Stability for initial value problems of fuzzy Volterra integro-differential equation with fractional order derivative

Abstract

The results of this paper is motivated from some recent papers treating the problem of the existence and stability of a solution for Volterra integro-differential equations in fuzzy setting with fractional order derivative (FFVIDEs). By constructing successive approximation method in the space of fuzzy functions, we establish the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for the given problems with two concepts of fuzzy-type fractional derivative.

Keywords

Ulam-hyers stability ulam-hyers-rassias stability fuzzy fractional integro-differential equations

1 Introduction

During the past decades, fuzzy analysis and fuzzy differential equations have attracted many authors and they have recently proved to be a valuable instrument to handle the uncertainties raised by impreciseness, vagueness and incomplete information in mathematical or computer models of some deterministic real-world phenomena. With this advantage, many dynamical systems can be described more precisely by using differential equations in fuzzy setting and so the theory of fuzzy analysis and fuzzy differential equations has been further developed and a wide number of applications of this theory have been considered in [4 , 62] and references therein. The fact that many practical dynamical systems including evolutionary processes have a memory effect, thus, the problems of differential equations with integer order derivative may not explain this phenomenon very well. Recently, fractional calculus and fractional differential equations have been proved to be a powerful tool for the description of hereditary properties of various materials and memory processes. In particular, integral equations and differential equations involving fractional integral and differential operator appear naturally in many fields and become strong tools in the modeling of many real-world phenomena. For more details on fractional calculus theory and interesting applications, one can see the monograph of Diethelm [18], Kilbas et al. [31], Podlubny [60] and the references therein. With the advantage of this theory, in recent years, the theory of fractional analysis in the area of fuzzy differential equations gained much attention and began to be studied, mainly due to the variety of results, from the stability, existence, and uniqueness to solving methods of fuzzy differential equations with fractional derivative. Specifically, for some fundamental results in theory of fuzzy-type fractional calculus basing on the concepts of generalized Hukuhara derivative of fuzzy functions and some applications to fuzzy fractional differential equations are investigated and developed by Alikhani et al. [5], Allahviranloo et al. [6], Lupulescu et al. [24 , 35] and Hoa et al. [26]. The studies on the existence and uniqueness of solutions to fuzzy fractional differential equations under Hukuhara fractional Riemann-Liouville differentiability are established by Arshad and Lupulescu in [10], Khastan, Nieto and Rodríguez-López [32, 33] and Allahviranloo et al. [7]. In addition, the problems of fuzzy differential equations and partial fuzzy differential equations with Caputo fractional derivative concept are discussed by An et al. [12, 13], Fard et al. [19], Hoa [23, 27], Long et al. [38 –44], Mazan [46], Salahshour et al. [56] and the references therein. For solving initial value problems of fuzzy fractional differential equations, some new methods which are extend and developed to find the exact and numerical solutions of the given problems are proposed by Mazandarani et. al. [45] with a modified fractional Euler method, Hoa et al. [22, 24] with the modified fractional Euler method and the modified Adams-Bashforth-Moulton method, Ahmadian et al. [2, 3] with the methods based on operational matrix of shifted Chebyshev polynomials and the spectral tau, Allahviranloo et al. [9] with the method of fuzzy Laplace transforms.

The theory of stability is always an interesting topic for fuzzy differential equations, and in the past seventy years Ulam-Hyers’s type stability problems of functional equations that was originally raised by Ulam [59] and Hyers [30]. Thereafter, this type of stability is called the Ulam-Hyers stability. In 1978, Rassias [52] gave a remarkable generalization of the Ulam-Hyers stability of mappings by considering variables. Then, the concept of stability for a functional equation arise when we replace the functional equation by an inequality which acts as a perturbation of the equation. During the past two decades, the above concepts have received great attention by a large number of mathematicians since it is quite useful in many applications such as numerical analysis, optimization, biology and economics, where finding the exact solutions is quite difficult. More details from historical point of view, and recent developments of such stabilities are reported in [1 , 63] for the field of real-valued differential equations and in [38 , 55] for the area of fuzzy differential equations. However, the results on stability of fractional fuzzy differential equations under Caputo-type and Riemann-Liouville-type fractional derivative have not been studied. With the expansion of the Ulam-Hyers’s types stability concepts and increasing number of paper about the field of fractional fuzzy differential equations, the studies about the existence, uniqueness, and stability of the Ulam-Hyers-Rassias’s type of solutions the above problems are interesting subjects. So, the motivation for the elaboration of this paper is the study of the stability of Hyers-Ulam and Ulam-Hyers-Rassias of the following fractional Volterra fuzzy integro-differential equation:

${\begin{matrix} ^{*} D_{a^{+}}^{α} u (t) = f (t, u (t)) + \int_{a}^{t} g (t, s, u (s)) ds, t \in J \\ u (a) = u_{0}, \end{matrix}$ (1.1) with t ∈ J, where α ∈ (0, 1), u₀ ∈ E, f : J × E → E, g : J × J × E → E are continuous fuzzy functions with respect to t, s and u on J × E, $^{*} D_{a^{+}}^{α} u$ denotes two kinds of the fractional order derivative of fuzzy function u such as fuzzy-type Riemann-Liouville and fuzzy-type Caputo derivatives. In this work, by using the method of fixed point and constructing successive approximation method in the space of fuzzy functions, we present the results of Ulam-Hyers’s type of the problem (1.1) with two fractional derivative concepts in Section 3 and Section 4.

2 Preliminaries

In this section, we will briefly review some of definitions and results which will use throughout the paper. The space of fuzzy numbers is denoted by E defined as follows: $E = {ω : ℝ \to [0, 1] | ω satisfies (i) - (iv)}$ , where (i) ω is normal; (ii) ω is fuzzy convex; (iii) ω is upper semi-continuous; (iv) $[ω]^{0} = \bar{{z \in ℝ | ω (z) > 0}}$ is compact. For r ∈ (0, 1], the r-level set of u is defined by $[ω]^{r} = {z \in ℝ | ω (z) \geq r}$ . Then it is well-known that the r-level set of ω, $[ω]^{r} = [\underline{ω} (r), \bar{ω} (r)]$ , is a bounded closed interval, for any r ∈ [0, 1].

Definition 2.1. Let ω₁, ω₂ ∈ E and $λ \in ℝ$ . Basing on Zadeh’s extension principle, we have addition, ω₁ + ω₂, and scalar multiplication, λ · ω₁, in fuzzy number space E as follows: $\begin{matrix} [ω_{1} + ω_{2}]^{r} = [ω_{1}]^{r} + [ω_{2}]^{r}, \\ [λ \cdot ω_{1}]^{r} = λ [ω_{1}]^{r}, \forall r \in [0, 1], \end{matrix}$ where [ω₁] ^r + [ω₂] ^r means the usual addition of two intervals of $ℝ$ and λ [ω₁] ^r means the usual product between a scalar and a real interval number. For ω ∈ E, the diameter of the r-level set of ω is defined by $d ([ω]^{r}) = \bar{ω} (r) - \underline{ω} (r) .$ Let ω₁, ω₂ ∈ E. If there exists ω₃ ∈ E such that ω₁ = ω₂ + ω₃, then ω₃ is called the Hukuhara difference of ω₁ and ω₂ and it is denoted by ω₁ ⊖ ω₂ . We note that ω₁ ⊖ ω₂ ≠ ω₁ + (-1) ω₂.

Definition 2.2. [15] The generalized Hukuhara difference of two fuzzy numbers ω₁, ω₂ ∈ E (gH-difference for short) is defined as follows: $ω_{1} \underset{gH}{⊖} ω_{2} = ω_{3} \Leftrightarrow {(i) ω1 = ω2 + ω3, or (ii) ω2 = ω1 + (- 1)ω3.$ (2.2)

It is well-known that in the r-levels we have that for all r ∈ [0, 1] $\begin{matrix} [ω_{1} \underset{gH}{⊖} ω_{2}]^{r} = [min {{\underline{ω}}_{1} (r) - {\underline{ω}}_{2} (r), {\bar{ω}}_{1} (r) \\ - {\bar{ω}}_{2} (r)}, max {{\underline{ω}}_{1} (r) - {\underline{ω}}_{2} (r), {\bar{ω}}_{1} (r) - {\bar{ω}}_{2} (r)}], \end{matrix}$ and the condition for the existence of $ω_{1} \underset{gH}{⊖} ω_{2}$ in the case (i) is d ([ω₁] ^r) ≥ d ([ω₂] ^r) and the condition for the existence of $ω_{1} \underset{gH}{⊖} ω_{2}$ in the case (ii) is d ([ω₂] ^r) ≥ d ([ω₁] ^r) .

Definition 2.3. [15] The distance D₀ [ω₁, ω₂] between two fuzzy numbers is defined as $D_{0} [ω_{1}, ω_{2}] = sup_{0 \leq r \leq 1} H ([ω_{1}]^{r}, [ω_{2}]^{r}), \forall ω_{1}, ω_{2} \in E,$ where $H ([ω_{1}]^{r}, [ω_{2}]^{r}) = max {| {\underline{ω}}_{1} (r) - {\underline{ω}}_{2} (r) |,$ $| {\bar{ω}}_{1} (r) - {\bar{ω}}_{2} (r) |}$ is the Hausdorff distance between [ω₁] ^r and [ω₂] ^r.

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

Definition 2.5.[15] Let u : (a, b) → E and t ∈ (a, b). We say that the fuzzy function u is generalized Hukuhara differentiable at t if there exists an element u′ (t) ∈ E such that the following statement is true

$u^{'} (t) = lim_{h \to 0} \frac{u (t + h) ⊖_{gH} u (t)}{h} .$ (2.3)

Denote by

- C ([a, b], E) the set of all continuous fuzzy functions on the interval [a, b] with value in E ; - AC ([a, b], E) the set of all absolutely continuous fuzzy functions on [a, b] with value in E; - L ([a, b], E) 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 [a, b] (the space of all Lebesgue integrable real-valued functions on the bounded interval [a, b]);

- C_γ ([a, b], E), where 0 < γ < 1, the weight space of fuzzy continuous functions u on (a, b] defined by $\begin{matrix} C_{γ} ([a, b], E) = {u : (a, b] \to E; \\ (t - a)^{γ} u (t) \in C ([a, b], E)} \end{matrix}$ with the norm $∥ u ∥_{γ} = D_{γ} [u (t), \hat{0}] = max_{t \in [a, b]} D_{0} [(t - a)^{γ} u (t), \hat{0}] .$

Definition 2.6. (see [28]) Let u ∈ L ([a, b], E). The Riemann-Liouville fractional integral of order α > 0 of the fuzzy-valued function u is defined as follows: $(ℑ_{a^{+}}^{α} u) (t) = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} u (s) ds, t > a,$ where Γ (α) is the well-known Gamma function.

We set $\begin{matrix} u_{1 - α} (t) : = (ℑ_{a^{+}}^{1 - α} u) (t) = \frac{1}{Γ (1 - α)} \\ \int_{a}^{t} (t - s)^{- α} u (s) ds, for t \in (a, b] . \end{matrix}$

Definition 2.7. (see [28]) Let u ∈ L ([a, b], E) and α ∈ (0, 1). The fuzzy function u : (a, b) → E is said to be Riemann-Liouville generalized Hukuhara fractional differentiable at t ∈ (a, b) (or Riemann-Liouville gH-fractional differentiable) of order α ∈ (0, 1), if there exists an element $(^{RL} D_{a^{+}}^{α} u) (t) \in E$ such that

$(^{RL} D_{a^{+}}^{α} u) (t) = u_{1 - α}^{'} (t) .$ (2.4)

Definition 2.8. (see [28]) Let u ∈ L ([a, b], E) be a fuzzy function such that $^{RL} D_{a^{+}}^{α} u$ , α ∈ (0, 1), exists on (a, b). Then, the Caputo generalized Hukuhara fractional derivative of the fuzzy function u is defined as follows: $(^{C} D_{a^{+}}^{α} u) (t) = (^{RL} D_{a^{+}}^{α} [u (\cdot) \underset{gH}{⊖} u (a)]) (t) .$

Theorem 2.1. (see [28]) If u ∈ AC ([a, b], E) is a d-monotone fuzzy function and α ∈ (0, 1), then

$\begin{array}{l} (^{C} D_{a^{+}}^{α} u) (t) \\ = (ℑ_{a^{+}}^{1 - α} u^{'}) (t) = \int_{a}^{t} \frac{{(t - s)}^{- α}}{Γ (1 - α)} u^{'} (s) d s, t \in (a, b] . \end{array}$ (2.5)

Next sections, the results of existence, uniqueness, and the stability of solution of fractional fuzzy integro-differential equations are presented by using Banach fixed point theorem. Here it is recalled.

Theorem 2.2. [17] Let (X, d) be a generalized complete metric space. Assume that the operator $ℙ : X \to X$ is a strictly contractive with Lipschitz constant L < 1 . If there exists a nonnegative integer n such that $d (ℙ^{n + 1} x, ℙ^{n} x) < \infty$ for any x ∈ X, then the following are true:

The sequence ${ℙ^{n} x}$ converges to a fixed point x^* of the operator $ℙ;$

x^* is the unique fixed point of the operator $ℙ$ in $X^{*} = {y \in X | d (ℙ^{n} x, y) < \infty}$ ;

If y ∈ X^*, then $d (y, x^{*}) \leq \frac{1}{1 - L} d (P y, y) .$

3 Stability for initial value problems with Caputo fractional derivative concept

This section deals with the existence and the Ulam stability of solution for the following fuzzy Volterra integro-differential equations with the Caputo fractional concept of the form

${\begin{matrix} ^{C} D_{a^{+}}^{α} u (t) = f (t, u (t)) + \int_{a}^{t} g (t, s, u (s)) ds, t \in J \\ u (a) = u_{0}, \end{matrix}$ (3.1) where J : = [a, b], α ∈ (0, 1), f : J × E → E, g : J × J × E → E are continuous fuzzy functions with respect to t, s and u on J × E, $^{C} D_{a^{+}}^{α} u$ denotes Caputo fractional derivative given in Definition 2. A fuzzy function u is called a solution of the problem (3.1) if and only if $^{C} D_{a^{+}}^{α} u (t) = f (t, u (t)) + \int_{a}^{t} g (t, s, u (s)) ds$ and u (a) = u₀.

Definition 3.1. Basing on Definition 2, a solution u of the problem (3.1) is said to be d-monotone on [a, b] if u is d-increasing or d-decreasing on [a, b] .

In the sequel, the Ulam stability for the problem (3.1) is investigated by using successive approximations. Let ɛ > 0 and $φ : J \to ℝ^{+}$ be a continuous function. Consider the following inequalities $\begin{matrix} D_{0} [^{C} D_{a^{+}}^{α} w (t), f (t, w (t)) \\ + \int_{a}^{t} g (t, s, w (s)) ds] \leq ɛ, t \in J . \end{matrix}$ (3.2) $\begin{matrix} D_{0} [^{C} D_{a^{+}}^{α} w (t), f (t, w (t)) \\ + \int_{a}^{t} g (t, s, w (s)) ds] \leq ɛ φ (t), t \in J . \end{matrix}$ (3.3)

Definition 3.2. We say that the problem (3.1) has

Ulam-Hyers stability if there exists a real number K_f > 0 such that for each ɛ > 0 and for each solution v ∈ C (J, E) of the inequality (3.2) there exists a solution u ∈ C (J, E) of (3.1) with $\begin{matrix} D_{0} [u (t), v (t)] \leq ɛ K_{f}, t \in J . \end{matrix}$

Ulam-Hyers-Rassias stable with respect to φ if there exists K_f,φ > 0 such that for each ɛ and for each solution v ∈ C (J, E) of the inequality (3.3) there exists a solution u ∈ C (J, E) of (3.1) with $\begin{matrix} D_{0} [u (t), v (t)] \leq ɛ K_{f, φ} φ (t), t \in J . \end{matrix}$

Remark 3.1. From the inequalities (3.2) and (3.3) we notice that:

A fuzzy function v ∈ C (J, E) is a solution of (3.2) if and only if there exists a fuzzy function δ ∈ C (J, E) such that $D_{0} [δ (t), \hat{0}] \leq ɛ$ for any t ∈ J, and $^{C} D_{a^{+}}^{α} v (t) = f (t, v (t)) + \int_{a}^{t} g (t, s, v (s)) ds + δ (t)$ for any t ∈ J.

A fuzzy function v ∈ C (J, E) is a solution of (3.3) if and only if there exists a fuzzy function δ ∈ C (J, E) such that $D_{0} [δ (t), \hat{0}] \leq φ (t)$ for any t ∈ J and $φ \in C (J, ℝ^{+})$ , and $^{C} D_{a^{+}}^{α} v (t) = f (t, v (t)) + \int_{a}^{t} g (t, s, v (s)) ds + δ (t)$ for any t ∈ J.

Theorem 3.1. Let f : J × E → E and g : J × J × E → E be continuous fuzzy functions that satisfy the following condition: there exists a constant L > 0 such that $\begin{matrix} max {D_{0} [f (t, u), f (t, v)], D_{0} [g (t, s, u), \\ g (t, s, v)]} \leq L D_{0} [u, v] \end{matrix}$ for any (t, u), (t, v) ∈ J × E, (t, s, u), (t, s, v) ∈ J × J × E. For every ɛ > 0, if a continuously differentiable fuzzy function v : J → E satisfies the inequality (3.2) for all t ∈ J, then there exists a unique solution u : J → E of the problem (3.1) with initial condition u₀ = v₀ such that

$\begin{matrix} D_{0} [v (t), u (t)] \leq \\ \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) E_{α, α} (L (t - a)^{α}) . \end{matrix}$ (3.4)

Proof. For every ɛ > 0 let us consider a continuously differentiable fuzzy function v : J → E which satisfies the inequality (3.2) for any t ∈ J, v ∈ E. By Remark 3, there exists a fuzzy function δ (t) ∈ E such that $D_{0} [δ (t), \hat{0}] \leq ɛ$ and for any t ∈ J

$^{C} D_{a^{+}}^{α} v (t) = f (t, v (t)) + \int_{a}^{t} g (t, s, v (s)) ds + δ (t) .$ (3.5)

If a function v : J → E satisfies (3.5), then in view of Theorem 3 in [28] it satisfies the following integral equation $\begin{matrix} v (t) \underset{gH}{⊖} v_{0} & = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (f (s, v (s)) + \int_{a}^{s} g (s, τ, v (τ)) d τ) ds \\ + \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} δ (s) ds, \forall t \in J . \end{matrix}$ (3.6)

Define u₀ (t) = v (t), ∀ t ∈ J, and let us consider the sequence u_n : J → E, n = 1, 2, … of successive approximations defined as follows: $\begin{matrix} u_{n} (t) \underset{gH}{⊖} v_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \end{matrix}$ $(f (s, u_{n - 1} (s)) + \int_{a}^{s} g (s, τ, u_{n - 1} (τ)) d τ) ds, \forall t \in J .$ (3.7)

Then, for n = 1 and from (3.6), we get $\begin{matrix} D_{0} [u_{1} (t), u_{0} (t)] \\ = D_{0} [u_{1} (t) \underset{gH}{⊖} v_{0}, u_{0} (t) \underset{gH}{⊖} v_{0}] \\ = D_{0} [\frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (f (s, v (s)) + \int_{a}^{s} g (s, τ, v (τ)) d τ) ds, v (t) \underset{gH}{⊖} v_{0}] \\ \leq D_{0} [\hat{0}, \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} δ (s) ds] \\ \leq ɛ \frac{(t - a)^{α}}{Γ (α + 1)} . \end{matrix}$

By using Lipschitz condition of f and g, from the definition of successive approximations and for any t ∈ J, for n = 1, 2, 3, . . . we obtain $\begin{matrix} D_{0} [u_{n + 1} (t), u_{n} (t)] \\ = D_{0} [u_{n + 1} (t) \underset{gH}{⊖} v_{0}, u_{n} (t) \underset{gH}{⊖} v_{0}] \\ \leq \frac{L}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (D_{0} [u_{n} (s), u_{n - 1} (s)] + \int_{a}^{s} D_{0} [u_{n} (τ), u_{n - 1} (τ)] d τ) ds . \end{matrix}$

Then, for n = 1 one has $\begin{matrix} D_{0} [u_{2} (t), u_{1} (t)] \\ \leq \frac{L ɛ}{Γ (α) Γ (α + 1)} \int_{a}^{t} (t - s)^{α - 1} \\ ((s - a)^{α} + \int_{a}^{s} (τ - a)^{α} d τ) ds \\ = L ɛ (\frac{(t - a)^{2 α}}{Γ (2 α + 1)} + \frac{(t - a)^{2 α + 1}}{Γ (2 α + 2)}) . \end{matrix}$ and for n = 2 one also has $\begin{matrix} D_{0} [u_{3} (t), u_{2} (t)] \\ \leq \frac{L^{2} ɛ}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} ([\frac{(s - a)^{2 α}}{Γ (2 α + 1)} + \frac{(s - a)^{2 α + 1}}{Γ (2 α + 2)}] \\ + \int_{a}^{s} [\frac{(τ - a)^{2 α}}{Γ (2 α + 1)} + \frac{(τ - a)^{2 α + 1}}{Γ (2 α + 2)}] d τ) ds \\ = L^{2} ɛ (\frac{(t - a)^{3 α}}{Γ (3 α + 1)} + 2 \frac{(t - a)^{3 α + 1}}{Γ (3 α + 2)} + \frac{(t - a)^{3 α + 2}}{Γ (3 α + 3)}) \\ \leq 3 L^{2} ɛ (\frac{(t - a)^{3 α}}{Γ (3 α + 1)} + \frac{(t - a)^{3 α + 1}}{Γ (3 α + 2)} + \frac{(t - a)^{3 α + 2}}{Γ (3 α + 3)}) . \end{matrix}$

By using mathematical induction method, for n ≥ 4 we have $\begin{matrix} D_{0} [u_{n} (t), u_{n - 1} (t)] \leq ɛ n L^{n - 1} \sum_{k = 1}^{n} \frac{{(t - a)}^{n α + k - 1}}{Γ (n α + k)} \\ = ɛ n L^{n - 1} \frac{(t - a)^{n α}}{Γ (n α + 1)} [1 + \frac{(t - a)}{(n α + 1)} + \frac{(t - a)^{2}}{(n α + 1) (n α + 2)} \\ + \dots + \frac{(t - a)^{n - 1}}{(n α + 1) (n α + 2) \dots (n α + n - 1)}] \\ \leq \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) \frac{(L (t - a)^{α})^{n - 1}}{Γ ((n - 1) α + α)} . \end{matrix}$

Furthermore, if we assume that

$\begin{matrix} D_{0} [u_{n} (t), u_{n - 1} (t)] \leq \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) \\ \frac{(L (t - a)^{α})^{n - 1}}{Γ ((n - 1) α + α)}, \forall t \in J, \end{matrix}$ (3.8) then one obtains $\begin{matrix} D_{0} [u_{n + 1} (t), u_{n} (t)] \leq \\ \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) \frac{(L (t - a)^{α})^{n}}{Γ (n α + α)}, \forall t \in J . \end{matrix}$

By the principle of mathematical induction that (3.8) holds for every n ≥ 1. Now using the estimation (3.8) for any t ∈ J, one has

$\begin{matrix} \sum_{n = 0}^{\infty} D_{0} [u_{n + 1} (t) \underset{gH}{⊖} u_{n} (t), \hat{0}] \\ = \sum_{n = 0}^{\infty} D_{0} [u_{n + 1} (t), u_{n} (t)] \\ \leq \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) \sum_{n = 0}^{\infty} \frac{(L (t - a)^{α})^{n}}{Γ (n α + α)} . \end{matrix}$ (3.9)

It follows that for every ɛ > 0 the series $u_{0} (t) + \sum_{n = 0}^{\infty} [u_{n + 1} (t) \underset{gH}{⊖} u_{n} (t)]$ is uniformly convergent on J with respect to the norm D₀ since the series of right-hand side of (3.9) is convergent to the Mittag-Leffler function E_α,α (L (t - a) ^α), that is,

$\begin{matrix} \sum_{n = 0}^{\infty} D_{0} [u_{n + 1} (t), u_{n} (t)] \\ \leq \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) E_{α, α} (L (t - a)^{α}) . \end{matrix}$ (3.10)

Now, we assume that

$\tilde{u} (t) = u_{0} (t) + \sum_{n = 0}^{\infty} [u_{n + 1} (t) \underset{gH}{⊖} u_{n} (t)], \forall t \in J .$ (3.11)

So,

$u_{j} (t) = u_{0} (t) + \sum_{n = 0}^{j} [u_{n + 1} (t) \underset{gH}{⊖} u_{n} (t)], \forall t \in J$ (3.12) is the j^th partial of the series (3.11). By (3.11) and (3.12), one has $\begin{matrix} lim_{j \to \infty} D_{0} [u_{j} (t), \tilde{u} (t)] \\ = lim_{j \to \infty} D_{0} [u_{j} (t) \underset{gH}{⊖} \tilde{u} (t), \hat{0}] = 0, \forall t \in J . \end{matrix}$

Define $u (t) = \tilde{u} (t)$ . Next, we show that this limit function is the solution of the following integral equation

$\begin{matrix} u (t) \underset{gH}{⊖} v_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (f (s, u (s)) + \int_{a}^{s} g (s, τ, u (τ)) d τ) ds, \forall t \in J . \end{matrix}$ (3.13) We set $\begin{matrix} (Fu) (t) = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (f (s, u (s)) + \int_{a}^{s} g (s, τ, u (τ)) d τ) ds . \end{matrix}$

Then, by using definition of successive approximation for any t ∈ J, we have $\begin{matrix} D_{0} [u (t) \underset{gH}{⊖} v_{0}, (Fu) (t)] \\ = D_{0} [u (t) \underset{gH}{⊖} v_{0} \underset{gH}{⊖} (u_{j} (t) \underset{gH}{⊖} v_{0}), (Fu) (t) \underset{gH}{⊖} ({Fu}_{j - 1}) (t)] \\ \leq D_{0} [\tilde{u} (t) \underset{gH}{⊖} u_{j} (t), \hat{0}] + D_{0} [(Fu) (t) \underset{gH}{⊖} ({Fu}_{j - 1}) (t), \hat{0}] \\ \leq D_{0} [\tilde{u} (t), u_{j} (t)] + \frac{L}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (D_{0} [u (s), u_{j - 1} (s)] + \int_{a}^{s} D_{0} [u (τ), u_{j - 1} (τ)] d τ) ds \end{matrix}$ (3.14)

From (3.11) and (3.12), one obtains, for any t ∈ J, $\begin{matrix} D_{0} [\tilde{u} (t), u_{j} (t)] \leq \sum_{n = j + 1}^{\infty} D_{0} [u_{n + 1} (t), u_{n} (t)] \end{matrix}$ and by (3.9), one has $\begin{matrix} D_{0} [u (t), u_{j} (t)] = D_{0} [\tilde{u} (t), u_{j} (t)] \\ \leq \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) \sum_{n = j + 1}^{\infty} \frac{(L (t - a)^{α})^{n}}{Γ (n α + α)}, \forall t \in J . \end{matrix}$ (3.15)

From the inequalities (3.14) and (3.15), we have $\begin{matrix} D_{0} [u (t) \underset{gH}{⊖} v_{0}, (Fu) (t)] \\ \leq \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) \sum_{n = j + 1}^{\infty} \frac{(L (t - a)^{α})^{n}}{Γ (n α + α)} \\ + E_{1, 1} (t - a) \frac{ɛ L (b - a)^{α}}{α Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (\sum_{n = j + 1}^{\infty} \frac{(L (s - a)^{α})^{n}}{Γ (n α + α)} + \int_{a}^{s} \sum_{n = j + 1}^{\infty} \frac{(L (τ - a)^{α})^{n}}{Γ (n α + α)} d τ) ds \\ \leq \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) \sum_{n = j + 1}^{\infty} \frac{(L (t - a)^{α})^{n}}{Γ (n α + α)} \end{matrix}$ $\begin{matrix} + E_{1, 1} (t - a) \frac{ɛ (b - a)^{α}}{α} \sum_{n = j + 1}^{\infty} \frac{(L (t - a)^{α})^{n + 1}}{Γ ((n + 1) α + α)} \\ + E_{1, 1} (t - a) \frac{ɛ (b - a)^{α + 1}}{α} \sum_{n = j + 1}^{\infty} \frac{(L (t - a)^{α})^{n + 1}}{Γ ((n + 1) α + α)} . \end{matrix}$

By taking limit as n→ ∞, we observe that the series on the right hand side of the inequality are convergent. This yields $\begin{matrix} D_{0} [u (t) \underset{gH}{⊖} v_{0}, (Fu) (t)] \leq 0, \forall t \in J . \end{matrix}$

This means that (3.13) is valid. In addition, by (3.10), (3.11) and (3.13) we have, for any t ∈ J, $\begin{matrix} D_{0} [v (t), u (t)] \leq \frac{ɛ (b - a)^{α}}{α} E_{1, 1} (t - a) E_{α, α} (L (t - a)^{α}) . \end{matrix}$

This yields an assertion that the problem (3.1) is Ulam-Hyers stable. Finally, we shall show the problem (3.1) has a unique solution. Assume that $\hat{u} (t)$ is another solution of (3.1) with initial condition $\hat{u} (a) = u (a)$ ,

$\begin{matrix} \hat{u} (t) \underset{gH}{⊖} v_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (f (s, \hat{u} (s)) + \int_{a}^{s} g (s, τ, \hat{u} (τ)) d τ) ds, \forall t \in J . \end{matrix}$ (3.16)

By Lipschitz condition of f and g, one has $\begin{matrix} D_{0} [u (t), \hat{u} (t)] \leq \frac{L}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (D_{0} [u (s), \hat{u} (s)] + \int_{a}^{s} D_{0} [u (τ), \hat{u} (τ)] d τ) ds \end{matrix}$

By putting $k (t) = D_{0} [u (t), \hat{u} (t)]$ , the above inequality yields $\begin{matrix} k (t) \leq \frac{L}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (k (s) + \int_{a}^{s} k (τ) d τ) ds \end{matrix}$

It follows, by a generalized Gronwall inequality (see Theorem 1 in [64]), that k (t) =0 for any t ∈ J. Therefore, $\begin{matrix} D_{0} [u (t), \hat{u} (t)] = 0, \end{matrix}$ for any t ∈ J. It yields that the problem (3.1) has a unique solution. □

Theorem 3.2. Let f : J × E → E and g : J × J × E → E be continuous fuzzy functions that satisfy the following conditions:

There exists a constant L > 0 such that $\begin{matrix} max {D_{0} [f (t, u), f (t, v)], D_{0} [g (t, s, u), g (t, s, v)]} \\ \leq L D_{0} [u, v] \end{matrix}$ for any (t, u), (t, v) ∈ J × E, (t, s, u), (t, s, v) ∈ J × J × E.

There exists a positive constant C ∈ (0, 1) such that kC^k ≤ (b - a) C^k-1, for all k ≥ 1, and 0 < CL < 1 .

For every ɛ > 0, if a fuzzy function v : J → E satisfies the inequality (3.3) for all t ∈ J provided that $\begin{matrix} \frac{1}{Γ (α^{*})} \int_{a}^{t} (t - s)^{α^{*} - 1} φ (s) ds \leq C φ (t), \forall t \in J, \end{matrix}$ where α^* ∈ (0, 1], then there exists a unique solution u : J → E of the problem (3.1) with initial condition u₀ = v₀ such that

$D_{0} [v (t), u (t)] \leq ɛ \frac{(b - a)}{(1 - C) (1 - CL)} φ (t), \forall t \in J .$ (3.17)

Proof. For each ɛ > 0 we consider a continuously differentiable function v : J → E which satisfies (3.3) for any t ∈ J, v ∈ E. By Remark 3, there exists a fuzzy function ζ ∈ E such that $D_{0} [ζ (t), \hat{0}] \leq ɛ φ (t)$ and for any t ∈ J $^{C} D_{a^{+}}^{α} v (t) = f (t, v (t)) + \int_{a}^{t} g (t, s, v (s)) ds + ζ (t) .$ (3.18)

Similar to the proof of Theorem 3, the solution of (3.18) satisfies the following integral equation $\begin{matrix} v (t) \underset{gH}{⊖} v_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \end{matrix}$

$\begin{matrix} (f (s, v (s)) + \int_{a}^{s} g (s, τ, v (τ)) d τ) ds \\ + \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} ζ (s) ds, \forall t \in J . \end{matrix}$ (3.19)

We also define u₀ (t) = v (t) and a sequence u_n : J→ E, n = 1, 2, … of successive approximations given by $\begin{matrix} u_{n} (t) \underset{gH}{⊖} v_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (f (s, u_{n - 1} (s)) + \int_{a}^{s} g (s, τ, u_{n - 1} (τ)) d τ) ds, \forall t \in J . \end{matrix}$ (3.20)

As in the proof of Theorem 3, for n = 1 and from (3.19), one has $\begin{matrix} D_{0} [u_{1} (t), u_{0} (t)] \\ = D_{0} [\frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, v (s)) \\ + \int_{a}^{s} g (s, τ, v (τ)) d τ) ds, v (t) \underset{gH}{⊖} v_{0}] \\ \leq D_{0} [\hat{0}, \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} ζ (s) ds] \\ \leq ɛ C φ (t) . \end{matrix}$

By assumption (i) and from the definition of successive approximations, one obtains for n = 1, 2, 3, …, $\begin{matrix} D_{0} [u_{n + 1} (t), u_{n} (t)] \leq \frac{L}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \\ (D_{0} [u_{n} (s), u_{n - 1} (s)] + \int_{a}^{s} D_{0} [u_{n} (τ), u_{n - 1} (τ)] d τ) ds . \end{matrix}$

Then one also has, for n = 1 and t ∈ J, $\begin{matrix} D_{0} [u_{2} (t), u_{1} (t)] \leq \frac{ɛ LC}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (φ (s) + \int_{a}^{s} φ (τ) d τ) ds \leq ɛ L (C^{2} + C^{3}) φ (t) . \end{matrix}$ This yields that $\begin{matrix} D_{0} [u_{3} (t), u_{2} (t)] \leq \frac{ɛ L^{2} (C^{2} + C^{3})}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (φ (s) + \int_{a}^{s} φ (τ) d τ) ds \leq 3 ɛ L^{2} (C^{3} + C^{4} + C^{5}) φ (t) . \end{matrix}$ By using mathematical induction method for n ≥ 4 one also gets $D_{0} [u_{n} (t), u_{n - 1} (t)] \leq n ɛ L^{n - 1} (C^{n} + C^{n + 1} + \dots + C^{2 n} + C^{2 n + 1}) φ (t) .$ (3.21) Then, by assumption (ii) the estimation (3.21) can be rewritten as follows $\begin{matrix} D_{0} [u_{n} (t), u_{n - 1} (t)] & \leq ɛ (b - a) (CL)^{n - 1} (1 + C^{1} + \dots + C^{n + 1}) φ (t) \\ \leq ɛ (b - a) (\frac{1 - C^{n + 1}}{1 - C}) (CL)^{n - 1} φ (t), \forall t \in J . \end{matrix}$ In addition, if we assume that $D_{0} [u_{n} (t), u_{n - 1} (t)] \leq ɛ (b - a) (\frac{1 - C^{n + 1}}{1 - C}) (CL)^{n - 1} φ (t), \forall t \in J,$ (3.22) then we also obtain $\begin{matrix} D_{0} [u_{n + 1} (t), u_{n} (t)] \leq ɛ (b - a) (\frac{1 - C^{n + 2}}{1 - C}) (CL)^{n} φ (t), \forall t \in J . \end{matrix}$ By the principle of mathematical induction, (3.22) holds for every n ≥ 1. Now using the estimation (3.22) for any t ∈ J, we get $\sum_{n = 0}^{\infty} D_{0} [u_{n + 1} (t), u_{n} (t)] \leq ɛ (b - a) (\frac{1}{1 - C}) \sum_{n = 0}^{\infty} (CL)^{n} φ (t)$ (3.23) By assumption (ii), we infer that the series $\sum_{n = 0}^{\infty} (CL)^{n}$ is convergent and equal to $\frac{1}{1 - CL}$ . Hence, for every ɛ > 0 we infer that the series $u_{0} (t) + \sum_{n = 0}^{\infty} [u_{n + 1} (t) \underset{gH}{⊖} u_{n} (t)]$ is uniformly convergent on J and $\sum_{n = 0}^{\infty} D_{0} [u_{n + 1} (t), u_{n} (t)] \leq ɛ \frac{(b - a)}{(1 - C) (1 - CL)} φ (t), \forall t \in J .$ (3.24) Similar to the proof of Theorem 3, we can show that the limit of integral equation (3.20) is the following integral equation $\begin{matrix} u (t) \underset{gH}{⊖} v_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, u (s)) + \int_{a}^{s} g (s, τ, u (τ)) d τ) ds, \forall t \in J . \end{matrix}$ and u (t) is also a solution of (3.1) with initial condition u₀. In addition, it holds $\begin{matrix} D_{0} [v (t), u (t)] \leq ɛ \frac{(b - a)}{(1 - C) (1 - CL)} φ (t), \forall t \in J . \end{matrix}$ This yields an assertion that (3.1) is Ulam-Hyers-Rassias stable. □

4 Stability for initial value problems with Riemann-Liouville fractional derivative concept

This section deals with the existence and the Ulam stability of solution for the following fuzzy Volterra integro-differential equations with the Riemann-Liouville fractional concept of the form

${\begin{matrix} ^{RL} D_{a^{+}}^{α} u (t) = f (t, u (t)) + \int_{a}^{t} g (t, s, u (s)) ds, t \in J \\ (ℑ_{a^{+}}^{1 - α} u) (t) |_{t = a} = u_{0}, \end{matrix}$ (4.1) where α ∈ (0, 1), f : J × E → E, g : J × J × E → E are continuous fuzzy functions with respect to t, s and u on J × E, $^{RL} D_{a^{+}}^{α} u$ denotes Riemann-Liouville fractional derivative given in Definition 2. A fuzzy function u is called a solution of the problem (3.1) if and only if $^{RL} D_{a^{+}}^{α} u (t) = f (t, u (t)) + \int_{a}^{t} g (t, s, u (s)) ds$ and $(ℑ_{a^{+}}^{1 - α} u) (t) |_{t = a} = u_{0}$ . Now, we consider the Ulam stability for the problem (4.1). Let ɛ > 0 and $φ : J \to ℝ^{+}$ be a continuous function. Consider the following inequalities $D_{0} [^{RL} D_{a^{+}}^{α} w (t), f (t, w (t)) + \int_{a}^{t} g (t, s, w (s)) ds] \leq ɛ, t \in J .$ (4.2) $D_{0} [^{RL} D_{a^{+}}^{α} w (t), f (t, w (t)) + \int_{a}^{t} g (t, s, w (s)) ds] \leq φ (t), t \in J .$ (4.3) $D_{0} [^{RL} D_{a^{+}}^{α} u (t), f (t, w (t)) + \int_{a}^{t} g (t, s, w (s)) ds] \leq ɛ φ (t), t \in J .$ (4.4)

Definition 4.1. The problem (4.1) is

Ulam-Hyers stable if there exists a positive real number K_f such that for each ɛ and for each solution v ∈ C (J, E) of the inequality (4.2) there exists a solution u ∈ C (J, E) of (4.1) with $\begin{matrix} D_{0} [u (t), v (t)] \leq ɛ K_{f}, t \in J . \end{matrix}$

generalized Ulam-Hyers stable if there exists a positive-valued continuous functions K_f with K_f (0) =0 such that for each ɛ and for each solution v ∈ C (J, E) of the inequality (4.2) there exists a solution u ∈ C (J, E) of (4.1) with $\begin{matrix} D_{0} [u (t), v (t)] \leq K_{f} (ɛ), t \in J . \end{matrix}$

Ulam-Hyers-Rassias stable with respect to φ if there exists a real number K_f,φ > 0 such that for each ɛ and for each solution v ∈ C (J, E) of the inequality (4.4) there exists a solution u ∈ C (J, E) of (4.1) with $\begin{matrix} D_{0} [u (t), v (t)] \leq ɛ K_{f, φ} φ (t), t \in J . \end{matrix}$

generalized Ulam-Hyers-Rassias stable with respect to φ if there exists K_f,φ > 0 such that for each ɛ and for each solution v ∈ C (J, E) of the inequality (4.3) there exists a solution u ∈ C (J, E) of (4.1) with $\begin{matrix} D_{0} [u (t), v (t)] \leq K_{f, φ} φ (t), t \in J . \end{matrix}$

In Definition 4, we observe that (i) ⇒ (ii); (iii) ⇒ (iv); (iii) ⇒ (i) if φ (·) =1.

Remark 4.1. If v ∈ C (J, E) is a solution of the inequality (4.3) then we have the following estimation: $\begin{matrix} D_{0} [v (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} v_{0}, \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, v (s)) + \int_{a}^{s} g (s, τ, v (τ)) d τ) ds] \leq \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} φ (s) ds, \end{matrix}$ where v₀ = u₀, with u₀ is the initial value of the problem (4.1).

Proof. From the inequality (4.3), we observe that a fuzzy function v ∈ C (J, E) is a solution of (4.3) if and only if there exists a fuzzy function δ ∈ C (J, E) such that $D_{0} [δ (t), \hat{0}] \leq φ (t)$ for any t ∈ J, and $^{RL} D_{a^{+}}^{α} v (t) = f (t, v (t)) + \int_{a}^{t} g (t, s, v (s)) ds + δ (t),$ (4.5) for any t ∈ J. So, from Appendix section, the solution of (4.5) with the initial condition $(ℑ_{a^{+}}^{1 - α} v) (t) |_{t = a} = v_{0}$ is formulated by $\begin{matrix} v (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} v_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, v (s)) + \int_{a}^{s} g (s, τ, v (τ)) d τ + δ (s)) ds . \end{matrix}$

This yields $\begin{matrix} D_{0} [v (t) \underset{gH}{⊖} I (t), F (t)] & = D_{0} [v (t) \underset{gH}{⊖} I (t) + \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} δ (s) ds, F (t) + \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} δ (s) ds] \\ = D_{0} [v (t) \underset{gH}{⊖} I (t) + \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} δ (s) ds, v (t) \underset{gH}{⊖} I (t)] \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} φ (s) ds, \end{matrix}$ where $I (t) : = \frac{(t - a)^{α - 1}}{Γ (α)} v_{0}, F (t) : = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, v (s)) + \int_{a}^{s} g (s, τ, v (τ)) d τ) ds .$ □

The generalized Ulam-Hyers-Rassias stability of the problem (4.1) is given by the below theorem.

Theorem 4.1. Assume that f : J × E → E, g : J × J × E → E are continuous fuzzy functions which satisfy the following hypotheses:

(i) There exists real-valued continuous functions $P_{f} : J \to ℝ^{+}, P_{g} : J \to ℝ^{+}$ such that for t, s ∈ J and for each w ∈ E $\begin{matrix} D_{0} [f (t, w), \hat{0}] \leq P_{f} (t) \frac{D_{0} [w, \hat{0}]}{1 + D_{0} [w, \hat{0}]} and D_{0} [g (t, s, w), \hat{0}] \leq P_{g} (t) \frac{D_{0} [w, \hat{0}]}{1 + D_{0} [w, \hat{0}]}; \end{matrix}$ (ii) There exists a positive constants λ_φ and a continuous function $q : J \to ℝ^{+}$ such that for each t ∈ J and for all α^* ∈ (0, 1] we have $\begin{matrix} \frac{1}{Γ (α^{*})} \int_{a}^{t} (t - s)^{α^{*} - 1} φ (s) ds \leq λ_{φ} φ (t) and max {P_{f} (t), P_{g} (t)} \leq q (t) φ (t); \end{matrix}$ (iii) There exists a continuous function $ψ : J \to ℝ^{+}$ such that for each t, s ∈ J, and for all z, w ∈ E, we have $\begin{matrix} max {D_{0} [f (t, z), f (t, w)], D_{0} [g (t, s, z), g (t, s, w)]} \leq ψ (t) (t - a)^{1 - α} D_{0} [z, w] . \end{matrix}$

If $L : = [\frac{b - a}{Γ (α + 1)} (1 + \frac{b - a}{α + 1}) sup_{t \in J} ψ (t)] < 1$ , then there exists a unique solution u^* of problem (4.1), and the problem (4.1) is generalized Ulam-Hyers-Rassias stable.

Proof. Based on hypotheses (i)-(ii) and from an application of Schauder’s theorem, we shall show that the problem (4.1) has at least one solution defined on J, and the problem (4.1) is generalized Ulam-Hyers-Rassias stable. Let γ : =1 - α. Consider the operator $ℚ : C_{γ} (J, E) \to C_{γ} (J, E)$ defined as follows:

$(ℚ u) (t) \underset{gH}{⊖} I (t) = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, u (s)) + \int_{a}^{s} g (s, τ, u (τ)) d τ) ds,$ (4.6) where $I (t) : = \frac{(t - a)^{α - 1}}{Γ (α)} u_{0} .$ Set $r = \frac{1}{Γ (α)} (D_{0} [u_{0}, \hat{0}] + P^{*} \frac{b}{α} (1 + \frac{b}{α + 1}))$ , where $P^{*} = max {sup_{t \in J}$ $P_{f} (t), sup_{t \in J} P_{g} (t)}$ . Consider the ball $B_{r} : = B (\hat{0}, r) = {u \in C_{γ} (J, E) : ∥ u ∥_{γ} \leq r} .$ For each t ∈ J and any u ∈ C_γ (J, E), by hypothesis (i) one has $\begin{matrix} D_{0} [(t - a)^{γ} (ℚ u) (t), \hat{0}] \leq & \frac{D_{0} [u_{0}, \hat{0}]}{Γ (α)} \\ + \frac{(t - a)^{γ}}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (D_{0} [f (s, u (s)), \hat{0}] + \int_{a}^{s} D_{0} [g (s, τ, u (τ)), \hat{0}] d τ) ds \\ \leq \frac{D_{0} [u_{0}, \hat{0}]}{Γ (α)} + \frac{(b - a)^{γ} P^{*}}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (1 + (s - a)) ds \\ \leq \frac{1}{Γ (α)} (D_{0} [u_{0}, \hat{0}] + P^{*} \frac{(b - a)^{γ + α}}{α} (1 + \frac{(b - a)}{α + 1})) \\ \leq r . \end{matrix}$ where γ = 1 - α . This proves that the operator $ℚ$ transforms the ball B_r into itself. In the sequel, we shall show that the operator $ℚ : B_{r} \to B_{r}$ satisfies all the assumptions of Schauder’s theorem.

+ The operator $ℚ : B_{r} \to B_{r}$ is continuous and uniformly bounded. Indeed, assume that {u_n} _n≥1 be a sequence such that u_n → u in B_r. Then, we have for each t ∈ J $\begin{matrix} D_{0} [(t - a)^{γ} ((ℚ u_{n}) (t) \underset{gH}{⊖} I (t)) & , (t - a)^{γ} ((ℚ u) (t) \underset{gH}{⊖} I (t))] \\ \leq \frac{(t - a)^{γ}}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} D_{0} [f (s, u_{n} (s)), f (s, u (s))] ds \\ + \frac{(t - a)^{γ}}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \int_{a}^{s} D_{0} [g (s, τ, u_{n} (τ)), g (s, τ, u (τ))] d τ ds . \end{matrix}$

Since f, g are continuous fuzzy functions and u_n → u in B_r, by the Lebesgue dominated convergence Theorem the above inequality implies $\begin{matrix} D_{γ} [(ℚ u_{n}) (t), (ℚ u) (t)] \to 0 as n \to + \infty . \end{matrix}$

In addition, since $ℚ (B_{r}) \subset B_{r}$ and B_r is bounded, it follows that the operator $ℚ : B_{r} \to B_{r}$ is uniformly bounded.

+ Next, $ℚ (B_{r})$ is equicontinuous. Indeed, let u ∈ B_r and a ≤ t₁ < t₂ ≤ b, then one has $\begin{matrix} D_{0} [(t_{2} - a)^{γ} (ℚ u) (t_{2}), (t_{1} - a)^{γ} (ℚ u) (t_{1})] = D_{0} [(t_{2} - a)^{γ} ((ℚ u) (t_{2}) \underset{gH}{⊖} I (t_{2})), (t_{1} - a)^{γ} ((ℚ u) (t_{1}) \underset{gH}{⊖} I (t_{1}))] \\ \leq \frac{1}{Γ (α)} D_{0} [(t_{2} - a)^{γ} \int_{a}^{t_{2}} (t_{2} - s)^{α - 1} f (s, u (s)) ds, (t_{1} - a)^{γ} \int_{a}^{t_{1}} (t_{1} - s)^{α - 1} f (s, u (s)) ds] \\ + \frac{1}{Γ (α)} D_{0} [(t_{2} - a)^{γ} \int_{a}^{t_{2}} (t_{2} - s)^{α - 1} \int_{a}^{s} g (t, τ, u (τ)) d τ ds, (t_{1} - a)^{γ} \int_{a}^{t_{1}} (t_{1} - s)^{α - 1} \int_{a}^{s} g (t, τ, u (τ)) d τ ds] \\ : = E_{1} + E_{2}, \end{matrix}$ where $\begin{matrix} E_{1} & \leq \frac{P^{*} (t_{2} - a)^{γ}}{Γ (α)} \int_{t_{1}}^{t_{2}} (t_{2} - s)^{α - 1} ds + \frac{P^{*}}{Γ (α)} | \int_{a}^{t_{1}} [(t_{2} - a)^{γ} (t_{2} - s)^{α - 1} - (t_{1} - a)^{γ} (t_{1} - s)^{α - 1}] ds | \\ = \frac{P^{*} (t_{2} - a)^{γ}}{Γ (α + 1)} (t_{2} - t_{1})^{α} + \frac{P^{*}}{Γ (α + 1)} | (t_{2} - t_{1}) - (t_{2} - a)^{α} (t_{2} - t_{1})^{α} |, \\ E_{2} & \leq \frac{P^{*} (t_{2} - a)^{γ}}{Γ (α)} \int_{t_{1}}^{t_{2}} (t_{2} - s)^{α - 1} (s - a) ds \\ + \frac{P^{*}}{Γ (α)} | \int_{a}^{t_{1}} [(t_{2} - a)^{γ} (t_{2} - s)^{α - 1} (s - a) - (t_{1} - a)^{γ} (t_{1} - s)^{α - 1} (s - a)] ds | \\ = \frac{P^{*} (1 / (α + 1) + t_{1} - a) (t_{2} - a)^{γ}}{Γ (α + 1)} (t_{2} - t_{1})^{α} + \frac{P^{*} (t_{2} + t_{1} - 2 a)}{Γ (α + 2)} (t_{2} - t_{1}) \\ + \frac{P^{*} (α (t_{2} - a)^{1 - α}) - (α + 1) (t_{2} - a)^{2 - α}}{Γ (α + 2)} (t_{2} - t_{1})^{α}, \end{matrix}$ by using assumption (i). Thus, as t₁ → t₂, the right-hand sides of the above inequality tends to zero. As a consequence of the operator $ℚ$ together with the Arzelá-Ascoli Theorem, we can infer that the operator $ℚ : B_{r} \to B_{r}$ is continuous and compact. From an application of Schauder’s Theorem we conclude that the operator $ℚ$ has a fixed point u^* which is solution of the problem (4.1). Now, based on the hypothesis (ii) the generalized Ulam-Hyers-Rassias stability of the problem (4.1) will be proven. Let u^* is a solution of the problem (4.1), then one has

$u^{*} (t) \underset{gH}{⊖} I (t) = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, u^{*} (s)) + \int_{a}^{s} g (s, τ, u^{*} (τ)) d τ) ds .$ (4.7)

Let v is a solution of the inequality (4.3) provided that v₀ = u₀, then from Remark 4 v is a solution of the following integral inequality, for each t ∈ J, $\begin{matrix} D_{0} [v (t) \underset{gH}{⊖} I (t), \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, v (s)) + \int_{a}^{s} g (s, τ, v (τ)) d τ) ds] \leq \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} φ (s) ds . \end{matrix}$

Therefore, from assumptions (i)-(ii) and (4.7), one gets $\begin{matrix} D_{0} [u^{*} (t), v (t)] & = D_{0} [u^{*} (t) \underset{gH}{⊖} I (t), v (t) \underset{gH}{⊖} I (t)] \\ = D_{0} [\frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, u^{*} (s)) + \int_{a}^{s} g (s, τ, u^{*} (τ)) d τ) ds, v (t) \underset{gH}{⊖} I (t)] \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (D_{0} [f (s, u^{*} (s)), f (s, v (s))] + \int_{a}^{s} D_{0} [g (s, τ, u^{*} (τ)), g (s, τ, v (τ))] d τ) ds \\ + \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} φ (s) ds \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (2 P_{f} (s) + \int_{a}^{s} 2 P_{g} (τ) d τ) ds + λ_{φ} φ (t) \end{matrix}$ $\begin{matrix} \leq λ_{φ} φ (t) + \frac{2 q^{*}}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} φ (s) ds + \frac{2 q^{*}}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \int_{a}^{s} φ (τ) d τ ds \\ \leq [λ_{φ} + 2 q^{*} λ_{φ} + 2 q^{*} λ_{φ}^{2}] φ (t), \end{matrix}$ where $q^{*} : = sup_{t \in J} q (t)$ . Then, if we set $K_{f, φ} = (λ_{φ} + 2 q^{*} λ_{φ} + 2 q^{*} λ_{φ}^{2})$ , the problem (4.1) is generalized Ulam-Hyers-Rassias stable. Finally, we shall prove there exists a unique solution u^* of the problem (4.1) with assumption (iii). Indeed, let $ℚ : C_{γ} (J, E) \to C_{γ} (J, E)$ be the operator defined in (4.6), then one has $\begin{matrix} D_{0} [({Q u}^{*}) (t) & , (Q u) (t)] = D_{0} [({Q u}^{*}) (t) \underset{gH}{⊖} I (t), (Q u) (t) \underset{gH}{⊖} I (t)] \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (D_{0} [f (s, u^{*} (s)), f (s, u (s))] + \int_{a}^{s} D_{0} [g (s, τ, u^{*} (τ)), g (s, τ, u (τ))] d τ) ds \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} ψ (s) D_{0} [(s - a)^{1 - α} u^{*} (s), (s - a)^{1 - α} u (s)] ds \\ + \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \int_{a}^{s} ψ (τ) D_{0} [(τ - a)^{1 - α} u^{*} (τ), (τ - a)^{1 - α} u (τ)] d τ ds \\ \leq \frac{ψ^{*} D_{γ} [u^{*}, v]}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} ds + \frac{ψ^{*} D_{γ} [u^{*}, v]}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (s - a) ds \\ \leq ψ^{*} \frac{b - a}{Γ (α + 1)} (1 + \frac{b - a}{α + 1}) D_{γ} [u^{*}, v], \end{matrix}$ where $ψ^{*} = sup_{t \in J} ψ (t) .$ By the hypothesis $L : = [ψ^{*} \frac{b - a}{Γ (α + 1)} (1 + \frac{b - a}{α + 1})] < 1$ , the operator $ℚ : C_{γ} (J, E) \to C_{γ} (J, E)$ is contractive. The proof is completed. □

The Ulam-Hyers-Rassias stability of the problem (4.1) is presented as follows.

Theorem 4.2. Assume that f : J × E → E, g : J × J × E → E are continuous fuzzy functions which satisfy the following hypotheses:

There exist positive constants L_f, L_g such that, for all t ∈ J and z, w ∈ E, $\begin{matrix} D_{0} [f (t, z), f (t, w)] \leq L_{f} D_{0} [z, w], D_{0} [g (t, s, z), g (t, s, w)] \leq L_{g} D_{0} [z, w] . \end{matrix}$

Let $φ \in C (J, ℝ^{+})$ and α^* ∈ (0, 1], there exists C_φ > 0 such that for each t ∈ J $\begin{matrix} \frac{1}{Γ (α^{*})} \int_{a}^{t} (t - s)^{α^{*} - 1} φ (s) ds \leq C_{φ} φ (s) . \end{matrix}$

Then, for every ɛ > 0, if a fuzzy continuous function v : J → E satisfies the inequality (4.4) for all t ∈ J, there exists a unique solution u : J → E of the problem (4.1) with initial condition u₀ = v₀ as follows:

$u (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, u (s)) + \int_{a}^{s} g (s, τ, u (τ)) d τ) ds$ (4.8) provided that $L : = (C_{φ} L_{f} + C_{φ}^{2} L_{g}) \in (0, 1) .$ In addition, the problem (4.1) is Ulam-Hyers-Rassias stable, i.e., $\begin{matrix} D_{0} [u (t), v (t)] \leq \frac{ɛ C_{φ}}{1 - L} φ (t), t \in J . \end{matrix}$

Proof. In this proof, we reconsider the operator $ℚ : X \to X$ defined by $\begin{matrix} (Q u) (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, u (s)) + \int_{a}^{s} g (s, τ, u (τ)) d τ) ds, \end{matrix}$ where X is the set of all fuzzy continuous functions on J defined by X = {w : J → E | w ∈ C (J, E) and w is d - monotone}, with the generalized metric on X defined by $\begin{matrix} D_{X} [z, w] = inf {C \in [0, + \infty] | D_{0} [z (t), w (t)] \leq C φ (t), \forall t \in J} . \end{matrix}$

Obviously, the operator is well-defined operator according to the continuous hypothesis of the fuzzy function f. In order to use Theorem 2, first of all we shall show that the operator is strictly contractive on X. According to definition of the space (X, D_X), we observe that for any z, w ∈ X, it is possible to find C ∈ [0, ∞] such that $D_{0} [z (t), w (t)] \leq C φ (t), \forall t \in J .$ (4.9)

From (4.9) and the hypotheses (i)-(ii), one has

$\begin{matrix} D_{0} [Q z (t), ℚ w (t)] & = D_{0} [Q z (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{0}, ℚ w (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{0}] \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (L_{f} D_{0} [z (s), w (s)] + \int_{a}^{s} L_{g} D_{0} [z (τ), w (τ)] d τ) ds \\ \leq (C_{φ} L_{f} + C_{φ}^{2} L_{g}) C φ (t) . \end{matrix}$ (4.10)

Hence, we conclude $D_{X} [Q z, ℚ w] \leq (C_{φ} L_{f} + C_{φ}^{2} L_{g}) D_{X} [z, w]$ for any z, w ∈ X. This yields the operator $ℚ$ is strictly contractive on X, because $L : = (C_{φ} L_{f} + C_{φ}^{2} L_{g}) \in (0, 1)$ . By the above estimate, from the continuous property of $Q_{w 0}$ on [a, b], f (s, w₀ (s)), g (s, τ, w₀ (τ)), w₀ are bounded on their domains and $min_{t \in [a, b]} φ (t) > 0$ , then there exist a constant W ∈ (0, ∞) such that for t ∈ [a, b] $\begin{matrix} D_{0} [{Q w}_{0} (t) & , w_{0} (t)] = D_{0} [Q w (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{0}, w_{0} (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{0}] \\ = D_{0} [\frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, w_{0} (s)) + \int_{a}^{s} g (s, τ, w_{0} (τ)) d τ) ds, w_{0} (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{0}] \\ \leq W φ (t), \end{matrix}$ where w₀ ∈ X is arbitrary. Thus, by definition of D_X we infer that $D_{X} [{Q w}_{0} (t), w_{0} (t)] < \infty$ . Also, by Theorem 2, there exist a continuous fuzzy function u such that $ℚ^{n} u \to u$ in (X, D_X) and $ℚ u = u_{0}$ , that is, u corresponds to (4.8) on [a, b] . According to (ii) of Theorem 2, in order to show that u given by (4.8) is a unique solution, we shall prove that X = {w ∈ X|D_X [w₀, w] < ∞} . Indeed, let w ∈ X . Because w and w₀ are bounded on [a, b] and $min_{t \in [a, b]} φ (t) > 0$ , there exist a constant C_w ∈ (0, ∞) such that for any t ∈ [a, b] $\begin{matrix} D_{0} [w_{0} (t), w_{(} t)] \leq C_{w} φ (t), t \in [a, b] . \end{matrix}$

This yields that D_X [w₀, w]< ∞ for all w ∈ X. Next, let a fuzzy continuous function v : J → E satisfies the inequality (4.4) for all t ∈ J, then by Remark 4 we have that $\begin{matrix} D_{0} & [v (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} v_{0}, \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, v (s)) + \int_{a}^{s} g (s, τ, v (τ)) d τ) ds] \\ \leq \frac{ɛ}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} φ (s) ds, \end{matrix}$ and this yields $\begin{matrix} D_{0} [v (t), ℚ v (t)] \\ = D_{0} [v (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} v_{0}, ℚ v (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} v_{0}] \\ \leq ɛ C_{φ} φ (t), \forall t \in [a, b] . \end{matrix}$

By definition of D_X, we obtain $D_{X} [v, ℚ v] \leq ɛ C_{φ} .$ Again by Theorem 2, we deduce that $\begin{matrix} D_{X} [v, u] \leq \frac{1}{1 - L} D_{X} [ℚ v, v] \leq \frac{ɛ C_{φ}}{1 - L}, \end{matrix}$ where $L : = (C_{φ} L_{f} + C_{φ}^{2} L_{g}) \in (0, 1) .$ The proof is complete. □

In the sequel, the following corollary is the result of the Ulam-Hyers stability of the problem (4.1).

Corollary 4.1. Let f : J × E → E, g : J × J × E → E are continuous fuzzy functions. Assume that the hypothesis (i) in Theorem 4 are satisfied. If for ɛ > 0 a continuous fuzzy function v satisfies the inequality (4.2) for all t ∈ J, there exists a unique solution u : J → E of the problem (4.1) with initial condition u₀ = v₀, and the problem (4.1) is Ulam-Hyers stable, i.e., $\begin{matrix} D_{0} [u (t), v (t)] \leq \frac{(b - a)^{α}}{Γ (α + 1) - (b - a)^{α} (L_{f} + \frac{(b - a)}{2} L_{g})} ɛ \end{matrix}$ provided that $L : = \frac{(b - a)^{α}}{Γ (α + 1)} [L_{f} + \frac{(b - a)}{2} L_{g}] \in (0, 1) .$

Proof. With the same argument as in the proof of Theorem 4 by taking φ (·) =1, ∀ t ∈ [a, b], we shall obtain the assertion of this corollary.

5 Conclusion

About the novelty of this paper, we emphasize that we initially introduce a new concept of Ulam-Hyers-Rassias’s types stability of fractional differential equations with the fuzzy concept and analyze the existence and stability of solutions, which fits into recently developing theory of fuzzy differential equations with fractional derivative.

Footnotes

Appendix

Theorem A. Let u be a fuzzy function such that u_1-α (t) ∈ AC ([a, b], E) and d-monotone on [a, b] . If u is d-monotone and satisfies the initial value problem (4.1), then u satisfies the following fractional Volterra integral equation:

(5.1) $u (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{0} = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (f (s, u (s)) + \int_{a}^{s} g (s, τ, u (τ)) d τ) ds .$

Proof. Let u be d-monotone and satisfy the initial value problem (4.1). We notice that for a real-valued function ψ ∈ L [a, b] such that ψ ∈ AC [a, b], one has for t ∈ [a, b] $\begin{matrix} I_{a^{+}}^{α} D_{a^{+}}^{α} ψ (t) = ψ (t) - \frac{(t - a)^{α - 1}}{Γ (α)} ψ_{1 - α} (a), \end{matrix}$ where $I_{a^{+}}^{α}, D_{a^{+}}^{α}$ denote Riemann-Liouville integral and derivative of the real-valued functions, respectively. This implies that $\begin{matrix} I_{a^{+}}^{α} D_{a^{+}}^{α} d [u (t)]^{r} = d [u (t)]^{r} - \frac{(t - a)^{α - 1}}{Γ (α)} d [u_{1 - α} (a)]^{r} \end{matrix}$ for every r ∈ [0, 1] and for t ∈ (a, b] . On the other hand, since $I_{a^{+}}^{α} D_{a^{+}}^{α} d [u (t)]^{r} = I_{a^{+}}^{α} \frac{d}{dt} I_{a^{+}}^{1 - α} d [u (t)]^{r} = I_{a^{+}}^{α} \frac{d}{dt} d [u_{1 - α} (t)]^{r},$ it follows that $d [u (t)]^{r} - \frac{(t - a)^{α - 1}}{Γ (α)} d [u_{1 - α} (a)]^{r} = I_{a^{+}}^{α} \frac{d}{dt} d [u_{1 - α} (t)]^{r}, t \in (a, b] .$ + As u_1-α is d-decreasing on [a, b], we have $\frac{d}{dt} d [u_{1 - α} (t)]^{r} \leq 0$ for t ∈ (a, b] and for every r ∈ [0, 1] and thus we get $I_{a^{+}}^{α} \frac{d}{dt} d [u_{1 - α} (t)]^{r} \leq 0$ for t ∈ (a, b] and for every r ∈ [0, 1]. Hence, we infer that d [u (t)] ^r≤ $\frac{(t - a)^{α - 1}}{Γ (α)} d [u_{1 - α} (a)]^{r}$ for t ∈ (a, b] and for every r ∈ [0, 1]. Moreover, we get for t ∈ (a, b]

(5.2) $\begin{matrix} [(ℑ_{a^{+}}^{α}^{RL} D_{a^{+}}^{α} u) (t)]^{r} & = & [I_{a^{+}}^{α} D_{a^{+}}^{α} \underline{u} (t, r), I_{a^{+}}^{α} D_{a^{+}}^{α} \bar{u} (t, r)] \\ = & [\underline{u} (t, r) - \frac{(t - a)^{α - 1}}{Γ (α)} {\underline{u}}_{1 - α} (a, r), \bar{u} (t, r) - \frac{(t - a)^{α - 1}}{Γ (α)} {\bar{u}}_{1 - α} (a, r)] \\ = & [\underline{u} (t, r), \bar{u} (t, r)] ⊖ \frac{(t - a)^{α - 1}}{Γ (α)} [{\underline{u}}_{1 - α} (a, r), {\bar{u}}_{1 - α} (a, r)] . \end{matrix}$ + As u_1-α is d-increasing on [a, b], it implies that $I_{a^{+}}^{α} \frac{d}{dt} d [u_{1 - α} (t)]^{r}$ ≥0 for t ∈ (a, b] or $d [u (t)]^{r} \geq \frac{(t - a)^{α - 1}}{Γ (α)} d [u_{1 - α} (a)]^{r}$ for t ∈ (a, b] and for every r ∈ [0, 1]. Moreover, we obtain

(5.3) $[(ℑ_{a^{+}}^{α}^{RL} D_{a^{+}}^{α} u) (t)]^{r} = - \frac{(t - a)^{α - 1}}{Γ (α)} [{\underline{u}}_{1 - α} (a), {\bar{u}}_{1 - α} (a)] ⊖ (- [\underline{u} (t, r), \bar{u} (t, r)])$ for t ∈ (a, b] and for every r ∈ [0, 1] .

Therefore, from (5.2), (5.3) and $(^{RL} D_{a^{+}}^{α - 1} u) (a) : = u_{1 - α} (a) = u_{0}$ , we get

(5.4) $\begin{matrix} (ℑ_{a^{+}}^{α}^{RL} D_{a^{+}}^{α} u) (t) & = u (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{1 - α} (a) \\ = u (t) \underset{gH}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} u_{0} . \end{matrix}$

Applying the operator $ℑ_{a^{+}}^{α}$ to both sides of problem (4.1) and from (5.4), we obtain the integral Equation (5.1). □

Acknowledgment

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.319.

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