Hadamard-type fractional calculus for fuzzy functions and existence theory for fuzzy fractional functional integro-differential equations

Abstract

The result of this paper presents the fundamental theory of fuzzy fractional calculus in the Caputo-Hadamard setting. The existence and uniqueness of solution of the initial value problem for fuzzy functional fractional integro-differential equations involving Caputo-Hadamard fractional derivative are investigated.

Keywords

Fractional calculus fuzzy fractional Hadamard derivative fuzzy Caputo-Hadamard fractional derivative

1 Introduction

In recent studies, fractional calculus and fractional differential equation models have been applied in various areas of engineering, mathematics, physics and bioengineering, and other applied sciences. For some fundamental results in the theory of fractional differential equations involving Riemann-Liouville fractional derivative and Caputo derivative, we refer the reader to monographs of Kilbas [27]. Recently, fractional differential equations with Hadamard derivative and Hadamard-Caputo derivative have attracted the attention of several researchers (see [1 , 43] and the references therein). In particular, the Caputo-Hadamard fractional derivatives were introduced by Jarad et al. [23], and it was shown that there are many advantages over the usual Hadamard fractional derivative. Gambo et al. [17] presented the fundamental theorem of fractional calculus in the Caputo-Hadamard setting based on the previous concept in [23], and very recently Almeida [10] proposed three types of Caputo-Hadamard derivatives of variable fractional order, and study the relations between them. Adjabi et al. [1] investigated the existence and uniqueness of the solution to the initial value problem of Caputo-Hadamard fractional differential equations by using Banach’s fixed point theorem. Meanwhile, Yukunthorn et al. [43] studied the existence of solutions for an impulsive hybrid systems of Caputo-Hadamard fractional differential equations equipped with integral boundary conditions by using the fixed point theorems.

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 [11–14 , 38–40] 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, fuzzy fractional calculus and fuzzy fractional differential equations have emerged as the significant topic, and the topic of the fractional calculus and fractional dynamic systems in a fuzzy setting can be applied as an important mathematical tool for modeling of practical systems with the effects of uncertainties. Therefore, the consideration and analysis of fractional order uncertain dynamical systems are essential in both research and practice, and so this field has attracted significant interests among the researchers in the last years [2–9 , 26] and the references therein. For some fundametal results in the theory of fuzzy fractional differential equations involving fuzzy-type Riemann-Liouville fractional derivative and fuzzy-type Caputo fractional derivative, the existence and uniqueness results of solution to fuzzy fractional differential equations and the method to find analytical, numerical solutions for initial value problems of fuzzy fractional differential equations, we refer the reader to the recent papers [18–21 , 42] and the references therein.

Most problems studied in the previous works involve only concepts of Riemann-Liouville fuzzy fractional derivative and Caputo fuzzy fractional derivative with non-integer α ∈ (0, 1), and there is no other study for an initial value problem of fuzzy fractional integro-differential equations in the case of a fractional derivative involving Caputo-Hadamard fuzzy fractional derivative. We, therefore, in this study introduce the Caputo-Hadamard fuzzy fractional derivative concept in the case of α ∈ (0, 1), and investigate the existence and uniqueness results of the solution for an initial value problem of fuzzy fractional functional integro-differential equations of the form ${^{C - H} D_{a^{+}}^{α} u (t) = f (t, u_{t}) + \int_{a}^{t} g (t, s, u_{s}) d s, \in [a, T], u (t) = φ (t), t \in [a - σ, a],$ (1.1) where $C - H D_{a^{+}}^{α, p}$ is the fuzzy Caputo-Hadamard fractional generalized Hukuhara derivative, f : [a, T] × C _σ → E, g : [a, T] × [a, T] × C _σ → E are fuzzy functions. Our aims are to introduce the mathematical foundations for studies of fuzzy Caputo-Hadamard fractional Hukuhara derivative which involves a fuzzy derivative of non-integer order α ∈ (0, 1), and prove the existence and uniqueness results of solution for the initial value problems (1).

The remainder of this article is arranged as follows. In Section 2, the fuzzy Hadamard fractional integral and fuzzy Caputo-Hadamard fractional derivative with non-integer order α ∈ (0, 1) for fuzzy functions are introduced. Some properties of fractional integral and derivative for a fuzzy function are also presented. We present the existence and uniqueness of solution of the initial value problem (1) in Section 3.

2 Fuzzy Hadamard fractional calculus

This section is devoted to the basic concepts of Caputo-Hadamard type fractional calculus in fuzzy setting. We also establish some auxiliary lemma to define the solution for the given problem. Fisrt of all, we recall definitions of Hadamard and Caputo-Hadamard of real-valued functions. Let C [a, b] be the space of all continuous functions ψ from [a, b] into $ℝ$ , AC [a, b] denotes the space of absolutely continuous functions from [a, b] into $ℝ$ . The Hadamard fractional integral of order α ∈ (0, 1) for a real function ψ ∈ L ¹ [a, b] is defined by $(H I_{a^{+}}^{α} ψ) (t) = \frac{1}{Γ (α)} \int_{a}^{t} {(ln \frac{t}{s})}^{α - 1} ψ (s) \frac{ds}{s} .$

The Hadamard fractional derivative of order α ∈ (0, 1) for a real function ψ ∈ AC [a, b] is defined by $(H D_{a^{+}}^{α} ψ) (t) = \frac{1}{Γ (1 - α)} (t \frac{d}{dt}) \int_{a}^{t} {(ln \frac{t}{s})}^{- α} ψ (s) \frac{ds}{s} .$

If ψ ∈ L ¹ [a, b] is a real function such that $(H D_{a^{+}}^{α} ψ) (t)$ exists on [a, b], then the Caputo-Hadamard fractional derivative $(C - H D_{a^{+}}^{α} ψ)$ of order α ∈ (0, 1) is defined by $\begin{matrix} (C - H D_{a^{+}}^{α} ψ) (t) = (H D_{a^{+}}^{α, p} [ψ (\cdot) - ψ (a)]) (t) . \end{matrix}$

If ψ ∈ AC [a, b], then we have $(C - H D_{a^{+}}^{α} ψ) (t) = \frac{1}{Γ (1 - α)} \int_{a}^{t} (ln t - ln s)^{- α} \frac{d}{ds} ψ (s) ds .$

Remark 2.1. Corresponding to upper Dini derivative D ⁺, one can define Caputo-Hadamard fractional upper Dini derivative from the relation, namely,

$(C - H D_{a^{+}}^{+ α} ψ) (t) = \frac{1}{Γ (1 - α)} \int_{a}^{t} (ln t - ln s)^{- α} D^{+} ψ (s) ds,$ where $D^{+} ψ (t) = \underset{h \to 0^{+}}{lim sup} \frac{ψ (t + h) - ψ (t)}{h} .$ Furthermore, we can obtain the relation $C - H D_{a^{+}}^{+ α} | ψ (t) | \leq | C - H D_{a^{+}}^{α} ψ (t) |,$ when $C - H D_{a^{+}}^{α} ψ (t)$ exists.

Next, we recall some preliminaries about the fuzzy numbers mentioned in [14, 29]. Let E be the class of fuzzy numbers, i.e. normal, convex, upper semicontinuous and compactly supported fuzzy subsets of the real numbers. For r ∈ (0, 1], denote $[u]^{r} = {z \in ℝ | u (z) \geq r}$ and $[u]^{0} = \bar{{z \in ℝ | u (z) > 0}} .$ For u, v ∈ E, and $λ \in ℝ$ , the sum u + v and the product λ · u are defined by [u + v] = [u] ^r + [v] ^r, [λ · u] ^r = λ [u] ^r, ∀ r ∈ [0, 1], where [u] ^r + [v] ^r means the usual addition of two intervals of $ℝ$ and λ [u] ^r means the usual product between a scalar and a real interval number. For u ∈ E, we define the diameter of the r-level set of u as $d ([u]^{r}) = \bar{u} (r) - \underline{u} (r) .$ Let u ₁, u ₂ ∈ E.

Definition 2.1. [29] Let u, v ∈ E. If there exists w ∈ E such that u = v + w, then w is called the Hukuhara difference of u and v and it is denoted by u ⊖ v .

Definition 2.2. [29] Let u, v ∈ E. The Hausdorff distance between u and v is defined by $D_{0} [u, v] = sup_{0 \leq r \leq 1} max {| \underline{u} (r) - \underline{v} (r) |, | \bar{u} (r) - \bar{v} (r) |} .$

Definition 2.3. [14] Let u : (a, b) → E and t ∈ (a, b). The fuzzy function u is said to be generalized Hukuhara differentiable (gH-differentiable) at t, if there exists an element u′ (t) ∈ E such that $\begin{matrix} u^{'} (t) = lim_{h \to 0} \frac{u (t + h) ⊖_{gH} u (t)}{h}, \end{matrix}$ where $\underset{gH}{⊖}$ denotes the generalized Hukuhara difference of two fuzzy numbers u, v ∈ E (gH-difference for short) defined as follows [14]: $u \underset{gH}{⊖} v = w \Leftrightarrow {tabular l < row 1 > < Col 1 > (i) u = v + w, < / Col 1 > < / row 1 > < row 1 > < Col 1 > or (ii) v = u + (- 1)w. < / Col 1 > < / row 1 > tabular$

Definition 2.4. Let u : [a, b] → E. We say that u is d-increasing (d-decreasing) on [a, b] if for every r ∈ [0, 1] the function t ↦ d ([u (t)] ^r) is nondecreasing (nonincreasing) on [a, b]. Furthermore, if u is d-increasing or d-decreasing on [a, b], then we say that u is d-monotone on [a, b].

We define some notations which are used throughout the paper as follows: C ([a, b], E) denotes the set of all continuous fuzzy functions and AC ([a, b], E) denotes the set of all absolutely continuous fuzzy functions on [a, b]. Let L ([a, b], E) be the set of all fuzzy functions u : [a, b] → E such that the functions $t \mapsto D_{0} [u (t), \hat{0}]$ belongs to L ¹ [a, b].

Definition 2.5. Let u ∈ L ([a, b], E), then the Hadamard fractional integral of order α > 0 of the fuzzy function u is defined as follows $(H ℑ_{a^{+}}^{α} u) (t) = \frac{1}{Γ (α)} \int_{a}^{t} (ln t - ln s)^{α - 1} u (s) \frac{ds}{s}, t \geq a,$ where Γ (α) is the well-known Gamma function.

Example 2.1. Let t > a > 0, α ∈ (0, 1), γ > 0 and a fuzzy function u (t) = (- (ln t - ln a) ^γ, 0, (ln t - ln a) ^γ). Then, by Definition 2.5 the Hadamard fractional integral of u (t) is $(H ℑ_{a^{+}}^{α} u) (t) = \frac{Γ (γ + 1)}{Γ (α + γ + 1)} (ln t - ln a)^{α + γ} (- 1, 0, 1) .$

Remark 2.2. Denote $∥ u ∥_{0} = sup_{t \in [a, b]} D_{0} [u (t), \hat{0}] .$ We observe that that the operator $(H ℑ_{a^{+}}^{α} u) (t)$ is linear and bounded. Indeed, we have $\begin{matrix} ∥ (H ℑ_{a^{+}}^{α} u) (t) ∥_{0} & \leq \frac{∥ u ∥_{0}}{Γ (α)} \int_{a}^{t} (ln t - ln s)^{α - 1} \frac{ds}{s} \\ \leq \frac{∥ u ∥_{0}}{Γ (α + 1)} (ln b - ln a)^{α} . \end{matrix}$ (2.2)

Remark 2.3. Let α, β > 0. Similar to the Hadamard fractional integral of the real-valued functions, the following properties holds: if u, v ∈ L ([a, b], E) and α, β > 0, then it is obvious that:

$H ℑ_{a^{+}}^{α} H ℑ_{a^{+}}^{β} u (t) = H ℑ_{a^{+}}^{α + β} u (t),$ for t∈ [a, b];

$H ℑ_{a^{+}}^{α} (u + v) (t) = H ℑ_{a^{+}}^{α} u (t) + H ℑ_{a^{+}}^{α} v (t),$ for t∈ [a, b];

$d ([H ℑ_{a^{+}}^{α} u (t)]^{r}) = H ℑ_{a^{+}}^{α} d ([u (t)]^{r})$ .

Remark 2.4. Let u : [a, b] → E be such that u ∈ L ([a, b], E) and t ↦ d ([u (t)] ^r), r ∈ [0, 1] is increasing on [a, b] . Similar to the proof of Lemma 2 in [21], we can show that the diameters of the r-level set of $H ℑ_{a^{+}}^{α} u (t)$ and $H ℑ_{a^{+}}^{1 - α} u (t)$ are also increasing on [a, b] for each r ∈ [0, 1], where α ∈ (0, 1) .

Definition 2.6. The fuzzy Hadamard generalized Hukuhara fractional derivative (or Hadamard gH-fractional derivative) of order α ∈ (0, 1) of u is defined $(H D_{a^{+}}^{α} u) (t) : = t {(H ℑ_{a^{+}}^{1 - α} u)}^{'} (t), t \in [a, b],$ if the gH-derivative ${(H ℑ_{a^{+}}^{1 - α} u)}^{'} (t)$ exists for t ∈ [a, b].

The following assertions can be inferred from Remark 4 in [21].

Remark 2.5. Let u, v ∈ AC ([a, b], E) be d-monotone on [a, b], and let α ∈ (0, 1). We have the following properties:

If $(H ℑ_{a^{+}}^{1 - α} u) (t)$ and $(H ℑ_{a^{+}}^{1 - α} v) (t)$ are equally d-monotonic on (a, b], then for t ∈ (a, b] $D_{a^{+}}^{α} (u \underset{gH}{⊖} v)) (t) = (H D_{a^{+}}^{α} u) (t) \underset{gH}{⊖} (H D_{a^{+}}^{α} v) (t) .$

If $(H ℑ_{a^{+}}^{1 - α} u) (t)$ and $(H ℑ_{a^{+}}^{1 - α} v) (t)$ are differently d-monotonic on (a, b], then $H D_{a^{+}}^{α} (u \underset{gH}{⊖} v)) (t) = (H D_{a^{+}}^{α} u) (t) + (- 1) (H D_{a^{+}}^{α} v) (t) .$

The following assertions can be inferred from Remark 3 in [21] and Theorem 2.4 in [14].

Remark 2.6. Let u ∈ AC ([a, b], E). Then:

If either u is d-increasing on [a, b] or u is d-decreasing and $(H ℑ_{a^{+}}^{1 - α} u) (t)$ is d-increasing on [a, b], then for r ∈ [0, 1] $[(H D_{a^{+}}^{α} u) (t)]^{r} = [H D_{a^{+}}^{α} \underline{u} (t, r), H D_{a^{+}}^{α} \bar{u} (t, r)] .$

If $(H ℑ_{a^{+}}^{1 - α} u) (t)$ is d-decreasing on [a, b], then for r ∈ [0, 1] $[(H D_{a^{+}}^{α} u) (t)]^{r} = [H D_{a^{+}}^{α} \bar{u} (t, r), H D_{a^{+}}^{α} \underline{u} (t, r)] .$

Example 2.2. Let a = 1, α ∈ (0, 1), and fuzzy functions u (t) = (ln t, 2 ln t, 3 ln t), $v (t) = (1 / \sqrt{ln t} - 1,$ $0, 1 - 1 / \sqrt{ln t}), t \in (1, 2]$ . Since d ([u (t)] ^r) = (2 - 2r) ln t is increasing for all t ∈ (1, 2], r ∈ [0, 1], and by Remark 2.6, we have that (see Fig. 1) $\begin{matrix} (H D_{1^{+}}^{α} u) (t) = \frac{(ln t)^{1 - α}}{Γ (2 - α)} (1, 2, 3) . \end{matrix}$ Similarly, because $(H ℑ_{1^{+}}^{1 - α} v) (t)$ is d-decreasing on (1, 2) (see Fig. 2), by Remark 2.6, one has (see Fig. 3) $(^{H} D_{1^{+}}^{α} v) (t) = (\begin{matrix} \frac{Γ (1 / 2) (1 / 2 - α)}{Γ (3 / 2 - α)} (ln t)^{- (1 / 2 + α)} - \frac{{(ln t)}^{- α}}{Γ (1 - α)}, \\ 0, \\ \frac{{(ln t)}^{- α}}{Γ (1 - α)} - \frac{Γ (1 / 2) (1 / 2 - α)}{Γ (3 / 2 - α)} (ln t)^{- (1 / 2 + α)} \end{matrix}) .$

Fig. 1.

Hadamard fractional derivative of order α = 0.75.

Fig. 2.

Graph of $(H ℑ_{1^{+}}^{1 - α} v) (t)$ .

Fig. 3.

Hadamard fractional derivative of order α = 0.75.

Proposition 2.1. Let u ∈ L ([a, b], E) and α, β ∈ (0, 1), then for 0 < α < β < 1 we have $\begin{matrix} (H D_{a^{+}}^{α} H ℑ_{a^{+}}^{α} u) (t) = u (t), t \in (a, b], \end{matrix}$ (2.3) and $\begin{matrix} (H D_{a^{+}}^{α} H ℑ_{a^{+}}^{β} u) (t) = (H ℑ_{a^{+}}^{(β - α)} u) (t), t \in (a, b] . \end{matrix}$ (2.4)

proof. For t ∈ (a, b], by Dirichlet formula, the known formula for the Beta function, and by setting x = (ln τ - ln s)/(ln t - ln s) we have $\begin{matrix} (H D_{a^{+}}^{α} H ℑ_{a^{+}}^{α} u) (t) \\ = \frac{t}{Γ (1 - α)} {(\begin{matrix} \int_{a}^{t} {(ln t - ln s)}^{- α} \\ \cdot (\frac{1}{Γ (α)} \int_{a}^{s} {(ln s - ln τ)}^{α - 1} u (τ) \frac{d τ}{τ}) \frac{ds}{s} \end{matrix})}^{'} \\ = \frac{t}{Γ (α) Γ (1 - α)} {(\int_{a}^{t} u (s) (\int_{s}^{t} \frac{(ln τ - ln s)^{α - 1}}{(ln t - ln τ)^{α}} \frac{d τ}{τ}) \frac{ds}{s})}^{'} \end{matrix}$ $\begin{matrix} = \frac{t}{Γ (α) Γ (1 - α)} {(\begin{matrix} \int_{a}^{t} u (s) \frac{ds}{s} \\ \cdot \int_{0}^{1} {(1 - x)}^{1 - α - 1} x^{α - 1} dx \end{matrix})}^{'} \\ = \frac{B (1 - α, α)}{Γ (α) Γ (1 - α)} t {(\int_{a}^{t} \frac{u (s)}{s} ds)}^{'} \\ = u (t) . \end{matrix}$ This completes the proof of (2.3). The proof of (2.4) is similar. Indeed, $\begin{matrix} (H D_{a^{+}}^{α} H ℑ_{a^{+}}^{β} u) (t) \\ = \frac{t}{Γ (β) Γ (1 - α)} {(\begin{matrix} \int_{a}^{t} {(ln t - ln s)}^{β - α} u (s) \frac{ds}{s} \\ \cdot \int_{0}^{1} {(1 - x)}^{1 - α - 1} x^{β - 1} dx \end{matrix})}^{'} \\ = \frac{(β - α)}{Γ (β - α + 1)} \int_{a}^{t} (ln t - ln s)^{β - α - 1} u (s) \frac{ds}{s} \\ = (H ℑ_{a^{+}}^{(β - α)} u) (t) . \end{matrix}$ □

Definition 2.7. Let u ∈ L ([a, b], E) be a fuzzy function such that $H D_{a^{+}}^{α} u$ exists on [a, b]. The fuzzy Caputo-Hadamard fractional derivative of order α ∈ (0, 1) of u is defined $\begin{matrix} (C - H D_{a^{+}}^{α} u) (t) : = \\ (H D_{a^{+}}^{α} [u (\cdot) \underset{gH}{⊖} u (a)]) (t), t \in [a, b] . \end{matrix}$

Similar to Theorem 1 in [14], we can show the relations between fuzzy Hadamard gH-fractional derivative and fuzzy Caputo-Hadamard fractional as in the following theorem.

Theorem 2.3. If u ∈ AC ([a, b], E) is a d-monotone fuzzy function and α ∈ (0, 1), then $(C - H D_{a^{+}}^{α} u) (t) = \frac{1}{Γ (1 - α)} \int_{a}^{t} {(\ln t - \ln s)}^{- α} u^{'} (s) d s .$ (2.5)

proof. Fist, by Lemma 6 in [17] we have $\begin{matrix} C - H D_{a^{+}}^{α} ψ) (t) = H D_{a^{+}}^{α} ψ (t) - \frac{ψ (a)}{Γ (1 - α)} (ln t - ln a)^{- α}, \end{matrix}$ (2.6) where $(C - H D_{a^{+}}^{α} ψ) (t)$ and $H D_{a^{+}}^{α} ψ (t)$ are Caputo-Hadamard fractional derivative and Hadamard fractional derivative of the real-valued function ψ (t) . In addition, the Caputo-Hadamard fractional derivative of the real function ψ (t) is defined by $\begin{matrix} (C - H D_{a^{+}}^{α} ψ) (t) = (H I_{a^{+}}^{1 - α} (s \frac{d}{ds} ψ)) (t) \\ = \frac{1}{Γ (1 - α)} \int_{a}^{t} (ln t - ln s)^{- α} \frac{d}{ds} ψ (s) ds . \end{matrix}$ (2.7)

Because u ∈ AC ([a, b], E), it yields $(H ℑ_{a^{+}}^{1 - α} u) (t) \in AC ([a, b], E)$ . Therefore, $(H ℑ_{a^{+}}^{1 - α} u)^{'} (t)$ exists for t ∈ (a, b] and $(H D_{a^{+}}^{α} u) (t) : = t (H ℑ_{a^{+}}^{1 - α} u)^{'} (t)$ exists for t ∈ (a, b]. By putting z (t) : = u (a), one has $(H ℑ_{a^{+}}^{1 - α} z) (t) = \frac{(ln t - ln a)^{1 - α}}{Γ (2 - α)} u (a)$ , and so $(H ℑ_{a^{+}}^{1 - α} z) (t)$ is d-increasing on (a, b].

If u is d-increasing on [a, b] or u is d-decreasing on [a, b] and $(H ℑ_{a^{+}}^{1 - α} u) (t)$ is d-increasing on (a, b], then by Remark 2.5 it follows that $\begin{matrix} (C - H D_{a^{+}}^{α} u) (t) = (H D_{a^{+}}^{α} [u (\cdot) \underset{gH}{⊖} u (a)]) (t) \\ = (H D_{a^{+}}^{α} (u \underset{gH}{⊖} z)) (t) \\ = (H D_{a^{+}}^{α} u) (t) \underset{gH}{⊖} (H D_{a^{+}}^{α} z) (t) \\ = (H D_{a^{+}}^{α} u) (t) \underset{gH}{⊖} \frac{(ln t - ln a)^{- α}}{Γ (1 - α)} u (a) . \end{matrix}$ (2.8) Moreover, by Remark 2.6 and (2.6) for every r ∈ [0, 1] we have that $\begin{matrix} D_{a^{+}}^{α} u (t)]^{r} & = & [(H D_{a^{+}}^{α} \underline{u}) (r, t), (H D_{a^{+}}^{α} \bar{u}) (r, t)] \\ = & [(C - H D_{a^{+}}^{α} \underline{u}) (r, t), (C - H D_{a^{+}}^{α} \bar{u}) (r, t)] \\ + & \frac{(ln t - ln a)^{- α}}{Γ (1 - α)} [\underline{u} (a, r), \bar{u} (a, r)] \end{matrix}$ (2.9) for t ∈ (a, b] and for every r ∈ [0, 1], that is, $(H D_{a^{+}}^{α} u) (t) = (H ℑ_{a^{+}}^{1 - α} (s u^{'})) (t) + \frac{(ln t - ln a)^{- α}}{Γ (1 - α)} u (a) .$ (2.10)

From (2.8) and (2.10) we get (2.5). If u is d-decreasing on [a, b] and $(H ℑ_{a^{+}}^{1 - α} u) (t)$ is d-decreasing on (a, b], then by Remark 2.5 it yields that $(C - H D_{a^{+}}^{α} u) (t) = (^{H} D_{a^{+}}^{α} [u (\cdot) ⊖_{g H} u (a)]) (t) - 1.5 p c = (^{H} D_{a^{+}}^{α} (u ⊖_{g H} z)) (t) - 1.5 p c = (^{H} D_{a^{+}}^{α} u) (t) + (- 1) (^{H} D_{a^{+}}^{α} z) (t) - 1.5 p c = (^{H} D_{a^{+}}^{α} u) (t) + (- 1) \frac{{(\ln t - \ln a)}^{- α}}{Γ (1 - α)} u (a) .$ (2.11) Moreover, by Remark 2.6 and (2.6) for every r ∈ [0, 1] we get for t ∈ (a, b] $[(H D_{a^{+}}^{α} u) (t)]^{r} = [(H D_{a^{+}}^{α} \bar{u}) (r, t), (H D_{a^{+}}^{α} \underline{u}) (r, t)]$ (2.12) and $\begin{matrix} [(H I_{a^{+}}^{1 - α} (s \frac{d}{ds} \bar{u})) (r, t), (H I_{a^{+}}^{1 - α} (s \frac{d}{dt} \underline{u})) (t)] \\ = [(H D_{a^{+}}^{α} \bar{u}) (r, t), (H D_{a^{+}}^{α} \underline{u}) (r, t)] \\ + (- 1) [\frac{(ln t - ln a)^{- α}}{Γ (1 - α)} \underline{u} (r, a), \frac{(ln t - ln a)^{- α}}{Γ (1 - α)} \bar{u} (r, a)] . \end{matrix}$ Then, we obtain $\begin{matrix} (H ℑ_{a^{+}}^{1 - α} (s u^{'})) (t) = (H D_{a^{+}}^{α} u) (t) \\ + (- 1) \frac{(ln t - ln a)^{- α}}{Γ (1 - α)} u (a) . \end{matrix}$ (2.13)

From (2.11) and (2.13) we get (2.5). The proof is complete. □

Theorem 2.4. Let u ∈ AC ([a, b], E). If u is a d-monotone fuzzy function, where $[u (t)]^{r} = [\underline{u} (r, t), \bar{u} (r, t)]$ for r ∈ [0, 1], t ∈ [a, b], then for α ∈ (0, 1) we have that

${[(C - H D_{a^{+}}^{α} u) (t)]}^{r} = [(C - H D_{a^{+}}^{α} \underline{u}) (r, t), (C - H D_{a^{+}}^{α} \bar{u}) (r, t)]$ , if u is d-increasing on [a, b];

${[(C - H D_{a^{+}}^{α} u) (t)]}^{r} = [(C - H D_{a^{+}}^{α} \bar{u}) (r, t), (C - H D_{a^{+}}^{α} \underline{u}) (r, t)]$ , if u is d-decreasing on [a, b].

proof. It is well-known that if u is d-increasing, then $[u^{'} (t)]^{r} = [\frac{d}{dt} \underline{u} (r, t), \frac{d}{dt} \bar{u} (r, t)]$ . Therefore, by Theorem 2.3 one has $\begin{matrix} {[(C - H D_{a^{+}}^{α} u) (t)]}^{r} & = [\begin{matrix} \int_{a}^{t} \frac{{(ln t - ln s)}^{- α}}{Γ (1 - α)} \frac{d}{ds} \underset{0.3 em -}{u} (r, s) ds, \\ \int_{a}^{t} \frac{{(ln t - ln s)}^{- α}}{Γ (1 - α)} \frac{d}{ds} \bar{u} (r, s) ds \end{matrix}] \\ = [(C - H D_{a^{+}}^{α} \underline{u}) (r, t), (C - H D_{a^{+}}^{α} \bar{u}) (r, t)] . \end{matrix}$ If u is d-decreasing, then $[u^{'} (t)]^{r} = [\frac{d}{dt} \bar{u} (r, t), \frac{d}{dt} \underline{u} (r, t)]$ and therefore it follows that $\begin{matrix} {[(C - H D_{a^{+}}^{α} u) (t)]}^{r} & = [\begin{matrix} \int_{a}^{t} \frac{{(ln t - ln s)}^{- α}}{Γ (1 - α)} \frac{d}{ds} \bar{u} (r, s) ds \\ , \int_{a}^{t} \frac{{(ln t - ln s)}^{- α}}{Γ (1 - α)} \frac{d}{ds} \underset{0.3 em -}{u} (r, s) ds \end{matrix}] \\ = [(C - H D_{a^{+}}^{α} \bar{u}) (r, t), (C - H D_{a^{+}}^{α} \underline{u}) (r, t)] . \end{matrix}$ The proof is complete. □

Example 2.5. Let α ∈ (0, 1), t > a > 0, and a fuzzy function u (t) = (k ₁, k ₂, k ₃) (ln t - ln a) ^β, β > 0. By Theorem 2.3 and by putting z = (ln s - ln a)/(ln t - ln a), it follows that $\begin{matrix} (C - H D_{a^{+}}^{α} u) (t) \\ = \frac{(k_{1}, k_{2}, k_{3})}{Γ (1 - α)} \int_{a}^{t} (ln t - ln s)^{- α} \frac{d}{ds} (ln s - ln a)^{β} ds \\ = \frac{β (k_{1}, k_{2}, k_{3})}{Γ (1 - α)} (ln t - ln a)^{β - α} \int_{0}^{1} (1 - z)^{1 - α - 1} z^{β - 1} dz \\ = \frac{β (k_{1}, k_{2}, k_{3})}{Γ (1 - α)} (ln t - ln a)^{β - α} B (1 - α, β) \\ = \frac{Γ (β + 1)}{Γ (1 - α + β)} (ln t - ln a)^{β - α} (k_{1}, k_{2}, k_{3}) . \end{matrix}$ In addition, let α = 0.75, and a fuzzy function $v (t) = (\sqrt{ln t} - 1, 0, 1 - \sqrt{ln t}), t \in (1, 2] .$ Then, since v (t) is d-decreasing on (1, 2] (see Fig. 4), by Theorem 2.3 we also obtain (see Fig. 5) $\begin{matrix} (C - H D_{1^{+}}^{0.75} v) (t) \\ = (- \frac{Γ (3 / 2)}{Γ (3 / 4)} (ln t)^{- 1 / 4}, 0, \frac{Γ (3 / 2)}{Γ (3 / 4)} (ln t)^{- 1 / 4}) . \end{matrix}$

Fig. 4.

Graph of v (t).

Fig. 5.

Caputo-Hadamard fractional derivative of v (t).

Theorem 2.6. If u ∈ AC ([a, b], E) is a d-monotone fuzzy function and α ∈ (0, 1), then for t ∈ (a, b] $(H ℑ_{a^{+}}^{α} C - H D_{a^{+}}^{α} u) (t) = u (t) \underset{gH}{⊖} u (a) . (2.14)$ (2.14)

proof. By using the property (iii) of the fuzzy Hadamard fractional integral in Remark 2.3 and from Theorem 2.3 we have that $\begin{matrix} (H ℑ_{a^{+}}^{α} C - H D_{a^{+}}^{α} u) (t) & = (H ℑ_{a^{+}}^{α} (H ℑ_{a^{+}}^{1 - α} (s u^{'}))) (t) \\ = (H ℑ_{a^{+}}^{1} (s u^{'})) (t) \\ = \int_{a}^{t} u^{'} (s) ds \\ = u (t) \underset{gH}{⊖} u (a) . \end{matrix}$ □

In the below theorem, the existence of solution to an initial value problem for Caputo-Hadamard fractional functional integro-differential equation is presented. The result of this theorem will be used to investigated the existence and uniqueness of solution for Caputo-Hadamard fractional functional integro-differential equations in fuzzy setting in next section. Let η > 0 be a given constant, $B (ψ_{0}, η) = {ψ \in ℝ : | ψ - ψ_{0} | \leq η}$ , and $D = {(t, s) | a < s \leq t \leq b}$ . Consider the initial value problem as follows: ${\begin{matrix} (^{C - H} D_{a^{+}}^{α} ψ) (t) = p (t, ψ) + \int_{a}^{t} q (t, s, ψ) ds, \\ ψ (a) = ψ_{0} = 0, \end{matrix} (2.15)$ where $p : [a, b] \times [0, η] \to ℝ^{+}$ , $q : D \times [0, η] \to ℝ^{+}$ are the real-valued functions.

Theorem 2.7. Assume that the real-valued functions p, q satisfy the following conditions: (i) $p \in C ([a, b] \times [0, η], ℝ^{+})$ , $q \in C (D \times [0, η], ℝ^{+})$ , p (t, 0) ≡0, q (t, s, 0) ≡0, 0 ≤ p (t, ψ) ≤ M _p, 0 ≤ q (t, s, ψ) ≤ M _q for all (t, ψ)∈ [a, b] × [0, η]; (ii) p (t, ψ), q (t, s, ψ) are nondecreasing in ψ for every $t \in [a, b], (t, s) \in D$ . Then, the problem (2.15) has at least one solution defined on [a, b] and ψ (t) ∈ B (ψ ₀, η) .

proof. It is well-known that the problem (2.15) is equivalent to the following fractional integral equation: $\begin{matrix} ψ (t) = & \frac{1}{Γ (α)} \int_{a}^{t} {(ln t - ln s)}^{α - 1} \\ \cdot [p (s, ψ (s)) + \int_{a}^{s} q (s, τ, ψ (τ)) d τ] \frac{ds}{s} . \end{matrix}$ Choose t ^* > a such that $t^{*} \leq a exp (\frac{η Γ (1 + α)}{M})^{1 / α}$ , where $M = sup_{t \in [a, b]} [M_{p} + \frac{M_{q} (t - a)}{α + 1}]$ and put b ^* : = min {t ^*, b} . Let us define a sequence ${ψ}_{n = 0}^{\infty}$ of successive approximations of IVP (2.15) on [a, b ^*] as follows: $ψ^{0} (t) = \frac{M}{Γ (α + 1)} (ln t - ln a)^{α},$ $\begin{matrix} ψ^{n + 1} (t) = & \frac{1}{Γ (α)} \int_{a}^{t} {(ln t - ln s)}^{α - 1} \\ \cdot [p (s, ψ^{n} (s)) + \int_{a}^{s} q (s, τ, ψ^{n} (τ)) d τ] \frac{ds}{s} . \end{matrix}$ Then, for n = 0 we have $\begin{matrix} ψ^{1} (t) & = \frac{1}{Γ (α)} \int_{a}^{t} {(ln t - ln s)}^{α - 1} \\ \cdot [p (s, ψ^{0} (s)) + \int_{a}^{s} q (s, τ, ψ^{0} (τ)) d τ] \frac{ds}{s} \\ \leq \frac{M}{Γ (α + 1)} (ln t - ln a)^{α} = ψ^{0} (t) \leq η, \end{matrix}$

for t ∈ [a, b ^*] . Hence, by using the hypothese (ii) (p (t, λ), q (t, s, λ) is nondecreasing in ψ for every t ∈ [a, b ^*]) and proceeding recursively, we obtain $\begin{matrix} 0 \leq ψ^{n + 1} (t) \leq ψ^{n} (t) \leq . . . \leq ψ^{0} (t) \leq η, n = 1, 2, 3 . . ., \end{matrix}$ and it follows that the sequence {ψ ⁿ} is uniformly bounded for all n ≥ 0 . Next, we set $F (t) : = p (t, ψ (t)) + \int_{a}^{t} q (t, s, ψ (s)) ds$ and for any t ₁, t ₂ ∈ [a, b ^*] with t ₁ < t ₂, we have $\begin{matrix} | ψ^{n} (t_{1}) - ψ^{n} (t_{2}) | \\ \leq \frac{1}{Γ (α)} \int_{a}^{t_{1}} [{(ln t_{1} - ln s)}^{α - 1} - {(ln t_{2} - ln s)}^{α - 1}] \\ \cdot | F^{n - 1} (s) | \frac{ds}{s} \\ + \frac{1}{Γ (α)} \int_{t_{1}}^{t_{2}} (ln t_{2} - ln s)^{α - 1} | F^{n - 1} (s) | \frac{ds}{s} \\ \leq \frac{M}{Γ (α + 1)} [\begin{matrix} 2 (ln t_{2} - ln t_{1})^{α} + \\ (ln t_{1} - ln a)^{α} - (ln t_{2} - ln a)^{α} \end{matrix}] \\ \leq \frac{2 M}{Γ (α + 1)} (ln t_{2} - ln t_{1})^{α} \end{matrix}$ or $\begin{matrix} | ψ^{n} (t_{1}) - ψ^{n} (t_{2}) | \leq \frac{2 M}{Γ (α + 1)} (ln t_{2} - ln t_{1})^{α} . \end{matrix}$

Therefore, for any ɛ > 0 and any n ≥ 1, by using Mean Value Theorem we have that if |t ₂ - t ₁| < δ ₀, then it follows |ψ ⁿ (t ₁) - ψ ⁿ (t ₂) | < ɛ, where $δ_{0} : = a (\frac{ɛ Γ (1 + α)}{2 K})^{1 / α} .$ It then follows that the family ${ψ^{n} (\cdot), n \in ℕ}$ is equicontinuous and uniformly bounded. Hence, by the Arzela-Ascoli Theorem and the monotonicity of the sequence {ψ ⁿ} we can conclude that $lim_{n \to \infty} ψ^{n} (\cdot) = ψ (\cdot)$ uniformly on [a, b ^*]. Thus, ψ ∈ C ([a, b ^*], [0, η]) and ψ (t) is the solution of the initial value problem (2.15). □

Given σ > 0, let $C ([- σ, 0], ℝ)$ denote the space of continuous functions on [- σ, 0] . For any element $ψ \in C ([- σ, 0], ℝ)$ define the norm $| ψ |_{σ} = max_{s \in [- σ, 0]} | ψ (s) | .$ Suppose that $ψ \in C ([a - σ, b], ℝ), a > 0 .$ For any t ≥ a, we let ψ _t denote a translation of the restriction of ψ to the interval [t - σ, t], that is, ψ _t is an element of $C ([- σ, 0], ℝ)$ defined by ψ _t (s) = ψ (t + s), s ∈ [- σ, 0 .]

Theorem 2.8. Let $ψ : [a - σ, b] \to ℝ^{+}$ be a continuous function on the interval [a - σ, b] and satisfy the inequality

$\begin{matrix} C - H D_{a^{+}}^{α} ψ (t) \leq p (t, | ψ_{t} |_{σ}) + \int_{a}^{t} q (t, s, | ψ_{s} |_{σ}) ds, t \geq a, \end{matrix}$ where $p \in C ([a, b] \times ℝ^{+}, ℝ^{+})$ , $q \in C (D \times ℝ^{+}, ℝ^{+})$ , and p (t, ψ), q (t, s, ψ) are nondecreasing in ψ for each $t \in [a, b], (t, s) \in D$ . Assume that m (t) = m (t, a, ξ ₀) is the maximal solution of the initial value problem $^{C - H} D_{a^{+}}^{α} ξ (t) = p (t, ξ) + \int_{a}^{t} q (t, s, ξ) d s, ξ (a) = ξ_{0} ⩾ 0$ on [a, b]. Then, if |ψ _a|_σ ≤ ξ ₀, we have ψ (t) ≤ m (t), t ∈ [a, b] .

proof. We call ξ (t, ɛ) is any solution of the initial value problem $^{C - H} D_{a^{+}}^{α} ξ (t) = p (t, ξ) + \int_{a}^{t} q (t, s, ξ) d s + ε, ξ (a) = ξ_{0} + ε ⩾ 0. (2.16)$ (2.16)

Then, from (2.16) we infer that $C - H D_{a^{+}}^{α} ξ (t, ɛ) > p (t, ξ (t, ɛ)) + \int_{a}^{t} q (t, s, ξ) ds$ on [a, b]. Now, we prove that ψ (t) < ξ (t, ɛ) on [a, b]. Indeed, suppose that there exists t ₁ ∈ (a, b] such that ψ (t ₁) = ξ (t ₁, ɛ). Then, by Theorem 2.1 in [22], we get $C - H D_{a^{+}}^{α} ψ (t_{1}) \geq C - H D_{a^{+}}^{α} ξ (t_{1}, ɛ),$ which yields, $\begin{matrix} p (t_{1}, | ψ_{t_{1}} |_{σ}) + \int_{a}^{t_{1}} q (t, s, | ψ_{s} |_{σ}) ds \\ \geq C - H D_{a^{+}}^{α} ψ (t_{1}) \\ \geq C - H D_{a^{+}}^{α} ξ (t_{1}, ɛ) \\ > p (t_{1}, ξ (t_{1}, ɛ)) + \int_{a}^{t_{1}} q (t_{1}, s, ξ) ds . \end{matrix}$

This is a contradiction because ψ (t ₁) = ξ (t ₁, ɛ), ψ _{t
₁} ≤ ξ (t ₁, ɛ) and the nondecreasing property of the functions f, g. Hence, the conclusion ψ (t) < ξ (t, ɛ) on [a, b] is valid. Furthermore, because $lim_{ɛ \to 0} ξ (t, ɛ) = m (t)$ uniformly on each t ∈ [a, b], we infer that ψ (t) ≤ m (t), t ∈ [a, b] . The proof is complete. □ Based on Remark 3.2 and Remark 3.3 of paper [22], we have the following remark.

Remark 2.7. Let α ∈ (0, 1) and consider the linear fuzzy Caputo-Hadamard fractional differential equation $^{C - H} D_{a^{+}}^{α} u (t) = λ u (t) + h (t), t \in (a, b] u (a) = u_{0}, (2.17)$ Suppose that λ > 0 and the solution of (2.17) is d-increasing. Then, we obtain the following solution of problem (2.17) with the Caputo-Hadamard fractional derivative

$\begin{matrix} u (t) = u_{0} E_{α, 1} (λ {(ln \frac{t}{a})}^{α}) \\ + \int_{a}^{t} {(ln \frac{t}{s})}^{α - 1} E_{α, α} (λ {(ln \frac{t}{s})}^{α}) \frac{h (s)}{s} ds . \end{matrix}$

Suppose that λ < 0 and the solution of (2.17) is d-decreasing. Then, we obtain the following solution of problem (2.17) with the Caputo-Hadamard fractional derivative $\begin{matrix} u (t) = u_{0} E_{α, 1} (λ {(ln \frac{t}{a})}^{α}) \\ ⊖ (- 1) \int_{a}^{t} {(ln \frac{t}{s})}^{α - 1} E_{α, α} (λ {(ln \frac{t}{s})}^{α}) \frac{h (s)}{s} ds . \end{matrix}$

3 The existence and uniqueness of solution

We denote by C _σ the space C ([- σ, 0], E) equipped with the metric defined by $D_{σ} [u, v] = sup_{t \in [- σ, 0]} D_{0} [u (t), v (t)] .$

For each t ∈ [a, b] we denote by u _t the element of C _σ defined by u _t (s) = u (t + s), s ∈ [- σ, 0] . We consider the following initial-type problem of Caputo-Hadamard-type fractional fuzzy differential equation: $^{C - H} D_{a^{+}}^{α} u (t) = f (t, u_{t}) + \int_{a}^{t} g (t, s, u_{s}) d s, t \in [a, b], u (t) = φ (t), t \in [a - σ, a],$ (3.1)

where 0 < a ≤ t ≤ b, f : [a, b] × C _σ → Eg : [a, b] × [a, b] × C _σ → E are fuzzy functions. In the sequel, we denote $f (t, u_{t}) + \int_{a}^{t} g (t, s, u_{s}) ds$ by F (t). Next, we denote C ^1,F ([a, b], E) as the space of fuzzy functions that are continuous Caputo-Hadamard differentiable on [a, b].

Lemma 3.1. Let f, g be fuzzy functions such that t ↦ F (t) belongs to C ([a, b], E) for any u ∈ E . Then a d-monotone fuzzy function u ∈ C ([a, b], E) is a solution of initial value problem (3.1), if and only if u satisfies the integral equation $\begin{matrix} \begin{matrix} \begin{matrix} u (t) = φ (t), t \in [a - σ, a], \\ u (t) \underset{gH}{⊖} φ (a) = \frac{1}{Γ (α)} \int_{a}^{t} {(ln t - ln s)}^{α - 1} \\ \cdot [f (s, u_{s}) + \int_{a}^{s} g (t, s, u_{s})] \frac{ds}{s}, t \in [a, b] . \end{matrix} \end{matrix} \end{matrix}$ (3.2) and the fuzzy function $t \mapsto H ℑ_{a^{+}}^{α} F (t)$ is d-increasing on (a, b] .

proof. Let u ∈ C ([a, b], E) be a d-monotone solution of (3.1) and let $z (t) : = u (t) \underset{gH}{⊖} u (a)$ , t ∈ (a, b]. Because u is d-monotone on [a, b], it follows that t ↦ z (t) is d-increasing on [a, b] (see [21]). From (3.1) and Theorem 2.6 we have that $\begin{matrix} (H ℑ_{a^{+}}^{α} C - H D_{a^{+}}^{α} u) (t) = u (t) \underset{gH}{⊖} u (a), \end{matrix}$ (3.3) for t ∈ [a, b] . Because F (t) ∈ C ([a, b], E) for any u ∈ E, it yields that $\begin{matrix} (H ℑ_{a^{+}}^{α} C - H D_{a^{+}}^{α} u) (t) = H ℑ_{a^{+}}^{α} F (t) \\ = \frac{1}{Γ (α)} \int_{a}^{t} {(ln t - ln s)}^{α - 1} \\ \cdot [f (s, u_{s}) + \int_{a}^{s} g (t, s, u_{s})] \frac{ds}{s}, \end{matrix}$ (3.4) for t ∈ [a, b] . Furthermore, because $u (t) \underset{gH}{⊖} φ (a)$ is d-increasing on (a, b], it follows that $H ℑ_{a^{+}}^{α} F (t)$ is also d-increasing on (a, b]. Next, we prove the sufficiency. Because $t \mapsto H ℑ_{a^{+}}^{α} F (t)$ is d-increasing on (a, b], acting on the two sides of (3.2) by the operator $H D_{a^{+}}^{α}$ and by Proposition 2.1 we obtain $\begin{matrix} H D_{a^{+}}^{α} [u (\cdot) \underset{gH}{⊖} u (a)] (t) = F (t), t \in (a, b] . \end{matrix}$ The proof is complete. □ In the following theorem, the existence and uniqueness results of solution to problem (3.1) by using an idea of successive approximations are investigated by using the results of Theorem 2.7 and Theorem 3. Given ρ > 0, and let $S (u_{0}, ρ) = {u \in E : D_{0} [u, u_{0}] \leq ρ}$ and $S_{σ} (u_{0}, ρ) = {v \in C ([- σ, 0], E) : D_{σ} [v, u_{0}] \leq ρ}$ , where u ₀ (t) is defined by $\begin{matrix} u_{0} (t) = {\begin{matrix} φ (t), t \in [a - σ, a], \\ φ (a), t \in [a, b] . \end{matrix} \end{matrix}$

Theorem 3.1. Let $f : [a, b] \times S_{σ} (u_{0}, ρ) \to E, g : D \times S_{σ} (u_{0}, ρ) \to E$ be continuous fuzzy functions. Assume that the following conditions are satisfied:

there exist positive constants K _f, K _g such that $D_{0} [f (t, z), \hat{0}] \leq K_{f}, D_{0} [g (t, s, z), \hat{0}] \leq K_{g},$ $\forall (t, z) \in [a, b] \times S_{σ} (u_{0}, ρ)$ ;

for every t ∈ [a, b] and every $z, w \in B (u_{0}, ρ)$ it holds $\begin{matrix} D_{0} [f (t, z), f (t, w)] \leq p (t, D_{σ} [z, w]), \\ D_{0} [g (t, s, z), g (t, s, w)] \leq q (t, s, D_{σ} [z, w]), \end{matrix}$ $p (t, \cdot) \in C ([a, b] \times [0, ρ], ℝ^{+}), q (t, s, \cdot)$ $\in C (D \times [0, ρ], ℝ^{+})$ satisfy the conditions in Theorem 2.7 provided that the problem (2.15) has only the solution ψ (t) ≡0 on [a, b].

Then, the following successive approximations given by u ⁰ (t) = u ₀ (t) and for n = 1, 2, … $\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} u^{n} (t) = φ (t), t \in [a - σ, a], \\ u^{n} (t) \underset{gH}{⊖} φ (a) = \frac{1}{Γ (α)} \int_{a}^{t} {(ln t - ln s)}^{α - 1} \end{matrix} \end{matrix} \\ \cdot [f (s, u_{s}^{n - 1}) + \int_{a}^{s} g (s, τ, u_{τ}^{n - 1}) ds] \frac{ds}{s}, t \in [a, b] . \end{matrix} \end{matrix}$ (3.5) converge uniformly to a unique solution of problem (3.1) on some intervals [a, T ^*] for some T ^* ∈ (a, b] provided that the function $t \mapsto H ℑ_{a^{+}}^{α}$ $F^{n - 1} (t) : = H ℑ_{a^{+}}^{α} [f (s, u_{s}^{n - 1}) + \int_{a}^{s} g (s, τ, u_{τ}^{n - 1}) ds]$ is d - increasing on [a, T ^*].

proof. Choose t ^* > a such that t ^* ≤ a exp $(\frac{ρ Γ (1 + α)}{K})^{1 / α}$ , where $K = sup_{t \in [a, b]} [K_{f} + \frac{K_{g} (t - a)}{α + 1}]$ , and put T : = min {t ^*, b} .

First of all, for all n ≥ 1, we show that $u^{n} (t) \in C ([a, T], S (u_{0}, ρ))$ . For any t ₁, t ₂ ∈ [a, T] with t ₁ < t ₂, one has $\begin{matrix} D_{0} [u^{n} (t_{1}) \underset{gH}{⊖} φ (a), u^{n} (t_{2}) \underset{gH}{⊖} φ (a)] \\ \leq \frac{1}{Γ (α)} \int_{a}^{t_{1}} [{(ln t_{1} - ln s)}^{α - 1} - {(ln t_{2} - ln s)}^{α - 1}] \\ \cdot D_{0} [F^{n - 1} (s), \hat{0}] \frac{ds}{s} \\ + \frac{1}{Γ (α)} \int_{t_{1}}^{t_{2}} (ln t_{2} - ln s)^{α - 1} D_{0} [F^{n - 1} (s), \hat{0}] \frac{ds}{s} \\ \leq \frac{K}{Γ (α + 1)} [(ln t_{1} - ln a)^{α} - (ln t_{2} - ln a)^{α}] \\ + \frac{K}{Γ (α + 1)} (ln t_{2} - ln t_{1})^{α} \\ \leq \frac{2 K}{Γ (α + 1)} (ln t_{2} - ln t_{1})^{α} \to 0, as t_{1} \to t_{2}, \end{matrix}$ which proves that u ⁿ ∈ C ([a, T], E) for all n ≥ 1 . In addition, we observe that $u^{n} (t) \in S (u_{0}, ρ)$ for all t ∈ [a, T] and for all n ≥ 0. Indeed, if we suppose that $u^{n - 1} (t) \in S (u_{0}, ρ)$ for all t ∈ [a, T] and for a given n ≥ 2, then from $\begin{matrix} D_{0} [u^{n} (t) \underset{gH}{⊖} φ (a), \hat{0}] \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} {(ln t - ln s)}^{α - 1} \\ \cdot (D_{0} [f (s, u_{s}^{n - 1}), \hat{0}] + \int_{a}^{s} D_{0} [g (s, τ, x_{τ}^{n - 1}), \hat{0}] d τ) \frac{ds}{s} \\ \leq \frac{K (ln t - ln a)^{α}}{Γ (1 + α)} \leq ρ . \end{matrix}$ Hence, we have that $u^{n} (t) \in C ([a, T], S (u_{0}, ρ))$ . Next, let T ^* = min {T, b ^*}, where b ^* is defined as in Theorem 2.7, then we have for t ∈ [a, T ^*] $\begin{matrix} D_{0} [u^{1} (t), u^{0} (t)] = D_{0} [u^{1} (t) \underset{gH}{⊖} φ (a), \hat{0}] \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (ln t - ln s)^{α - 1} \\ \cdot (D_{0} [f (s, u_{s}^{0}), \hat{0}] + \int_{a}^{s} D_{0} [g (s, τ, x_{τ}^{0}), \hat{0}] d τ) \frac{ds}{s} \\ \leq \frac{K (ln t - ln a)^{α}}{Γ (1 + α)} \leq ψ^{0} (t), \end{matrix}$ where $ψ^{0} (t) = \frac{max {M, K} (ln t - ln a)^{α}}{Γ (1 + α)},$ and $\begin{matrix} sup_{ϑ \in [- σ, 0]} D_{0} [u^{1} (t + ϑ), u^{0} (t + ϑ)] \\ \leq \frac{1}{Γ (α)} sup_{ϑ \in [- σ, 0]} \int_{a}^{t + ϑ} (ln (t + ϑ) - ln s)^{α - 1} \\ \cdot (D_{0} [f (s, u_{s}^{0}), \hat{0}] + \int_{a}^{s} D_{0} [g (s, τ, x_{τ}^{0}), \hat{0}] d τ) \frac{ds}{s} \\ \leq \frac{K}{Γ (1 + α)} sup_{ϑ \in [- σ, 0]} (ln (t + ϑ) - ln a)^{α} \\ \leq ψ^{0} (T) . \end{matrix}$

Suppose D ₀ [u ⁿ (t), u ^n-1 (t)] ≤ ψ ^n-1 (t) and $sup_{ϑ \in [- σ, 0]} D_{0} [u^{n} (t + ϑ), u^{n - 1} (t + ϑ)] \leq ψ^{n - 1} (t)$ , where ψ ^n-1 (t) is defined as in Theorem 2.7 for n = 2, 3, 4, …, then by the assumption (ii), we get $\begin{matrix} D_{0} [u^{n + 1} (t), u^{n} (t)] \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (ln t - ln s)^{α - 1} \\ \cdot p (s, sup_{ϑ \in [- σ, 0]} D_{0} [u^{n} (s + ϑ), u^{n - 1} (s + ϑ)]) \frac{ds}{s} \\ + \frac{1}{Γ (α)} \int_{a}^{t} (ln t - ln s)^{α - 1} \\ \cdot (\int_{a}^{s} q (s, τ, sup_{ϑ \in [- σ, 0]} D_{0} [u^{n} (τ + ϑ), u^{n - 1} (τ + ϑ)]) d τ) \frac{ds}{s} \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (ln t - ln s)^{α - 1} p (s, ψ^{n - 1} (s)) \frac{ds}{s} \\ + \frac{1}{Γ (α)} \int_{a}^{t} (ln t - ln s)^{α - 1} (\int_{a}^{s} q (s, τ, ψ^{n - 1} (s)) d τ) \frac{ds}{s} \\ = ψ^{n} (t) . \end{matrix}$

Thus, by mathematical induction, for t ∈ [a, T ^*] one obtains for n = 0, 1, 2, … $\begin{matrix} D_{0} [u^{n + 1} (t) \underset{gH}{⊖} φ (a), u^{n} (t) \underset{gH}{⊖} φ (a)] \leq ψ^{n} (t) . \end{matrix}$ (3.6)

Applying this property, we have, for t ∈ [a, T ^*] and for n = 0, 1, 2, …, $\begin{matrix} D_{0} [C - H D_{a^{+}}^{α} u^{n + 1} (t), C - H D_{a^{+}}^{α} u^{n} (t)] \\ \leq D_{0} [f (t, u_{t}^{n}), f (t, u_{t}^{n - 1})] \\ + \int_{a}^{t} D_{0} [g (t, s, u_{s}^{n}), g (t, s, u_{s}^{n - 1})] ds \\ \leq p (t, ψ^{n - 1} (t)) + \int_{a}^{t} q (t, s, ψ^{n - 1} (s)) ds . \end{matrix}$

Let m ≥ n and t ∈ [a, T ^*], then from Remark 2.1 one can obtain $\begin{matrix} C - H D_{a^{+}}^{+ α} D_{0} [u^{n} (t), u^{m} (t)] \\ \leq D_{0} [C - H D_{a^{+}}^{α} u^{n} (t), C - H D_{a^{+}}^{α} u^{m} (t)] \\ \leq D_{0} [C - H D_{a^{+}}^{α} u^{n} (t), C - H D_{a^{+}}^{α} u^{n + 1} (t)] \\ + D_{0} [C - H D_{a^{+}}^{α} u^{n + 1} (t), C - H D_{a^{+}}^{α} u^{m + 1} (t)] \\ + D_{0} [C - H D_{a^{+}}^{α} u^{m + 1} (t), C - H D_{a^{+}}^{α} u^{m} (t)] \\ \leq 2 (p (t, ψ^{n - 1} (t)) + \int_{a}^{t} q (t, s, ψ^{n - 1} (t)) ds) \\ + p (t, D_{σ} [u_{t}^{n}, u_{t}^{m}]) + \int_{a}^{t} q (t, s, D_{σ} [u_{s}^{n}, u_{s}^{m}]) ds . \end{matrix}$

From (ii), because we have that the solution ψ (t) =0 is a unique solution of problem (2.15) and f (·, ψ ^n-1 (·)) : [a, T] → [0, M _f], g (·, ·, ψ ^n-1 (·)) : [a, T ^*] × [a, T ^*] → [0, M _g] uniformly converge to 0, for every ɛ > 0 there exists a natural number n ₀ such that for m ≥ n ≥ n ₀ $\begin{matrix} C D_{a}^{+ α} & D_{0} [u^{n} (t), u^{m} (t)] \leq p (t, D_{σ} [u_{t}^{n}, u_{t}^{m}]) \\ + \int_{a}^{t} q (t, s, D_{σ} [u_{s}^{n}, u_{s}^{m}]) ds + ɛ . \end{matrix}$

From the fact that D ₀ [u ⁿ (a), u ^m (a)] =0 < ɛ and by using Theorem 2.8, we have for t ∈ [a, T ^*] $\begin{matrix} D_{0} [u^{n} (t), u^{m} (t)] \leq λ_{ɛ} (t), m \geq n \geq n_{0}, \end{matrix}$ (3.7) where λ _ɛ (t) is the maximal solution to the following IVP: ${\begin{matrix} (^{C - H} D_{a^{+}}^{α} λ_{ɛ}) (t) = p (t, λ_{ɛ} (t)) + \int_{a}^{t} q (t, s, λ_{ɛ} (s)) ds + ɛ, \\ λ_{ɛ} (a) = ɛ . \end{matrix}$ (3.8)

Due to Theorem 2.8 one can infer that {ψ _ɛ (·)} converges uniformly to the maximal solution ψ (t) ≡0 of (2.15) on [a, T ^*] as ɛ → 0 . Hence, by virtue of (3.7), we can find $n_{0} \in ℕ$ large enough such that for n, m > n ₀ $\begin{matrix} sup_{t \in [a, T^{*}]} D_{0} [u^{n} (t) \underset{gH}{⊖} φ (a), u^{m} (t) \underset{gH}{⊖} φ (a)] \leq ɛ . (3.9) \end{matrix}$

Since (E, D ₀) is a complete metric space and (3.9) holds, it follows that {u ⁿ (t)} converges uniformly to $u \in C ([a, b], S (u_{0}, ρ))$ . Hence, we obtain $\begin{matrix} u (t) \underset{gH}{⊖} φ (a) = lim_{n \to \infty} (u^{n} (t) \underset{gH}{⊖} φ (a)) \\ = \frac{1}{Γ (α)} \int_{a}^{t} {(ln t - ln s)}^{α - 1} \\ \cdot [f (s, u_{s}) + \int_{a}^{s} g (s, τ, u_{τ}) ds] \frac{ds}{s}, t \in [a, T^{*}] . \end{matrix}$ Due to Lemma 3.1 the function u (t) is the solution to (3.1) on [a, T ^*] .

Finally, to show the uniqueness of the solution, we assume that v : [a, T ^*] → E is another solution of problem (3.1) on [a, T ^*]. Set k (t) = D ₀ [u (t), v (t)]. Then, k (a) =0 and we shall prove that k (t) =0 on [a, T ^*]. Indeed, one has

$\begin{matrix} C - H D_{a^{+}}^{α} k (t) \leq p (t, D_{σ} [u_{t}, v_{t}]) \\ + \int_{a}^{t} q (t, s, D_{σ} [u_{s}, v_{s}]) ds \\ \leq p (t, | k_{t} |_{σ}) + \int_{a}^{t} q (t, s, | k_{s} |_{σ}) ds . \end{matrix}$

By Theorem 2.8, we obtain k (t) ≤ ψ (t), where ψ is a maximal solution of (3.8). By assumption (ii), we have ψ (t) =0 and therefore u (t) = v (t), ∀ t ∈ [a, T ^*].

Corollary 3.1. Let φ (t) ∈ C ([- σ, 0], E) and suppose that $f \in C ([a, b] \times S_{σ} (u_{0}, ρ), E),$ $g \in C (D \times S_{σ} (u_{0}, ρ), E)$ satisfy the conditions: there exist positive constants L _f, K _f, L _g, K _g such that for every z, w ∈ E it holds $\begin{matrix} D_{0} [f (t, z), f (t, w)] \leq L_{f} D_{σ} [z, w], \\ D_{0} [f (t, z), \hat{0}] \leq K_{f}, \end{matrix}$ $\begin{matrix} D_{0} [g (t, s, z), g (t, s, w)] \leq L_{g} D_{σ} [z, w], \\ D_{0} [g (t, s, z), \hat{0}] \leq K_{g} . \end{matrix}$ Then, the successive approximations given by (3.5) converge uniformly to a unique solution of problem (3.1) on some intervals [a, T ^*] for some T ^* ∈ (a, b] provided that the function $t \mapsto H ℑ_{a^{+}}^{α} F^{n - 1} (t) : = H ℑ_{a^{+}}^{α} [f (s, u_{s}^{n}) + \int_{a}^{s} g (s, τ, u_{τ}^{n}) ds]$ is d-increasing on [a, T ^*].

proof. The proof is obtained immediately by taking in Theorem 3.1, p (t, D _σ [z, w]) = L _f D _σ [z, w], q (t, s, D _σ [z, w]) = L _g D _σ [z, w] . □

Example 3.2. We consider the following initial-type problem of Caputo-Hadamard-type fractional fuzzy differential equation: $\begin{matrix} \begin{matrix} C - H D_{1^{+}}^{α} u (t) = λ u (t) + u (t - 1) \\ + \int_{1}^{t} u (s - 1) ds, t \in [1, 2], \\ u (t) = (t + 1, t + 2, t + 3), t \in [0, 1], \end{matrix} \end{matrix}$ (3.10) where λ ≠ 0. By using the method of steps, we get the following problem: $\begin{matrix} \begin{matrix} C - H D_{1^{+}}^{α} u (t) = λ u (t) + (t, t + 1, t + 2) \\ + \int_{1}^{t} (s, s + 1, s + 2) ds, t \in [1, 2], \\ u (1) = (2, 3, 4), t \in [0, 1], \end{matrix} \end{matrix}$ (3.11)

We set h (t) = (t ²/2 + t - 1/2, t ²/2 +2t - 1/2, t ²/2 +3t - 1/2) . Then, by applying Remark 2.7, we obtain the solution of (3.10) as follows: + Suppose that λ > 0 and the solution of (3.11) is d-increasing. Then, the solution is given by $\begin{matrix} u (t) & = (2, 3, 4) E_{α, 1} (λ {(ln t)}^{α}) \\ + \int_{1}^{t} {(ln \frac{t}{s})}^{α - 1} E_{α, α} (λ {(ln \frac{t}{s})}^{α}) \frac{h (s)}{s} ds . \end{matrix}$ + Suppose that λ < 0 and the solution of (3.11) is d-decreasing. Then, the solution is given by $\begin{matrix} u (t) & = (2, 3, 4) E_{α, 1} (λ {(ln t)}^{α}) \\ ⊖ (- 1) \int_{1}^{t} {(ln \frac{t}{s})}^{α - 1} E_{α, α} (λ {(ln \frac{t}{s})}^{α}) \frac{h (s)}{s} ds . \end{matrix}$

4 Conclusion

The main aim of this study is to establish the fundamental theory of fuzzy fractional calculus in the Caputo-Hadamard setting, and to investigate the existence and uniqueness of solution of the initial value problem for fuzzy functional fractional integro-differential equations involving Caputo-Hadamard fractional derivative. Our point is that the basic theory of this fractional derivative concept can be of great help in the study of fuzzy fractional differential equations. We observe that it is not easy to derive analytical solutions to most the given problem under Caputo-Hadamard fractional derivative (see Example 3.2). Therefore, it is vital to develop some reliable and efficient techniques to solve the above problems next time. In the recent time, the results of fuzzy fractional differential equations are based on the well-known notions of Hukuhara differentiability, strongly Hukuhara differentiability and generalized Hukuhara differentiability for investigating the fuzzy problems with fractional order. In general, these proposed approaches suffer from some limitations namely: the existence of these kinds of derivative cannot be guaranteed; the monotony of the solution fuzziness is necessary for solving the proposed fuzzy problems. Following the above drawbacks the new concepts of fuzzy differentiability which are based on the assumption of generalized difference [14, 15] and granular difference [36] between fuzzy sets have been proposed and developed in some papers to investigate solutions of fuzzy differential equations, and these approaches can overcome the restrictions imposed by the approaches which are based on fuzzy derivatives with respect to Hukuhara difference. Therefore, in a future research one could investigate the fuzzy fractional differential equations under Hadamard derivative by using the proposed approaches in [14 , 36].

Footnotes

Acknowledgements

The authors are very grateful to the referees for their valuable suggestions, which helped to improve the paper significantly. The first named author would like to thank the University of Technical Education, Ho Chi Minh City, Vietnam. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.319.

References

Adjabi ,

Jarad ,

Baleanu and

Abdeljawad , On Cauchy problems with Caputo Hadamard fractional derivatives, Journal Computational Analysis and Applications 21 (2016), 661–681.

R.P.

Agarwal ,

Lakshmikantham and

J.J.

Nieto , On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis: (TMA) 72(6) (2010), 2859–2862.

Ahmadian ,

Suleiman ,

Salahshour and

Baleanu , A Jacobi operational matrix for solving a fuzzy linear fractional differential equation, Advances in Difference Equations 2013 (2013), 104.

Ahmadian ,

Allahviranloo ,

Abbasbandy and

Gouyandeh , The fuzzy generalized Taylor's expansion with application in fractional differential equations, Iranian Journal of Fuzzy Systems (2018).

Allahviranloo ,

Gouyandeh and

Armand , Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent & Fuzzy Systems 26 (2014), 1481–1490.

Allahviranloo ,

Salahshour and

Abbasbandy , Explicit solutions of fractional differential equations with uncertainty, Soft Computing 16 (2012), 297–302.

Allahviranloo ,

Salahshour and

Abbasbandy , Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simulation 17 (2012), 1372–1381.

Arshad and

Lupulescu , On the fractional differential equations with uncertainty, Nonlinear Analysis: (TMA) 74 (2011), 85–93.

Arshad and

Lupulescu , Fractional differential equation with the fuzzy initial condition, Electronic Journal of Differential Equations 2011 (2011).

10.

Almeida , Caputo-Hadamard fractional derivatives of variable order, Numerical Functional Analysis and Optimization 38 (2017), 1–19.

11.

M.Z.

Ahmad ,

M.K.

Hasan and

De Baets , Analytical and numerical solutions of fuzzy differential equations, Information Sciences 236 (2013), 156–167.

12.

Bede ,

I.J.

Rudas and

A.L.

Bencsik , First order linear fuzzy differential equations under generalized differentiability, Information Sciences 177 (2007), 1648–1662.

13.

Bede and

S.G.

Gal , Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems 151 (2005), 581–599.

14.

Bede and

Stefanini , Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems 230 (2013), 119–141.

15.

L.T.

Gomes and

L.C.

Barros , A note on the generalized difference and the generalized differentiability, Fuzzy Sets and Systems 280 (2015), 142–145.

16.

O.S.

Fard and

Salehi , A survey on fuzzy fractional vari-ational problems, Journal of Computational and Applied Mathematics 271 (2014), 71–82.

17.

Y.Y.

Gambo ,

Jarad ,

Baleanu and

Abdeljawad , On Caputo modification of the Hadamard fractional derivatives, Advances in Difference Equations 2014 (2014), 10.

18.

N.V.

Hoa , Fuzzy fractional functional integral and differential equations, Fuzzy Sets and Systems 280 (2015), 58–90.

19.

N.V.

Hoa , Fuzzy fractional functional differential equations under Caputo gH-differentiability, Communications in Nonlinear Science and Numerical Simulation 22 (2015), 1134–1157.

20.

N.V.

Hoa ,

Lupulescu and

O'Regan , Solving inter-valvalued fractional initial value problems under Caputo gH-fractional differentiability, Fuzzy Sets and Systems 309 (2017), 1–34.

21.

N.V.

Hoa ,

Lupulescu and

O’Regan , Anote on initial value problems for fractional fuzzy differential equations, Fuzzy Sets and Systems 347 (2018), 54–69.

22.

N.V.

Hoa ,

Vu and

T.M.

Duc , Fuzzy fractional differential equations under Caputo-Katugampola fractional derivative approach, Fuzzy Sets and Systems (2018).

23.

Jarad ,

Baleanu and

Abdeljawad , Caputo-type modiiňAcation of the Hadamard fractional derivatives, Advances Differential Equations 2012 (2012), 142.

24.

Khastan ,

J.J.

Nieto and

Rodriguez-Lopez , Fuzzy delay differential equations under generalized differentiability, Information Sciences 275 (2014), 145–167.

25.

Khastan and

Rodriguez-Lopez , On the solutions to first order linear fuzzy differential equations, Fuzzy Sets and Systems 295 (2016), 114–135.

26.

Khastan ,

J.J.

Nieto and

Rodriguez-Lopez , Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty, Fixed Point Theory and Applications 2014 (2014), 21.

27.

A.A.

Kilbas ,

H.M.

Srivastava and

J.J.

Trujillo , Theory and applications of fractional differential equations, Elsevier Science B.V. (2006).

28.

Klimek , Sequential fractional differential equations with Hadamard derivative, Communication Nonlinear Science Numerical Simulation 16 (2011), 4689–4697.

29.

Lakshmikantham and

R.N.

Mohapatra , Theory of Fuzzy Differential Equations and Applications, Taylor & Francis, London, 2003.

30.

H.V.

Long ,

N.T.K.

Son and

N.V.

Hoa , Fuzzy fractional partial differential equations in partially ordered metric spaces, Iranian Journal of Fuzzy Systems 14 (2017), 107–126.

31.

H.V.

Long ,

N.K.

Son ,

H.T.

Tam and

J.C.

Yao , Ulam Stability for fractional partial integro-differential equation with uncertainty, Acta Math Vietnam 42 (2017), 675–700.

32.

H.V.

Long ,

N.K.

Son and

H.T.

Tam , The solvability of fuzzy fractional partial differential equations under Caputo gH-di fferentiability, Fuzzy Sets Systems 309 (2017), 35–63.

33.

H.V.

Long and

N.P.

Dong , An extension of Krasnoselskii's fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty, Journal of Fixed Point Theory and Applications 20 (2018), 37.

34.

Lupulescu , Fractional calculus for interval-valued functions, Fuzzy Sets and Systems 265 (2015), 63–85.

35.

Mazandarani and

A.V.

Kamyad , Modified fractional Euler method for solving fuzzy fractional initial value problem, Communications in Nonlinear Science and Numerical Simulation 18 (2013), 12–21.

36.

Mazandarani ,

Pariz and

A.V.

Kamyad , Granular differentiability of fuzzy-number-valued functions, IEEE Trans Fuzzy Syst (2017). https://dx-doi-org.web.bisu.edu.cn/10.1109/ TFUZZ.2017.2659731 (in press)

37.

Mazandarani and

Najariyan , Type-2 fuzzy fractional derivatives, Communications in Nonlinear Science and Numerical Simulation 19 (2014), 2354–2372.

38.

Qiu and

Wei , Symmetric fuzzy numbers and additive equivalence of fuzzy numbers, Soft Computing 17 (2013), 1471–1477.

39.

Qiu ,

Chongxia ,

Wei and

Yaoyao , Algebraic properties and topological properties of the quotient space of fuzzy numbers based on Mares equivalence relation, Fuzzy Sets and Systems 245 (2014), 63–82.

40.

Qiu ,

Dong and

Wei , Chongxia, On fuzzy differential equations in the quotient space of fuzzy numbers, Fuzzy Sets and Systems 295 (2016), 72–98.

41.

Salahshour ,

Allahviranloo ,

Abbasbandy and

Baleanu , Existence and uniqueness results for fractional differential equations with uncertainty, Advances In Difference Equations 2012 (2012), 1–12.

42.

Salahshour ,

Allahviranloo and

Abbasbandy , Solving fuzzy fractional differential equations by fuzzy laplace transforms, Commun Nonlinear Sci Numer Simulat 17 (2012), 1372–1381.

43.

Yukunthorn ,

Ahmad ,

S.K.

Ntouyas and

Tariboon , On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Analysis: Hybrid Systems 19 (2016), 77–92.