Existence and stability results for quaternion fuzzy fractional differential equations in the sense of Hilfer

1 Introduction

The concept of fuzzy complex number was first proposed by Buckley in [1]. In order to extend fuzzy complex number, the concept of the fuzzy quaternion number was introduced by Moura et al., who in [2] discussed some concepts such as their arithmetic properties, infimum, supremum, distance, and so on. The quaternion membership function was defined as a fuzzy mapping μ : ℍ → [ 0 , 1 ] such that μ ( a + bi + cj + dk ) = min { μ 0 ( a ) , μ 1 ( b ) , μ 2 ( c ) , μ 3 ( d ) } , where μ _i , i = 0, 1, 2, 3 are real fuzzy numbers. Yang et al. proposed a different definition of quaternion fuzzy sets and discussed entailed results which paralleled those of regular real fuzzy numbers in [3].

Fuzzy differential equations have received increasing attentions in recent years [4–6]. Agarwal et al. considered a differential equation of fractional order with uncertainty and presented a new concept of solution in [7]. They introduced the notion of the fuzzy Riemann-Liouville differentiability which was a combination of Riemann-Liouville derivative and Hukuhara difference. Some results on existence and uniqueness of the solution were later established in [8, 9], and see [10, 11] for more on this topic. Salahshour et al. applied fuzzy Laplace transform method to solve fuzzy differential equations [12, 13]. Allahviranloo et al. obtained some new properties such as switching point, the univariate and multivariate fuzzy chain rules, fuzzy mean value theorem, and then they showed the uniqueness of a solution for fuzzy heat equation with fuzzy initial values [14]. By using fuzzy Fourier transform method, Gouyandeh et al. investigated the analytical solution of a fuzzy heat equation under generalized Hukuhara partial differentiability [15]. Alinezhad and Allahviranloo presented an extension to fractional optimal control problems with ambiguity under some necessary conditions [16]. The numerical solution of the differential equation was obtained in [17–20].

The study of stability problems for functional equations is related to a question of Ulam [21] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [22]. Shen gave some results on the Ulam stability problems of three variants of first order linear fuzzy differential equations under some suitable conditions [23]. A Laplace transform method was proposed to prove the Hyers-Ulam stability of linear fuzzy differential equations of first order with constant coefficients in [24]. Jung et al. showed the stability of the time independent Schr_ger equation with a potential box of finite walls by applying some properties of approximate solutions of the second-order inhomogeneous linear differential equations [25]. Some fixed point method were used to study the Hyers-Ulam stability of the differential and functional equations [26, 27]. In fact, the fixed point technology is an important tool for Hyers-Ulam stability. However, as far as we known, we can only use the Banach’s fixed theorem to solve this problem, and by using the other fixed theorem such as Brouwer fixed point theorem we cannot obtain the similar results. One can see [28–30] for more details about these topic. In this paper, we give some results on the existence and stability of solutions for quaternion fuzzy fractional differential equations in the sense of Hilfer. By the Piccard operator theorem (fixed point method) and associate space theorem, some sufficient conditions for the Hyers–Ulam stability of quaternion fuzzy fractional differential equations are given.

2 Notation and basic results

The set of nonempty convex compact subsets on ℝ 3 is denoted by C K ( R 3 ) . For μ , ν ∈ C K ( ℝ 3 ) , the Hausdorff distance is defined as d ( μ , ν ) = inf { ɛ | μ ⊂ N ( ν , ɛ ) and ν ⊂ N ( μ , ɛ ) } , where N ( μ , ɛ ) = { a ∈ R 3 | ∥ a - b ∥ < ɛ for some a ∈ μ}.

For convenience, we denote Λ : = {0, 1, 2, 3} and e₀ = 1,  e₁ = i,  e₂ = j,  e₃ = k, where i,  j,  k are units of the real quaternion algebra ℍ .

In [3], Yang et al. introduced a new notion of quaternion fuzzy sets on ℝ 3 and defined the quaternion grades of membership as follows.

Definition 1. [3] The quaternion membership function f is defined by f ( V , μ ) = e 0 f 0 ( V ) + e 1 f 1 ( μ ) + e 2 f 2 ( μ ) + e 3 f 3 ( μ ) , where V is to be interpreted as a set in a fuzzy set of sets and μ ∈ V.

In particluar, for μ ∈ ℝ 3 , we have f ( μ ) = e 0 f 0 ( μ ) + e 1 f 1 ( μ ) + e 2 f 2 ( μ ) + e 3 f 3 ( μ ) , where f 0 , f 1 , f 2 , f 3 : ℝ 3 → [ 0 , 1 ] . Denote f by (f₀, f₁, f₂, f₃). Then, the α ¯ = ( α 0 , α 1 , α 2 , α 3 ) -level set of f = (f₀, f₁, f₂, f₃) is defined as [ f ] α ¯ = [ f 0 ] α 0 ∩ [ f 1 ] α 1 ∩ [ f 2 ] α 2 ∩ [ f 3 ] α 3 .(2.1)

Denote F n the set of all ν : ℝ n → [ 0 , 1 ] satisfying of the following conditions:

ν is normal, i.e., there exists x 0 ∈ ℝ n such that ν (x₀) =1;

Moreover, we define F ˆ 4 n as follows: F ˆ 4 n = { ( ν 0 , ν 1 , ν 2 , ν 3 ) ∈ F n × F n × F n × F n | ∃ t 0 , s . t . , v l ( t 0 ) = 1 , l ∈ Λ } .

Then, for ν = ( ν 0 , ν 1 , ν 2 , ν 3 ) ∈ F ˆ 4 n , [ g ] α ¯ = ∩ l ∈ Λ [ ν l ] α l ∈ C K ( ℝ 3 ) where α _l  ∈ [0, 1] , l ∈ Λ.

For g , f ∈ F ˆ 4 n , where f = (f₀, f₁, f₂, f₃) and g = (g₀, g₁, g₂, g₃), and λ is a scalar, let f + g = ( f 0 + g 0 , f 1 + g 1 , f 2 + g 2 , f 3 + g 3 ) , λ g = ( λ g 0 , λ g 1 , λ g 2 , λ g 3 ) .

Let us define a metric D : F n × F n → [ 0 , ∞ ) by D ( ν 1 , ν 2 ) = sup { d ( [ ν 1 ] r , [ ν 2 ] r ) | r ∈ [ 0 , 1 ] } ,(2.2) where d is the Hausdorff distance. The metric space ( F n , D ) as a cone can be embedded isomorphically in a Banach space [31]. However, D is not a suitable metric for our space of interest, F ˆ 4 n , as we quickly see that linearity is violated. Instead, let us consider the product metric D′ on F 4 n = F n × F n × F n × F n . For f = ( f 0 , f 1 , f 2 , f 3 ) ∈ F 4 n and g = ( g 0 , g 1 , g 2 , g 3 ) ∈ F 4 n , we modify the metric as D ′ : F 4 n × F 4 n → [ 0 , ∞ )

D ′ ( f , g ) = D ′ ( ( f 0 , f 1 , f 2 , f 3 ) , ( g 0 , g 1 , g 2 , g 3 ) ) = max l ∈ Λ { D ( f l , g l ) } .(2.3)

Then, the zero element on F ˆ 4 n is denoted as 0 ˆ 4 ( x ) = ( 0 ˆ ( x ) , 0 ˆ ( x ) , 0 ˆ ( x ) , 0 ˆ ( x ) ) ∈ F 4 n . It is easy to verify that D ′ is a linearity preserving metric for F 4 n . Since F ˆ 4 n ⊂ F 4 n , D ′ is also a metric for F ˆ 4 n . Hence, ( F ˆ 4 n , D ′ ) is a complete metric space. The metric space ( F ˆ 4 n , D ′ ) can be embedded into a Banach space by the Arens-Eells theorem.

We introduce the strongly generalized differentiability in terms of the generalized Hukuhara difference. For a , b ∈ F ˆ 4 n , if there exists c ∈ F ˆ 4 n such that a = c + b or b = a + (-1) c, then we call c the difference of a and b and denote it as a ⊖ b = c.

A fuzzy-valued function F defined on the bounded, simply connected domain Ω ⊂ ℝ 3 is a mapping F : Ω → F ˆ 4 n , and F can be represented in a form F = ∑ j = 0 3 e j F j ( x ) . Its conjugate F ¯ is defined by F ¯ = e 0 F 0 ( x ) ⊖ ∑ j = 1 3 e j F j ( x ) , where x = (x₁, x₂, x₃) ∈ Ω and F _j  (x) , j = Λ are continuous fuzzy-valued functions.

Definition 2. [3] Let Ω ⊂ ℝ 3 be a bounded, simply connected domain. The mapping F : Ω → F ˆ 4 n is called strongly generalized partial derivative at x = (x₁, x₂, x₃) ∈ Ω if there exists some ∂ F ∂ x i ∈ F ˆ 4 n such that

there exists the differences F (· , x _i  + h, ·) ⊖ F (· , x _i , ·), F (· , x _i , ·) ⊖ F (· , x _i  - h, ·) and

∂ F ∂ x i = lim h → 0 + F ( · , x i + h , · ) ⊖ F ( · , x i , · ) h = lim h → 0 + F ( · , x i , · ) ⊖ F ( · , x i - h , · ) h ,(2.4) or

We can obtain the following connection between the strongly generalized partial derivative of F and its endpoint functions F l α and F r α .

Let F : Ω → F ˆ 4 n be a quaternion fuzzy function. If F is strongly generalized partial derivable at x ∈ Ω, then we have the following case:

If F has strongly generalized partial derivative at x ∈ Ω in (i), then, for each α _i  ∈ [0, 1] , F _il and F _ir are strongly generalized partial derivable functions at x and [ ∂ F ∂ x i ] α = [ ( ∂ F ∂ x i ) l α , ( ∂ F ∂ x i ) r α ] , where

( ∂ F ∂ x i ) l α = [ ( ∂ F ∂ x i ) 0 l α 0 , ( ∂ F ∂ x i ) 1 l α 1 , ( ∂ F ∂ x i ) 2 l α 2 , ( ∂ F ∂ x i ) 3 l α 3 ](2.8) and

( ∂ F ∂ x i ) r α = [ ( ∂ F ∂ x i ) 0 r α 0 , ( ∂ F ∂ x i ) 1 r α 1 , ( ∂ F ∂ x i ) 2 r α 2 , ( ∂ F ∂ x i ) 3 r α 3 ] .(2.9)

Definition 3. [3] Let F : Ω → F ˆ 4 n be a continuous mapping. The fuzzy Riemann-Liouville integral of F is defined by ( I 0 + β F ) ( x ) = 1 Γ ( β ) ∫ 0 x i ( x i - τ ) β - 1 F ( · , τ , · ) d τ ,(2.10) where x ∈ Ω, x _i  > 0, 0 < β < 1.

Moreover, the Riemann-Liouville integral of a quaternion fuzzy-valued function F can be expressed as follows: ( I 0 + β F α ) ( x ) = [ ( I 0 + β F l α ) ( x ) , ( I 0 + β F r α ) ( x ) ] , where ( I 0 + β F l α ) ( x ) = 1 Γ ( β ) ∫ 0 x i ( x i - τ ) β - 1 F l α ( · , τ , · ) d τ and ( I 0 + β F r α ) ( x ) = 1 Γ ( β ) ∫ 0 x i ( x i - τ ) β - 1 F r α ( · , τ , · ) d τ .

Definition 4. [3] The fuzzy Riemann-Liouville fractional derivative of order n - 1 < β < n for fuzzy-valued function F is defined by

( RL D 0 + β F ) ( x ) = 1 Γ ( n - β ) ∂ n ∂ x i n ∫ 0 x i ( x i - τ ) n - β - 1 F ( · , τ , · ) d τ .(2.11)

Similarly, we have

( RL D 0 + β F α ) ( x ) = [ ( RL D 0 + β F l α ) ( x ) , ( RL D 0 + β F r α ) ( x ) ] ,(2.12) where

( RL D 0 + β F l α ) ( x ) = 1 Γ ( n - β ) ∂ n ∂ x i n ∫ 0 x i ( x i - τ ) n - β - 1 F l α ( · , τ , · ) d τ ,(2.13) and

( RL D 0 + β F r α ) ( x ) = 1 Γ ( n - β ) ∂ n ∂ x i n ∫ 0 x i ( x i - τ ) n - β - 1 F r α ( · , τ , · ) d τ .(2.14)

Definition 5. [3] The fuzzy Caputo derivative of F for n - 1 < β < n and x ∈ Ω is denoted as ( C D 0 + β F ) ( x ) and defined by

( C D 0 + β F ) ( x ) = 1 Γ ( n - β ) ∫ 0 x i ( x i - τ ) n - β - 1 ∂ n ∂ τ n F ( · , τ , · ) d τ .(2.15)

Then, ( C D 0 + β F α ) ( x ) = [ ( C D 0 + β F l α ) ( x ) , ( C D 0 + β F r α ) ( x ) ] , where

( C D 0 + β F l α ) ( x ) = 1 Γ ( n - β ) ∫ 0 x i ( x i - τ ) n - β - 1 ∂ n ∂ τ n F l α ( · , τ , · ) d τ(2.16) and

( C D 0 + β F r α ) ( x ) = 1 Γ ( n - β ) ∫ 0 x i ( x i - τ ) n - β - 1 ∂ n ∂ τ n F r α ( · , τ , · ) d τ .(2.17)

Definition 6. The fuzzy Hilfer fractional derivative of order 0 ≤ α ≤ 1 and 0 < β < 1 is defined as D 0 + x i α , β F ( x ) = I 0 + x i α ( 1 - β ) ∂ ∂ x i I 0 + x i ( 1 - α ) ( 1 - β ) F ( x )(2.18) for a function F : Ω → F ˆ 4 n such that the expression on the right side exists.

Then,

( D 0 + x i α , β F α ) ( x ) = [ ( D 0 + x i α , β F l α ) ( x ) , ( D 0 + x i α , β F r α ) ( x ) ] = [ ( I 0 + x i α ( 1 - β ) ∂ ∂ x i I 0 + x i ( 1 - α ) ( 1 - β ) F l α ) ( x ) , ( I 0 + x i α ( 1 - β ) ∂ ∂ x i I 0 + x i ( 1 - α ) ( 1 - β ) F r α ) ( x ) ] .(2.19)

Remark 1. (i) When α = 0 and 0 < β < 1, the Hilfer fractional derivative corresponds to the fuzzy Riemann-Liouville fractional derivative: D 0 + x i α , β F ( x ) = ∂ ∂ x i I 0 + x i ( 1 - β ) F ( x ) = RL D 0 + x i α F ( x ) .

(ii) When α = 1 and 0 < β < 1, the fuzzy Hilfer fractional derivative corresponds to the fuzzy Caputo fractional derivative: D 0 + x i α , β F ( x ) = I 0 + x i ( 1 - β ) ∂ ∂ x i F ( x ) = C D 0 + x i α F ( x ) .

The fuzzy Dirac operator of F is defined as D ( F ) = ∑ k = 1 , j = 0 3 e k e j ∂ F j ∂ x k .

Definition 7. Let γ be a real number. The disturbed fuzzy Dirac operator is defined as D γ μ = D μ + γ μ .

Definition 8. A fuzzy function F : Ω → F ˆ 4 n is called a generalized regular fuzzy function if it satisfies D γ F = 0 ˆ 4 .

Definition 9. Assume that L ( t , x , μ ) is a first order differential operator depending on the first order derivative ∂ μ ∂ x j and t,  x,  μ, and that l (t, x, u) is a differential operator on the time variate t. If L transforms solutions of l μ = 0 ˆ 4 into solutions of the same equation for fixed t (i.e. l μ = 0 ˆ 4 implies l [ L μ ] = 0 ˆ 4 ), then L is called “associated” to l.

Let O : Y → X be an abstract operator. Consider the fixed point equation τ = O ( τ ) , τ ∈ Y(2.20) and the inequation D ′ ( τ , O ( τ ) ) ≤ ɛ .(2.21)

Definition 10. Assume that φ : ℝ + → ℝ + is an increasing function (continuous at 0 and φ (0) =0). If for each ɛ > 0 and for each solution y^* of (2.21) there exists a solution x^* of the fixed point equation (2.20) such that D ′ ( y * , x * ) ≤ φ ( ɛ ) , then the equation (2.20) is generalized Hyers–Ulam stable. For each τ ∈ ℝ + , if there exists k > 0 such that φ (τ) : = kτ, then the equation (2.20) is Hyers–Ulam stable.

Consider the following initial value problem { D 0 + t α , β μ = ∑ j = 1 3 A ( j ) ∂ μ ∂ x j + B μ + C : = L ( μ ) I 0 + t ( 1 - α ) ( 1 - β ) μ ( 0 , x ) = φ ( x )(3.1) where Ω is a bounded, simply connected domain in ℝ 3 and x = (x₁, x₂, x₃) ∈ Ω; C D 0 + t α , β is the Hilfer fractional derivative of t; t ∈ [0, T] is the time variate; μ = μ (t, x) is a quaternion fuzzy-valued function defined in [0, T] × Ω.  B = B (t, x) , A^(j) = A^(j) (t, x) and C = C (t, x) are quaternion-valued functions defined in [0, T] × Ω. The initial function φ (x) is a generalized regular fuzzy function.

The fixed points of the following operator equation (3.2) are the solutions of (3.1),

O ( μ ) : = μ ( t , x ) = φ ( x ) Γ ( α ( 1 - β ) + β ) t ( α - 1 ) ( 1 - β ) + 1 Γ ( β ) ∫ 0 t ( t - τ ) β - 1 L ( μ ) d τ .(3.2)

Theorem 1. Assume that A^(j) (t, x) (j = 1, 2, 3) , B (t, x) and C (t, x) are quaternion-valued functions for t ∈ [0, T]. The operator L is associated with D _γ , if the following hypotheses are satisfied:

A 0 ( 1 ) = A 3 ( 2 ) = - A 2 ( 3 ) , A 1 ( 1 ) = A 2 ( 2 ) = - A 3 ( 3 ) , A 2 ( 1 ) = - A 1 ( 2 ) = A 0 ( 3 ) , A 3 ( 1 ) = - A 0 ( 2 ) = - A 1 ( 3 ) ;

Example 1. If γ = 1, A^(j) = f (t, x₁, x₂, x₃) e _j  (j = 1, 2, 3), B = f (t, x₁, x₂, x₃) e₀ and C (t, x) =0, where real-valued function f (t, x₁, x₂, x₃) ∈ C² (Ω) for each t ∈ [0, T]. Then the operator L is associated with the operatot D _γ .

Moreover, we can get the interior estimate of a generalized fuzzy regular function by the associated function space theory.

Theorem 2. Suppose that Ω _{s 1}  ⊂ Ω _{s 2} and Ω ¯ s 2 ⊂ Ω . Assume that μ is a generalized fuzzy regular function and mΩ is the finite measure of Ω ⊂ ℝ n . We obtain the interior estimate of generalized fuzzy regular function

D ′ ( ∂ μ ∂ x i , 0 ˆ 4 ) ≤ γ 2 ( 3 m Ω 4 π ) 1 3 [ 3 + 1 2 ( 3 m Ω 4 π ) 1 3 ] dist ( Ω s 1 , ∂ Ω s 2 ) D ′ ( μ , 0 ˆ 4 ) = η D ′ ( μ , 0 ˆ 4 ) .(3.3)

Proof. Assume that μ is a quaternion-valued function. By Theorem 5 in [32], we have

∥ ∂ μ ∂ x i ∥ s 1 ≤ γ 2 ( 3 m Ω 4 π ) 1 3 [ 3 + 1 2 ( 3 m Ω 4 π ) 1 3 ] dist ( Ω s 1 , ∂ Ω s 2 ) ∥ μ ∥ s 2 = η ∥ μ ∥ s 2 .(3.4)

Now, for a generalized fuzzy regular function μ, we consider its endpoint functions μ l α and μ r α . It easy to see that u l α and u r α are generalized regular functions. We have ∥ ∂ μ l α ∂ x i ∥ s 1 ≤ γ 2 ( 3 m Ω 4 π ) 1 3 [ 3 + 1 2 ( 3 m Ω 4 π ) 1 3 ] dist ( Ω s 1 , ∂ Ω s 2 ) ∥ μ l α ∥ s 2(3.5) and ∥ ∂ μ r α ∂ x i ∥ s 1 ≤ γ 2 ( 3 m Ω 4 π ) 1 3 [ 3 + 1 2 ( 3 m Ω 4 π ) 1 3 ] dist ( Ω s 1 , ∂ Ω s 2 ) ∥ μ r α ∥ s 2 .(3.6)

Moreover, we can obtain

D ′ ( ∂ μ ∂ x i , 0 ˆ 4 ) = sup 0 ≤ α ≤ 1 { d ( [ ∂ μ ∂ x i ] α , 0 ˆ 4 ) } = sup 0 ≤ α ≤ 1 { d ( [ ( ∂ μ ∂ x i ) l α , ( ∂ μ ∂ x i ) r α ] , 0 ˆ 4 ) } ≤ γ 2 ( 3 m Ω 4 π ) 1 3 [ 3 + 1 2 ( 3 m Ω 4 π ) 1 3 ] dist ( Ω s 1 , ∂ Ω s 2 ) sup 0 ≤ α ≤ 1 { d ( [ μ l α , μ r α ] , 0 ˆ 4 ) } = η D ′ ( μ , 0 ˆ 4 ) ,(3.7) where η is a fixed constant. □

For our subsequent results, we need the following hypotheses.

For μ ∈ F ˆ 4 n and α = (α₀, α₁, α₂, α₃) ∈ [0, 1] × [0, 1] × [0, 1] × [0, 1],

len i ( μ α ) ≥ len i ( μ 0 α Γ ( α ( 1 - β ) + β ) t ( α - 1 ) ( 1 - β ) + 1 Γ ( β ) ∫ 0 t ( t - τ ) β - 1 L ( μ ) d τ ) ,(3.8) where i ∈ Λ and len i ( μ α ) = μ ir α - μ il α ;

Theorem 3. Assume that L satisfies the hypotheses of Theorem 1 and the hypotheses (h1)-(h4). The solution of the initial value problem (refI - 1) μ (t, x) is in the conical domain M _σ  = {(t, x) : x ∈ Ω, 0 ≤ t ≤ σ · dist (x, ∂Ω)}(σ is small enough), and is also generalized fuzzy regular for each t. Moreover, the fixed point equation μ = O ( μ ) is Hyers–Ulam stable.

Proof. By (h1)-(h4), the initial value system (3.1) can be transformed the following Volterra equation

O ( μ ) : = μ ( t , x ) = φ ( x ) Γ ( α ( 1 - β ) + β ) t ( α - 1 ) ( 1 - β ) + 1 Γ ( β ) ∫ 0 t ( t - τ ) β - 1 L ( μ ) d τ .(3.9)

We will prove that the operator O has a fixed point. It is easy to see that O maps C ( [ 0 , T ] × Ω , F ˆ 4 n ) to itself. Moreover, we have

D ′ ( O ( μ ) ⊖ O ( ν ) , 0 ˆ 4 ) = D ′ ( 1 Γ ( β ) ∫ 0 t ( t - τ ) β - 1 L ( μ ) d τ , 1 Γ ( β ) ∫ 0 t ( t - τ ) β - 1 L ( ν ) d τ ) = 1 Γ ( β ) ∫ 0 t ( t - τ ) β - 1 ( ∑ j = 1 3 A ( j ) ∂ ∂ x j D ′ ( μ ⊖ ν , 0 ˆ 4 ) + B D ′ ( μ ⊖ ν , 0 ˆ 4 ) ) d τ ≤ 1 Γ ( β ) ( M + 3 η N ) D ′ ( μ ⊖ ν , 0 ˆ 4 ) ∫ 0 t ( t - τ ) β - 1 d τ = 1 Γ ( β + 1 ) ( M + 3 η N ) t β D ′ ( μ ⊖ ν , 0 ˆ 4 ) : = γ D ′ ( μ ⊖ ν , 0 ˆ 4 ) ,(3.10) where M =∥ B ∥, N = max j = 1 , 2 , 3 { ∥ A ( j ) ∥ } . We can choose a number τ > 0 such that γ = 1 Γ ( β + 1 ) ( M + 3 η N ) τ α < 1 .

Then in the domain M _σ  = {(t, x) : x ∈ Ω, 0 ≤ t ≤ σ · dist (x, ∂Ω) ≤ τ}, O is a contraction mapping. Thus, by the Banach’s fixed point theorem, we obtain the desired uniqueness of the solution of the differential equation. Theorem 2.10 in [27] implies that the operator O is a c-weakly Picard operator with the positive constant c = 1 1 - γ and the fixed point equation μ = O ( μ ) is Hyers–Ulam stable.

Moreover, the solution μ (t, x) belongs to the associated space for each t. The solution μ (t, x) is also generalized regular. □

Example 2. Suppose that γ is any real number, C ( t , x ) ∈ C 1 ( Ω , ℍ ) is any generalized regular function, and A ( 1 ) ( t , x ) ∈ C 2 ( Ω , ℍ ) is any quaternion-valued function for each t ∈ [0, T]. Suppose, further, that A⁽²⁾ (t, x) = - A⁽¹⁾ (t, x) e₃, A⁽³⁾ (t, x) = A⁽¹⁾ (t, x) e₂, B (t, x) = - γA⁽¹⁾ (t, x) e₁ . It is easy to verify that L is associated with D γ under the above conditions. Then, by Theorem 3, there eixsts a unique solution of the initial value problem (refI - 1), and the solution μ (t, x) is also generalized fuzzy regular for each t. Moreover, the fixed point equation μ = O ( μ ) is Hyers–Ulam stable.

In fact, the fixed point theorem plays a key role in the proof of Theorem 3, and this is called the fixed point method. By using the fixed point method, we continue to consider the existence and stability of the solution for the abstract Cauchy problem { D 0 + t α , β μ ( t ) = F ( t , μ ( t ) ) , 0 < α , β < 1 , t ∈ [ 0 , T ] , μ ( 0 ) = μ 0 ,(3.11) where μ (x) is a quaternion fuzzy function.

We denote the integral operator O by

O ( μ ( t ) ) = μ 0 Γ ( α ( 1 - β ) + β ) t ( α - 1 ) ( 1 - β ) + 1 Γ ( β ) ∫ 0 t ( t - τ ) β - 1 F ( τ , μ ( τ ) ) d τ .(3.12)

Recall that μ is a solution to the Cauchy problem (3.11) if and only if μ is a solution to the integral equation (3.12). Moreover, μ satisfies the integral equation (3.12) if only if μ satisfies the fixed point equation μ = O ( μ ) . In other words, μ is a solution to the Cauchy problem (3.11) if and only if μ is a fixed point of the operator O .

O ( μ ) : = μ ( t ) = μ 0 Γ ( α ( 1 - β ) + β ) t ( α - 1 ) ( 1 - β ) + 1 Γ ( β ) ∫ 0 t ( t - τ ) β - 1 F ( τ , μ ( τ ) ) d τ .(3.13)

For our subsequent results, we need the following hypotheses.

There exists a constant M for which D ′ ( F ( t , μ ) , 0 ^ 4 ) < M holds for all t ∈ I and all μ ∈ F ˆ 4 n .

Theorem 4. Assume that hypotheses (H1)–(H4) are satisfied. Let F : I × F ˆ 4 n → F ˆ 4 n be Lipschitz continuous and bounded, with Lipschlitz constant L. Then, there exists a unique solution to the Cauchy problem (refadd - 1) on a neighborhood of a ∈ I. Moreover, the fixed point equation μ = O ( μ ) is Hyers–Ulam stable.

Proof. By definition and hypotheses (H1)–(H3), it is easy to verify that the Hilfer derivative of μ is well-defined. Let [ 0 , ɛ ] × [ μ 0 - δ , μ 0 + δ ] ⊂ I × F ˆ 4 n be a compact subset on which F is defined. Then O to be well-defined, D ′ ( O μ − μ 0 , 0 ^ 4 ) < δ for all μ ∈ [μ₀ - δ, μ₀ + δ], and

D ′ ( O μ ⊖ μ 0 , 0 ^ 4 ) = D ′ ( 1 Γ ( β ) ∫ 0 t ( t − τ ) β − 1 F ( τ , μ ( τ ) ) d τ , 0 ^ 4 ) ≤ 1 Γ ( β ) ∫ 0 t D ′ ( ( t − τ ) β − 1 F ( τ , μ ( τ ) ) , 0 ^ 4 ) d τ ≤ M Γ ( β ) ∫ 0 t ( t − τ ) β − 1 d τ ≤ M Γ ( β + 1 ) ϵ β .(3.15)

Hence we require M Γ ( β + 1 ) ɛ β < δ , i.e., we must chose ɛ > 0 such that ɛ < [ δ Γ ( β + 1 ) M ] 1 β . For k = 0, 1, 2, …, define Y₀ (t) = μ₀,

Y k + 1 ( t ) = μ 0 Γ ( α ( 1 - β ) + β ) t ( α - 1 ) ( 1 - β ) + 1 Γ ( β ) ∫ 0 t ( t - τ ) β - 1 F ( τ , Y k ( τ ) ) d τ .(3.16)

Then,

D ′ ( O ( Y m ⊖ Y n ) , 0 ^ 4 ) = 1 Γ ( β ) D ′ ( ∫ 0 t ( t − τ ) β − 1 ( F ( τ , Y m ( τ ) ) ⊖ F ( τ , Y n ( τ ) ) ) d τ , 0 ^ 4 ) ≤ L Γ ( β ) | ∫ 0 t ( t − τ ) β − 1 D ′ ( Y m ( τ ) ⊖ Y n ( τ ) , 0 ^ 4 ) d τ | ≤ L Γ ( β + 1 ) ϵ β D ′ ( Y m ⊖ Y n , 0 ^ 4 ) .(3.17)

If ɛ < min { [ δ Γ ( β + 1 ) M ] 1 β , [ Γ ( β + 1 ) L ] 1 β } , the mapping is a contraction. In such a case, O is a contraction, and by the Banach fixed point theorem, O has a unique fixed point. Thus, there exists a unique Y^* ∈ C ([0, ɛ] × [μ₀ - δ, μ₀ + δ]) such that O ( Y * ) = Y * . One may construct this function by Y * ( t ) = lim k → ∞ Y k ( t ) . This function is the unique solution to the Cauchy problem (3.11) on the interval [0, ɛ], where ɛ < min { [ δ Γ ( β + 1 ) M ] 1 β , [ Γ ( β + 1 ) L ] 1 β } . Theorem 2.10 in [27] implies that the operator O is a c-weakly Picard operator with the positive constant c = 1 1 - L Γ ( β + 1 ) ɛ β and the fixed point equation μ = O ( μ ) is Hyers–Ulam stable. □

Example 3. { D 0 + t α , β μ ( t ) = F ( t , μ ( t ) ) = | μ ( t ) | + 9 ⊖ ( i + j + k ) sin ( | μ ( t ) | ) μ ( 0 ) = μ 0 , 0 < α , β < 1 .(3.18)

Let μ , ν ∈ F ˆ 4 n , and note that

D ′ ( F ( μ ) ⊖ F ( ν ) , 0 ^ 4 ) = D ′ ( | μ | 2 + 9 ⊖ | ν | 2 + 9 ⊖ ( i + j + k ) ( sin ( | μ | ) ⊖ sin ( | ν | ) ) , 0 ^ 4 ) ≤ D ′ ( | μ | 2 + 9 ⊖ | ν | 2 + 9 , 0 ^ 4 ) + 3 D ′ ( sin ( | μ | ) ⊖ sin ( | ν | ) , 0 ^ 4 ) ≤ | μ | + | ν | | μ | 2 + 9 + | ν | 2 + 9 D ′ ( | μ | ⊖ | ν | , 0 ^ 4 ) + 3 D ′ ( | μ | ⊖ | ν | , 0 ^ 4 ) ≤ 4 D ′ ( μ ⊖ ν , 0 ^ 4 ) ,(3.19) so F is Lipschitz continuous with Lipschitz constant L = 4. Furthermore, since μ ∈ F ˆ 4 n by assumption, μ (t) = (μ₀ (t) , μ₁ (t) , μ₂ (t) , μ₃ (t)) and |μ _i |≤1, (i = 0, 1, 2, 3), hence | F ( μ ) | ≤ 13 + 3 . So F is continuous and bounded. Assume that hypotheses (H1)–(H4) are satisfied. By Theorem 4, there exists a unique local solution to (3.18). Moreover, the fixed point equation of (3.18) is Hyers–Ulam stable.