Fuzzy Laplace transform method for the Ulam stability of linear fuzzy differential equations of first order with constant coefficients

Abstract

In the present paper, the notion of fuzzy multiplication-convolution is introduced. Meantime, an important property is derived, which is similar to the Laplace transform of an ordinary convolution. By using the fuzzy Laplace transform and the preceding property, this paper deals with the Ulam stability of three variants of first order linear fuzzy differential equations with constant coefficients.

Keywords

Fuzzy Laplace transform fuzzy multiplication-convolution Ulam stability linear fuzzy differential equation generalized differentiability

1 Introduction

With the rapid development of the Ulam stability of functional equations, such stability problems of differential equations have gradually received attention among scholars. Obloza [16] seems to be the first author to consider the Ulam stability problem of differential equations. Afterwards, Alsina and Ger [3] studied the Hyers-Ulam stability of the differential equation y′ = y. Hereafter, the deep research on the Ulam stability in various abstract spaces carried out by Miura and Takahasi [14 , 22]. Since then, the Ulam stability problems of different types of differential equations, especially linear differential equations, have been systematically and extensively investigated by various authors. It is worth noting that several methods for solving differential equations have been applied to study the Ulam stability of differential equations, such as the integrating factor method [23], the power series method [11] and the Laplace transform method [17] and so on.

At present, there are many different interpretations for fuzzy differential equations, such as fuzzy differential equations based on Hukuhara differentiability (i.e., H-differentiability) [12], fuzzy differential inclusions based on the theory of differential inclusion for set-valued functions [5], fuzzy differential equations using the derivative and integral operators defined by Zadeh’s extension principle [6], and so on. However, each interpretation contains some flaws. Until now, most studies associated with fuzzy differential equations are based on H-differentiability. The main shortcoming of this approach is the length of support or the diameter of the solution is non-decreasing as the time increases. Consequently, non periodic behavior can be modeled except in the non-fuzzy case. To overcome this shortcoming, Bede and Gal [7] proposed the strongly generalized differentiability (GH-differentiability) which allows a decreasing diameter of the solution of a fuzzy differential equation. GH-derivative can be viewed as the first generalization of H-derivative. Later, the other more general derivatives, the generalized Hukuhara derivative (gH-derivative) and the most general so far, the fuzzy generalized derivative (g-derivative) [9], were proposed to enlarge the class of differentiable fuzzy number-valued functions. Although these generalized derivatives have been extensively studied, the theory of fuzzy differential equations using the generalizations of H-derivative has still been limited to the version of GH-derivative. Recently, Shen [19] considered the Ulam stability of three variants of first order linear fuzzy differential equations under GH-differentiability using the direct method. Soon after, he [20] further studied the Ulam stability of two types of linear partial fuzzy differential equation of first order under GH-differentiability. Especially, Shen and Wang [21] studied the Ulam stability of a general fuzzy differential equation under generalized differentiability by using the fixed point technique.

In the theory of differential equations, the Laplace transform method, which can convert a differential equation into an algebraic equation, is an important and effective tool for solving linear differential equations with constant coefficients. For this reason, Allahviranloo and Ahmadi [2] proposed the fuzzy Laplace transform and applied it to solve first order fuzzy differential equations under GH-differentiability. Subsequently, Salahshour and Allahviranloo [18] further studied the existence of the fuzzy Laplace transform and expanded its applications. Lately, Ahmadi et al. [1] proved a Laplace transform formula of an n-th order fuzzy derivative and showed its application in fuzzy ordinary differential equations of second order under GH-differentiability.

The purpose of the present paper is to reconsider the Ulam stability of first order linear fuzzy differential equations with constant coefficients under GH-differentiability by using the fuzzy Laplace transform method. Three variants of first order linear fuzzy differential equations with constant coefficients are given as follows: $u^{'} (t) \oplus δ ⊙ u (t) = σ (t),$ (1) $u^{'} (t) = δ ⊙ u (t) \oplus σ (t),$ (2) $u^{'} (t) \oplus σ (t) = δ ⊙ u (t) .$ (3)

2 Preliminaries

Let $ℝ$ denote the set of all real numbers. Denote by $ℝ_{F}$ the set of fuzzy sets $u : ℝ \to [0, 1]$ with the following properties:

u is normal, i.e., there exists $x_{0} \in ℝ$ such that u (x₀) =1;

u is fuzzy convex, that is, u (λx + (1 - λ) y) ≥ min {u (x) , u (y)} for any $x, y \in ℝ$ and λ ∈ [0, 1];

u is upper semicontinuous;

$cl {x \in ℝ : u (x) > 0}$ is compact, where cl denotes the closure of a set.

Usually, the set $ℝ_{F}$ is called the space of fuzzy numbers. If every real number $a \in ℝ$ is equivalently represented by its characteristic function χ_{a}, then it is easy to know that $ℝ \subset ℝ_{F}$ . For 0 < α ≤ 1, we denote $[u]^{α} = {x \in ℝ : u (x) \geq α}$ and $[u]^{0} = cl {x \in ℝ : u (x) > 0}$ . Then it follows from the conditions (i)-(iv) that the α-level set [u] ^α is a nonempty compact interval for all α ∈ [0, 1] and each $u \in ℝ_{F}$ .

For $u, v \in ℝ_{F}$ , $λ \in ℝ$ , the addition u ⊕ v and scalar multiplication λ ⊙ u can be defined, levelwise, by $[u \oplus v]^{α} = [u]^{α} + [v]^{α} and [λ ⊙ u]^{α} = λ [u]^{α}$ for all α ∈ [0, 1].

The supremum metric between two fuzzy numbers u and v is defined by $\begin{matrix} D : ℝ_{F} \times ℝ_{F} \to ℝ_{+} \cup {0}, \\ D (u, v) = sup_{α \in [0, 1]} d_{H} ([u]^{α}, [v]^{α}), \end{matrix}$ where d_H is the Hausdorff metric. It is well known that the metric space $(ℝ_{F}, D)$ is a complete metric space and the following properties for the metric D are satisfied:

D (u ⊕ w, v ⊕ w) = D (u, v), $\forall u, v, w \in ℝ_{F}$ ;

D (λ ⊙ u, λ ⊙ v) = |λ|D (u, v), $\forall λ \in ℝ, u, v \in ℝ_{F}$ ;

D (u ⊕ v, w ⊕ e) ≤ D (u, w) + D (v, e), $\forall u, v, w, e \in ℝ_{F}$ .

Let $u, v \in ℝ_{F}$ . If there exists $w \in ℝ_{F}$ such that u = v ⊕ w, then w is called the H-difference of u and v, and it is denoted by u ⊖ v.

Throughout this paper, the symbol “⊖” always stands for the H-difference. In general, u ⊖ v ≠ u ⊕ (-1) ⊙ v, (-1) ⊙ v = - v.

Theorem 2.1. (Anastassiou and Gal [4])

If we denote $\tilde{0} = χ_{{0}}$ , then $\tilde{0} \in ℝ_{F}$ is a neutral element with respect to the Minkowski addtition ⊕, i.e., $u \oplus \tilde{0} = \tilde{0} \oplus u = u$ for all $u \in ℝ_{F}$ .

None of $u \in ℝ_{F} \ ℝ$ has inverse in $ℝ_{F}$ with respect to $\tilde{0}$ .

For any $a, b \in ℝ$ with a, b ≥ 0 or a, b ≤ 0 and any $u \in ℝ_{F}$ , we have (a + b) ⊙ u = a ⊙ u ⊕ b ⊙ u. In generally, the above property does not hold for $a, b \in ℝ$ .

For any $λ \in ℝ$ and any $u, v \in ℝ_{F}$ , we have λ ⊙ (u ⊕ v) = λ ⊙ u ⊕ λ ⊙ v.

For any $λ, μ \in ℝ$ and any $y \in ℝ_{F}$ , we have λ ⊙ (μ ⊙ u) = (λ · μ) ⊙ u.

Lemma 2.2. Let $u \in ℝ_{F}$ and let $a, b \in ℝ$ with a · b < 0. The following equalities hold:

if a > 0, b < 0 and a + b > 0, then - (a + b) ⊙ u = (- a) ⊙ u ⊖ b ⊙ u;

if a < 0, b > 0 and a + b < 0, then - (a + b) ⊙ u = (- a) ⊙ u ⊖ b ⊙ u.

Proof. The proof follows immediately from (iii) of Theorem 2.1. □

The concept of strongly generalized differentiability of fuzzy number-valued functions was introduced by Bede et al. [7] and further studied by Chalco-Cano and Román-Flores [10].

Definition 2.1. Let I = (a, b) be an open interval, where $a, b \in ℝ \cup {\pm \infty}$ with a < b, and let $F : I \to ℝ_{F}$ be a fuzzy number-valued mapping and t₀ ∈ I. We say that F is differentiable at t₀ if there exists an element $F^{'} (t_{0}) \in ℝ_{F}$ such that either

(i) for all h > 0 sufficiently small, the H-differences F (t₀ + h) ⊖ F (t₀), F (t₀) ⊖ F (t₀ - h) exist and the limits (in the metric D) $\begin{matrix} lim_{h \to 0^{+}} \frac{F (t_{0} + h) ⊖ F (t_{0})}{h} \\ = lim_{h \to 0^{+}} \frac{F (t_{0}) ⊖ F (t_{0} - h)}{h} \\ = F^{'} (t_{0}) \end{matrix}$ or

(ii) for all h > 0 sufficiently small, the H-differences F (t₀) ⊖ F (t₀ + h), F (t₀ - h) ⊖ F (t₀) exist and the limits (in the metric D) $\begin{matrix} lim_{h \to 0^{+}} \frac{F (t_{0}) ⊖ F (t_{0} + h)}{- h} \\ = lim_{h \to 0^{+}} \frac{F (t_{0} - h) ⊖ F (t_{0})}{- h} = F^{'} (t_{0}), \end{matrix}$ where h and -h at denominators denote $\frac{1}{h} ⊙$ and $- \frac{1}{h} ⊙$ , respectively.

A mapping $F : I \to ℝ_{F}$ is said to be (i)-differentiable (or (ii)-differentiable) on I if it is differentiable in the sense (i) (or (ii)) of Definition 2.1.

Theorem 2.3. (Kaleva [12], Khastan [13]) Let $F : [0, \infty) \to ℝ_{F}$ be a differentiable fuzzy number-valued mapping and assume that the derivative F′ is integrable over I. Then for each t ∈ [0, ∞) we have

If F is (i)-differentiable, then $F (t) = F (a) \oplus \int_{a}^{t} F^{'} (s) ds;$

If F is (ii)-differentiable, then $F (t) = F (a) ⊖ \int_{a}^{t} - F^{'} (s) ds .$

Theorem 2.4. (Kaleva [12]) Let $F : [0, \infty) \to ℝ_{F}$ be continuous. Then for any t ∈ [0, ∞) the integral $H (t) = \int_{a}^{t} F (τ) d τ$ is (i)-differentiable and H′ (t) = F (t).

Theorem 2.5. (Khastan et al. [13]) Let $F : [0, \infty) \to ℝ_{F}$ continuous. Define the integral $H (t) : = γ ⊖ \int_{a}^{t} - F (τ) d τ, t \in [0, \infty),$ where $γ \in ℝ_{F}$ is such that the preceding H-difference exists on [0, ∞). Then H(t) is (ii)-differentiable and H′ (t) = F (t).

Here, we cite two important conditions introduced by Bede et al. [8] in order to guarantee the differentiability of the product function between a real-valued function and a fuzzy number-valued function.

For a given t ∈ I, if f (t + h) ⊖ f (t) and f (t) ⊖ f (t - h) exist for sufficiently small h > 0;

For a given t ∈ I, if f (t) ⊖ f (t + h) and f (t - h) ⊖ f (t) exist for sufficiently small h > 0.

Theorem 2.6. (Bede et al. [8]) Let $f : ℝ \to ℝ$ and $G : ℝ \to ℝ_{F}$ be two differentiable functions. Then (i) If f (t) · f′ (t) >0 and g is (i)-differentiable, then f ⊙ g is (i)-differentiable and $(f ⊙ g)^{'} (t) = f^{'} (t) ⊙ g (t) \oplus f (t) ⊙ g^{'} (t);$ (ii) If f (t) · f′ (t) <0, g is (i)-differentiable and f ⊙ g satisfies the condition (H1) at t, then f ⊙ g is (i)-differentiable and $(f ⊙ g)^{'} (t) = f (t) ⊙ g^{'} (t) ⊖ (- f^{'} (t)) ⊙ g (t);$ (iii) If f (t) · f′ (t) >0, g is (ii)-differentiable and f ⊙ g satisfies the condition (H1) at t, then f ⊙ g is (i)-differentiable and $(f ⊙ g)^{'} (t) = f^{'} (t) ⊙ g (t) ⊖ (- f (t)) ⊙ g^{'} (t);$ (iv) If f (t) · f′ (t) <0, g is (i)-differentiable and f ⊙ g satisfies the condition (H2) at t, then f ⊙ g is (ii)-differentiable and $(f ⊙ g)^{'} (t) = f^{'} (t) ⊙ g (t) ⊖ (- f (t)) ⊙ g^{'} (t);$ (v) If f (t) · f′ (t) >0, g is (ii)-differentiable and f ⊙ g satisfies the condition (H2) at t, then f ⊙ g is (ii)-differentiable and $(f ⊙ g)^{'} (t) = f (t) ⊙ g^{'} (t) ⊖ (- f^{'} (t)) ⊙ g (t);$ (vi) If f (t) · f′ (t) <0, g is (ii)-differentiable, then f ⊙ g is (ii)-differentiable and $(f ⊙ g)^{'} (t) = f^{'} (t) ⊙ g (t) \oplus f (t) ⊙ g^{'} (t) .$

Definition 2.2. (Allahviranloo and Ahmadi [2]) Let f be a continuous fuzzy number-valued function defined on [0, ∞). Suppose that e^-st ⊙ f (t) is improper fuzzy Riemann integrable on [0, ∞), then $\int_{0}^{\infty} e^{- st} ⊙ f (t) dt$ is called fuzzy Laplace transform and it is denoted by $L [f (t)] (s) = \int_{0}^{\infty} e^{- st} ⊙ f (t) dt (s > 0 and integer) .$

Theorem 2.7. (Allahviranloo and Ahmadi [2]) Let f (t) , g (t) be continuous fuzzy number-valued functions and let c₁, c₂ be two constants. Then $\begin{matrix} L [(c_{1} ⊙ f (t)) \oplus (c_{2} ⊙ g (t))] (s) \\ = (c_{1} ⊙ L [f (t)] (s)) \oplus (c_{2} ⊙ L [g (t)] (s)) . \end{matrix}$

Theorem 2.8. (Allahviranloo and Ahmadi [2]) Let f′ (t) be an integrable fuzzy number-valued function and let f (t) be the primitive of f′ (t) on [0, ∞). Then $L [f^{'} (t)] (s) = s ⊙ L [f (t)] (s) ⊖ f (0)$ where f is (i)-differentiable

or $L [f^{'} (t)] (s) = (- f (0)) ⊖ (- s ⊙ L [f (t)] (s))$ where f is (ii)-differentiable.

Definition 2.3. Let $f : [0, \infty) \to ℝ$ and $g : [0, \infty) \to ℝ_{F}$ be two piecewise continuous real valued and fuzzy number-valued functions, respectively. The fuzzy multiplication-convolution f ⊗ g of f and g is defined by $(f \otimes g) (t) = \int_{0}^{t} f (t - τ) ⊙ g (τ) d τ .$

The following result is concerned with the fuzzy Laplace transform of the fuzzy multiplication-convolution function.

Theorem 2.9. Let $f : [0, \infty) \to ℝ$ and $g : [0, \infty) \to ℝ_{F}$ be two piecewise continuous real valued and fuzzy number-valued functions, respectively. If $L [f (t)] (s) = F (s)$ , $L [g (t)] (s) = \hat{G} (s)$ , then $L [(f \otimes g) (t)] (s) = F (s) ⊙ \hat{G} (s),$ where $L$ denotes Laplace transform of an ordinary real-valued function.

Proof. According to Definitions 2.1 and 2.2, we know that $\begin{matrix} L [(f \otimes g) (t)] (s) \\ = \int_{0}^{\infty} e^{- st} ⊙ {\int_{0}^{t} f (t - τ) ⊙ g (τ) d τ} dt \\ = \int_{0}^{\infty} \int_{0}^{t} (e^{- st} \cdot f (t - τ)) ⊙ g (τ) d τ dt . \end{matrix}$ If we set t = ξ + τ, then the preceding equality becomes $\begin{matrix} L [(f \otimes g) (t)] (s) \\ = \int_{0}^{\infty} \int_{0}^{\infty} (e^{- s (ξ + τ)} \cdot f (ξ)) ⊙ g (τ) d τ d ξ \\ = \int_{0}^{\infty} \int_{0}^{\infty} (e^{- s ξ} \cdot f (ξ)) ⊙ (e^{- s τ} ⊙ g (τ)) d τ d ξ \\ = \int_{0}^{\infty} e^{- s ξ} \cdot f (ξ) d ξ ⊙ \int_{0}^{\infty} e^{- s τ} ⊙ g (τ) d τ \\ = L [f (t)] (s) ⊙ L [g (t)] (s) \\ = F (s) ⊙ \hat{G} (s) . □ \end{matrix}$

3 Ulam stability of linear fuzzy differential equation of first order with constant coefficients

In this section, we shall establish the Ulam stability of three variants of first order linear fuzzy differential equations with constant coefficients.

3.1 Stability of Equation (1) under generalized differentiability

Case I: u(t) is (i)-differentiable

Theorem 3.1. Let $σ : [0, \infty) \to ℝ_{F}$ be a continuous fuzzy number-valued function and let δ be a positive number. Let $φ : [0, \infty) \to ℝ_{+} \cup {0}$ be a function such that φ (t) e^δt is integrable on [0, ∞). Suppose that a continuously (i)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies $D (u^{'} (t) \oplus δ ⊙ u (t), σ (t)) \leq φ (t)$ (4) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊕ δ ⊙ u (t) ⊖ σ (t) exists for each t ∈ [0, ∞). Furthermore, assume that $e^{- δ t} ⊙ (u (0) \oplus \int_{0}^{t} e^{δ τ} ⊙ σ (τ) d τ)$ satisfies the condition (H1) on [0, ∞). Then there exists a (i)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (1) such that $D (u (t), \tilde{u} (t)) \leq e^{- δ t} \int_{0}^{t} e^{δ τ} φ (τ) d τ$ (5) for all t ∈ [0, ∞).

Proof. Setting w (t) ⊕ σ (t) = u′ (t) ⊕ δ ⊙ u (t) for each t ∈ [0, ∞). By Theorems 2.7 and 2.8, we have

$\begin{matrix} L [w (t)] (s) \oplus L [σ (t)] (s) \\ = L [u^{'} (t)] (s) \oplus δ ⊙ L [u (t)] (s) \\ = s ⊙ L [u (t)] (s) ⊖ u (0) \oplus δ ⊙ L [u (t)] (s) \\ = (s + δ) ⊙ L [u (t)] (s) ⊖ u (0) . \end{matrix}$ (6) If we set $\tilde{u} (t) = e^{- δ t} ⊙ u (0) \oplus (E_{- δ} \otimes σ) (t),$ where $(E_{- δ} \otimes σ) (t) = \int_{0}^{t} e^{- δ (t - τ)} ⊙ σ (τ) d τ .$ Obviously, $\tilde{u} (0) = u (0)$ . Then, it follows from Theorems 2.7 and 2.9 that

$\begin{matrix} L [\tilde{u} (t)] (s) \\ = L [e^{- δ t}] (s) ⊙ u (0) \oplus L [e^{- δ t}] (s) ⊙ L [σ (t)] (s) \\ = \frac{1}{s + δ} ⊙ (u (0) \oplus L [σ (t)] (s)) . \end{matrix}$ (7) Notice that $\tilde{u} (t) = e^{- δ t} ⊙ (u (0) \oplus \int_{0}^{t} e^{δ τ} ⊙ σ (τ) d τ) .$ According to Theorems 2.4 and 2.6 (ii), it is easy to know that $\tilde{u} (t)$ is (i)-differentiable on [0, ∞), since $\tilde{u} (t)$ satisfies the condition (H1) on [0, ∞) and e^-δt · (e^-δt) ′ < 0 for every t ∈ [0, ∞). Therefore, by Theorem 2.8 and (6), we can infer that $\begin{matrix} L [{\tilde{u}}^{'} (t) \oplus δ ⊙ \tilde{u} (t)] (s) \\ = s ⊙ L [\tilde{u} (t)] (s) ⊖ \tilde{u} (0) \oplus δ ⊙ L [\tilde{u} (t)] (s) \\ = L [σ (t)] (s) . \end{matrix}$ Since L is one-to-one, it follows that ${\tilde{u}}^{'} (t) \oplus δ ⊙ \tilde{u} (t) = σ (t)$ . This implies that $\tilde{u} (t)$ is a (i)-differentiable solution of Equation (1). Using the formulas (6), (7) and Theorem 2.9, we can obtain $\begin{matrix} L [u (t)] (s) & = & L [\tilde{u} (t)] (s) \oplus \frac{1}{s + δ} ⊙ L [w (t)] (s) \\ = & L [\tilde{u} (t)] (s) \oplus L [e^{- δ t}] (s) ⊙ L [w (t)] (s) \\ = & L [\tilde{u} (t)] (s) \oplus L [(E_{- δ} \otimes w) (t)] (s) \\ = & L [\tilde{u} (t) \oplus (E_{- δ} \otimes w) (t)] (s), \end{matrix}$ which means that $u (t) = \tilde{u} (t) \oplus (E_{- δ} \otimes w) (t)$ . By (4), we get $\begin{matrix} D (u (t), \tilde{u} (t)) \\ = D (\tilde{u} (t) \oplus (E_{- δ} \otimes w) (t), \tilde{u} (t)) \\ = D ((E_{- δ} \otimes w) (t), \tilde{0}) \\ = D (\int_{0}^{t} e^{- δ (t - τ)} ⊙ w (τ) d τ, \tilde{0}) \\ \leq e^{- δ t} \int_{0}^{t} e^{δ τ} D (w (τ), \tilde{0}) d τ \\ \leq e^{- δ t} \int_{0}^{t} e^{δ τ} φ (τ) d τ, \end{matrix}$ which completes the proof. □

As a direct consequence of Theorem 3.1, we can obtain the following stability result of Equation (1).

Corollary 3.2. Let σ and δ be given as in Theorem 3.1. For a given ε > 0, if a continuously (i)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies $D (u^{'} (t) \oplus δ ⊙ u (t), σ (t)) \leq ε$ (8) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊕ δ ⊙ u (t) ⊖ σ (t) exists for each t ∈ [0, ∞). Moreover, assume that $e^{- δ t} ⊙ (u (0) \oplus \int_{0}^{t} e^{τ} ⊙ σ (τ) d τ)$ satisfies the condition (H1) on [0, ∞). Then there exists a (i)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (1) such that $D (u (t), \tilde{u} (t)) \leq \frac{ε}{δ} (1 - e^{- δ t})$ (9) for all t ∈ [0, ∞).

Remark 1. Since δ > 0, t ∈ [0, ∞), we know that $\frac{ε}{δ} (1 - e^{- δ t}) < \frac{ε}{δ}$ . Consequently, Corollary 3.2 implies the Hyers-Ulam stability of Equation (1) under (i)-differentiability.

Case II: u(t) is (ii)-differentiable

Theorem 3.3. Let $σ : [0, \infty) \to ℝ_{F}$ be a continuous fuzzy number-valued function and let δ be a negative number. Let $φ : [0, \infty) \to ℝ_{+} \cup {0}$ be a function such that φ (t) e^δt is integrable on [0, ∞). Suppose that a continuously (ii)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies the inequality (4) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊕ δ ⊙ u (t) ⊖ σ (t) exists for each t ∈ [0, ∞). Furthermore, assume that the H-difference $e^{- δ t} ⊙ (u (0) ⊖ \int_{0}^{t} e^{δ τ} ⊙ (- σ (τ)) d τ)$ exists and satisfies the condition (H2) on [0, ∞). Then there exists a (ii)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (1) such that the inequality (5) holds for all t ∈ [0, ∞).

Proof. Similar to Theorem 3.1, let -w (t) ⊕ σ (t) = u′ (t) ⊕ δ ⊙ u (t). From Theorems 2.5 and 2.6, we can obtain $\begin{matrix} L [- w (t)] (s) \oplus L [σ (t)] (s) \\ = L [u^{'} (t)] (s) \oplus δ ⊙ L [u (t)] (s) \\ = (- u (0)) ⊖ (- s ⊙ L [u (t)] (s)) \oplus δ ⊙ L [u (t)] (s) . \end{matrix}$ Then, we get $\begin{matrix} L [- w (t)] (s) & \oplus & L [σ (t)] (s) \oplus (- s ⊙ L [u (t)] (s)) \\ ⊖ & δ ⊙ L [u (t)] (s) = - u (0) . \end{matrix}$ By Lemma 2.2, for s > - δ, we have

$\begin{matrix} L [u (t)] (s) & = & \frac{1}{s + δ} ⊙ u (0) ⊖ \frac{1}{s + δ} ⊙ L [- σ (t)] (s) \\ ⊖ & \frac{1}{s + δ} ⊙ L [w (t)] (s) . \end{matrix}$ (10) Define $\tilde{u} (t) = e^{- δ t} ⊙ u (0) ⊖ (E_{- δ} \otimes (- σ)) (t),$ where $(E_{- δ} \otimes (- σ)) (t) = \int_{0}^{t} e^{- δ (t - τ)} ⊙ (- σ (τ)) d τ .$ It is easy to know that $\tilde{u} (0) = u (0)$ . By Theorems 2.7, 2.8 and 2.9, we can obtain

$\begin{matrix} L [\tilde{u} (t)] (s) \\ = L [e^{- δ t}] (s) ⊙ u (0) ⊖ L [e^{- δ t}] (s) ⊙ L [- σ (t)] (s) \\ = \frac{1}{s + δ} ⊙ u (0) ⊖ \frac{1}{s + δ} ⊙ L [- σ (t)] (s) . \end{matrix}$ (11) By the definition of $\tilde{u} (t)$ , we have $\tilde{u} (t) = e^{- δ t} ⊙ (u (0) ⊖ \int_{0}^{t} e^{δ τ} ⊙ (- σ (τ)) d τ) .$ In view of Theorems 2.5 and 2.6 (v), we know that $\tilde{u} (t)$ is (ii)-differentiable on [0, ∞), since $\tilde{u} (t)$ satisfies the condition (H2) on [0, ∞) and e^-δt · (e^-δt) ′ > 0 for every t ∈ [0, ∞). For s > - δ, we can infer from Lemma 2.2 and (11) that

$\begin{matrix} - u (0) ⊖ L [σ (t)] (s) \\ = - (s + δ) ⊙ L [\tilde{u} (t)] (s) \\ = - s ⊙ L [\tilde{u} (t)] (s) ⊖ δ ⊙ L [\tilde{u} (t)] (s) . \end{matrix}$ (12) From (12), it follows that $\begin{matrix} L [{\tilde{u}}^{'} (t) \oplus δ ⊙ \tilde{u} (t)] (s) \\ = (- u (0)) ⊖ (- s ⊙ L [\tilde{u} (t)] (s)) \oplus δ ⊙ L [\tilde{u} (t)] (s) \\ = L [σ (t)] (s) . \end{matrix}$ This implies that ${\tilde{u}}^{'} (t) \oplus δ ⊙ \tilde{u} (t) = σ (t)$ , since L is one-to-one. That is to say, $\tilde{u} (t)$ is (ii)-differentiable solution of Equation (1).

Applying the formulas (10), (11) and Theorem 2.9, we can obtain $\begin{matrix} L [\tilde{u} (t)] (s) \\ = L [u (t)] (s) \oplus \frac{1}{s + δ} ⊙ L [w (t)] (s) \\ = L [u (t)] (s) \oplus L [e^{- δ t}] (s) ⊙ L [w (t)] (s) \\ = L [u (t)] (s) \oplus L [(E_{- δ} \otimes w) (t)] (s) \\ = L [u (t) \oplus (E_{- δ} \otimes w) (t)] (s), \end{matrix}$ which implies that $\tilde{u} (t) = u (t) \oplus (E_{- δ} \otimes w) (t)$ . Then, by (4), we have $\begin{matrix} D (u (t), \tilde{u} (t)) \\ = D (u (t), u (t) \oplus (E_{- δ} \otimes w) (t)) \\ = D ((E_{- δ} \otimes w) (t), \tilde{0}) \\ = D (\int_{0}^{t} e^{- δ (t - τ)} ⊙ w (τ) d τ, \tilde{0}) \\ \leq e^{- δ t} \int_{0}^{t} e^{δ τ} D (w (τ), \tilde{0}) d τ \\ \leq e^{- δ t} \int_{0}^{t} e^{δ τ} φ (τ) d τ . \end{matrix}$ The proof of the theorem is now completed. □

Based on Theorem 3.3, we can establish the following stability result of Equation (1) under (ii)-differentiability.

Corollary 3.4. Let σ and δ be given as in Theorem 3.3. For a given ε > 0, if a continuously (ii)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies the inequality (8) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊕ δ ⊙ u (t) ⊖ σ (t) exists for each t ∈ [0, ∞). Moreover, assume that the H-difference $e^{- δ t} ⊙ (u (0) ⊖ \int_{0}^{t} e^{δ τ} ⊙ (- σ (τ)) d τ)$ exists and satisfies the condition (H2) on [0, ∞). Then there exists a (ii)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (1) such that the inequality (9) holds for all t ∈ [0, ∞).

Remark 2. Different from Corollary 3.2, Corollary 3.4 can not imply the Hyers-Ulam stability of Equation (1) under (ii)-differentiability.

3.2 Stability of Equation (2) under generalized differentiability

Case I: u(t) is (i)-differentiable

Theorem 3.5. Let $σ : [0, \infty) \to ℝ_{F}$ be a continuous fuzzy number-valued function and let δ be a positive number. Let $φ : [0, \infty) \to ℝ_{+} \cup {0}$ be a function such that φ (t) e^-δt is integrable on [0, ∞). Suppose that a continuously (i)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies $D (u^{'} (t), δ ⊙ u (t) \oplus σ (t)) \leq φ (t)$ (13) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊖ (δ ⊙ u (t) ⊕ σ (t)) exists for each t ∈ [0, ∞). Then there exists a (i)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (2) such that $D (u (t), \tilde{u} (t)) \leq e^{δ t} \int_{0}^{t} e^{- δ τ} φ (τ) d τ$ (14) for all t ∈ [0, ∞).

Proof. Setting w (t) ⊕ δ ⊙ u (t) ⊕ σ (t) = u′ (t). Using the similar argument as in Theorem 3.1, we can obtain $\begin{matrix} L [w (t)] (s) \oplus δ ⊙ L [u (t)] (s) \oplus L [σ (t)] (s) \\ = s ⊙ L [u (t)] (s) ⊖ u (0) . \end{matrix}$ For s > δ, we can infer that

$\begin{matrix} L [u (t)] (s) & = & \frac{1}{s - δ} ⊙ u (0) \oplus \frac{1}{s - δ} ⊙ L [σ (t)] (s) \\ \oplus \frac{1}{s - δ} ⊙ L [w (t)] (s) . \end{matrix}$ (15) Define $\tilde{u} (t) = e^{δ t} ⊙ u (0) \oplus (E_{δ} \otimes σ) (t),$ where $(E_{δ} \otimes σ) (t) = \int_{0}^{t} e^{- δ τ} ⊙ σ (τ) d τ .$ It is clear that $\tilde{u} (0) = u (0)$ . Furthermore, we can obtain $L [\tilde{u} (t)] (s) = \frac{1}{s - δ} ⊙ u (0) \oplus \frac{1}{s - δ} ⊙ L [σ (t)] (s) .$ (16) Similar to Theorem 3.1, it is easy to check that $\tilde{u} (t)$ is (i)-differentiable on [0, ∞) and it is a solution of Equation (2). From (15) and (16), it follows that $u (t) = \tilde{u} (t) \oplus (E_{δ} \otimes w) (t)$ . Then, by using the inequality (13), we can obtain the inequality (14), which completes the proof. □

From Theorem 3.5, the following stability result of Equation (2) can be obtained as follows.

Corollary 3.6. Let σ and δ be given as in Theorem 3.5. For a given ε > 0, if a continuously (i)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies $D (u^{'} (t), δ ⊙ u (t) \oplus σ (t)) \leq ε$ (17) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊖ (δ ⊙ u (t) ⊕ σ (t)) exists for each t ∈ [0, ∞). Then there exists a (i)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (2) such that $D (u (t), \tilde{u} (t)) \leq \frac{ε}{δ} (e^{δ t} - 1)$ (18) for all t ∈ [0, ∞).

Remark 3. Similar to Corollary 3.4, it is easy to see that Corollary 3.6 can not imply the Hyers-Ulam stability of Equation (2) under (i)-differentiability.

Case II: u(t) is (ii)-differentiable

Theorem 3.7. Let $σ : [0, \infty) \to ℝ_{F}$ be a continuous fuzzy number-valued function and let δ be a negative number. Let $φ : [0, \infty) \to ℝ_{+} \cup {0}$ be a function such that φ (t) e^δt is integrable on [0, ∞). Suppose that a continuously (ii)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies the inequality (13) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊖ (δ ⊙ u (t) ⊕ σ (t)) exists for each t ∈ [0, ∞). Furthermore, assume that the H-difference $e^{δ t} ⊙ (u (0) ⊖ \int_{0}^{t} e^{- δ τ} ⊙ (- σ (τ)) d τ)$ exists on [0, ∞). Then there exists a (ii)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (2) such that the inequality (14) holds for all t ∈ [0, ∞).

Proof. Setting -w (t) ⊕ δ ⊙ u (t) ⊕ σ (t) = u′ (t). By Theorems 2.7 and 2.8, we can obtain $\begin{matrix} L [- w (t)] (s) \oplus δ ⊙ L [u (t)] (s) \oplus L [σ (t)] (s) \\ = - u (0) ⊖ (- s ⊙ L [u (t)] (s)) . \end{matrix}$ Then, we can infer that

$\begin{matrix} L [u (t)] (s) & = & \frac{1}{s - δ} ⊙ u (0) ⊖ \frac{1}{s - δ} ⊙ L [- σ (t)] (s) \\ ⊖ \frac{1}{s - δ} ⊙ L [w (t)] (s) . \end{matrix}$ (19) Define $\tilde{u} (t) = e^{δ t} ⊙ u (0) ⊖ (E_{δ} \otimes (- σ)) (t),$ where $(E_{δ} \otimes (- σ)) (t) = \int_{0}^{t} e^{δ (t - τ)} ⊙ (- σ (τ)) d τ .$ Clearly, $\tilde{u} (0) = u (0)$ . From Theorems 2.7 and 2.9, it follows that $L [\tilde{u} (t)] (s) = \frac{1}{s - δ} ⊙ u (0) ⊖ \frac{1}{s - δ} ⊙ L [- σ (t)] (s) .$ (20) Using the same argument as in Theorem 3.3, it can easily be verified that $\tilde{u} (t)$ is a (ii)-differentiable solution of Equation (2) on [0, ∞). By (19) and (20), we can obtain $\tilde{u} (t) = u (t) \oplus (E_{δ} \otimes w) (t)$ . Thus, we can infer that the inequality (14) by using (13). This completes the proof. □ As a direct consequence of Theorem 3.7, the following stability result of Equation (2) under (ii)-differentiability can be obtained.

Corollary 3.8. Let σ and δ be given as in Theorem 3.7. For a given ε > 0, if a continuously (ii)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies the inequality (17) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊖ (δ ⊙ u (t) ⊕ σ (t)) exists for each t ∈ [0, ∞). Assume that the H-difference $e^{δ t} ⊙ (u (0) ⊖ \int_{0}^{t} e^{- δ τ} ⊙ (- σ (τ)) d τ)$ exists on [0, ∞). Then there exists a (ii)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (2) such that the inequality (18) holds for all t ∈ [0, ∞).

Remark 4. Since δ < 0, it follows that $\frac{ε}{δ} (e^{δ t} - 1) < - \frac{ε}{δ}$ for any t ∈ [0, ∞). Then, Corollary 3.8 implies the Hyers-Ulam stability of Equation (2) under (ii)-differentiability.

3.3 Stability of Equation (3) under generalized differentiability

Case I: u(t) is (i)-differentiable

Theorem 3.9. Let $σ : [0, \infty) \to ℝ_{F}$ be a continuous fuzzy number-valued function and let δ be a positive number. Let $φ : [0, \infty) \to ℝ_{+} \cup {0}$ be a function such thatφ (t) e^-δt is integrable on [0, ∞). Suppose that a continuously (i)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies $D (u^{'} (t) \oplus σ (t), δ ⊙ u (t)) \leq φ (t)$ (21) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊕ σ (t) ⊖ δ ⊙ u (t) exists for each t ∈ [0, ∞). Furthermore, assume that the H-difference $e^{δ t} ⊙ (u (0) ⊖ \int_{0}^{t} e^{- δ τ} ⊙ σ (τ) d τ)$ exists and satisfies the condition (H1) on [0, ∞). Then there exists a (i)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (3) such that the inequality (14) holds for all t ∈ [0, ∞).

Proof. Putting w (t) ⊕ δ ⊙ u (t) = u′ (t) ⊕ σ (t). Theorems 2.7 and 2.8 yield $\begin{matrix} L [w (t)] (s) \oplus δ ⊙ L [u (t)] (s) \\ = s ⊙ L [w (t)] (s) ⊖ u (0) \oplus L [σ (t)] (s) . \end{matrix}$ For s > δ, we can obtain

$\begin{matrix} L [u (t)] (s) & = & \frac{1}{s - δ} ⊙ u (0) ⊖ \frac{1}{s - δ} \\ ⊙ L [σ (t)] (s) \oplus L [w (t)] (s) . \end{matrix}$ (22) Define $\tilde{u} (t) = e^{δ t} ⊙ u (0) ⊖ (E_{δ} \otimes σ) (t),$ where $(E_{δ} \otimes σ) (t) = \int_{0}^{t} e^{δ (t - τ)} ⊙ σ (τ) d τ .$ Obviously, $\tilde{u} (0) = u (0)$ . By Theorems 2.7 and 2.9, we can infer that $L [\tilde{u} (t)] (s) = \frac{1}{s - δ} ⊙ u (0) ⊖ \frac{1}{s - δ} ⊙ L [σ (t)] (s) .$ (23) Using the same argument as in Theorem 3.1, it is easy to verify that $\tilde{u} (t)$ is a (i)-differentiable solution of Equation (3) on [0, ∞). From (22) and (23), it follows that $u (t) = \tilde{u} (t) \oplus (E_{δ} \otimes w) (t)$ . Consequently, we can obtain the inequality (14) by using (21). □

According to Theorem 3.9, we can obtain the following stability result of Equation (3) under (i)-differentiability.

Corollary 3.10. Let σ and δ be given as in Theorem 3.9. For a given ε > 0, if a continuously (i)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies $D (u^{'} (t) \oplus σ (t), δ ⊙ u (t)) \leq ε$ (24) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊕ σ (t) ⊖ δ ⊙ u (t) exists for each t ∈ [0, ∞). Assume that the H-difference $e^{δ t} ⊙ (u (0) ⊖ \int_{0}^{t} e^{- δ τ} ⊙ σ (τ) d τ)$ exists and satisfies the condition (H1) on [0, ∞). Then there exists a (i)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (3) such that the inequality (18) holds for all t ∈ [0, ∞).

Remark 5. Similar to Corollary 3.6, the above result given in Corollary 3.10 can not imply the Hyers-Ulam stability of Equation (3) under (i)-differentiability.

Case II: u(t) is (ii)-differentiable

Theorem 3.11. Let $σ : [0, \infty) \to ℝ_{F}$ be a continuous fuzzy number-valued function and let δ be a negative number. Let $φ : [0, \infty) \to ℝ_{+} \cup {0}$ be a function such that φ (t) e^δt is integrable on [0, ∞). Suppose that a continuously (ii)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies the inequality (21) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊕ σ (t) ⊖ δ ⊙ u (t) exists for each t ∈ [0, ∞). Furthermore, assume that $e^{δ t} ⊙ (u (0) \oplus \int_{0}^{t} e^{- δ τ} ⊙ (- σ (τ)) d τ)$ satisfies the condition (H2) on [0, ∞). Then there exists a (ii)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (3) such that the inequality (14) holds for all t ∈ [0, ∞).

Proof. Putting -w (t) ⊕ δ ⊙ u (t) = u′ (t) ⊕ σ (t). Using Theorems 2.7 and 2.8, we obtain $\begin{matrix} L [- w (t)] (s) \oplus δ ⊙ L [u (t)] (s) \\ = - u (0) ⊖ (- s ⊙ L [u (t)] (s)) \oplus L [σ (t)] (s) . \end{matrix}$ Equivalently, we get

$\begin{matrix} L [u (t)] (s) & = & \frac{1}{s - δ} ⊙ u (0) \oplus \frac{1}{s - δ} ⊙ L [- σ (t)] (s) \\ ⊖ \frac{1}{s - δ} ⊙ L [w (t)] (s) . \end{matrix}$ (25) Define $\tilde{u} (t) = e^{δ t} ⊙ u (0) \oplus (E_{δ} \otimes (- σ)) (t),$ where $(E_{δ} \otimes (- σ)) (t) = \int_{0}^{t} e^{δ (t - τ)} ⊙ (- σ (τ)) d τ .$ Clearly, $\tilde{u} (0) = u (0)$ . By Theorems 2.7 and 2.9, we can infer that $L [\tilde{u} (t)] (s) = \frac{1}{s - δ} ⊙ u (0) \oplus \frac{1}{s - δ} ⊙ L [- σ (t)] (s) .$ (26) In a similar way as in Theorem 3.3, it can easily be checked that $\tilde{u} (t)$ is a (ii)-differentiable solution of Equation (3) on [0, ∞). From (25) and (26), it follows that $\tilde{u} (t) = u (t) \oplus (E_{δ} \otimes (- w)) (t)$ . Therefore, the inequality (14) can be obtained by using the condition (21). □

Based on Theorem 3.11, we can obtain the following stability result of Equation (3) under (ii)-differentiability.

Corollary 3.12. Let σ and δ be given as in Theorem 3.11. For a given ε > 0, if a continuously (ii)-differentiable mapping $u : [0, \infty) \to ℝ_{F}$ satisfies the inequality (21) for all t ∈ [0, ∞), and the H-difference u′ (t) ⊕ σ (t) ⊖ δ ⊙ u (t) exists for each t ∈ [0, ∞). Moreover, assume that $e^{δ t} ⊙ (u (0) \oplus \int_{0}^{t} e^{- δ τ} ⊙ (- σ (τ)) d τ)$ satisfies the condition (H2) on [0, ∞). Then there exists a (ii)-differentiable solution $\tilde{u} : [0, \infty) \to ℝ_{F}$ of Equation (3) such that the inequality (18) holds for all t ∈ [0, ∞).

Remark 6. Similar to Corollary 3.8, the above result obtained in Corollary 3.12 yields the Hyers-Ulam stability of Equation (3) under (ii)-differentiability.
4 Conclusion
By using the fuzzy Laplace transform method, we consider the Ulam stability of three variants of first order linear fuzzy differential equations with constant coefficients under generalized differentiability. In essence, the stability results obtained in this paper can be regarded as special cases of the general results established in [19]. However, this paper shows that the fuzzy Laplace transform method has more advantages than the direct method used in [19] to study the Ulam stability of linear fuzzy differential equations of first order with constant coefficients. More importantly, this approach provides the possibility for the further study of the Ulam stability of second order or n-th order linear fuzzy differential equations with constant coefficients.

Footnotes

Acknowledgments

This work was supported by “Qing Lan” Talent Engineering Funds by Tianshui Normal University, the Research Project of Higher Learning of Gansu Province (No. 2014B-080) and the Key Subjects Construction of Tianshui Normal University.

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