k℘β,where j,℘∈Em,k∈Rm and 0<β<1. Assume EN denotes a set of all numbers upper semicontinuously convex fuzzy on Rm.Definition 1 (see [35]).
Given a complete metricDL by
(7)DLj,y=sup0<β<1dLjβ,℘β=sup0<β<1maxjlβ−℘lβ,jlβ−℘rβ,for any u,v∈EN, which satisfies DLj+z,℘+z=DLj,℘ for each z∈EN and jα=jlβ,urβ, for each β∈j,℘ where χlβ,urβ∈Rm with jlβ≤urβ.Definition 2 (see [36]).
The fractional derivative of RL is stated as(8)Daνλfν=ddνn+1∫aνν−τn−λfτdτ,wheren≤λ≤n+1.Definition 3 (see [37]).
The fractional derivatives in the sense of CaputoDaCνσfν of order α∈Rm+ are described by
(9)DaCνσfν=Daνσfν−∑k=0n−1fkak!ν−ak,
wheren=σ+1 for σ∉N0;n=σ for σ∈N0.Definition 4 (see [37]).
The Wright functionψσ is defined by
(10)ψσϖ=∑n=0∞−ϖnn!Γ−σn+1−σ=1π∑n=1∞−ϖnn−1!Γnσsinnπσ,where ϖ∈ℂ with 0<σ<1.Definition 5 (see [38]).
For anyj,℘∈C0,T,EN, metric H1χ,℘ on C0,T,EN is defined by
(11)H1j,℘=sup0≤ν≤TDLjν,℘ν.Consider thatΘ1 is a nonempty set and Pm denotes power set on Θ1. A case is a label given to each element of Pm. To present an axiomatic credibility, an idea based on the consideration of Ai will occur. To validate that the number CrAi is applied to each Ai event, representing the probability of Ai happens. We accept the four main axioms to ensure that the number CrAi has certain mathematical features that we predict:
(a)
Normality propertyCrΘ1=1,(b)
Monotonicity propertyCrAi≤CrBi, whenever Ai⊂Bi,(c)
Self-duality propertyCrAi+CrAic=1 for any event Ai,(d)
Maximality propertyCr∪iAi=supiCrAi for any events Ai with supiCrAi<0.5.Definition 6 (see [39]).
TakeΘ be the nonempty set, Pm be the power set of Θ1, and Cr be the credibility measure. After that, the triplet Θ1,Pm,Cr is assigned to the set of real numbers.Definition 7 (see [39]).
A fuzzy variable is a function that is generated from a set of real numbersΘ1,Pm,Cr to credibility space Θ1,Pm,Cr.Definition 8 (see [39]).
IfΘ1,Pm,Cr be credibility space and Θ1,Pm,Cr be an index set, a fuzzy process is a function that takes a set of real numbers and multiplies them by T×Θ1,Pm,Cr.It is a fuzzy method.uν,ζ is a two-variable function in which uν,ζ∗ represents a fuzzy variable for each ν∗. For each fixed ζ∗, the function uν,ζ is termed a sample path of fuzzy process. The fuzzy process uν,ζ is said to be sample continuous if sample ping is continuous for almost all ζ. Alternately of uν,ζ, we frequently use the notation uν.Definition 9 (see [39]).
Θ1,Pm,Cr is the symbol of a credibility space. The β-level set is applied for the fuzzy random variable uν in credibility space for each β∈0,1.
(12)uνβ=uνlβ,uνrβ,is defined by
(13)uνlβ=infuνβ=infa∈Rm;uνa≥β,uνrβ=supuνβ=infa∈Rm;uνa≥β,where uνlβ,uνrβ∈Rm with uνlβ≤uνrβ when β∈0,1.Definition 10 (see [5]).
Suppose thatϖ is a fuzzy variable and that r is a real number. Then, ϖ’s expected value is defined:
(14)Eϖ=∫0+∞Crϖ≥rdr−∫−∞0Crϖ≤rdr,if at least one of the integrals is finite.Lemma 11 (see [5]).
Ifϖ is a fuzzy vector, then the following are properties of expected value operator E:
(a)
Iff≤g, Efϖ≤Egϖ(b)
E−fϖ=−Efϖ(c)
Iff and g are comonotonic, we have for any nonnegative real numbers ai and bi,
(a)
(15)Eaifϖ+bigϖ=a1Efϖ+b1Egϖ,where fϖ and gϖ are fuzzy variables, respectively.Definition 12 (see [5]).
A fuzzy processCν is Liu process, if
(a)
C0=0,(b)
theCν has independent and stationary increments,(c)
any incrementCν+s−Cs is normally distributed fuzzy variable with expected value eν and variance ϕ2ν2, with membership function.(16)ξu=21+expπu−eν6ϕν−1,u∈RmThe parametersϕ and e represent the diffusion and drift coefficients, respectively. If e=0 and ϕ=1, the Liu process is standard.Definition 13 (see [40]).
Suppose thatCν is a standard Liu process and uν is a fuzzy process. The mesh is fixed as c=ν0<⋯<νn=d for any partition of the closed interval c,d with c=ν0<⋯<νn=d,
(17)Δ=max1≤i≤nνi−νi−1.
After that, the fuzzy integral ofuν with regard to Cν is calculated:
(18)∫cduνdCν=limΔ⟶0∑i=1nμνi−1Cνi−Cνi−1,determined by the limit exists almost positively and is a fuzzy variable.Lemma 14 (see [40]).
Consider thatCν represent the standard Liu process with Crζ>0, and the direction Cν is Lipschitz continuous, employing the below inequality:
(19)Cν1−Cν2<Kζν1−ν2,
whereKζ is Lipschitz, which is a fuzzy variable described by
(20)Kζ=sup0≤s≤νCν−Csν−s,Crζ>1,∞,otherwise,and EKp<∞ for all p>1.Lemma 15 (see [40]).
Assume thathν;c is a continuously differentiable function and that Cν is a standard Liu process. The function is defined as uν=hν;Cν. Then, there is the chain rule, which is as follows:
(21)duν=∂hν;Cν∂νdν+∂hν;Cν∂CdCν.Lemma 16 (see [40]).
The fuzzy integral inequality exists iffnu is a continuous fuzzy process:
(22)∫cdfνdCν≤K∫cdfνdν.
In Lemma14, the term K=Kζ is defined.


## 3. Existence of Solutions

This part applies the symboluν instead of the lengthy notation uν,ζ, as defined by Definition 8. The existence-uniqueness of solutions to FDE 1 x≡0 has been investigated.
(23)Dνβ0Cuν=giuν+Auν+∫0νfν,uν+∫0sKs,uνdCν,β∈0,1,u0=u0+hν1,ν2,⋯,νi,u.,∈EN,whereuν is state that includes values from the U⊂EN set of values. The set of all upper semicontinuously convex fuzzy numbers on Rm is called EN, credibility space is Θ1,Pm,Cr, fuzzy coefficient is A, and state function u:0,I×Θ1,Pm,Cr⟶U is fuzzy process, f:0,I×U⟶U is regular fuzzy function, Cν is standard Liu process, and u0∈EN is initial value.Lemma 17.
Ifuν is the solution of equation (3) for u0=u0+gν1,ν2,⋯,νp,u., then uν is given by
(24)uν=νβ−1u0+gν1,ν2,⋯,νp,u.+1q∫0νν−sβ−1gis,xsds+∫0νν−sβ−1Aus,ζ+∫0νfs,us,ζ,∫0sKs,us,ζdCs+BsCsds,holds, and then,
(25)uν=νβ−1Pβνu0+gν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,xsds+∫0νν−sβ−1Pβν−sAus,ζ+∫0νfs,us,ζ,∫0sKs,us,ζdCs+BsCsds,where
(26)Pqν=∫0∞qζMqζQνqζdζ.
Suppose that the statements below are correct:
(J1) For uν,vν∈C0,I×Θ1,Pm,Cr,U,ν∈0,I. There exist positive number m that is
(27)dLfν,uνβ,fν,vνβ≤mdLuνβ,vνβf0,X00≡0.
(J2) 2cmKI≤1. Because of Lemma 17, (23) has the solution uν. As a result, we establish in Theorem 18 that the solution to (23) is unique.Theorem 18.
For (u0+gν1,ν2,⋯,νp,u.∈EN, if (J1) and (J2) are hold, (23) has an unique solution uν∈C0,I×Θ1,Pm,Cr,U).Proof.
For allϖν∈C0,I×Θ1,Pm,Cr,U,ν∈0,I, define
(28)ϕϖν=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,ϖsds+∫0νν−sβ−1Pβν−sAϖs+∫0νfs,ϖs,∫0sKs,ϖsdCs+BsCsds.As a result, theϕϖ:0,I×Θ1,Pm,Cr⟶0,I×Θ1,Pm,Cr,U can be established as
(29)ϕ:C0,I×Θ1,Pm,Cr,U⟶C0,I×Θ1,Pm,Cr,U.For equation (23), ϕ is a fixed point which is likewise an obvious solution. ϖν,μν∈C0,I×Θ1,Pm,Cr,U, according to hypothesis (J1) and Lemma 16.
(30)dLϕϖνβ,ϕμνβ=dL∫0νν−sβ−1Pβν−sgis,ϖs+∫0νν−sβ−1Pβν−sAs,ϖs+fs,ϖs,∫0sKs,ϖsdCsβ,∫0νν−sβ−1Pβν−sgis,μs+∫0νν−sβ−1Pβν−sAμs+fs,μs,∫0sKs,μsdCsβ≤cmK∫0νdLθsβ,μsβds.Therefore, we obtain(31)DLϕϖν,ϕμν=supβ∈0,1dLϕϖνβ,ϕμνβ≤cmK∫0νsupβ∈0,1dLϖνβ,μνβds=cmK∫0νDLϖs,μsds.As a result, according to Lemma11, for a.s. ϖ∈Θ1,
(32)EH1ϕϖ,ϕμ=Esupν∈0,TDLϕϖν,ϕμν≤EcmKsupν∈0,I∫0νDLϖν,μν≤cmKIEH1ϖ,μ.A contraction mapping isϕ according to hypothesis (J2). The Banach fixed point theorem equation (23) has unique fixed point xν∈C0,I×Θ1,Pm,Cr,U.


### 3.1. Exact Controllability

In this section, we will study exact controllability for differential equation in the context of Caputo operator (3). We investigate a solution for equation (3) x in V⊂EN.
(33)ϕϖν=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+∫0νfs,us,∫0sKs,usdCs+BusCusds,u0=u0+hν1,ν2,⋯,νi,u.,whereSν continuous, such that S0=I=S′0 and Sν≤c,c>0,ν∈0,I. The term of controllability is defined for Caputo fuzzy differential equations.Definition 19.
Equation (3) is called a controllable on 0,I, if there is control uν∈V for every u0∈EN where the solution u of (3) satisfies the condition uν=u−1∈U, a.s. ζ, that is, uνβ=u1β.Given fuzzyG~:P~Rm⟶U mapping such that
(34)G~βv=∫0Tν−sβ−1Pβν−sBvsCvsds,℘⊂Γ¯j,0,otherwise,where Γ¯x is closure of support x and a nonempty fuzzy subset P~Rm of Rm.After that, there is aG~iβi=m,n,
(35)G~mβ℘m=∫0Tν−sq−1Pmβν−sB℘smC℘smds,℘sm∈℘smβ,℘s1,G~nβ℘n=∫0Tν−sq−1Pnβν−sB℘snC℘snds,℘sn∈℘s1,℘snβ.We assume thatG~mβ,G~nβ are bijective functions. A β-level set of xs can be presented as below:
(36)xsβ=xsmβ,xsnβ=G~mβ−1u1mβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.mβ−∫0νν−sβ−1Pβν−sgimβs,usds+∫0νν−sβ−1Pβν−sAus+∫0νfmβs,us,∫0sKmβs,usdCs+BmβusCmβusdsds,G~nβ−1−u1nβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ−∫0νν−sβ−1Pβν−sginβs,usds+∫0νν−sβ−1Pβν−sAus+∫0νfnβs,us,∫0sKnβs,usdCs+BnβusCnβusds.This expression is substituted into (33) to get the β-level of xν.
(37)uνβ=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+fs,us,∫0sKs,usdCs+BusCusdsβ=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.mβ+∫0νν−sβ−1Pβν−sgmiβs,usds+∫0νν−sβ−1Pβν−sAus+fmβs,us,∫0sKmβs,usdCs+∫0νν−sβ−1Pβν−sBG~mβ−1u1mβ−νβ−1Pβνu0+gν1,ν2,⋯,νp,u.mβ−∫0νν−sβ−1Pβν−sgmiβs,usds−∫0νν−sβ−1Pβν−sAus−fmβs,us,∫0sKmβs,usdCs−BusCusds,νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ+∫0νν−sβ−1Pβν−sgniβs,usds+∫0νν−sβ−1Pβν−sAus+fnβs,us,∫0sKnβs,usdCs+∫0νν−sβ−1Pβν−sBG~nβ−1u1nβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ−∫0νν−sβ−1Pβν−sgniβs,usds−∫0νν−sβ−1Pβν−sAus−fnβs,us,∫0sKnβs,usdCs−BusCusds=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.mβ+∫0νν−sβ−1Pβν−sgmiβs,usds+∫0νν−sβ−1Pβν−sAus+fmβs,us,∫0sKmβs,usdCs+∫0νν−sβ−1Pβν−sG~nβG~nβ−1u1mβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.mβ−∫0νν−sβ−1Pβν−sgmiβs,usds−∫0νν−sβ−1Pβν−sAus−fmβs,us,∫0sKmβs,usdCs−BusCusds,νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ+∫0νν−sβ−1Pβν−sgniβs,usds+∫0νν−sβ−1Pβν−sAus+fnβs,us,∫0sKnβs,usdCs+∫0νν−sβ−1Pβν−sG~nβG~nβ−1u1nβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ−∫0νν−sβ−1Pβν−sgniβs,usds−∫0νν−sβ−1Pβν−sAus−fnβs,us,∫0sKnβs,usdCsBusCusds=u1mβ,u1nβ=u1α.Hence, this controlxν satisfies uν=u1, a.s. ζ.We now set(38)ψuν=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+fs,us,∫0sKs,usdCs+∫0νν−sβ−1Pβν−sBG~−1u1−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.−∫0νν−sβ−1Pβν−sgis,usds−∫0νν−sβ−1Pβν−sAus−fs,us,∫0sKs,usdCs−BusCusds.Fuzzy mappingsG~−1 holds the above equation.
(39)dLψuνβ,ψvνβ=dLνβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+fs,us,∫0sKs,usdCs+∫0νν−sβ−1Pβν−sBG~−1u1−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.−∫0νν−sβ−1Pβν−sgis,usds−∫0νν−sβ−1Pβν−sAus−fs,us,∫0sKs,usdCs−BusCusdsβ,νβ−1Pβνv0+hν1,ν2,⋯,νp,v.+∫0νν−sβ−1Pβν−sgis,vsds+∫0νν−sβ−1Pβν−sAvs+fs,vs,∫0sKs,vsdCs+∫0νν−sβ−1Pβν−sBG~−1v1−νβ−1Pβνv0+hν1,ν2,⋯,νp,v.−∫0νν−sβ−1Pβν−sgis,vsds−∫0νν−sβ−1Pβν−sAvs−fs,vs,∫0sKs,vsdCs−BvsCvsds≤dL∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+fs,us,∫0sKs,usdCsβ,∫0νν−sβ−1Pβν−sgis,vsds+∫0νν−sβ−1Pβν−sAvs+fs,vs,∫0sKs,vsdCτsβ+dL∫0νν−sβ−1Pβν−sBG~−1×∫0νν−sβ−1Pβν−sgis,usds−∫0νν−sβ−1Pβν−sfs,us,∫0sKs,usdCτsds,∫0νν−sβ−1Pβν−sgis,vsds+∫0νν−sβ−1Pβν−sAvs+fs,vs,∫0sKs,vsdCsβ+dL∫0νν−sβ−1Pβν−sBG~−1×∫0νν−sβ−1Pβν−sgis,usds−∫0νν−sβ−1Pβν−sfs,us,∫0sKs,usdCτsdsβ≤cmK∫0νdLusβ,vsβds+dLG~G~−1∫0νν−sβ−1Pβν−sgis,usds+∫0νfs,us,∫0sKs,usdCsβG~G~−1∫0νν−sβ−1Pβν−sgis,vsds+∫0νfs,vs,∫0sKs,vsdCsβ≤cmK∫0νdLusβ,vsβds+cmK∫0νdLfs,usβ,fs,vsβds≤2cmK∫0νdLusβ,vsβds.Theorem 20.
If Lemma16 and hypotheses (J1) and (J2) are hold, then equation (3) is controllable on 0,I.Proof.
FromC0,I×Θ1,Pm,U to C0,I, we can clearly see that ψ is continuous. We have Lemma 16 and hypotheses (J1) and (J2) for any given ζ with Crζ>0,xν,℘ν∈C0,I×Θ1,Pm,Cr,U.Hence, by Lemma11,
(40)EH1ψu,ψv=Esupν∈0,IDLψuν,ψvν=Esupν∈0,Isup0<β≤1DLψuνβ,ψvνβds≤Esupν∈0,Isup0<β≤12cmK∫0νDLusβ,vsβds≤Esupν∈0,I2cmK∫0νDLus,vsds≤2cmKIFH1u,v.As a consequence,2cmKI<1 is a Ã‚ sufficient I. As a result, ψ stands for contraction. The Banach fixed point theorem is now being applied to show that (33) has a single fixed point. 0,I can be used to control (3).Example 1.
We investigate FFDE in credibility space:(41)Dνβ0Cuν,ζ=giν,uν+Auν,ζ+∫0νfν,uν,ζ+∫0sks,uν,ζdCν+BxνCxνdν,u0=u0+hν1,ν2,⋯,νi,u.,∈EN,,
where states consider values fromU⊂EN and space V⊂EN two bounded spaces. The set of all, upper semicontinuously convex, fuzzy numbers on Rm is EN and Θ1,Pm,Cr denotes credibility space.The state functionu:0,I×Θ1,Pm,Cr⟶U is fuzzy coefficient. Fuzzy process f:0,I×U⟶U. x:0,I×Θ1,Pm,Cr⟶V is a regular fuzzy function, x:0,I×Θ1,Pm,Cr⟶V is a control function, and B is a V to U linear bounded operator. u0∈EN is an initial value, and Cν is standard Liu process.Assumefν,uν=2~νuν,S−1ν=e−2~ν, defining wν=S−1νuν. Then, the equations of balance become
(42)uν=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,xsds+∫0νν−sβ−1Pβν−sAus,ζ+∫0νfs,us,ζ,∫0sKs,us,ζdCs+BsCsds,u0=u0+hν1,ν2,⋯,νi,u.∈EN.Therefore, Lemma17 is satisfied.2β=β+1,3−β is the β-level, set of fuzzy, number 2~, for all β∈0,1. β-level set of fν,uν is
(43)fν,uνβ=νβ+1uνmβ,3−βuνmβ.Further, we have(44)dLfν,uνβ,fν,vνβ=dLνβ+1uνmβ,3−βuνnβ,νβ+1mβ,3−βvνnβ=νmaxβ+1uνmβ−vνmβ,3−βuνnβ−vνnβ≤3Imaxuνmβ−vνmβ,uνnβ−vνnβ=mdLuνβ,vνβ,wherem=3I satisfies an inequality in the (J1) and (J2) hypotheses. All conditions given in Theorem 18 are fulfilled. Assume that 1~ is the initial value for u0. The plan set u1=2~. 1~ is 1~=β−1,1−β,β∈0,1 is β-level set of fuzzy numbers 1~. The xs of (41)’s β-level set is presented.
(45)xs=xsmβ,xsnβ=G~mβ−1β+1−SmβI−sβ−1−∫0ISmβI−ssβ+1usmβdCs,G~nβ−13−β−SnβI3−β−∫0ISnβI−ss3−βusnβdCs.This expression is then substituted into (42) to get the β-level of uν:
(46)uνβ=SmβIβ−1+∫0ISmβI−ssβ+1usmβdCs+∫0ISmβI−sBG~mβ−1β+1−SmβIβ−1−∫0ISmβI−ssβ+1usmβdCsds,SnβI1−β+∫0ISnβI−ss1−βusnβdCs+∫0ISnβI−sBG~nβ−13−β−SrβI1−β−∫0ISnβI−ss3−βusnβdCsds=β+1,3,−β=2~β.Following that, conditions in Theorem20 have been fulfilled. As a result, (41) on 0,T can be controlled.


## 3.1. Exact Controllability

In this section, we will study exact controllability for differential equation in the context of Caputo operator (3). We investigate a solution for equation (3) x in V⊂EN.
(33)ϕϖν=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+∫0νfs,us,∫0sKs,usdCs+BusCusds,u0=u0+hν1,ν2,⋯,νi,u.,whereSν continuous, such that S0=I=S′0 and Sν≤c,c>0,ν∈0,I. The term of controllability is defined for Caputo fuzzy differential equations.Definition 19.
Equation (3) is called a controllable on 0,I, if there is control uν∈V for every u0∈EN where the solution u of (3) satisfies the condition uν=u−1∈U, a.s. ζ, that is, uνβ=u1β.Given fuzzyG~:P~Rm⟶U mapping such that
(34)G~βv=∫0Tν−sβ−1Pβν−sBvsCvsds,℘⊂Γ¯j,0,otherwise,where Γ¯x is closure of support x and a nonempty fuzzy subset P~Rm of Rm.After that, there is aG~iβi=m,n,
(35)G~mβ℘m=∫0Tν−sq−1Pmβν−sB℘smC℘smds,℘sm∈℘smβ,℘s1,G~nβ℘n=∫0Tν−sq−1Pnβν−sB℘snC℘snds,℘sn∈℘s1,℘snβ.We assume thatG~mβ,G~nβ are bijective functions. A β-level set of xs can be presented as below:
(36)xsβ=xsmβ,xsnβ=G~mβ−1u1mβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.mβ−∫0νν−sβ−1Pβν−sgimβs,usds+∫0νν−sβ−1Pβν−sAus+∫0νfmβs,us,∫0sKmβs,usdCs+BmβusCmβusdsds,G~nβ−1−u1nβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ−∫0νν−sβ−1Pβν−sginβs,usds+∫0νν−sβ−1Pβν−sAus+∫0νfnβs,us,∫0sKnβs,usdCs+BnβusCnβusds.This expression is substituted into (33) to get the β-level of xν.
(37)uνβ=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+fs,us,∫0sKs,usdCs+BusCusdsβ=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.mβ+∫0νν−sβ−1Pβν−sgmiβs,usds+∫0νν−sβ−1Pβν−sAus+fmβs,us,∫0sKmβs,usdCs+∫0νν−sβ−1Pβν−sBG~mβ−1u1mβ−νβ−1Pβνu0+gν1,ν2,⋯,νp,u.mβ−∫0νν−sβ−1Pβν−sgmiβs,usds−∫0νν−sβ−1Pβν−sAus−fmβs,us,∫0sKmβs,usdCs−BusCusds,νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ+∫0νν−sβ−1Pβν−sgniβs,usds+∫0νν−sβ−1Pβν−sAus+fnβs,us,∫0sKnβs,usdCs+∫0νν−sβ−1Pβν−sBG~nβ−1u1nβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ−∫0νν−sβ−1Pβν−sgniβs,usds−∫0νν−sβ−1Pβν−sAus−fnβs,us,∫0sKnβs,usdCs−BusCusds=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.mβ+∫0νν−sβ−1Pβν−sgmiβs,usds+∫0νν−sβ−1Pβν−sAus+fmβs,us,∫0sKmβs,usdCs+∫0νν−sβ−1Pβν−sG~nβG~nβ−1u1mβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.mβ−∫0νν−sβ−1Pβν−sgmiβs,usds−∫0νν−sβ−1Pβν−sAus−fmβs,us,∫0sKmβs,usdCs−BusCusds,νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ+∫0νν−sβ−1Pβν−sgniβs,usds+∫0νν−sβ−1Pβν−sAus+fnβs,us,∫0sKnβs,usdCs+∫0νν−sβ−1Pβν−sG~nβG~nβ−1u1nβ−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.nβ−∫0νν−sβ−1Pβν−sgniβs,usds−∫0νν−sβ−1Pβν−sAus−fnβs,us,∫0sKnβs,usdCsBusCusds=u1mβ,u1nβ=u1α.Hence, this controlxν satisfies uν=u1, a.s. ζ.We now set(38)ψuν=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+fs,us,∫0sKs,usdCs+∫0νν−sβ−1Pβν−sBG~−1u1−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.−∫0νν−sβ−1Pβν−sgis,usds−∫0νν−sβ−1Pβν−sAus−fs,us,∫0sKs,usdCs−BusCusds.Fuzzy mappingsG~−1 holds the above equation.
(39)dLψuνβ,ψvνβ=dLνβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+fs,us,∫0sKs,usdCs+∫0νν−sβ−1Pβν−sBG~−1u1−νβ−1Pβνu0+hν1,ν2,⋯,νp,u.−∫0νν−sβ−1Pβν−sgis,usds−∫0νν−sβ−1Pβν−sAus−fs,us,∫0sKs,usdCs−BusCusdsβ,νβ−1Pβνv0+hν1,ν2,⋯,νp,v.+∫0νν−sβ−1Pβν−sgis,vsds+∫0νν−sβ−1Pβν−sAvs+fs,vs,∫0sKs,vsdCs+∫0νν−sβ−1Pβν−sBG~−1v1−νβ−1Pβνv0+hν1,ν2,⋯,νp,v.−∫0νν−sβ−1Pβν−sgis,vsds−∫0νν−sβ−1Pβν−sAvs−fs,vs,∫0sKs,vsdCs−BvsCvsds≤dL∫0νν−sβ−1Pβν−sgis,usds+∫0νν−sβ−1Pβν−sAus+fs,us,∫0sKs,usdCsβ,∫0νν−sβ−1Pβν−sgis,vsds+∫0νν−sβ−1Pβν−sAvs+fs,vs,∫0sKs,vsdCτsβ+dL∫0νν−sβ−1Pβν−sBG~−1×∫0νν−sβ−1Pβν−sgis,usds−∫0νν−sβ−1Pβν−sfs,us,∫0sKs,usdCτsds,∫0νν−sβ−1Pβν−sgis,vsds+∫0νν−sβ−1Pβν−sAvs+fs,vs,∫0sKs,vsdCsβ+dL∫0νν−sβ−1Pβν−sBG~−1×∫0νν−sβ−1Pβν−sgis,usds−∫0νν−sβ−1Pβν−sfs,us,∫0sKs,usdCτsdsβ≤cmK∫0νdLusβ,vsβds+dLG~G~−1∫0νν−sβ−1Pβν−sgis,usds+∫0νfs,us,∫0sKs,usdCsβG~G~−1∫0νν−sβ−1Pβν−sgis,vsds+∫0νfs,vs,∫0sKs,vsdCsβ≤cmK∫0νdLusβ,vsβds+cmK∫0νdLfs,usβ,fs,vsβds≤2cmK∫0νdLusβ,vsβds.Theorem 20.
If Lemma16 and hypotheses (J1) and (J2) are hold, then equation (3) is controllable on 0,I.Proof.
FromC0,I×Θ1,Pm,U to C0,I, we can clearly see that ψ is continuous. We have Lemma 16 and hypotheses (J1) and (J2) for any given ζ with Crζ>0,xν,℘ν∈C0,I×Θ1,Pm,Cr,U.Hence, by Lemma11,
(40)EH1ψu,ψv=Esupν∈0,IDLψuν,ψvν=Esupν∈0,Isup0<β≤1DLψuνβ,ψvνβds≤Esupν∈0,Isup0<β≤12cmK∫0νDLusβ,vsβds≤Esupν∈0,I2cmK∫0νDLus,vsds≤2cmKIFH1u,v.As a consequence,2cmKI<1 is a Ã‚ sufficient I. As a result, ψ stands for contraction. The Banach fixed point theorem is now being applied to show that (33) has a single fixed point. 0,I can be used to control (3).Example 1.
We investigate FFDE in credibility space:(41)Dνβ0Cuν,ζ=giν,uν+Auν,ζ+∫0νfν,uν,ζ+∫0sks,uν,ζdCν+BxνCxνdν,u0=u0+hν1,ν2,⋯,νi,u.,∈EN,,
where states consider values fromU⊂EN and space V⊂EN two bounded spaces. The set of all, upper semicontinuously convex, fuzzy numbers on Rm is EN and Θ1,Pm,Cr denotes credibility space.The state functionu:0,I×Θ1,Pm,Cr⟶U is fuzzy coefficient. Fuzzy process f:0,I×U⟶U. x:0,I×Θ1,Pm,Cr⟶V is a regular fuzzy function, x:0,I×Θ1,Pm,Cr⟶V is a control function, and B is a V to U linear bounded operator. u0∈EN is an initial value, and Cν is standard Liu process.Assumefν,uν=2~νuν,S−1ν=e−2~ν, defining wν=S−1νuν. Then, the equations of balance become
(42)uν=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,xsds+∫0νν−sβ−1Pβν−sAus,ζ+∫0νfs,us,ζ,∫0sKs,us,ζdCs+BsCsds,u0=u0+hν1,ν2,⋯,νi,u.∈EN.Therefore, Lemma17 is satisfied.2β=β+1,3−β is the β-level, set of fuzzy, number 2~, for all β∈0,1. β-level set of fν,uν is
(43)fν,uνβ=νβ+1uνmβ,3−βuνmβ.Further, we have(44)dLfν,uνβ,fν,vνβ=dLνβ+1uνmβ,3−βuνnβ,νβ+1mβ,3−βvνnβ=νmaxβ+1uνmβ−vνmβ,3−βuνnβ−vνnβ≤3Imaxuνmβ−vνmβ,uνnβ−vνnβ=mdLuνβ,vνβ,wherem=3I satisfies an inequality in the (J1) and (J2) hypotheses. All conditions given in Theorem 18 are fulfilled. Assume that 1~ is the initial value for u0. The plan set u1=2~. 1~ is 1~=β−1,1−β,β∈0,1 is β-level set of fuzzy numbers 1~. The xs of (41)’s β-level set is presented.
(45)xs=xsmβ,xsnβ=G~mβ−1β+1−SmβI−sβ−1−∫0ISmβI−ssβ+1usmβdCs,G~nβ−13−β−SnβI3−β−∫0ISnβI−ss3−βusnβdCs.This expression is then substituted into (42) to get the β-level of uν:
(46)uνβ=SmβIβ−1+∫0ISmβI−ssβ+1usmβdCs+∫0ISmβI−sBG~mβ−1β+1−SmβIβ−1−∫0ISmβI−ssβ+1usmβdCsds,SnβI1−β+∫0ISnβI−ss1−βusnβdCs+∫0ISnβI−sBG~nβ−13−β−SrβI1−β−∫0ISnβI−ss3−βusnβdCsds=β+1,3,−β=2~β.Following that, conditions in Theorem20 have been fulfilled. As a result, (41) on 0,T can be controlled.


## 4. Definition of Stability in Credibility

We shall provide a concept of credibility stability for FFDEs driven by the Liu process in this part.Definition 21.
The FDE 1 is said to be stability in credibility if for, any two, solutionsuν and vν corresponding to different initial values u0+hν1,ν2,⋯,νp,u. and v0+hν1,ν2,⋯,νp,v., we have
(47)limu0−v0⟶0Cruν−vν<ε=1,forallν≥0,where ε is any given number and ε>0.Example 2.
Take the FFDE to better understand the concept of credibility stability.(48)uν=νβ−1Pβνu0+hν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,xsds+∫0νν−sβ−1Pβν−sAus,ζ+∫0νfs,us,ζ,∫0sKs,us,ζdCs+BsCsds,vν=νβ−1Pβνv0+hν1,ν2,⋯,νp,v.+∫0νν−sβ−1Pβν−sgis,xsds+∫0νν−sβ−1Pβν−sAvs,ζ+∫0νfs,vs,ζ,∫0sKs,vs,ζdCs+BsCsds,
respectively. Then, we have(49)uν−vν=u0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v..
Deduce to, for any givenε>0, we always have
(50)limu0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.⟶0Cruν−vν<ε=limu0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.⟶0Cru0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.<ε=1,∀ν≥0.
As a result, the credibility of FFDE is stable.Definition 22.
Then-dimensional FDE 1 is called stable in credibility, if for any two solutions uν and vν corresponding to different initial values u0+hν1,ν2,⋯,νp,u. and v0+hν1,ν2,⋯,νp,v., we have
(51)limu0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.⟶0Cruν−vν<ε=1,∀ν≥0.Example 3.
Take anm-dimensional FFDE:
(52)D0Cνβuν,ζ=giν,uν+Auν,ζ+∫0νfν,uν,ζ,∫0sks,uν,ζdCν+BxνCxνdν.The two solutions corresponding to different initial values are(53)uν=νβ−1Pβνu0+gν1,ν2,⋯,νp,u.+∫0νν−sβ−1Pβν−sgis,xsds+∫0νν−sβ−1Pβν−sAus,ζ+∫0νfs,us,ζ,∫0sKs,us,ζdCs+BsCsdsvν=νβ−1Pβνv0+hν1,ν2,⋯,νp,v.+∫0νν−sβ−1Pβν−sgis,xsds+∫0νν−sβ−1Pβν−sAvs,ζ+∫0νfs,vs,ζ,∫0sKs,vs,ζdCs+BsCsds,respectively. Then, we have
(54)uν−vν=u0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v..As a result, we always have(55)limu0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.⟶0Cruν−vν<ε=limu0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.⟶0Cru0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.<ε=1,∀ν≥0.Thus,m-dimensional FFDE is stability in credibility.Note that some fuzzy differential equations driven by the Liu process are not stable in credibility. It will be demonstrated in the following example.


## 5. Theorems of Stability in Credibility

In this part, we will discuss the necessary criteria for a FFDE driven by the Liu process to achieve credibility stability.Theorem 23.
Assume the FFDE 1 for each initial value has a unique solution. Then, it is stable in credibility space, if coefficientsfν,u and gν,u satisfy strongly Lipschitz condition
(56)Dfν,u−fν,v+gν,u+gν,v≤LνDu−v,∀u,v∈Rm,ν≥0,for some integrable function Lν on 0,+∞.Proof.
Letuν and vν be two solutions corresponding to differential initial values u0+hν1,ν2,⋯,νp,u. and v0+hν1,ν2,⋯,νp,v., respectively. Then, for each ϑ∈Θ1,
(57)Duν−vν=Dfν,utdν−fν,vνdν+Dgν,uνdCν−gν,vνdCν=Dfν,uν−fν,vνdν+Dgν,uν−gν,vνdCν≤Dfν,uν−fν,vνdν+Dgν,uν−gν,vνdCν≤LνDuν−vνdν+DLtut−vνdCν≤LνDut−vνdν+DLνKϑuν−vνdν=Lt1+KϑDuν−vν,
whereKϑ is the Lipschitz constant of the Liu process. When we take integral on both sides of equation (57),
(58)Duν−vν≤Du0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.exp1+Kϑ∫0νLsds.For any givenε>0, we always have
(59)Cruν−vν<ε≥u0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.exp1+Kϑ∫0νLsds<ε.Since(60)Cru0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.exp1+Kϑ∫0νLsds<ε⟶1,asu0−v0⟶0, we obtain
(61)limu0+hν1,ν2,⋯,νp,u.−v0+hν1,ν2,⋯,νp,v.⟶0Cruν−vν<ε=1.Hence, the FFDE is stability in credibility. If it is not easy to determine whether or notfν,u and gν,u satisfy strong Lipschitz condition, the following corollary can be used to determine whether the FFDE is stable in credibility space.Corollary 24.
Assumefν,u and gν,u be bounded real value functions on 0,+∞. If fν,u and gν,u have derivatives with respect to u and satisfy
(62)fu′ν,u+gu′ν,u≤Lν,∀≥0,
for some integrable functionLν on 0,+∞, then FFDE 1 is stability in credibility.Proof.
For the bounded real valued functionsfν,u and gν,u,
(63)fν,u+gν,u<K1+u,
whereK is constant which satisfy fν,u+gν,u<K. We can derive from the mean value theorem that
(64)fν,u′−fν,u′′+gν,u′−gν,u′′=fu′ν,ξu′−u′′+gu′ν,ηu′−u′′≤Lνu′−u′′+Lνu′−u′′=2Lνu′−u′′,
whereξ,η∈u′−u existence-uniqueness theorem demonstrates that FFDE has a unique solution. We can deduce from Theorem 23 that FFDE is stable in credibility. Different from Theorem 23 and Corollary 24, we have below corollary when FFDE is general linear FFDE driven by the Liu process.Corollary 25.
Suppose thatu1ν, u2ν, v1ν, and v2ν are bounded functions, with respect to ν on 0,+∞. If u1ν and v1ν are integrable, on 0,+∞, then linear FDE driven by Liu process
(65)duν=u1νuν+u2νdν+v1νuν+v2νdCν,is stability in credibility.Proof.
For the linear FFDE 7, we havefν,x=u1νx+u2ν and gν,x=v1νx+v2ν, since
(66)u1νuν+u2ν+v1νvν+v2ν≤u1νuν+u2ν+v1νuν+v2ν<Kuν+K+Kuν+K=2Kuν+1,u1νuν+u2ν−u1νvν+u2ν+v1νuν+v2ν−v1νvν+v2ν=u1νuν−vν+v1νuν+vν≤u1νuν−vν+v1νuν+vν=u1ν+v1νuν−vν≤2Kuν−vν,
whereK is a constant which make u1ν<K,u2ν<K,v1ν<K,v2ν<K hold. The existence-uniqueness theorem shows that FDE 7 has a unique solution. Since Lν=u1ν+v1ν is integrable function on 0,+∞, from Theorem 23, the credibility of FFDE can be determined.According to Definition22, Theorem 23 can be used to n-dimensional FFDEs driven by the Liu process.Theorem 26.
Assume that each initial value of then-dimensional FFDE 1 has a unique solution. If coefficients fν,u and gν,u satisfy Lipschitz’s strong condition,then it is stable in credibility space:
(67)fν,u−fν,v+gν,u−gν,v≤Lνu−v,for∀u,v∈Rm,ν≥0,for some integrable function Lν on 0,+∞.


## 6. Conclusion

Accurate controllability for FFDEs can be used as a standard when analyzing controllability for semilinear integro-differential equations in the credibility space and fuzzy delay integro-differential equations. Therefore, the research’s theoretical conclusions can be applied to construct stochastic extensions on credibility space. The FFDEs driven by the Liu process have an important role in both theory and practice as a technique for dealing with dynamic systems in a fuzzy environment. There have been some proposed stability approaches for FFDEs driven by the Liu process up until now. This is a rewarding field with numerous research projects that can lead to a variety of applications and theories. We hope to learn more about fuzzy fractional evolution problems in future projects. We can discover uniqueness and existence with uncertainty using the Caputo derivative. Future work could include expanding on the mission concept, including observability, and generalizing other activities. This is an interesting area with a lot of study going on that could lead to a lot of different applications and theories. This is a path in which we intend to invest significant resources.



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*Source: 4123949-2022-08-31.xml*