Ulam–Hyers stability of Caputo-type fractional fuzzy stochastic differential equations with delay

Ulam-Hyers stability of Caputo-type fractional fuzzy stochastic differential equations with delays Danfeng Luoa, Xue Wanga, Tom´as Caraballob, Quanxin Zhuc,d,∗ aDepartment of Mathematics, Guizhou University, Guiyang 550025, China bDpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Seville, Spain cMOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China dKey Laboratory of Control and Optimization of Complex Systems, College of Hunan Province, Hunan Normal University, Changsha 410081, China Abstract In this paper, we explore a new kind of Caputo-type fractional fuzzy stochastic differential equations (FFSDEs) with delays. We establish the existence result of FFSDEs with delays by the method of upper and lower solutions, and then the uniqueness for this considered system is proved with the aid of the method of contradiction. Subsequently, we study the U-H stability with the help of H¨older’s inequality and Gr¨onwall-Bellman inequality. Finally, we demonstrate the validity of the proposed conclusions through three examples with numerical simulations. Keywords: Fuzzy stochastic differential equations, Fractional calculus, Ulam-Hyers stability, Existence and uniqueness 1. Introduction In 1695, L’Hospital sent a letter asking Leibniz what dny dxnmeant when n=1 2. We know that when nis a positive integer, dny dxnrepresents the n-th derivative of function y. The researchers of that era basically considered the case where nwas a positive integer, so Leibniz was puzzled by this sudden question. He ∗Corresponding author Email address: [email protected] (Quanxin Zhu) Preprint submitted to Elsevier October 10, 2022 said a lot of vague things in his reply to L’Hospital, and he actually did not answer L’Hospital’s question correctly at that time. This was the eleventh year of calculus, and Leibniz was understandably confused, and then the concept of fractional calculus (FC) was born. However, due to the limitations of cognition at that time and the lack of corresponding practical applications, the development of FC was relatively slow. Later Laplace, Abel, Liouville, Letnikov, Pincherle, Davis and many other mathematicians made many important contributions to the development of FC. As a result, the definitions of FC have become more and more perfect, and its properties have become clearer, and its applications have become more and more extensive. Fractional differential equations are widely used in many fields, such as genetic law, disease control, viscoelastic mechanics, and the number of biological populations, because it is more suitable to describe some complex systems with memory and hereditary properties, and readers can refer to [1, 2, 3, 4, 5, 6, 7, 8]. With the progress and development, the deterministic differential equations can no longer solve some problems in related fields. Many real phenomena in control theory, physics, biology, and economics are characterized by uncertainty. Most of these uncertainties have two cases, and one is stochastic uncertainty caused by random disturbance, and the other is fuzzy uncertainty caused by observation, experiment and maintenance. The method of probability theory is used in the analysis of random uncertainty, and the fuzzy set theory is used in the analysis of fuzzy uncertainty. In 1951, Itˆo combined probability theory and ordinary differential equations and published his book: On Stochastic D- ifferential Equations [9]. He strictly described stochastic differential equations in mathematical language, which provided an important theoretical basis for dealing with stochastic problems in differential equations. Later in order to consider the advantages of fractional differential equations, researchers extended stochastic differential equations to fractional stochastic differential equations. In 1965, Zadeh [10] proposed fuzzy set theory, which provided a theoretical basis for dealing with fuzzy uncertainty in systems, and scholars combined fuzzy set theory and ordinary differential equations theory to consider fuzzy differ- 2 ential equations. In 2010, Agarwal et al. studied fractional fuzzy differential equations (FFDEs) in [11]. Nowadays, FFSDEs have received extensive attention because they can be used in the study of many physical and engineering problems affected by both randomness and ambiguity. Since the existence and uniqueness of solutions to differential equations are the premise subject of studying other properties, they have basic significance. Therefore, the existence and uniqueness of solutions to the fuzzy stochastic differential equations (FSDEs) is also a focus to researchers. In [12], the author obtained the existence and uniqueness of solutions to FSDEs under bounded conditions and Lipschitz conditions that are weaker than the linear growth conditions. Kim [13] proved the existence of the one-dimensional FSDEs by Picard iteration under the Lipschitz conditions, and proved its uniqueness by contradictory method. Malinowski [14] used the same method to prove the existence and uniqueness of FSDEs, but Malinowski explored n-dimensional FSDEs. The papers [15, 16] also studied the existence and uniqueness to FSDEs. So far, many papers have studied the existence and uniqueness of the solutions to FS- DEs, but there are relatively few papers on FFSDEs. We can only find that Priyadharsini et al. [17] proved the existence and uniqueness of the solution to the fuzzy fractional stochastic Pantograph differential system by using the compression mapping principle. In 1940, Ulam [18] proposed a question about the stability of group homomorphism. The following year, Hyers [19] gave the first affirmative answer to this question. Since then, the mathematical community has produced U-H stability. With the further exploration of U-H stability by scholars such as Aoki, Rassias, Sahoo and Jung [20, 21, 22, 23, 24], the U-H stability theory has been gradually developed. Vu et al. made important contributions to the U-H stability of FFDEs, and they proved the U-H stability of different types of FFDEs or fractional fuzzy integral equations by successive approximation [25, 26, 27, 28]. Vu and Hoa studied µ-Ulam-Hyers-Rassias stability and Ulam-Hyers-Rassias- Mittag-Leffler stability of FFDEs using the fixed point theorem in [29], and similar results can be found in reference [30, 31, 32, 33, 34, 35, 36, 37], but there 3 are few papers to study the U-H stability of FSDEs and FFSDEs. Therefore, there are still many questions in this field waiting to be discovered and answered by researchers. Due to the relative lack of results on FFSDEs, and inspired by the above articles, this paper will study the existence, uniqueness and U-H stability of solutions to FFSDEs with delays:      CDβ 0+y(τ) = U(τ, y(τ),y(τ−ρ)) + V(τ, y(τ),y(τ−ρ))dW(τ) dτ , τ ∈[0, T], y(τ) = P(τ), τ ∈[−ρ, 0], (1.1) where CDβ 0+denotes the Caputo fractional derivative, 1 2< β < 1,the state vector y∈F(R) is the fuzzy stochastic process, U: [0, T ]×F(R)×F(R)→ F(R), V: [0, T]×F(R)×F(R)→Rare continuous functions with respect to y that are nondecreasing, W(·) is one-dimensional Brownian motion on a complete probability space (Ω, A, P ), P: [−ρ, 0] −→ F(R) is a continuous fuzzy function with P(0) = P0, and Ed2 ∞[P0,h0i]<∞. The innovations and difficulties of this paper have at least the following aspects: (i) Since there are few papers on FFSDEs at present for this issue, the methods for our reference are also not enough. Therefore, many problems need to be solved by referring to the methods of integer FSDEs and FFDEs. For example, when proving the existence of solutions to system (1.1), we refer to the method of proving FFDEs in article [38], and use the upper and lower solutions method to get the existence results of FFSDEs. (ii) Although the authors also studied the existence and uniqueness of solutions to FFSDEs in [17], the authors considered the granular difference, which is different to the situation that we consider the generalized Hukuhara difference. Since the properties of generalized Hukuhara difference is not as good as that of granular difference [39], we will encounter more difficulties in the proof process. 4 (iii) We study the U-H stability of FFSDEs, which firstly promotes the further development of U-H stability in fuzzy space, and secondly fills the gap of U-H stability theory in the field of FFSDEs. The remainder of this article is organized as follows: we review several definitions and lemmas related to fuzzy sets, fractional calculus, and stochastic differential equations and give an assumption that will be used throughout the paper in Section 2. The existence and uniqueness of solution to FFSDEs are investigated in Section 3. In Section 4, the U-H stability of solution to Eq. (1.1) is presented. In Section 5, we give three examples to illustrate the validity of the theoretical results. 2. Preliminaries Let K(R) be the family of all nonempty, compact and convex subsets of R, then the Hausdorff metric of ∀B,S∈K(R) is defined as (see [14]) D0(B,S) := max sup b∈B inf s∈Skb−sk,sup s∈S inf b∈Bkb−sk, where k·kis the norm in R, and (K(R), D0) is a complete metric space. Definition 2.1. [40] Let F(R)denote the fuzzy sets X:R→[0,1] such that [X]r∈K(R)for every r∈[0,1], and Xsatisfies (1) For 0≤a≤1and ∀τ1, τ2∈R,X(aτ1+ (1 −a)τ2)≥min {X(τ1),X(τ2)}, which means that Xis fuzzy convex in R; (2) There is τ0∈Rsuch that X(τ0) = 1,which means that Xis normal; (3) [X]0={τ0∈R|X(τ0)>0}is compact; (4) Xis upper semi-continuous on R. Here [X]ris the r-level set of Xand defines as [X]r=   {τ0∈R|X(τ0)≥r},∀0< r ≤1, {τ0∈R|X(τ0)>0}, r = 0. 5 Let h·i :R→F(R) denote the embedding of Rinto F(R), i.e. for X∈R we have hXi(τ) =      1,if τ=X, 0,if τ∈R\{X}. Definition 2.2. [40] The summation and scalar multiplication in F(R)are defined as [X1+X2]r= [X1]r+ [X2]r={τ1+τ2|τ1∈[X1]r, τ2∈[X2]r}; [k·X1]r=k[X1]r={kτ |τ∈[X]r},∀r∈[0,1], k ∈R, where [X]r= [X(r),X(r)] (∀r∈[0,1]) is a bounded, closed interval and the diameter of [X]ris defined by d([X]r) = X(r)−X(r). Definition 2.3. [41] Let X1,X2∈F(R). If there exists a X3∈F(R)such that X1=X2+X3, then X3is the Hukuhara difference of X1,X2, and we define X1X2=X3. Definition 2.4. [41] The generalized Hukuhara difference between X1∈F(R) and X2∈F(R)is given as: X1gH X2=X3⇔   (i) X1=X2+X3, d ([X1]r)≥d([X2]r), (ii) X2=X1+ (−1)X3, d ([X1]r)≤d([X2]r). Definition 2.5. [41] If for every r∈[0,1], the real function d([G(·)]r)is nondecreasing (nonincreasing) on [a, b], then the fuzzy function G: [a, b]→ F(R)is called d-increasing (d-decreasing) on [a, b]. If Gis d-increasing or d-decreasing on [a, b], then we say that Gis d-monotone on [a, b]. Definition 2.6. [14] For X1,X2∈F(R), we denote d∞[X1,X2] = sup 0≤r≤1 D0([X1]r,[X2]r), where d∞is a metric in F(R). It is known that F(R)is a complete metric space with respect to d∞. For H1,H2,H3,H4∈F(R), l ∈R, the following properties are hold: (1) d∞(H1+H3,H2+H3) = d∞(H1,H2); 6 (2) d∞(H1+H2,H3+H4)≤d∞(H1,H3) + d∞(H2,H4); (3) d∞(lH1, lH2) = |l|d∞(H1,H2); (4) d∞(H1H2,h0i) = d∞(H1,H2); (5) d∞(H1H2,H1H3) = d∞(H2,H3); (6) d∞(H1H2,H3H4)≤d∞(H1,H3) + d∞(H2,H4). Definition 2.7. [41] The Riemann-Liouville fractional integral of order β∈ (0,1] of the fuzzy function J∈L([0, b],F(R)) is defined by =β 0+J(s) = 1 Γ(β)Zs 0 (s−t)β−1J(t)dt, s > 0. Definition 2.8. [42] Let O,Q∈F(R). We say that OQ(OQ)if and only if O(r)≥Q(r)and O(r)≤Q(r)O(r)≤Q(r)and O(r)≥Q(r), that is, [O]r⊆[Q]r([O]r⊇[Q]r),∀r∈[0,1]. Definition 2.9. [41] The Caputo type fractional derivative of order β∈(0,1] of a d-monotone fuzzy function J∈AC([0, T],F(R)) is defined by CDβ 0+J(s) = 1 Γ(1 −β)Zs 0 (s−t)−βd dtJ(t)dt, where AC([0, T ],F(R)) represents the set of all absolute continuous fuzzy functions from [0, T ]to F(R). Lemma 2.1. [38] Let W,R∈AC([0, T],F(R)). If there exists a s1∈ (0, T]such that W(s1) = R(s1)and W(s)R(s)on [0, T ], then we have CDβ 0+W(s1)CDβ 0+R(s1). Definition 2.10. [41] A d-monotone fuzzy function y∈AC ([0, T],F(R)) is a solution of the Eq. (1.1) if and only if y∈AC ([0, T ],F(R)) satisfies the following fractional interval integral equation y(τ)gH P0=1 Γ(β)Zτ 0 (τ−η)β−1U(η, y(η),y(η−ρ))dη +1 Γ(β)Zτ 0 (τ−η)β−1V(η, y(η),y(η−ρ))dW(η), (2.1) 7 and the fuzzy function τ7→ =β 0+W(τ)is d-increasing on [0, T],where W(τ) = U(τ, y(τ),y(τ−ρ)) + V(τ, y(τ),y(τ−ρ))dW(τ) dτ . Remark 2.1. If y∈AC ([0, T ],F(R)) such that d([y(τ)]r)≥d([P0]r), then (2.1) can be rewritten as y(τ) =P0+1 Γ(β)Zτ 0 (τ−η)β−1U(η, y(η),y(η−ρ))dη +1 Γ(β)Zτ 0 (τ−η)β−1V(η, y(η),y(η−ρ))dW(η). (2.2) If y∈AC ([0, T],F(R)) such that d([y(τ)]r)≤d([P0]r), then (2.1) can be rewritten as y(τ) =P0(−1) 1 Γ(β)Zτ 0 (τ−η)β−1U(η, y(η),y(η−ρ))dη +1 Γ(β)Zτ 0 (τ−η)β−1V(η, y(η),y(η−ρ))dW(η). (2.3) Here we can represent (2.2) and (2.3) as y(τ) =P0Θ1 Γ(β)Zτ 0 (τ−η)β−1U(η, y(η),y(η−ρ))dη +1 Γ(β)Zτ 0 (τ−η)β−1V(η, y(η),y(η−ρ))dW(η), where Θ := {+,(−1)}. Definition 2.11. [38] A d-monotone fuzzy function yλ∈AC([−ρ, T ],F(R)) is said to be a lower solution for (1.1) if      CDβ 0+yλ(τ)4U(τ, yλ(τ),yλ(τ−ρ)) + V(τ, yλ(τ),yλ(τ−ρ))dW(τ) dτ , τ ∈[0, T ], yλ(τ)4P(τ), τ ∈[−ρ, 0]. (2.4) Ad-monotone upper solution yω∈AC([−ρ, T ],F(R)) for (1.1) is defined analogously by reversing the inequalities in (2.4). Lemma 2.2. [43] For T > 0,2β > 1,y(·)is a stochastic process satisfying ZT 0 d2 ∞[y(η),h0i]dt < ∞. Then we can obtain sup 0≤τ≤T Ed2 ∞Zτ 0 (τ−η)β−1y(η)dW(η),h0i≤T2β−1 2β−1EZT 0 d2 ∞[y(τ),h0i]dτ. 8 Assumption 1. For y,˜y,x,ex∈F(R), η ∈[−ρ, T], and suppose that there exist some constants N1, N2and N3such that (a1)d2 ∞[P(η),h0i]≤N1; (a2)d2 ∞[U(η, 0,0),h0i]∨d2 ∞[V(η, 0,0),h0i]≤N2; (a3)d2 ∞[U(η, y(η),˜y(η)),U(η, x(η),ex(η))]∨d2 ∞[V(η, y(η),˜y(η)),V(η, x(η),˜x(η))] ≤ N3d2 ∞[y(η),x(η)] + d2 ∞[˜y(η),ex(η)], where ∨is defined as: K1∨K2= max {K1,K2}. 3. Existence and uniqueness For convenience, we use the notation y(τ)P.1 =z(τ) instead of P(y(τ) = z(τ)) = 1,where τ∈[−ρ, T],yand zare the stochastic processes. For any xi,yi∈F(R), i = 1,2, . . . , p, the following inequality is true d2 ∞(x1+x2+· · · +xp,y1+y2+· · · +yp)≤p p X i=1 d2 ∞(xi,yi).(3.1) Theorem 3.1. If Assumption 1 and T2βN3<Γ2(β)(2β−1) 32 are true, then there exists a unique d-monotone solution y∈[yλ,yω]for Eq. (1.1) in AC ([−ρ, T ],F(R)). Proof. Step 1. We consider the sequence of continuous fuzzy functions {yn, n = 0,1,2, . . .} given by :    y0(τ) = P0, τ ∈[0, T ], y0(τ) = P(τ), τ ∈[−ρ, 0], and for n∈N+                yn(τ) =P0Θ1 Γ(β)Zτ 0 (τ−η)β−1U(η, yn−1(η),yn−1(η−ρ)) dη +1 Γ(β)Zτ 0 (τ−η)β−1V(η, yn−1(η),yn−1(η−ρ)) dW(η), τ ∈[0, T ], yn(τ) =P(τ), τ ∈[−ρ, 0], (3.2) 9 where yλ(τ), yω(τ) are the lower and upper solutions of the Eq. (1.1). Hence, we derive that yλ ε(τ)yλ(τ)y(τ)yω(τ)yω ε(τ), τ ∈[−ρ, 0], and yλ ε(0) y(0) yω ε(0), where y(τ) is a solution of the Eq. (1.1). Next, we need to prove that yλ ε(τ)≺y(τ)≺yω ε(τ), τ ∈[0, T]. If the above assertion does not hold, then there exists τ1∈(0, T ) such that y(τ1) = yλ ε(τ1),yλ ε(τ)≺y(τ)≺yω ε(τ), τ ∈[0, T]\ {τ1}.(3.3) Therefore, based on Lemma 2.2, we infer that CDβ 0+y(τ1)CDβ 0+yω ε(τ1),which yields CDβ 0+yω ε(τ1) = U(τ1,yω ε(τ1),yω ε(τ1−ρ)) + V(τ1,yω ε(τ1),yω ε(τ1−ρ))dW(τ1) dτ1 CDβ 0+y(τ1) =U(τ1,y(τ1),y(τ1−ρ)) + V(τ1,y(τ1),y(τ1−ρ))dW(τ1) dτ1 . Moreover, from y(τ)yω ε(τ), τ ∈[−ρ, 0] and (3.3), we can conclude that y(τ1+η)≺yω ε(τ1+η), η ∈[−ρ, 0]. Given the nondecreasing property of the functions U(τ, y(τ),y(τ−ρ)),V(τ, y(τ),y(τ− ρ)) in y, it is easy to get U(τ1,yω ε(τ1),yω ε(τ1−ρ)) + V(τ1,yω ε(τ1),yω ε(τ1−ρ))dW(τ1) dτ1 U(τ1,y(τ1),y(τ1−ρ)) + V(τ1,y(τ1),y(τ1−ρ))dW(τ1) dτ1 , which is a contradiction, and then we know that y(τ)≺yω ε(τ), τ ∈[0, T]. Similarly, it can be proved that yλ ε(τ)≺y(τ), τ ∈[0, T].Then yλ ε(τ)≺y(τ)≺ yω ε(τ), τ ∈[0, T] holds. Let ε→0, we deduce that yλ ε(τ)y(τ)yω ε(τ). 16 Step 3. We shall show the uniqueness of the solution. Suppose yand ˜y are two different solutions of (1.1). For all τ∈[−ρ, 0],˜y(τ) = P(τ). For all τ∈[0, T], by the same technique as Step 2, we have Ed2 ∞[y(τ),˜y(τ)] ≤2T2β−1 Γ2(β)(2β−1)EZT 0 d2 ∞[U(η, y(η),y(η−ρ)) ,U(η, ˜y(η),˜y(η−ρ))] dη +2T2β−1 Γ2(β)(2β−1)EZT 0 d2 ∞[V(η, y(η),y(η−ρ)) ,V(η),˜y(η),˜y(η−ρ))#dη ≤4T2β−1N3 Γ2(β)(2β−1)EZT 0d2 ∞[y(η), yn−1(η)] + d2 ∞[y(η−ρ),˜y(η−ρ)]dη ≤4T2β−1N3 Γ2(β)(2β−1) ZT 0 E2d2 ∞[y(η),˜y(η)] + sup 0≤η≤ρ d2 ∞[y(η−ρ),˜y(η−ρ)]dη =8T2β−1N3 Γ2(β)(2β−1) ZT 0 Ed2 ∞[y(η),˜y(η)] dη, which, with the aid of Gr¨onwall-Bellman inequality, yields Ed2 ∞[y(τ),˜y(τ)] P.1 = 0. Therefore, the solution of (1.1) is unique. The proof is complete. 4. Ulam-Hyers stability results Let ε > 0, we consider:      Ed2 ∞CDβ 0+h(τ),U(τ, h(τ),h(τ−ρ)) + V(τ, h(τ),h(τ−ρ))dW(τ) dτ ≤ε, τ ∈[0, T ], h(τ) = P(τ), τ ∈[−ρ, 0]. (4.1) Definition 4.1. ([28]) If there is a real number δ > 0, such that for ∀ε > 0and for each solution h∈AC ([−ρ, T ],F(R)) of the (4.1), there exists a solution y∈AC ([−ρ, T],F(R)) of Eq. (1.1) with Ed2 ∞[y(τ),h(τ)] ≤εδ, then the solution to Eq. (1.1) is U-H stable. 17 Remark 4.1. An d-monotone function h∈AC ([−ρ, T ],F(R)) is a solution of (4.1) if and only if there is g∈AC ([−ρ, T ],F(R)) such that (1) Ed2 ∞[g(τ),ˆ 0] ≤ε; (2)      CDβ 0+h(τ) = U(τ, h(τ),h(τ−ρ)) + V(τ, h(τ),h(τ−ρ))dW(τ) dτ +g(τ), τ ∈[0, T ], h(τ) = P(τ), τ ∈[−ρ, 0]. Theorem 4.1. Under the Assumption 1(a3), the solution to Eq. (1.1) is U-H stable. Proof. Taking remark 4.1, Definition 2.10 and remark 2.1 into account, we can derive h(τ) =P0Θ1 Γ(β)Zτ 0 (τ−η)β−1U(η, h(η),h(η−ρ))dη +1 Γ(β)Zτ 0 (τ−η)β−1g(η)dη +1 Γ(β)Zτ 0 (τ−η)β−1V(η, h(η),h(η−ρ))dW(η), and τ7→ =β 0+˜ W(τ) is d-increasing on [0, T],where ˜ W(τ) = U(τ, h(τ),h(τ− ρ)) + DV(τ, h(τ),h(τ−ρ))dW (τ) dτ E+g(τ) and Θ := {+,(−1)}. Then, employing inequality (3.1), Assumption 1, H¨older’s inequality and Lemma 2.2, we attain Ed2 ∞[y(τ),h(τ)] =Ed2 ∞1 Γ(β)Zτ 0 (τ−η)β−1U(η, y(η),y(η−ρ))dη +1 Γ(β)Zτ 0 (τ−η)β−1V(η, y(η),y(η−ρ))dW(η), 1 Γ(β)Zτ 0 (τ−η)β−1U(η, h(η),h(η−ρ))dη +1 Γ(β)Zτ 0 (τ−η)β−1g(η)dη +1 Γ(β)Zτ 0 (τ−η)β−1V(η, h(η),h(η−ρ))dη ≤3 Γ2(β)Ed2 ∞Zτ 0 (τ−η)β−1[U(η, h(η),h(η−ρ)),U(η, h(η),h(η−ρ))]dη 18 +3 Γ2(β)Ed2 ∞Zτ 0 (τ−η)β−1[g(η),h0i]dη +3 Γ2(β)Ed2 ∞Zτ 0 (τ−η)β−1[V(η, y(η),y(η−ρ)),V(η, h(η),h(η−ρ))]dW (η) ≤3 Γ2(β)Zτ 0 (τ−η)2β−2dηEZτ 0 d2 ∞[U(η, y(η, y(η−ρ)),U(η, h(η),h(η−ρ))]dη +3 Γ2(β)Zτ 0 (τ−η)2β−2dηEZτ 0 d2 ∞[g(η),h0i]dη +3 Γ2(β)Esup 0≤τ≤T d2 ∞Zτ 0 (τ−η)β−1[V(η, y(η),y(η−ρ)),V(η, h(η),h(η−ρ))]dW (η) ≤3T2β−1 Γ2(β)(2β−1)EZT 0 d2 ∞[U(η, y(η),y(η−ρ)),U(η, h(η),h(η−ρ))]dη +3T2β Γ2(β)(2β−1)ε +3T2β−1 Γ2(β)(2β−1)EZT 0 d2 ∞[V(η, y(η),y(η−ρ)),V(η, h(η),h(η−ρ))]dη ≤6T2β−1N3 Γ2(β)(2β−1)EZT 0d2 ∞[y(η),h(η)] + d2 ∞[y(η−ρ),h(η−ρ)]dη +3T2β Γ2(β)(2β−1)ε ≤6T2β−1N3 Γ2(β)(2β−1) ZT 0 E2d2 ∞[y(η),h(η)] + sup 0≤η≤ρ d2 ∞[y(η−ρ),h(η−ρ)]dη +3T2β Γ2(β)(2β−1)ε =6T2β−1N3 Γ2(β)(2β−1) ZT 0 E2d2 ∞[y(η),h(η)] + sup 0≤η≤ρ d2 ∞[P(η−ρ),P(η−ρ)]dη +3T2β Γ2(β)(2β−1)ε =12T2β−1N3 Γ2(β)(2β−1) ZT 0 Ed2 ∞[y(η),h(η)]dη +3T2β Γ2(β)(2β−1)ε. In terms of Gr¨onwall-Bellman inequality, we get Ed2 ∞[y(τ),h(τ)] ≤3T2βε Γ2(β)(2β−1) exp ZT 0 12T2β−1N3 Γ2(β)(2β−1)dη! =3T2β Γ2(β)(2β−1) exp 12T2βN3 Γ2(β)(2β−1)ε. 19 Let δ=3T2β Γ2(β)(2β−1) exp 12T2βN3 Γ2(β)(2β−1),we have Ed2 ∞[y(τ),h(τ)] ≤εδ. Therefore, this Theorem is proved. 5. Examples We present three examples to support the U-H stability theory, in the section. Example 1 Consider the following FFSDEs on [0,8]:              CD 2 3 0+y(τ) = 1 50y(τ) + 1 15y(τ−1) + (−4,0,4) +sin(τ+τ2) π3 dW(τ) dτ , τ ∈[0,8], y(τ) = −τ2−20,0,2τ−τ2+ 20, τ ∈[−1,0], (5.1) where U(τ, y(τ),y(τ−1)) = 1 50y(τ) + 1 15y(τ−1) + (−4,0,4), V(τ, y(τ),y(τ−1)) = sin(τ+τ2) π3. Figure 1: The U-H stability of the solution to Eq. (5.1) with ε= 0.3. 20 Figure 2: The d-monotone of the fuzzy function determined by Eq. (5.1). From Figure 1, we can see that the solution yof Eq. (5.1) is d-increasing and from Figure 2 we know that the fuzzy function Ois d-increasing, where the fuzzy function τ7→ O(τ) = =β 0+W(τ),and W(τ) = U(τ, y(τ),y(τ−1)) + V(τ, y(τ),y(τ−1))dW(τ) dτ . Also, it is easy to verify that for y,h∈F(R) d2 ∞[U(τ, y(τ),y(τ−1)),U(τ, h(τ),h(τ−1))] ≤3 152d2 ∞[y(τ),h(τ)] + d2 ∞[y(τ−1),h(τ−1)], d2 ∞[V(τ, y(τ),y(τ−1)),V(τ, h(τ),h(τ−1))] ≤0d2 ∞[y(τ),h(τ)] + d2 ∞[y(τ−1),h(τ−1)], then d2 ∞[U(τ, y(τ),y(τ−1)),U(τ, h(τ),h(τ−1))] ∨d2 ∞[V(τ, y(τ),y(τ−1)),V(τ, h(τ),h(τ−1))] ≤3 152d2 ∞[y(τ),h(τ)] + d2 ∞[y(τ−1),h(τ−1)]. Therefore, the condition of Theorem 4.1 holds, system (5.1) is U-H stable. And Figure 1 also shows that system (5.1) is U-H stable. Example 2 Consider the following FFSDEs:                CD 5 6 0+y(τ) =(−1) 3.5πsin(y(τ) + y(τ−1)) + (−1) 11 y(τ)+(−1) −1 3τ, 0,1 3τ +(−1) 20 sin(2τ)dW(τ) dτ , τ ∈[0,8], y(τ) = τ1 2−100,2τ1 2+ 30,3τ1 2+ 100, τ ∈[−1,0], (5.2) 21 where U(τ, y(τ),y(τ−1)) = (−1) 3.5πsin(y(τ) + y(τ−1)) + (−1) 11 y(τ)+(−1) −1 3τ, 0,1 3τ, V(τ, y(τ),y(τ−1)) = (−1) 20 sin(2τ). Figure 3: The U-H stability of the solution to Eq. (5.2) with ε= 0.5. Figure 4: The d-monotone of the fuzzy function determined by Eq. (5.2). It is easy to see from Figure 3 that the solution of system (5.2) is d-decreasing. And it is not difficult to find from Figure 4 that the fuzzy function Ois dincreasing, where O(τ) = =β 0+W(τ),and W(τ) = U(τ, y(τ),y(τ−1)) + V(τ, y(τ),y(τ−1))dW(τ) dτ . 22 For y,h∈F(R), we can easily verify that d2 ∞[U(τ, y(τ),y(τ−1)),U(τ, h(τ),h(τ−1))] ∨d2 ∞[V(τ, y(τ),y(τ−1)),V(τ, h(τ),h(τ−1))] ≤9 (3.5π)2d2 ∞[y(τ),h(τ)] + d2 ∞[y(τ−1),h(τ−1)]. Hence, from Theorem 4.1, Eq. (5.2) is U-H stable on [0, 8]. It is not difficult to see from Figure 3 that Eq. (5.2) is U-H stable. Example 3 Consider the following FFSDEs:                CD 2 3 0+y(τ) =(−1) 50 y(τ) + (−1) 20 y(τ−1) + (−1)(−4,0,4) +(−1) π4sin(3τ)dW(τ) dτ , τ ∈[0,8], y(τ) = −τ2−20,0, τ2+ 20, τ ∈[−1,0], (5.3) where U(τ, y(τ),y(τ−1)) = (−1) 50 y(τ) + (−1) 20 y(τ−1) + (−1)(−4,0,4), V(τ, y(τ),y(τ−1)) = (−1) π4sin(3τ). Figure 5: The U-H stability of the solution to Eq. (5.3) with ε= 0.3. 23 Figure 6: The d-monotone of the fuzzy function determined by Eq. (5.3). Figure 5 shows that the solution of Eq. (5.3) is d-monotone, and Figure 6 implies that Ois d-increasing, where O(τ) = =β 0+W(τ),and W(τ) = U(τ, y(τ),y(τ−1)) + V(τ, y(τ),y(τ−1))dW(τ) dτ . Obviously, for y,h∈F(R) we can verify that d2 ∞[U(τ, y(τ),y(τ−1)),U(τ, h(τ),h(τ−1))] ∨d2 ∞[V(τ, y(τ),y(τ−1)),V(τ, h(τ),h(τ−1))] ≤3 202d2 ∞[y(τ),h(τ)] + d2 ∞[y(τ−1),h(τ−1)], which means that Eq. (5.3) satisfies the conditions of Theorem 4.1, so Eq. (5.3) is U-H stable on [0, 8]. Similarly, we can also know from Figure 5 that Eq. (5.3) is U-H stable on [0, 8]. 6. Conclusion This article dealt with a FFSDEs with delays in the sense of generalized Hukuhara differentiability. And the existence of the solutions to FFSDEs was proved by the upper and lower solution method, and then its uniqueness was proved by means of contradictory method. Next, we explored the U-H stability of FFSDEs with the help of H¨older’s inequality and Gr¨onwall-Bellman inequality. Finally, we demonstrate the validity of the proposed theory through three examples with numerical simulations. 24 Acknowledgements The authors wishes to thank the editors and reviewers for their suggestions and comments which greatly improved the manuscript. This work was supported by the National Natural Science Foundation of China (62173139). References [1] K.S. Miller, B. 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