Existence and finite-time stability results of fuzzy Hilfer-Katugampola fractional delay differential equations 1

1 Introduction

Fuzzy fractional integral and differential equations have drawn a large amount of attention, due to they not only provide a valuable tool for describing the memory and hereditary properties in various types of material and motion processes [1–4], but also can be describing uncertainty, vagueness, incomplete information in physics, chemistry, biology, economics, engineering systems, etc [5, 6]. For these reasons, fuzzy fractional calculus and fuzzy fractional differential equations has emerged as a significant topic (see [7–9]). In particular, some researchers have dedicated to the study of the existence, uniqueness, stability for fuzzy fractional integral and differential equations. For instance, Agarwal et al. [10] considered a Riemann-Liouville type fractional fuzzy differential equations and presented the concept of solution. Alikhani and Bahrami [11] employed the method of upper and lower solutions to obtain the existence of global solutions for nonlinear fuzzy fractional integral and integrodifferential equations involving Riemann-Liouville differential operators. Other contribution was also studied in [12]. Fard and Salehi [13] considered some fuzzy Caputo-type fractional variational problems. Mazandarani and Kamyad [14] solved fuzzy fractional initial value problem by modified fractional Euler method. Najariyan and Zhao [15] studied fuzzy fractional quadratic regulator problem under granular Caputo and Riemann-Liouville fuzzy fractional derivatives by approximation method. Salahshour et al. [16] solved fuzzy Riemann-Liouville fractional differential equations by fuzzy Laplace transforms. Hoa et al. [17] pointed out that the equivalence between fractional fuzzy differential equations and fractional fuzzy integral equations in paper [18] is not correct and gave an appropriate condition so that thisequivalence is valid. Following the idea in [17], Hoa et al. [19] proved the existence and uniqueness results of the solution to Caputo-Katugampola fractional differential equations in fuzzy setting by employing the method of successive approximation and by means of fixed point theorems. For some different classes of fuzzy fractional order differential equations, other contributions were also studied in [20–22]. Hoa and Vu [23] applied generalized Banach fixed point theorem to prove the initial value problems of fuzzy implicit fractional differential equations. Recently, Long et al. [24] investigated Ulam stability of partial fuzzy differential equations with Caputo fractional derivative concept. Ulam-Hyers stability and Ulam-Hyers-Rassias stability for various types of fuzzy fractional integral and differential equations were investigated in [25–28] and the cited references therein.

As mentioned above, fruitful results for Fuzzy fractional integral and differential equations are obtained, it seems that there are still many unanswered questions and interesting ideas in the making. The well known the concept of the finite time stability plays a crucial role in characterizing the nature of dynamic systems or differential equations. As pointed out by [29–31] that “ the finite time stability is different from concepts of Lyapunov asymptotic stability and exponential stability.”Furthermore, Oliveira and de Oliveira [32] recently proposed a Hilfer-Katugampola fractional derivatives and pointed out this fractional derivative is a generalization of the classical fractional derivatives. Therefore, it is of great importance to analyze the finite time stability for fuzzy fractional integral and differential equations with Hilfer-Katugampola fractional derivatives. However, to the best of the authors’ knowledge, there is no result on existence and finite time stability results of fuzzy Hilfer-Katugampola fractional delay differential equations. Consequently, this paper is devoted to exploring existence and finite time stability results of fuzzy Hilfer-Katugampola fractional delay differential equations given by

{ ( ρ D 0 + α , β ​ x ) ( t ) = f ( t , x t ) , ρ > 0 α ∈ ( 0 , 1 ) , β ∈ [ 0 , 1 ] t ∈ ( 0 , b ] , ( ρ I 0 + ( 1 − α ) ( 1 − β ) x ) ( 0 + ) = x 0 , x ( t ) = φ ( t ) , t ∈ [ - τ , 0 ] ,(1) where x ∈ ℝ , symbols ( ρ I 0 + ( 1 − α ) ( 1 − β ) x ) , ( ρ D 0 + α , β ​ ) stand for Riemann-Liouville generalized fractional integral of order (1 - α) (1 - β) of the fuzzy function x and the fuzzy Hilfer-Katugampola fractional derivative, respectively, which will be specified later, f : [0, b] × C ([- τ, 0] ; E) → E is a fuzzy function, φ ∈ C ([- τ, 0] ; E), E is the space of fuzzy sets.

The novelties of this article are twofold. The first one is to introduce a new class of fuzzy fractional delay differential equations and to investigate their existence of solutions and finite time stability. Here, compared with existing results in [10–17, 24–27], the novelty arises in the special structure of the problem we consider, which use Hilfer-Katugampola fractional derivatives admitting as particular cases the well-known fractional derivatives of Caputo [13–15, 27], generalized Caputo-type [19], Riemann-Liouville [10–12, 25], Hadamard, Caputo-Hadamard [26], Weyl and Liouville, Hilfer, Hilfer-Hadamard. As pointed out by authors [32] that ”in contrast to fractional derivatives mentioned above, the Hilfer-Katugampola fractional derivatives with regular variation function can widely describe the behaviour of fractional integrals and derivatives near their low limits.” On the other hand, authors of the literature [20, 22] mainly applied successive approximation method to consider fuzzy fractional differential equations, while we mainly use the generalized Gronwall inequality to investigate the finite time stability for fuzzy fractional delay differential equations. Therefore, the method used in this paper is different from that in [20, 22]. This is the second novelty of the present work.

The plan of this paper is organized into four sections. We list some necessary preliminaries in Section 2. In section 3, we obtain the existence of solutions, and the finite time stability for (1.1). Section 4 gives an example to illustrate our main results.

In this section, we will give some basic notations and preliminaries needed throughout the whole paper.

Definition 2.1.2.1[33] A fuzzy number E is a fuzzy convex and normal fuzzy subset in ℝ with upper semicontinuous membership function and compact support.

Let us denote by [ u ] l = { x ∈ ℝ : u ( x ) ≥ l } , for each l ∈ ( 0 , 1 ] , the l-level set of a fuzzy set u on ℝ , and the support [u] ⁰ of a fuzzy set is defined as [ u ] 0 = ⋃ l ∈ ( 0 , 1 ] [ u ] l ¯ , where ⋃ l ∈ ( 0 , 1 ] [ u ] l ¯ is the closure of ⋃_l∈(0,1] [u]  ^l . Then we know that [ u ] l = [ u _ ( l ) , u ¯ ( l ) ] is a bounded closed interval.

Definition 2.2. [33] The distance 𝔻 0 [ u 1 , u 2 ] between two fuzzy numbers is defined as 𝔻 0 [ u 1 , u 2 ] = sup 0 ≤ l ≤ 1 H ( [ u 1 ] l , [ u 2 ] l ) , u 1 , u 2 ∈ E , where H ( [ u 1 ] l , [ u 2 ] l ) = max { | u _ 1 ( l ) - u _ 2 ( l ) | , | u ¯ 1 ( l ) - u ¯ 2 ( l ) | } is the Hausdorff distance between [u₁]  ^l and [u₂]  ^l .

Definition 2.3.[34] Basing on Zadehs extension principle, the addition and scalar multiplication in E are defined by: [ u 1 + u 2 ] l = [ u 1 ] l + [ u 2 ] l = { x 1 + x 2 | x 1 ∈ [ u 1 ] l , x 2 ∈ [ u 2 ] l } , [ k · u 1 ] l = k [ u 1 ] l = { kx | x ∈ [ u ] l } , l ∈ [ 0 , 1 ] . Let us denote 0 ˆ ∈ E the zero element of E as follows: 0 ˆ ( x 0 ) is equal to 1 if x₀ = 0 and 0 ˆ ( x 0 ) is equal to 0 if x₀ = 0, where 0 is the zero element of ℝ . for any u 1 , u 2 , u 3 , u 4 , u ∈ E , k 1 ≥ 0 , k 2 ≥ 0 , k ∈ ℝ , the next properties hold:

u₁ + (u₂ + u₃) = (u₁ + u₂) + u₃ ; (k₁ + k₂) u₁ = k₁u₁ + k₂u₁ ; k (u₁ + u₂) = ku₁ + ku₂ .

Definition 2.4. [7] Let u₁, u₂ ∈ E. If there exists u₃ ∈ E such that u₁ = u₂ + u₃, then u₃ is said to be the Hukuhara difference of u₁ and u₂ and it is denoted by u₁ ⊖ u₂. It is clear, the existence of the difference is conditional and depends on the existence of fuzzy number u₃. We note that u₁ ⊖ u₂ ≠ u₁ + (-1) u₂. Then, a generalization of the Hukuhara difference aims to overcome this situation. The generalized Hukuhara difference of two fuzzy numbers u₁, u₂ ∈ E (for short gH-difference) 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 .

In terms of the l-levels, we notice (see [5]): [ u 1 ⊖ gH u 2 ] l = [ min { u _ 1 ( l ) - u _ 2 ( l ) , u ¯ 1 ( l ) - u ¯ 2 ( l ) } , max { u _ 1 ( l ) - u _ 2 ( l ) , u ¯ 1 ( l ) - u ¯ 2 ( l ) } ] , l ∈ [ 0 , 1 ] , and the conditions for the existence of u 1 ⊖ gH u 2 in the case (i) are

{ u _ 3 ( l ) = u _ 1 ( l ) - u _ 2 ( l ) and u ¯ 3 ( l ) = u ¯ 1 ( l ) - u ¯ 2 ( l ) , ∀ l ∈ [ 0 , 1 ] , with u _ 3 ( l ) increasing , u ¯ 3 ( l ) decreasing , u _ 3 ( l ) ≤ u ¯ 3 ( l ) .

and in the case (ii) are

{ u _ 3 ( l ) = u ¯ 1 ( l ) - u ¯ 2 ( l ) and u ¯ 3 ( l ) = u _ 1 ( l ) - u _ 2 ( l ) , ∀ l ∈ [ 0 , 1 ] , with u _ 3 ( l ) increasing , u ¯ 3 ( l ) decreasing , u _ 3 ( l ) ≤ u ¯ 3 ( l ) .

Definition 2.5. [17] Let u : [0, b] → E. The diameter of the l-level set of the fuzzy function u is defined as d ( [ u ( t ) ] l ) = u ¯ ( l , t ) - u _ ( l , t ) . The fuzzy function u is called d-increasing or d-decreasing on [0, b] if for every l ∈ [0, 1], the diameter of the fuzzy function u (t) is nondecreasing or nonincreasing, respectively, on [0, b]. If u is d-increasing or d-decreasing on [0, b], then we say that u is d-monotone on [0, b].

Definition 2.6. [7] Let u : (0, b) → E and t₀ ∈ (0, b). The fuzzy function u is called generalized Hukuhara differentiable at t₀, if there exists an element u′ (t₀) ∈ E such that u ′ ( t 0 ) = lim h → 0 u ( t 0 + h ) ⊖ gH u ( t 0 ) h .

Let C ([- τ, b] ; E) be the space of all continuous fuzzy functions from [- τ, b] into E, which endowed with the supremum metric: 𝔻 [ - τ , b ] [ u , 0 ˆ ] = sup t ∈ [ - τ , b ] 𝔻 0 [ u ( t ) , 0 ˆ ] , AC ([0, b] ; E) the set of all absolutely continuous fuzzy functions on the interval [0, b] with values in E and L ([0, b] ; E) be the set of all fuzzy functions u : [0, b] → E such that the functions t ↦ 𝔻 0 [ u ( t ) , 0 ˆ ] belongs to L [0, b]. Let γ = α + β - αβ, then 1 - γ = (1 - α) (1 - β), we let C γ ( [ 0 , b ] ; E ) = { u : ( t ρ ρ ) 1 - γ u ( t ) ∈ C ( [ 0 , b ] ; E ) } with the distance 𝔻 [ 0 , b ] γ [ u , 0 ˆ ] = sup t ∈ [ 0 , b ] 𝔻 0 [ ( t ρ ρ ) 1 - γ u ( t ) , 0 ˆ ] . We denote by A ([0, b] ; E) the family of all fuzzy functions u ∈ AC ([0, b] ; E)

We now recall some concepts of the fuzzy-type fractional calculus theory.

Definition 2.7. ([5], Definition 3.5.1) Let u ∈ L ([0, b] ; E), then the Riemann-Liouville generalized fractional integral of order α of the fuzzy function x is defined as follows: ( ρ I 0 + α u ) ( t ) = ρ 1 - α Γ ( α ) ∫ 0 t s ρ - 1 ( t ρ - s ρ ) α - 1 u ( s ) ds , α > 0 , t > 0 , where Γ is the gamma function defined by Γ ( α ) = ∫ 0 ∞ t α - 1 e - t dt .

Definition 2.8. 2.8[20] Let order α and type β satisfy 0 < α ≤ 1 and 0 ≤ β ≤ 1. The fuzzy Hilfer-Katugampola generalized Hukuhara fractional derivative (or Hilfer-Katugampola gH-fractional derivative) (left-sided/right-sided), with respect to t, with ρ > 0 of a fuzzy function u ∈ C_1-γ ([0, b] ; E) is defined by ( ρ D 0 + α , β u ) ( t ) = ( ρ I 0 + ( 1 − α ) β u ( s 1 - ρ d ds ) ( ρ I 0 + ( 1 - α ) ( 1 - β ) u ) ] ( t ) = [ ( ρ I 0 + ( 1 − α ) β δ ρ ) ( ρ I 0 + ( 1 - α ) ( 1 - β ) u ) ] ( t ) , if the gH-derivative u ( 1 - α ) , ρ ′ ( t ) exists for t ∈ [0, b], where u ( 1 - α ) , ρ ( t ) = ( ρ I 0 + ( 1 - α ) u ) ( t ) .

{Remark 2.1.

Similar to the Definition 3.5.2 in [5], Definition 2.8 can be defined.

3 Existence and finite-time stability results

In the section, we study existence of the solution, and the finite time stability for the fuzzy problem (1.1). Based on Lemma 2, similar the proof of Lemma 3.1 in [20], we are able to get the next lemma.

Lemma 3.1. Let f : [0, b] × C ([- τ, 0] ; E) → E be a continuous fuzzy function, φ ∈ C ([- τ, 0] ; E). Then a d-monotone fuzzy function x ∈ C ([0, b] ; E) is a solution of initial value problem (1.1), if and only if x satisfies the integral equation

x ( t ) ⊖ gH ( t ρ ρ ) γ - 1 Γ ( γ ) x 0 = ρ 1 - α Γ ( α ) ∫ 0 t s ρ - 1 ( t ρ - s ρ ) α - 1 f ( s , x s ) ds , t ∈ ( 0 , b ] ,(2) and x (t) = φ (t) ,   t ∈ [- τ, 0], and the fuzzy function t ↦ ρ I 0 + 1 - γ f ( t , x t ) is d-increasing on (0, b].

Remark 3.1. The main idea of the proof for Lemma 3.1 in [20] comes from Lemma 3.1 in [19] and section 4.2 in [5, pp.128–130].

Based on definition 2.4 in [35], we give the definition of finite time stability for fuzzy problem (1.1).

Definition 3.1. The fuzzy problem (1.1) is said to be finite time stable with respect to {0, J, τ, δ, ɛ}, 0 < δ < ɛ , δ , ɛ ∈ ℝ , such that for any solution x of fuzzy problem (1.1), if and only if 𝔻 0 [ x 0 , 0 ˆ ] < δ and 𝔻 0 [ φ , 0 ˆ ] < δ , implies a solution x of fuzzy problem (1.1) satisfying 𝔻 [ 0 , b ] γ [ x , 0 ˆ ] < ɛ . For simplicity, we put 𝕨 ( x ) = { x ∈ C γ ( [ 0 , b ] ; E ) : x satisfies ( ref e 3.1 ) } Lemma 3.2. Let f : [0, b] × C ([- τ, 0] ; E) → E be a continuous fuzzy function, φ ∈ C ([- τ, 0] ; E). Assume that the following hypotheses hold:

for all v ∈ C ([- τ, 0] ; E) and every t ∈ [0, b], there exist a positive constants L₁ such that 𝔻 0 [ f ( t , v ) , 0 ˆ ] ≤ L 1 ( t ρ ρ ) 1 - γ 𝔻 [ - τ , 0 ] [ v , 0 ˆ ] , ∀ v ∈ E , with L 1 ∈ [ 0 , Γ ( α + 1 ) ( ρ b ρ ) 1 + α - γ ) .

In this section, we will present the following example to illustrate our main results. We consider the following fuzzy problem:

{ ( ρ D 0 + α , β x ) ( t ) = λ 1 x ( t ) + λ 2 x t + d ( t ) = λ 1 x ( t ) + λ 2 x ( t - τ ) + d ( t ) , ρ > 0 , α ∈ ( 0 , 1 ) , β ∈ [ 0 , 1 ] t ∈ ( 0 , b ] , ( ρ I 0 + ( 1 - α ) , ( 1 - β ) x ) ( 0 + ) = x 0 , x ( t ) = φ ( t ) , t ∈ [ - τ , 0 ] ,(9) where λ 1 ∈ ℝ , λ 2 = L 1 ( t ρ ρ ) 1 - γ , d is continuous.

On the one hand, if we take λ₁ = 0, then we know f (t, x _t ) = λx (t - τ) + d (t). It is easy to check that f is a continuous fuzzy function and satisfies hypotheses (H1) and (H2). We now choose ρ = 1 2 , α = 1 2 , β = 1 3 , γ = 2 3 , b = 1 , τ = 1 , and L 1 = { 10 - 2 ρ 1 + α - γ , k = 0 , 10 - k + 2 k ρ 1 + α - γ , k > 0 . Then we have M ∗ E α ( L 1 ( b ρ ρ ) 1 - γ b α ρ ρ α ) = M ∗ × ∑ k = 0 ∞ 10 - ( k + 2 ) Γ ( k α + 1 ) ≈ 0.8198 < 1 , {where M_∗ is the same symbol as M_∗ of Theorem 3.2. Summarizing the above, all hypotheses of Theorems 3.1 3.2 are fulfilled. Consequently, we deduce that problem (4.1) has at least one solution and is finite time stable. On the other hand, by Theorem 4.3 in [21] and example 3.4 [20], then the solution x (t) of 4.1) expressed by x ( t ) ⊖ x 0 Γ ( γ ) ( t ρ ρ ) γ - 1 = E α , 1 ( λ 1 ( t ρ ρ ) α ) x 0 + 1 ρ α - 1 ∫ 0 t s ρ - 1 ( t ρ - s ρ ) α - 1 E α , α ( λ 1 ( t ρ ρ ) α ) [ λ 2 φ ( s - τ ) + d ( s ) ] ds ,(10) where E _α,β (·) is the Mittag-Leffler function defined by E α , β ( t ) = ∑ k = 0 ∞ t k Γ ( k α + β ) . We take λ₁ = 0, d (t) = t ^p  (d₁, d₂, d₃) , φ (t) = (1 - t², 2 - t², 3 - t²) , x₀ = (1, 2, 3), here p > 0, and (d₁, d₂, d₃) is a triangular fuzzy number. By (4.2), we obtain d-increasing solution of problem (4.1) on [0, 1] as follows: x ( t ) = ( Θ ( t ) + 1 , 2 Θ ( t ) + 2 , 3 Θ ( t ) + 3 ) + L 1 ρ α - 1 ∫ 0 t s ρ - 1 ( t ρ - s ρ ) α - 1 × ( s ρ ρ ) 1 - γ × ( 2 s - s 2 , 1 + 2 s - s 2 , 2 + 2 s - s 2 ) ds + 1 ρ α - 1 ∫ 0 t s ρ - 1 ( t ρ - s ρ ) α - 1 s p ( d 1 , d 2 , d 3 ) ds where Θ ( t ) = 1 Γ ( γ ) ( t ρ ρ ) γ - 1 . This solution is shown in Fig. 1.