Random fractional differential equations with Riemann-Liouville-type fuzzy differentiability concept

1 Introduction

Fractional differential equations are used in modeling many physical and chemical processes and in engineering; see the monographs of Podlubny [53] and Kilbas et al. [24] and the papers [28, 29] and the references therein. In the recent years, the theory of fuzzy analysis and fuzzy differential equations has been proposed and developed to handle uncertainty due to incomplete information that appears in many mathematical or computer models of some deterministic real-world phenomena, and a wide number of applications of this theory have been considered in [14, 52], [42–48], [57–59] and the references therein. Recently, because of applications of fractional calculus and fractional differential equations in real-world systems, and because of the existence of uncertainties and disturbances in dynamic systems subject, the issue of fuzzy fractional calculus and fuzzy fractional differential equations has emerged as the significant subject and this new theory becomes very attractive to many scientists. The concept of Riemann-Liouville differentiability of fuzzy functions based on the Hukuhara differentiability was initiated by Agarwal et al. in [3–5] with some applications to fractional order initial value problem of fuzzy differential equation. By using the Hausdorff measure of non-compactness and under compactness type conditions, authors proved the existence of solution of fuzzy fractional integral equation. Following this direction the concepts of fuzzy fractional differentiability have been developed and extended in some papers to investigate some results on the existence and uniqueness of solutions to fuzzy differential equations, and in a wide number of applications of this theory have been considered (see for instance [1, 60–63], and the references therein). These methods are based on the assumption of Hukuhara differentiability or generalized Hukuhara differentiability (see [14]). Specifically, Arshad and Lupulescu in [13] established some results on the existence and uniqueness of solutions to fuzzy fractional differential equations under Hukuhara fractional Riemann-Liouville differentiability. Khastan, Nieto and Rodríguez-López [23] also established the existence of solution for a fuzzy fractional initial value problem under Riemann-Liouville Hukuhara differentiability by using the Schauder fixed point. Allahviranloo et al. [8, 9] studied the concepts of generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions. Salahshour et al. [60] proposed some new results on existence and uniqueness of solutions of fuzzy fractional differential equation. In [21, 63], the authors studied the existence results for extremal solutions of interval fractional functional differential and integro-differential equations by using the monotone iterative technique combined with the method of upper and lower solutions. Long et al. [32–36] used the concepts of generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions to investigate the existence and uniqueness of the solutions to fractional partial differential equations with uncertainty and the stability properties of the solutions were presented. For solving initial value problems of fuzzy fractional differential equations, Mazandarani et. al. [40] studied the solution to fuzzy fractional initial value problem under Caputo generalized Hukuhara differentiability using a modified fractional Euler method. The authors in [41] discussed the existence and uniqueness of solution to fuzzy fractional differential equation under a Caputo type-2 fuzzy fractional derivative and presented a definition of the Laplace transform of type-2 fuzzy number-valued functions for solving fuzzy fractional differential equations with initial value conditions. By using some recent results of fixed point of weakly contractive mappings on the partially ordered space, the existence and uniqueness of solution for interval fractional integro-differential equations in the setting of the Caputo generalized Hukuhara fractional differentiability are studied in [11, 12]. In addition, authors proposed a new technique to find the exact solutions of initial value problems with fractional order by using the solutions of initial value problems with integer order. Hoa [16, 19] presented some existence and uniqueness results of solutions to fuzzy fractional differential equations with delay and the modi?ed fractional Euler method is investigated for solving the given problems. Furthermore, in [20] author proposed the modified fractional Euler method and the modified Adams-Bashforth-Moulton method for solving initial value problems of interval-valued differential equations with Caputo generalized Hukuhara differentiability. Some other methods for solving solutions of initial value problems of fuzzy fractional differential equations based on operational matrix of shifted Chebyshev polynomials, the spectral tau method and fuzzy Laplace transforms were proposed in [1, 2] by Ahmadian and in [10] by Allahviranloo et al.

Initial value problems for random fuzzy differential equations consider the phenomena of fuzziness and also randomness. Fuzzy set random variables were introduced by Puri and Ralescu [56] and authors proposed the concept of differentiability by the Hukuhara difference in [54]. Besides, the concept of a fuzzy random variable was proposed by Kwakernaak [27] and used by Kruse and Meyer [26]. The concepts of measurability of fuzzy mappings were given in [25, 56] and the relations of different concepts of measurability for fuzzy random variables were discussed in the investigations of Colubi et al. [15], Terán Agraz [6], Puri and Ralescu [54]. In the current years, applications of the fuzzy differential equations in modeling dynamical systems with imprecise, incomplete, vague and imperfect properties are considered. Later, this problem has been extended to the random fuzzy differential equation (RFDE) which combines two different kinds of uncertainties, i.e. fuzziness and randomness in order to describe the uncertain dynamical systems subjected to random forces. In [42], Malinowski considered random fuzzy differential equations with the fuzzy Hukuhara derivative and author [43, 44] proposed two different kinds of solutions to RFDE with respect to the geometrical properties, and under Lipschitz type conditions the existence and uniqueness results of solution for RFDE using the method of successive approximations were proved. In [46–48], RFDE was extended to stochastic fuzzy differential equations and for other results on random fuzzy equations with fractional order we refer the reader to [45, 62].

Currently, random fractional differential equations under Riemann-Liouville-type fractional derivative have not been studied. Therefore, in this paper, by using the definition of fuzzy random variables which was introduced by Puri and Ralescu [55] we investigate an initial value problem for random fuzzy fractional differential equations with Riemann-Liouville derivative. Since one of interesting subjects in this area, is the investigation of the existence and uniqueness of solutions and the solving methods of the above problems, in this paper our aim is to

show the equivalence between a random fuzzy fractional differential equation and a random fuzzy fractional integral equation under suitable conditions.

In Section 2, we present some fundamental theorems of fuzzy-valued fractional analysis. In Section 3, we present the existence and uniqueness of solution to an initial value problem of random fuzzy fractional differential equations with Riemann-Liouville generalized Hukuhara differentiability.

2 Preliminaries

We recall some preliminaries about the fuzzy numbers mentioned in [30]. A fuzzy number on ℝ is a mapping u : ℝ → [ 0 , 1 ] , where the value u (z₀) denotes the degree of membership of the element z₀ to the fuzzy number u. For r ∈ (0, 1], the r-level of u is defined by the set [ u ] r = { z ∈ ℝ | u ( z ) ≥ r } = [ u _ ( r ) , u ¯ ( r ) ] . For r = 0, the support of u is defined as the set [ u ] 0 = { z ∈ ℝ | u ( z ) ≥ 0 } ¯ . From now on, let us denote by E the collection of fuzzy numbers u on ℝ satisfying:

u is normal, that is, there exists z 0 ∈ ℝ such that u (z₀) =1;

For addition and scalar multiplication in fuzzy set space E, we have [u₁ + u₂]  ^r  = [u₁]  ^r  + [u₂]  ^r , [λu₁]  ^r  = λ [u₁]  ^r  . For u ∈ E, the diameter of the r-level set of u is defined as diam [ u ] r = u ¯ ( r ) - u _ ( r ) . The Hausdorff distance between fuzzy numbers is given by D 0 [ u 1 , u 2 ] = sup 0 ≤ r ≤ 1 { | u _ 1 ( r ) - u _ 2 ( r ) | , | u ¯ 1 ( r ) - u ¯ 2 ( r ) | } , ∀ u 1 , u 2 ∈ E .

The couple (E, D₀) is a complete metric space. The following properties of the metric D₀ are valid (see [30]): D₀ [u₁ + u₃, u₂ + u₃] = D₀ [u₁, u₂] , D₀ [λu₁, λu₂] = |λ|D₀[u₁, u₂] , D₀ [u₁, u₂]  ≤ D₀ [u₁, u₃]  + D₀ [u₃, u₂] , for all u₁, u₂, u₃ ∈ E and λ ∈ ℝ . Let u₁, u₂ ∈ E. An element u₃ is called the Hukuhara difference (H-difference for short) of u₁ and u₂, if it verifies the equation u₁ = u₂ + u₃. If the H-difference there exists, it will be denoted by u₁ ⊖ u₂ . Let us remark that u₁ ⊖ u₂ ≠ u₁ + (-1) u₂.

Definition 2.1. [14] The generalized Hukuhara difference of two fuzzy numbers u₁, u₂ ∈ E (gH-difference for short) is defined as follows u 1 ⊖ gH u 2 = u 3 ⇔ { ( i ) u 1 = u 2 + u 3 , or ( ii ) u 2 = u 1 + ( - 1 ) u 3 .

A function x : [a, b] → E is called d-increasing (d-decreasing) on [a, b] if for every r ∈ [0, 1] the function t ↦ diam [x (t)]  ^r is nondecreasing (nonincreasing) on [a, b]. If x is d-increasing or d-decreasing on [a, b], then we say that x is d-monotone on [a, b].

Definition 2.2. [14] Let x : (a, b) → E and t ∈ (a, b). The fuzzy function x is said generalized Hukuhara differentiable (gH-differentiable) at t, if there exists an element D _gH x (t) ∈ E such that

D gH x ( t ) = lim l → 0 x ( t + l ) ⊖ gH x ( t ) l .(2.1)

Denote by C ([a, b] , E) the set of all continuous fuzzy functions on the interval [a, b] with values in E . Let L¹ ([a, b] , E) be the set of all fuzzy functions x : [a, b] → E such that the functions t ↦ D 0 [ x ( t ) , 0 ˆ ] belongs to L¹ [a, b].

Riemann-Liouville fractional integral of real function. The Riemann-Liouville fractional integral of order α ∈ (0, 1) for a real function ψ ∈ L¹ [a, b] is defined by (see [24]) ψ α ( t ) : = ( I a + α φ ) ( t ) = 1 Γ ( α ) ∫ a t ( t - s ) α - 1 ψ ( s ) ds , for t > a .

The Riemann-Liouville derivative of order α ∈ (0, 1) for a real function ψ ∈ AC [a, b] is defined by (see [24]) ( D a + α ψ ) ( t ) = d dt I a + 1 - α ψ ( t ) = 1 Γ ( 1 - α ) d dt ∫ a t ( t - s ) - α ψ ( s ) ds , t > a .

In addition, we have the following property (see [24]) I a + α D a + α ψ ( t ) = ψ ( t ) - ( t - a ) α - 1 Γ ( α ) ψ 1 - α ( a ) , t ∈ ( a , b ] .(2.2)

Definition 2.3. Let x ∈ L¹ ([a, b] , E). The Riemann-Liouville fractional integral of order α of the fuzzy-valued function x is defined as follows: ( ℑ a + α x ) ( t ) = 1 Γ ( α ) ∫ a t ( t - s ) α - 1 x ( s ) ds , t > a , where Γ (α) is the well-known Gamma function.

For a given fuzzy function x ∈ L¹ ([a, b] , E) and α ∈ (0, 1), we define the fuzzy function x_1-α : [a, b] → E by x 1 - α ( t ) : = ( ℑ a + 1 - α x ) ( t ) = 1 Γ ( 1 - α ) ∫ a t ( t - s ) - α x ( s ) ds , for t ∈ ( a , b ] .

If the gH-derivative D _gH x_1-α (t) exists for t ∈ (a, b], then D _gH x_1-α (t) is called the fuzzy Riemann-Liouville generalized Hukuhara fractional derivative (or Riemann-Liouville gH-fractional derivative) of order α ∈ (0, 1). The Riemann-Liouville gH-fractional derivative of x will be denoted by RL D a + α x . Therefore, ( RL D a + α x ) ( t ) : = D gH ( ℑ a + 1 - α x ) ( t ) , t ∈ ( a , b ] .

Proposition 2.1. [17] If x ∈ L¹ ([a, b] , E), then ( RL D a + α ℑ a + α x ) ( t ) = x ( t ) , t ∈ ( a , b ] .(2.3)

Remark 2.1. (see [9]) Let x ∈ AC ([a, b] , E). Then:

If either x is d-increasing for all t ∈ [a, b] or x is d-increasing and x_1-α is d-increasing on [a, b], then [ ( RL D a + α x ) ( t ) ] r = [ D a + α x _ ( t , r ) , D a + α x ¯ ( t , r ) ] , r ∈ [ 0 , 1 ] .

Let ( Ω , F , ℙ ) be a complete probability space. A mapping x : Ω → E is said to be a fuzzy random variable, if for every r ∈ [0, 1] { ω ∈ Ω | [ x ( ω ) ] α ∩ B ≠ ∅ } ∈ F for every closed set B ⊂ ℝ .

A function x : [a, b] × Ω → E is called a fuzzy stochastic process, if x (t, ·) is a fuzzy random variable for any fixed t ∈ [a, b], and x (· , ω) is a fuzzy function with any fixed ω ∈ Ω. In [42], the x (· , ω) function is called a trajectory. A mapping x : [a, b] × Ω → E is called continuous, if for almost all ω ∈ Ω the trajectory x (· , ω) is a continuous function on [a, b]. Next, in this paper we use the notations x ( ω ) = ℙ . 1 y ( ω ) and x ( ω ) ≤ ℙ . 1 y ( ω ) instead of ℙ ( { ω ∈ Ω | x ( ω ) = y ( ω ) } ) = 1 and ℙ ( { ω ∈ Ω | x ( ω ) ≤ y ( ω ) } ) = 1 , respectively. Also the facts that ℙ ( { ω ∈ Ω | x ( t , ω ) = y ( t , ω ) , ∀ t ∈ [ a , b ] } ) = 1 and ℙ ( { ω ∈ Ω | x ( t , ω ) ≤ y ( t , ω ) , ∀ t ∈ [ a , b ] } ) = 1 will be denoted by x ( t , ω ) = [ a , b ] , ℙ . 1 y ( t , ω ) and x ( t , ω ) ≤ [ a , b ] , ℙ . 1 y ( t , ω ) , respectively.

In the below section, we prove the existence and uniqueness theorems for a solution to an initial value problem for random fuzzy fractional differential equations. We assume that f : Ω × I : = [ a , b ] ⊂ ℝ + × E → E satisfies the following assumptions: (A1) f_· (t, u) : Ω → E is a fuzzy random variable for (t, u) ∈ I × E and f _ω  (· , u) : I → E is measurable for any u ∈ E, (A2) with ℙ . 1 the mapping f _ω  (· , ·) : I × E → E belongs to L¹ ([a, b] , E), i.e., there exists Ω₀ ⊂ Ω with ℙ ( Ω 0 ) = 1 such that for every ω ∈ Ω₀ the real function t ↦ ∥ f ω ( t , x ) ∥ : = D 0 [ f ω ( t , x ) , 0 ˆ ] belongs to L¹ [a, b] . (A3) with ℙ . 1 the mapping f _ω  (· , ·) : [a, b] × E → E is a continuous fuzzy mapping at every point (t₀, u₀) ∈ I × E .

For α ∈ (0, 1), we consider the following initial-type problem:

{ RL D a + α x ( t , ω ) = [ a , b ] , ℙ . 1 f ω ( t , x ( t , ω ) ) , RL D a + α - 1 x ( a , ω ) = ℙ . 1 x 0 ( ω ) ,(3.1) where RL D a + α is the Riemann-Liouville’s generalized Hukuhara derivative, f : Ω × [a, b] × E → E is a fuzzy stochastic function. A function x : [a, b] × Ω → E is said to be a solution of (3.1) if x is a continuous fuzzy stochastic process, RL D a + α - 1 x ( a , ω ) = ℙ . 1 x 0 ( ω ) and RL D a + α x ( t , ω ) = [ a , b ] , ℙ . 1 f ω ( t , x ) , t ∈ [a, b] . A solution x of (3.1) is said to be d-monotone on [a, b] if it is d-increasing or d-decreasing on [a, b] for almost all ω ∈ Ω (or ℙ - a.a. ω .)

Consider the following random fuzzy fractional integral equation x ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) = ℙ . 1 1 Γ ( α ) ∫ a t ( t - s ) α - 1 f ω ( s , x ( s , ω ) ) ds , t ∈ ( a , b ] .(3.2)

Denote φ (t, ω) : = ((t - a) ^α-1/Γ (α)) x₀ (ω). If x is a continuous fuzzy stochastic process such that diam [x (t, ω)]  ^r  ≥ diam [φ (t, ω)]  ^r for all t ∈ [a, b], for ℙ - a.a. ω and for every r ∈ [0, 1], then (3.2) can be written as x ( t , ω ) = ℙ . 1 ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) + 1 Γ ( α ) ∫ a t ( t - s ) α - 1 f ω ( s , x ( s , ω ) ) ds , t ∈ ( a , b ] .(3.3)

If x is a continuous fuzzy stochastic process such that diam [x (t, ω)]  ^r  ≤ diam [φ (t, ω)]  ^r for all t ∈ [a, b], for ℙ - a.a. ω and for every r ∈ [0, 1], then (3.2) can be written as

x ( t , ω ) = ℙ . 1 ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) ⊖ ( - 1 ) Γ ( α ) ∫ a t ( t - s ) α - 1 f ω ( s , x ( s , ω ) ) ds , t ∈ ( a , b ] .(3.4)

The following lemma shows the equivalence between a random fuzzy fractional differential equation and a random fuzzy fractional integral equation.

Lemma 3.1. Let the function f satisfy (A1) and (A2). Then a d-monotone fuzzy stochastic process x satisfying x 1 - α ( t , ω ) : = ℑ a + 1 - α x ( t , ω ) ∈ AC ( [ a , b ] , E ) and d-monotone for ℙ - a.a. ω is a solution of initial value problem (3.1), if and only if x satisfies (3.2) and diam [x (t, ω)]  ^r  - diam [φ (t, ω)]  ^r has a constant sign on [a, b] for ℙ - a.a. ω .

Proof. Let x be a d-monotone fuzzy stochastic process satisfying initial value problem (3.1). As f satisfies conditions (A1) and (A2), RL D a + α x ∈ L 1 ( [ a , b ] , E ) with ℙ . 1 . From (3.1) and RL D a + α x ( t , ω ) = D gH ( ℑ a + 1 - α x ) ( t , ω ) , x 1 - α ( a , ω ) : = ( ℑ a + 1 - α x ) ( a , ω ) , where x_1-α (a, ω) exists in the sense that x 1 - α ( a , ω ) = ℙ . 1 lim t → a + ( ℑ a + 1 - α x ) ( t ) , and since x 1 - α ( t , ω ) : = ℑ a + 1 - α x ( t , ω ) ∈ AC ( [ a , b ] , E ) is d-monotone for ℙ - a.a. ω, the gH-difference x ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 1 - α ( a , ω ) exists for t ∈ (a, b] with ℙ - a.a. ω. Indeed, from (2.2) we obtain I a + α D a + α diam [ x ( t , ω ) ] r = ℙ . 1 diam [ x ( t , ω ) ] r - ( t - a ) α - 1 Γ ( α ) diam [ x 1 - α ( a , ω ) ] r for every r ∈ [0, 1] and for t ∈ (a, b] . On the other hand, since I a + α D a + α diam [ x ( t , ω ) ] r = I a + α d dt I a + 1 - α diam [ x ( t , ω ) ] r = I a + α d dt diam [ x 1 - α ( t , ω ) ] r ,

it follows that diam [ x ( t , ω ) ] r - ( t - a ) α - 1 Γ ( α ) diam [ x 1 - α ( a , ω ) ] r = ℙ . 1 I a + α d dt diam [ x 1 - α ( t , ω ) ] r , t ∈ ( a , b ] .

As x_1-α is d-decreasing for ℙ - a.a. ω, we have d dt diam [ x 1 - α ( t , ω ) ] r ≤ ℙ . 1 0 for t ∈ (a, b] and for every r ∈ [0, 1] and thus we get I a + α d dt diam [ x 1 - α ( t , ω ) ] r ≤ ℙ . 1 0 for t ∈ (a, b] and for every r ∈ [0, 1]. Hence, we infer that diam [ x ( t , ω ) ] r ≤ ℙ . 1 ( t - a ) α - 1 Γ ( α ) diam [ x 1 - α ( a , ω ) ] r for t ∈ (a, b] and for every r ∈ [0, 1]. Moreover, we get for t ∈ (a, b] [ ( ℑ a + α RL D a + α x ) ( t , ω ) ] r = [ I a + α D a + α x _ ( t , r , ω ) , I a + α D a + α x ¯ ( t , r , ω ) ] = ℙ . 1 [ x _ ( t , r , ω ) - ( t - a ) α - 1 Γ ( α ) x _ 1 - α ( a , r , ω ) ,

x ¯ ( t , r , ω ) - ( t - a ) α - 1 Γ ( α ) x ¯ 1 - α ( a , r , ω ) ] = ℙ . 1 [ x _ ( t , r , ω ) , x ¯ ( t , r , ω ) ] ⊖ ( t - a ) α - 1 Γ ( α ) [ x _ 1 - α ( a , r , ω ) , x ¯ 1 - α ( a , r , ω ) ] .(3.5)

Similarly, as x_1-α is d-increasing for ℙ - a.a. ω, it implies that I a + α d dt diam [ x 1 - α ( t , ω ) ] r ≥ ℙ . 1 0 for t ∈ (a, b] or diam [ x ( t , ω ) ] r ≥ ℙ . 1 ( t - a ) α - 1 Γ ( α ) diam [ x 1 - α ( a , ω ) ] r for t ∈ (a, b] and for every r ∈ [0, 1]. Moreover, we obtain [ ( ℑ a + α RL D a + α x ) ( t , ω ) ] r = ℙ . 1 - ( t - a ) α - 1 Γ ( α ) [ x _ 1 - α ( a , ω ) , x ¯ 1 - α ( a , ω ) ] ⊖ ( - [ x _ ( t , r , ω ) , x ¯ ( t , r , ω ) ] )(3.6) for t ∈ (a, b] and for every r ∈ [0, 1] .

Therefore, from (3.5), (3.6) and ( RL D a + α - 1 x ) ( a , ω ) : = x 1 - α ( a , ω ) = ℙ . 1 x 0 ( ω ) , we get

( ℑ a + α RL D a + α x ) ( t , ω ) = ( a , b ] , ℙ . 1 x ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 1 - α ( a , ω ) = x ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) .(3.7)

Applying the operator ℑ a + α to both sides of problem (3.1) and from (3.7), we obtain the integral equation (3.2).

Conversely, assume that x is a d-monotone fuzzy stochastic process satisfying integral equation (3.2) and x_1-α (t, ω) ∈ AC ([a, b] , E) is d-monotone for ℙ - a.a. ω. Firstly, since x_1-α is absolutely continuous for ℙ - a.a. ω, x_1-α is gH- differentiable on [a, b] and D _gH  (x_1-α) (t, ω) ∈ L¹ ([a, b] , E) for ℙ - a.a. ω. Let x 1 - α ( t , ω ) = 1 Γ ( 1 - α ) ∫ a t ( t - s ) - α x ( s , ω ) ds be d-decreasing on [a, b] for ℙ - a.a. ω. Let t ∈ [a, b) and h > 0 such that t + h ≤ b . By direct computation we have x 1 - α ( t + h , ω ) + ( - 1 ) ∫ t t + h f ω ( s , x ( s , ω ) ) ds = [ a , b ) , ℙ . 1 1 Γ ( 1 - α ) ∫ a t + h ( t + h - s ) - α ( ( s - a ) α - 1 Γ ( α ) x 0 ( ω ) ⊖ ( - 1 ) Γ ( α ) ∫ a s ( s - r ) α - 1 f ω ( r , x ( r , ω ) ) dr ) ds + ( - 1 ) ∫ t t + h f ω ( s , x ( s , ω ) ) ds = 1 Γ ( 1 - α ) ∫ a t ( t - s ) - α ( s - a ) α - 1 Γ ( α ) x 0 ( ω ) ds + ( - 1 ) ∫ t t + h f ω ( s , x ( s , ω ) ) ds ⊖ ( - 1 ) Γ ( 1 - α ) Γ ( α ) ∫ a t + h ( t + h - s ) - α ( ∫ a s ( s - r ) α - 1 f ω ( r , x ( r , ω ) ) dr ) ds .(3.8) Using the following properties ∫ t t + h f ω ( s , x ( s , ω ) ) ds = ∫ a t + h f ω ( s , x ( s , ω ) ) ds ⊖ ∫ a t f ω ( s , x ( s , ω ) ) ds , ( ℑ a + 1 - α ( ( ℑ a + α f ω ) ( · , x ( · , ω ) ) ) ) ( t + h ) = ( ℑ a + 1 f ω ( · , x ( · , ω ) ) ) ( t + h ) , from equation (3.8) we get x 1 - α ( t + h , ω ) + ( - 1 ) ∫ t t + h f ω ( s , x ( s , ω ) ) ds = [ a , b ) , ℙ . 1 1 Γ ( 1 - α ) ∫ a t ( t - s ) - α ( s - a ) α - 1 Γ ( α )

x 0 ( ω ) ds + ( - 1 ) ∫ a t + h f ω ( s , x ( s , ω ) ) ds ⊖ ( - 1 ) ∫ a t f ω ( s , x ( s , ω ) ) ds ⊖ ( - 1 ) ∫ a t + h f ω ( s , x ( s , ω ) ) ds = 1 Γ ( 1 - α ) Γ ( α ) ∫ a t ( t - s ) - α ( s - a ) α - 1 x 0 ( ω ) ds ⊖ ( - 1 ) Γ ( 1 - α ) Γ ( α ) ∫ a t ( t - s ) - α ( ∫ a s ( s - r ) α - 1 f ω ( r , x ( r , ω ) ) dr ) ds = x 1 - α ( t , ω ) .

Similarly, we can get x 1 - α ( t , ω ) + ( - 1 ) ∫ t - h t f ω ( s , x ( s , ω ) ) ds = ( a , b ] , ℙ . 1 x 1 - α ( t - h , ω ) .

Multiplying by (1/- h) and passing to limit with h → 0⁺, we have ( RL D a + α x ) ( t , ω ) : = ( D gH x 1 - α ) ( t , ω ) = lim h → 0 + x 1 - α ( t , ω ) ⊖ x 1 - α ( t + h , ω ) - h = x 1 - α ( t - h , ω ) ⊖ x 1 - α ( t , ω ) - h = [ a , b ] , ℙ . 1 f ω ( t , x ( t , ω ) ) .

Finally, we will show that x satisfies the initial condition of problem (3.1). It is sufficient to show that RL D a + α - 1 x ( a , ω ) = ℙ . 1 x 0 ( ω ) . Indeed, applying the operator ℑ a + 1 - α to both sides of integral equation (3.2) yields lim t → a ℑ a + 1 - α ( x ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) ) = ℙ . 1 lim t → a ℑ a + 1 - α ( 1 Γ ( α ) ∫ a t ( t - s ) α - 1 f ω ( s , x ( s , ω ) ) ds ) = ℙ . 1 lim t → a 1 Γ ( α ) ∫ a t f ω ( s , x ( s , ω ) ) ds = ℙ . 1 0 ˆ .(3.9)

Furthermore, since the difference diam [x (t, ω)]  ^r  - diam [φ (t, ω)]  ^r has a constant sign on [a, b] for ℙ - a.a. ω, we get (see [38] - Theorem 1) lim t → a ℑ a + 1 - α ( x ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) ) = ℙ . 1 lim t → a ℑ a + 1 - α x ( t , ω ) ⊖ gH x 0 ( ω ) by using the property I a + 1 - α ( t - a ) α - 1 = Γ ( α ) ( t - a ) 0 (see [24] - Property 2.1). Therefore, from (3.8q) we infer that D a + α - 1 x ( a , ω ) : = lim t → a ℑ a + 1 - α x ( t , ω ) = ℙ . 1 x 0 ( ω ) . The proof is completed. □

Denote S (u₀, ρ) = {u ∈ E | D₀ [u, u₀] ≤ ρ}, ρ > 0 and Q = {(t, u)  : t ∈ [a, b] , u ∈ S (u₀, ρ)}, where u 0 ( ω ) : = ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) .

Theorem 3.1. Let f : [a, b] × Ω × E ^d  → E ^d satisfies the conditions (A1)-(A2) and assume that there exists a non-negative constant M such that D 0 [ f ω ( t , u ) , 0 ˆ ] ≤ P .1 M on Q. Suppose that f satisfies the following conditions on Q with ℙ.1 for every t ∈ [a, b] and for every u, v ∈ S (u₀, ρ): ( F 1 ) D 0 [ f ω ( t , u ) , f ω ( t , v ) ] ≤ K η Γ ( α ) D 0 [ u , v ] ( t - a ) α , t ≠ a , with K η < α , K > 1 ; ( F 2 ) D 0 [ f ω ( t , u ) , f ω ( t , v ) ] ≤ β D 0 p [ u , v ] , p ∈ ( 0 , 1 ) , K ( 1 - p ) < 1 and β > 0 is a constant . Then, the following successive approximation given by u 0 ( t , ω ) = [ a , b ] , ℙ . 1 ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) and for n = 0, 1, 2, . . .

x n + 1 ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) = [ a , b ] , ℙ . 1 1 Γ ( α ) ∫ a t ( t - s ) α - 1 f ω ( s , x n ( s , ω ) ) ds(3.10) converges uniformly to unique solution x of (3.1) on [ a , 𝕋 ] , where 𝕋 ∈ ( a , b ] .

Proof. Choose t^* > a such that t * - a ≤ ( ρ Γ ( 1 + α ) M ) 1 / α and put 𝕋 : = min { t * , b } . From Lemma 3, we note that (3.10) is equivalent to

{ RL x n + 1 ( t , ω ) = [ a , T ] , f ω ( t , x n ( t , ω ) ) , RL D a + α - 1 x n + 1 ( t , ω ) | t = a = [ a , T ] , P .1 x 0 ( ω ) ,(3.11) for all n = 1, 2, 3, …. We wish to show that

lim n → ∞ D 0 [ RL x n ( t , ω ) , f ω ( t , x n ( t , ω ) ) ] = [ a , 𝕋 ] , ℙ . 1 0 .(3.12)

By assumptions p ∈ (0, 1) and K (1 - p) <1, we have K < 1 + p ∑ k = 0 ∞ p k . Therefore, there exists an integer N > 1 such that K < 1 + p ∑ k = 0 N - 1 p k .

For n = 0, from Eq (3.11) and assumption D 0 [ f ω ( t , u ) , 0 ˆ ] ≤ P .1 M on Q, we have the following estimate: D 0 [ RL x 1 ( t , ω ) , f ω ( t , x 1 ( t , ω ) ) ] = P .1 D 0 [ f ω ( t , x 0 ( t , ω ) ) , f ω ( t , x 1 ( t , ω ) ) ] ≤ P .1 2 M on [ a , 𝕋 ](3.13) and for n = 1, 2, …, D 0 [ RL x n ( t , ω ) , f ω ( t , x n ( t , ω ) ) ] = P .1 D 0 [ RL x n + 1 ( t , ω ) , RL x n ( t , ω ) ] on [ a , 𝕋 ] .

Using the assumption (F2) and Eq (3.11), we find that D 0 [ RL x k + 1 ( t , ω ) , f ω ( t , x k + 1 ( t , ω ) ) ] = [ a , T ] , D 0 [ f ω ( t , x k ( t , ω ) ) , f ω ( t , x k + 1 ( t , ω ) ) ] ≤ [ a , T ] , P .1 β D 0 p [ x k ( t , ω ) , x k + 1 ( t , ω ) ] = [ a , T ] , β D 0 p [ x k ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) , x k + 1 ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) ] ≤ [ a , T ] , P .1 β Γ ( α ) ( ∫ a t ( t - s ) α - 1 D 0 [ f ω ( s , x k - 1 ( s , ω ) ) , f ω ( s , x k ( s , ω ) ) ] ds ) p = [ a , T ] , β Γ ( α ) ( ∫ a t ( t - s ) α - 1 D 0 [ RL x k ( t , ω ) , f ω ( s , x k ( s , ω ) ) ] ds ) p ,(3.14) for k = 1, 2, 3, ….

From (3.13), (3.14) and proceeding recursively, we get

D 0 [ RL x N + 1 ( t , ω ) , f ω ( s , x N + 1 ( s , ω ) ) ] ds ≤ [ a , T ] , P .1 R ( t - a ) λ ,(3.15) where λ = αγ, γ = p ∑ k = 0 N - 1 p k , N > 1 such that K < 1 + γ and

R = ( β ∑ k = 0 N - 1 p k ) ( 2 M ) p N ( 1 / ( α Γ ( α ) ) ) p ∑ k = 1 N p k .(3.16)

Using the assumption (F2) and Eq (3.11), for j = 0, 1, 2, … and a < t < T, we have D 0 [ RL x N + j + 1 ( t , ω ) , f ω ( t , x N + j + 1 ( t , ω ) ) ] ≤ [ a , T ] , P .1 K η Γ ( α ) ( t - a ) α D 0 [ x N + j + 1 ( t , ω ) , x N + j ( t , ω ) ] ≤ [ a , T ] , P .1 K η ( t - a ) α ( ∫ a t ( t - s ) α - 1 D 0 [ f ω ( s , x N + j ( s , ω ) ) , f ω ( s , x N + j - 1 ( s , ω ) ) ] ds ) ≤ [ a , T ] , P .1 K η ( t - a ) α ( ∫ a t ( t - s ) α - 1 D 0 [ RL x N + j ( t , ω ) , f ω ( s , x N + j ( s , ω ) ) ] ds ) .(3.17)

Combining the estimate (3.15) with the estimate (3.17), for j = 0, 1, 2…, a < t < T and the induction argument, we obtain D 0 [ RL x N + j + 1 ( t , ω ) , f ω ( t , x N + j + 1 ( t , ω ) ) ] ≤ [ a , T ] , P .1 ( K η α ) j R ( t - a ) λ or

D 0 [ RL x N + j + 1 ( t , ω ) , f ω ( t , x N + j + 1 ( t , ω ) ) ] ≤ [ a , T ] , P .1 θ j Rt λ ,(3.18) where θ = Kη/α < 1 and R, λ and N are determined by (3.16). Since θ < 1, the claim (3.12) is proved.

In the sequel, we shall show that the sequence {x ⁿ  (t, ω)} converges uniformly on [a, 𝕋] with ℙ.1. Note that by (3.18), there exists a continuous fuzzy stochastic process Φ^N+j+1 (t, ω), j = 0, 1, 2…, such that

RL x N + j + 1 ( t , ω ) = [ a , T ] , f ω ( t , x N + j + 1 ( t , ω ) ) + Φ N + j + 1 ( t , ω ) ,(3.19) where

D 0 [ Φ N + j + 1 ( t , ω ) , 0 ˆ ] ≤ [ a , T ] , P .1 θ j R ( t - a ) λ ≤ R θ j ( T - a ) λ .(3.20)

Hence, from (3.10) we have x N + j + 1 ( t , ω ) ⊖ gH ( t - a ) α - 1 Γ ( α ) x 0 ( ω ) = 1 Γ ( α ) ∫ a t ( t - s ) α - 1 [ f ω ( s , x N + j ( s , ω ) ) + Φ N + j ( s , ω ) ] ds ,(3.21) and according to condition (F2) we obtain

D 0 [ x N + j + 1 ( t , ω ) , x N + i + 1 ( t , ω ) ] ≤ [ a , T ] , P .1 1 Γ ( α ) ∫ a t ( t - s ) α - 1 D 0 [ f ω ( s , x N + j ( s , ω ) ) , f ω ( s , x N + i ( s , ω ) ) ] ds + 1 Γ ( α ) ∫ a t ( t - s ) α - 1 D 0 [ Φ N + j ( s , ω ) , Φ N + i ( s , ω ) ] ds ≤ [ a , T ] , P .1 2 M + R ( θ i + θ j ) ( t - a ) λ Γ ( 1 + α ) ( T - a ) α .(3.22) As θ ⁱ  + θ ^j  ≤ 2 for i, j = 0, 1, 2, … and from (3.19)- (3.21), we have the following estimate D 0 [ RL x N + j + 1 ( t , ω ) , RL x N + i + 1 ( t , ω ) ] ≤ [ a , T ] , P .1 β ( 2 M + 2 R ( T - a ) λ Γ ( 1 + α ) ) p ( T - a ) α p + 2 R ( T - a ) λ .(3.23)