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

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


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 E N n 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: where C ν 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][30][31][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: 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, where Uð⊂E N Þ and Vð⊂E N Þ are two bounded spaces. E N is denoted for the set of numbers; all upper semicontinuously convex fuzzy on R m , and ðΘ 1 , P m , C r Þ, is the credibility space.
The fuzzy coefficient is defined by the state function u : ½0, I × ðΘ 1 , P m , C r Þ ⟶ U. f : ½0, I × U ⟶ U is a fuzzy process. x : ½0, I × ðΘ 1 , P m , C r Þ ⟶ V is regular fuzzy function, x : ½0, I × ðΘ 1 , P m , C r Þ ⟶ V is control function, and B is linear bounded operator on V to U. The initial value is u 0 ∈ E N , and C ν 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 order ð0, 1Þ 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 Section 2, 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.

Preliminary
If M k ðR m Þ be the family of all nonempty compact convex subsets of R m , then addition and scalar multiplication are commonly defined as M k ðR m Þ. Consider two nonempty bounded subsets of R m , A 1 and B 1 . The distance between A 1 and B 1 is measured using the Hausdorff metric as where k·k indicates the usual Euclidean norm in R m . It follows that ðM k ðR m Þ, dÞ is a separable and complete metric space [20]. Satisfy the below condition: where (a) j is normal; there exists an j 0 ∈ R m such that jðj 0 Þ = 1.
(c) j is upper semicontinuous function on R m , that is, j In R m [34], for 0 < β < 1, denote ½j β = fν ∈ R m juðνÞ ≥ βg and ½u 0 are nonempty compact convex sets. Then from (a) to (b), it concludes that β-level set ½j β ν ∈ M k ðR m Þ for all 0 < β < 1. Using Zadeh's extension principle, we can have scalar multiplication and addition in fuzzy number space E m as follows: where j, ℘ ∈ E m , k ∈ R m and 0 < β < 1. Assume E N denotes a set of all numbers upper semicontinuously convex fuzzy on R m .
Definition 5 (see [38]). For any j, Consider that Θ 1 is a nonempty set and P m denotes power set on Θ 1 . A case is a label given to each element of P m . To present an axiomatic credibility, an idea based on the consideration of A i will occur. To validate that the number C r fA i g is applied to each A i event, representing the probability of A i happens. We accept the four main axioms to ensure that the number C r fA i g has certain mathematical features that we predict: (a) Normality property C r fΘ 1 g = 1, (c) Self-duality property C r fA i g + C r fA c i g = 1 for any event A i , (d) Maximality property C r f∪ i A i g = sup i C r fA i g for any events fA i g with sup i C r fA i g < 0:5.
Definition 6 (see [39]). Take Θ be the nonempty set, P m be the power set of Θ 1 , and C r be the credibility measure. After that, the triplet ðΘ 1 , P m , C r Þ 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 numbers ðΘ 1 , P m , C r Þ to credibility space ðΘ 1 , P m , C r Þ.
Definition 8 (see [39]). If ðΘ 1 , P m , C r Þ be credibility space and ðΘ 1 , P m , C r Þ be an index set, a fuzzy process is a function that takes a set of real numbers and multiplies them by It is a fuzzy method. uðν, ζÞ is a two-variable function in which uðν, ζ * Þ represents a fuzzy variable for each ν * . For each fixed ζ * , the function uðν, ζÞ is termed a sample path of fuzzy process. The fuzzy process uðν, ζÞ is said to be sample continuous if sample ping is continuous for almost all ζ. Alternately of uðν, ζÞ, we frequently use the notation u ν .
Definition 9 (see [39]). ðΘ 1 , P m , C r Þ is the symbol of a credibility space. The β-level set is applied for the fuzzy random variable u ν in credibility space for each β ∈ ð0, 1Þ.
is defined by where Definition 10 (see [5]). Suppose that ϖ is a fuzzy variable and that r is a real number. Then, ϖ's expected value is defined: if at least one of the integrals is finite.
Lemma 11 (see [5]). If ϖ is a fuzzy vector, then the following are properties of expected value operator E: (c) If f and g are comonotonic, we have for any nonnegative real numbers a i and b i , where f ðϖÞ and gðϖÞ are fuzzy variables, respectively.
Definition 12 (see [5]). A fuzzy process C ν is Liu process, if (a) C 0 = 0, (b) the C ν has independent and stationary increments, (c) any increment C ν+s − C s is normally distributed fuzzy variable with expected value eν and variance ϕ 2 ν 2 , with membership function.
The parameters ϕ 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 that C ν is a standard Liu process and u ν is a fuzzy process. The mesh is fixed as c = ν 0 < ⋯ < ν n = d for any partition of the closed interval ½c, d with c = ν 0 < ⋯ < ν n = d, After that, the fuzzy integral of u ν with regard to C ν is calculated: determined by the limit exists almost positively and is a fuzzy variable.
Lemma 14 (see [40]). Consider that C ν represent the standard Liu process with C r fζg > 0, and the direction C ν is Lipschitz continuous, employing the below inequality: where KðζÞ is Lipschitz, which is a fuzzy variable described by and E½K p < ∞ for all p > 1.
Lemma 15 (see [40]). Assume that hðν ; cÞ is a continuously differentiable function and that C ν is a standard Liu process. The function is defined as u ν = hðν ; C ν Þ. Then, there is the chain rule, which is as follows: Lemma 16 (see [40]). The fuzzy integral inequality exists if f ðnuÞ is a continuous fuzzy process: In Lemma 14, the term K = KðζÞ is defined.

Existence of Solutions
This part applies the symbol u ν instead of the lengthy notation uðν, ζÞ, as defined by Definition 8. The existence-uniqueness of solutions to FDE 1 ðx ≡ 0Þ has been investigated.
where u ν is state that includes values from the Uð⊂E N Þ set of values. The set of all upper semicontinuously convex fuzzy numbers on R m is called E N , credibility space is ðΘ 1 , P m , C r Þ, fuzzy coefficient is A, and state function u : ½0, I × ðΘ 1 , P m , C r Þ ⟶ U is fuzzy process, f : ½0, I × U ⟶ U is regular fuzzy function, C ν is standard Liu process, and u 0 ∈ E N is initial value.

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 Vð⊂E N Þ.

Journal of Function Spaces
where SðνÞ continuous, such that Sð0Þ = I = S ′ ð0Þ and j SðνÞj ≤ c, c > 0, ν ∈ ½0, I. The term of controllability is defined for Caputo fuzzy differential equations. (3) is called a controllable on ½0, I, if there is control u ν ∈ V for every u 0 ∈ E N where the solution u of (3) satisfies the condition u ν = u −1 ∈ U, a.s. ζ, that is, ½u ν β = ½u 1 β .

Definition 19. Equation
Given fuzzyG :PðR m Þ ⟶ U mapping such that where Γ x is closure of support x and a nonempty fuzzy sub-setPðR m Þ of R m .
After that, there is aG We assume thatG β m ,G β n are bijective functions. A β-level set of x s can be presented as below: This expression is substituted into (33) to get the β-level of x ν .
Journal of Function Spaces Hence, this control x ν satisfies u ν = u 1 , a.s. ζ. We now set Fuzzy mappingsG −1 holds the above equation.

Journal of Function Spaces
As a consequence, ð2cmKIÞ < 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. ½0, I can be used to control (3).

Example 1. We investigate FFDE in credibility space:
where states consider values from Uð⊂E N Þ and space V ð⊂E N Þ two bounded spaces. The set of all, upper semicontinuously convex, fuzzy numbers on R m is E N and ðΘ 1 , P m , C r Þ denotes credibility space.
The state function u : ½0, I × ðΘ 1 , P m , C r Þ ⟶ U is fuzzy coefficient. Fuzzy process f : ½0, I × U ⟶ U. x : ½0, I × ðΘ 1 , P m , C r Þ ⟶ V is a regular fuzzy function, x : ½0, I × ðΘ 1 , P m , C r Þ ⟶ V is a control function, and B is a V to U linear bounded operator. u 0 ∈ E N is an initial value, and C ν is standard Liu process.
This expression is then substituted into (42) to get the β -level of u ν : Journal of Function Spaces Following that, conditions in Theorem 20 have been fulfilled. As a result, (41) on ½0, T can be controlled.

Definition of Stability in Credibility
We shall provide a concept of credibility stability for FFDEs driven by the Liu process in this part.