Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with Fractional Brownian Motions

&ere appears to be confusion of various kinds in the modeling of several real world systems, such as trying to characterize a physical system and opinions on its parameters. To deal with this ambiguity, the fuzzy set theory will be used [1]. It is able to handle such linguistic statements mathematically using this theory, such as “large” and “less.” &e capacity to investigate fuzzy differential equations (FDEs) in modeling numerous phenomena, including imprecision, is provided by a fuzzy set. In particular, the fuzzy stochastic differential equations (FSDEs), in instance, might be used to investigate a variety of economics and engineering problems that involve two types of uncertainty: randomness and fuzziness. &e fuzzy It 􏽢o stochastic integral was powered in [2, 3]. In [4, 5], the fuzzy stochastic integral is driven by the Wiener process as a fuzzy adapted stochastic process. In [6], Fei et al. studied the existence and uniqueness of solutions to the (FSDEs) under non-Lipschitzian condition. In [7], Jafari et al. study FSDEs driven by fBm. Jialu Zhu et al., in [8], prove existence of solutions to SDEs with fBm. Ding and Nieto [9] investigated analytical solutions of multitime-scale FSDEs driven by fBm. Vas’kovskii et al. [10] prove that the pth moments, p≥ 1, of strong solutions of a mixed-type SDEs are driven by a standard Brownian motion and a fBm. Despite the fact that some research exists on the problem of the uniqueness and existence of solutions to SDEs and FSDEs which are disturbed by Brownian motions or semimartingales [4, 11–15], a kind of the FFSDEs driven by an fBm has not been investigated. Agarwal et al. [16, 17] considered the concept of solution for FDEs with uncertainty and some results on FFDEs and optimal control nonlocal evolution equations. Recently, Zhou et al., in [18–20], gave some important works on the stability analysis of such SFDEs. Our results are inspired by the one in [21] where the existence and uniqueness results for the FSDEs with local martingales under the Lipschitzian conditions are studied. &e rest of this paper is given as follows. Section 2 summarizes the fundamental aspects. In Section 3, existence and uniqueness of solutions to the FFSDEs are proved. Moreover, the stability of solutions is studied in Section 4. Finally, in Section 5, a conclusion is given.


Introduction
ere appears to be confusion of various kinds in the modeling of several real world systems, such as trying to characterize a physical system and opinions on its parameters. To deal with this ambiguity, the fuzzy set theory will be used [1]. It is able to handle such linguistic statements mathematically using this theory, such as "large" and "less." e capacity to investigate fuzzy differential equations (FDEs) in modeling numerous phenomena, including imprecision, is provided by a fuzzy set. In particular, the fuzzy stochastic differential equations (FSDEs), in instance, might be used to investigate a variety of economics and engineering problems that involve two types of uncertainty: randomness and fuzziness. e fuzzy It o stochastic integral was powered in [2,3]. In [4,5], the fuzzy stochastic integral is driven by the Wiener process as a fuzzy adapted stochastic process. In [6], Fei et al. studied the existence and uniqueness of solutions to the (FSDEs) under non-Lipschitzian condition. In [7], Jafari et al. study FSDEs driven by fBm. Jialu Zhu et al., in [8], prove existence of solutions to SDEs with fBm. Ding and Nieto [9] investigated analytical solutions of multitime-scale FSDEs driven by fBm. Vas'kovskii et al. [10] prove that the pth moments, p ≥ 1, of strong solutions of a mixed-type SDEs are driven by a standard Brownian motion and a fBm.
Despite the fact that some research exists on the problem of the uniqueness and existence of solutions to SDEs and FSDEs which are disturbed by Brownian motions or semimartingales [4,[11][12][13][14][15], a kind of the FFSDEs driven by an fBm has not been investigated. Agarwal et al. [16,17] considered the concept of solution for FDEs with uncertainty and some results on FFDEs and optimal control nonlocal evolution equations. Recently, Zhou et al., in [18][19][20], gave some important works on the stability analysis of such SFDEs. Our results are inspired by the one in [21] where the existence and uniqueness results for the FSDEs with local martingales under the Lipschitzian conditions are studied. e rest of this paper is given as follows. Section 2 summarizes the fundamental aspects. In Section 3, existence and uniqueness of solutions to the FFSDEs are proved. Moreover, the stability of solutions is studied in Section 4. Finally, in Section 5, a conclusion is given.

Preliminaries
is part introduces the notations, definitions, and background information that will be utilized throughout the article.
Let K(R n ) be the family of nonempty convex and compact subsets of R n . In K(R n ), the distance d H is defined by (1) We denote by M(Ω, A; K(R n )) the family of A-measurable multifunction, taking value in K(R n ).
Definition 1 (see [21,22] (2) We denote by Let E n denote the set of the fuzzy x: Definition 2 (see [23]). Let f: [c, d] ⟶ E n ; the fuzzy Riemann-Liouville integral of f is given by Definition 3 (see [23]) e fuzzy fractional Caputo differentiability of f is given by Now, we define the Henry-Gronwall inequality [24], which can be used in the proof of our result.
Let L p (Ω, A, P; E n ) denote the set of all fuzzy random variables; they are L p -integrally bounded.
For the notion of an fBm, we referred to [25].
i )) converge to the same limit for all this sequences ψ m , m ∈ N , then this limit is said a Stratonovich-type stochastic integral and noted by Definition 5 (see [21,22]).
e function f: J × Ω ⟶ E n is said to be nonanticipating if it is A H t t∈J -adapted and measurable

Remark 2.
e process x is nonanticipating if and only if x is measurable with respect to N: Definition 7 (see [21,22]). A fuzzy process f: We denote by L p (J × Ω, N; E n ) the set of all L p -integrally bounded and nonanticipating fuzzy stochastic processes.

Main Result
Now, we investigate the FFSDEs driven by an fBm given by where and B H (s) s∈J is a fBm defined on (Ω, A, A H s s∈J , P) with Hirst index H ∈ (1/2, 1).

Definition 8.
A process x: J × Ω ⟶ E n is said to be a solution to equation (14) if the following holds: We will assume that all through this paper, f: Let the following assumptions be introduced.
(H2) For f(s, 0) and g, we have for every s ∈ J.
(H3) For all z, w ∈ E n ,

Advances in Fuzzy Systems
where c is equal to one in (H2).
Let us now introduce the main theorem in this part.
Proof. e method of successive approximations will be used to demonstrate the existence of a solution to (1). erefore, define a sequence x n : J × Ω ⟶ E n as follows: and for n � 1, . . ., It is clear that x n s are in L 2 (J × Ω, N; E n ) and d ∞ -continuous. Indeed, we have x 0 ∈ L 2 (J × Ω, N; E n ) and x 0 is d ∞ -continuous.