A fixed point approach to the Ulam stability of fuzzy differential equations under generalized differentiability

Abstract

Under some appropriate conditions and generalized differentiability, we study the Ulam stability of the fuzzy differential equation x′ (t) = F (t, x (t)) by employing the fixed point technique.

Keywords

Ulam stability fuzzy differential equations generalized differentiability fixed point

1 Introduction

In 1940, Ulam [25] posed the following problem:“Give condition in order for a linear mapping near an approximately linear mapping to exist”. Subsequently, Hyers [11] gave a partial answer to the question of Ulam. Later, Hyers’ theorem was generalized by Rassias [21] for linear mapping by allowing an unbounded Cauchy difference. Since then, various generalizations of Hyers’ theorem have been investigated by many authors. In the beginning, the main results are based on the direct method presented by Hyers. Until 2003, Radu [20] considered the Ulam stability problem of Cauchy functional equation via fixed point technique. So far, the direct method and the fixed point method, acted as two basic tools, have been widely used in the Ulam stability problem of functional equations.

Obloza [18] initiated the study of the Ulamstability prblem of differential equations. Several years later, Alsina and Ger [1] studied the Hyers-Ulam stability of the differential equation y′ = y. Further, the Ulam stability of the differential equation y′ = λy in various abstract spaces have been systematically investigated by Miura and Takahasi et al. [16 , 24]. Up to now, the Ulam stability problem of many types of differential equations, especially linear differential equations, have been studied. Compared with the study of the Ulam stability of functional equations, the methods used in the study of the stability of differential equations are more flexible and diverse. Usually, the original direct method can not be directly applied to differential equations. However, the fixed point method has been successfully used to study the Ulam stability of differential equations by Jung [13].

Generally speaking, for an n-order X-valued differential equation (Here X denotes a Banach space with the norm ∥· ∥) $F (t, y, y^{'}, \dots, y^{(n)}) = 0, t \in I,$ where I denotes a subinterval of $ℝ$ , we say that it has Hyers-Ulam stability or it is stable in the sense of Hyers-Ulam if for a given ɛ > 0 and an n times strongly differentiable mapping f : I → Xsatisfying $∥ F (t, f, f^{'}, \dots, f^{(n)}) ∥ \leq ɛ$ for all t ∈ I, then there exists an exact solutiong : I → X of the preceding differential equation such that ∥f (t) - g (t) ∥ ≤ K (ɛ) for all t ∈ I, where K (ɛ) depends only on ɛ and $lim_{ɛ \to 0} K (ɛ) = 0$ . More generally, if the ɛ and K (ɛ) are replaced by two control functions φ and Φ in t, respectively, then we say that the differential equation mentioned above has the Hyers-Ulam-Rassias stability or it is stable in the sense of Hyers-Ulam-Rassias.

At present, there are several different interpretations for fuzzy differential equations, such as fuzzy differential equations based on Hukuhara differentiability (i.e., H-differentiability) [14, 19], fuzzy differential inclusions based on the theory of differential inclusion for set-valued functions [2], fuzzy differential equations using the derivative and integral operators defined by Zadeh’s extension principle [3] and so on. However, each method exists some shortcomings. Up to 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 diameterof the solution is non-decreasing as the time increases. To overcome this shortcoming, Bede and Gal [4] introduced the concept of the strongly generalized differentiability (GH-differentiability). GH-differentiability can be viewed as a first improvement for H-differentiability and it allows a decreasing diameter of the solution of a fuzzy differential equation. Afterwards, the other more general differentiability, the generalized Hukuhara differentiability (gH-differentiability) and the most general so far, the fuzzy generalized differentiability(g-differentiability) [6], were proposed to further enlarge the class of differentiable fuzzy number-valued functions. Although these generalized differentiability have been extensively studied, the development of the theory of fuzzy differential equations using the generalizations of H-differentiability has still been limited to the version of GH-differentiability. As far as we know, the progress of the research on the stability problems of fuzzy differential equations is very slow. Diamond [9] seems to be the first person to discuss the Lyapunov stability of fuzzy differential equations by using the differential inclusions method. A dozen years later, Cecconello et al. [7] considered the asymptotic behavior of fuzzy solution obtained by Zadeh’s extension principle for a fuzzy dynamical system. Besides, there is little literature about the stability problems of fuzzy differential equations. Recently, the authors studied the Ulam stability of first order linear (partial) fuzzy differential equations under generalized differentiability [22, 23]. The purpose of the present paper is to study the Ulam stability of fuzzy differential equations under generalized differentiability, from a more general perspective, by employing the fixed point method.

2 Preliminaries

Let $ℕ$ , $ℝ$ , $ℝ_{+}$ and $ℝ_{-}$ denote the set of all natural numbers, the set of all real numbers, the set of all positive real numbers and the set of all negative real numbers, respectively. Denote by $ℝ_{F}$ the class of fuzzy sets $u : ℝ \to [0, 1]$ with the following properties: (i) u is normal, i.e., there exists $x_{0} \in ℝ$ such that u (x₀) =1; (ii) 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]; (iii) u is upper semicontinuous; (iv) $cl {x \in ℝ : u (x) > 0}$ 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 is equivalently represented by its characteristic function, then it is easy to know that $ℝ \subset ℝ_{F}$ . Especially, the fuzzy number zero $\hat{0} \in ℝ_{F}$ is defined by its characteristic function $\hat{0} (x) = 1$ if x = 0, and $\hat{0} (x) = 0$ if x ≠ 0. 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: (P1) D (u ⊕ w, v ⊕ w) = D (u, v), $\forall u, v, w \in ℝ_{F}$ ; (P2) D (λ ⊙ u, λ ⊙ v) = |λ|D (u, v), $\forall λ \in ℝ, u, v \in ℝ_{F}$ ; (P3) D (u ⊕ v, w ⊕ e) ≤ D (u, w) + D (v, e) , ∀ 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.

Here we recall the concept of strongly generalized differentiability introduced by Bede et al. [4, 5] and further studied by Chalco-Cano and Román-Flores [8].

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 function (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 $\frac{1}{h} \cdot$ and $- \frac{1}{h} \cdot$ , respectively.

Remark 1. In the preceding definition, the case (i) corresponds to the H-differentiability introduced by Puri and Ralescu [19]. So the differentiability presented in Definition 2.1 can be regarded as a generalization of the H-differentiability.

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.More generally, we call a mapping $F : I \to ℝ_{F}$ generalized differentiable if it is (i)- or (ii)-differentiable.

In what follows, let T = [a, b] be a closed interval, where $a, b \in ℝ$ with a < b.

Theorem 2.1. (Kaleva [14], Khastan [15]) Let $F : T \to ℝ_{F}$ be a generalized differentiable fuzzy-number valued function and assume that the derivativeF′ is integrable overT. Then for eacht ∈ Twe have

(i) If F is (i)-differentiable, then $F (t) = F (a) \oplus \int_{a}^{t} F^{'} (τ) d τ;$ (ii) If F is (ii)-differentiable, then $F (t) = F (a) ⊖ \int_{a}^{t} - F^{'} (τ) d τ .$

Theorem 2.2. (Kaleva [14]) Let $F : T \to ℝ_{F}$ be continuous. Then for anyt ∈ T, the integral $H (t) = \int_{a}^{t} F (τ) d τ$ is (i)-differentiable andH′ (t) = F (t).

Theorem 2.3. (Khastan et al. [15]) Let $F : T \to ℝ_{F}$ be continuous. Define the integral $H (t) : = γ ⊖ \int_{a}^{t} - F (τ) d τ, t \in T,$ where $γ \in ℝ_{F}$ is such that the preceding H-difference exists onT. Then H(t) is (ii)-differentiable andH′ (t) = F (t).

A fuzzy number-valued function $x : T \to ℝ_{F}$ is said to be bounded on T if there exists a positive number M such that $D (x (t), \hat{0}) \leq M$ for each t ∈ T. By Theorem 2.10 in [10], if a fuzzy number-valued function x is continuous on T, then x is bounded.

At the end of this section, we review a fundamental theorem in fixed point theory which will play a key role in proving our main results.

Theorem 2.4. (Diaz and Margolis [12]) Let (X, d) be a complete generalized metric space, i.e., one for whichdmay assume infinite values. Suppose thatJ : X → Xbe a strictly contractive mapping with Lipschitz constantL < 1. Then, for each given elementx ∈ X, either $d (J^{n} x, J^{n + 1} x) = \infty$ for alln ≥ 0 or there exists an $n_{0} \in ℕ$ such that

(i)d (Jⁿx, Jⁿ⁺¹x)< ∞ for alln ≥ n₀; (ii) The sequence {Jⁿx} converges to a fixed pointy^∗ofJ; (iii) y^∗is the unique fixed point ofJin the setY = {y ∈ X|d (J^{n
₀}x, y) < ∞}; (iv) $d (y, y^{*}) \leq \frac{1}{1 - L} d (y, Jy)$ for ally ∈ Y.

3 Hyers-Ulam-Rassias stability

In this section, we shall establish the Ulam stability (including Hyer-Ulam-Rassias stability and Hyers-Ulam stability) of the fuzzy differential equation x′ (t) = F (t, x (t)) under various types of differentiability by using the fixed point method.

3.1 Stability of the equation x′ (t) = F (t, x (t)) under (i)-differentiability

Lemma 3.1.Let $φ : T \to ℝ_{+}$ is a continuous function. Set $X = {x : T \to ℝ_{F} | x is continuous on T} .$ Define the mappingd : X × X → [0, + ∞] asfollows $\begin{matrix} d (x, y) & = inf {η \in ℝ_{+} \cup {+ \infty} : D (x (t), y (t)) \\ \leq η φ (t), \forall t \in T} . \end{matrix}$ (1)Then (X, d) is a complete generalized metric spaces.

Proof. It is immediate that d is symmetric due to the definition of d. Obviously, d (x, x) =0 for each x ∈ X. Now, we assume that d (x, y) =0. By (1), we obtain that D (x (t) , y (t)) ≤ ηφ (t) for all t ∈ T and all $η \in ℝ_{+}$ . This implies that D (x (t) , y (t)) =0 for any t ∈ T. That is, x = y. Next, it suffices to show that the triangle inequality holds. Assume that the triangleinequality does not hold. Then we have d (x, z) > d (x, y) + d (y, z) for some x, y, z ∈ X. Therefore, by (1), there exists t₀ ∈ T such that $\begin{matrix} D (x (t_{0}), z (t_{0})) & > (d (x, y) + d (y, z)) φ (t_{0}) \\ = d (x, y) φ (t_{0}) + d (y, z) φ (t_{0}) \\ \geq D (x (t_{0}), y (t_{0})) + D (y (t_{0}), z (t_{0})), \end{matrix}$ this leads to a contradiction.

Finally, we have to show that (X, d) is complete. Assume that {x_n} is a Cauchy sequence in X. Then, for each ɛ > 0, there exists N = N (ɛ) such that d (x_n, x_m) < ɛ for all m, n > N. Thus, it follows from (1) that $D (x_{n} (t), x_{m} (t)) \leq ɛ φ (t)$ (2) for all m, n > N and all t ∈ T. For a fixed t in T, the inequality (2) implies that the sequence {x_n} is a Cauchy sequence in $ℝ_{F}$ . In view of the completeness of $(ℝ_{F}, D)$ , there exists a mapping $x : T \to ℝ_{F}$ such that {x_n (t)} converges to x (t) for each t ∈ T. Now, it remains to show that x belongs to X. Letting m→ + ∞, it follows from (2) that $D (x_{n} (t), x (t)) \leq ɛ φ (t)$ (3) for all n > N and all t ∈ T. Since φ is continuous on T, there exists r > 0 such that φ (t) ≤ r. Then we obtain that D (x_n (t) , x (t)) ≤ ɛr for all n > N and all t ∈ T. This means that {x_n} converges uniformly to x,and then $x : T \to ℝ_{F}$ is continuous. Therefore, we conclude that x ∈ X. Combining the inequality (3) and the definition of d, we can obtain that d (x_n, x) ≤ ɛ for all n < N. That is to say, the Cauchy sequence {x_n} converges to x, and hence (X, d) is complete. This completes the proof. □

Theorem 3.2.LetKandLbe two positive numbers such that 0 < KL < 1. Assume that $F : T \times ℝ_{F} \to ℝ_{F}$ is a continuous fuzzy-number valued function which satisfies the following Lipschitz condition $D (F (t, x), F (t, y)) \leq L \cdot D (x, y)$ (4)for allt ∈ T, $x, y \in ℝ_{F}$ . Suppose that a continuously (i)-differentiable function $x : T \to ℝ_{F}$ satisfies $D (x^{'} (t), F (t, x (t))) \leq φ (t)$ (5)for allt ∈ T, where $φ : T \to ℝ_{+}$ is a continuous function with $\int_{a}^{t} φ (τ) d τ \leq K φ (t)$ for eacht ∈ T. Then there exists a unique continuous fuzzy-number valued function $\tilde{x} : T \to ℝ_{F}$ such that $\tilde{x} (t) = x (a) \oplus \int_{a}^{t} F (τ, \tilde{x} (τ)) d τ$ (6)and $D (x (t), \tilde{x} (t)) \leq \frac{K}{1 - KL} φ (t)$ (7)for allt ∈ T.

Proof. Consider the set $X = {f : T \to ℝ_{F} | f is continuous on T},$ and introduce the generalized metric d on X which is defined by $\begin{matrix} d (f, g) & = inf {η \in ℝ_{+} \cup {+ \infty} : D (f (t), g (t)) \\ \leq η φ (t), \forall t \in T} . \end{matrix}$ By Lemma 3.1, (X, d) is a complete generalizedmetric space.

Now, we define the operator J : X → X such that $(Jf) (t) = x (a) \oplus \int_{a}^{t} F (τ, f (τ)) d τ, t \in T,$ (8) for all f ∈ X. By Theorem 2.2 and Theorem 5.4 in [14], it is easy to see that Jf is (i)-differentiable and hence it is also continuous. Thus, we know thatJf ∈ X.

Next, we show that J is a strictly contractive mapping on X. Let f, g ∈ X be given such that d (f, g) = η, η ∈ [0, + ∞]. Then, by the definition of d, we have $D (f (t), g (t)) \leq η φ (t)$ for all t ∈ T. Therefore, we can obtain that $\begin{matrix} D ((Jf) (t), (Jg) (t)) \\ = D (x (a) \oplus \int_{a}^{t} F (τ, f (τ)) d τ, x (a) \\ \oplus \int_{a}^{t} F (τ, g (τ)) d τ) \\ = D (\int_{a}^{t} F (τ, f (τ)) d τ, \int_{a}^{t} F (τ, g (τ)) d τ) \\ \leq \int_{a}^{t} D (F (τ, f (τ)), F (τ, g (τ))) d τ \\ \leq L \int_{a}^{t} D (f (τ), g (τ)) d τ \\ \leq η L \int_{a}^{t} φ (τ) d τ \leq η LK φ (t) \end{matrix}$ for all t ∈ T. That is, $d (f, g) = η \Rightarrow d (Jf, Jg) \leq η LK,$ which means that d (Jf, Jg) ≤ LKd (f, g) for allf, g ∈ X. Since 0 < KL < 1, we know that J is a strictly contractive mapping on X.

For each f ∈ X, there exists a positive constant 0< α < + ∞ such that $\begin{matrix} D ((Jf) (t), f (t)) & = D (x (a) \oplus \int_{a}^{t} F (τ, f (τ)) d τ, f (t)) \\ \leq α φ (t) \end{matrix}$ for all t ∈ T, since F (t, f (t)) and f (t) are bounded on I, and $min_{t \in T} φ (t) > 0$ , i.e., the minimum of φ (t). Then, we know that d (Jf, f)≤ α < + ∞.

According to (i) and (ii) of Theorem 2.4, there exists a fixed point $\tilde{x}$ of J such that Jⁿf converges to $\tilde{x}$ , i.e., $J^{n} f \to \tilde{x}$ as n→ ∞. In addition, for any g ∈ X, there exists a positive 0< β < + ∞ such that D (f (t) , g (t)) ≤ βφ (t), since f and g are bounded on T, and $min_{t \in T} φ (t) > 0$ . It follows from the precedinginequality that d (f, g)< + ∞ for all g ∈ X. This implies that {g ∈ X : d (f, g) < + ∞} = X. By (iii) of Theorem 2.4, $\tilde{x}$ is the unique fixed point of J in X. It is obvious that $\tilde{x}$ is a unique fuzzy-number valued function in X which satisfies the equality $J \tilde{x} = \tilde{x}$ .

Moreover, it should be noticed that the H-difference x (t) ⊖ x (a) exists for any t ∈ T, since $x : T \to ℝ_{F}$ is (i)-differentiable. Then, it follows from (5) and Theorem 2.1 that $\begin{matrix} D (x (t), x (a) \oplus \int_{a}^{t} F (τ, x (τ)) d τ)) \\ = D (x (t) ⊖ x (a), \int_{a}^{t} F (τ, x (τ)) d τ)) \\ = D (\int_{a}^{t} x^{'} (τ) d τ, \int_{a}^{t} F (τ, x (τ)) d τ)) \\ \leq \int_{a}^{t} D (x^{'} (τ), F (τ, x (τ))) d τ \\ \leq \int_{a}^{t} φ (τ) d τ \leq K φ (t) \end{matrix}$ for each t ∈ T. That is, D ((Jx) (t) , x (t)) ≤ Kφ (t) for each t ∈ T. Immediately, we obtain that d (Jx, x) ≤ K. According to (iv) of Theorem 2.4, we can infer that $d (x, \tilde{x}) \leq \frac{1}{1 - KL} d (x, Jx) \leq \frac{K}{1 - KL},$ (9) which means that the inequality (7) holds for each t ∈ T. The proof of the theorem is now completed. □

Remark 2. By Theorem 2.2, it is easy to verifythat $\tilde{x} (t)$ defined in Theorem 3.2 is the unique continuously (i)-differentiable solution to the fuzzy differential equation x′ (t) = F (t, x (t)).

As a particular consequence of Theorem 3.2, the Hyers-Ulam stability of the fuzzy differential equation x′ (t) = F (t, x (t)) under (i)-differentiability can be formulated as follows.

Corollary 3.3.LetLbe a positive number such that 0 < (b - a) L < 1. Assume that $F : T \times ℝ_{F} \to ℝ_{F}$ is a continuous fuzzy-number valued function which satisfies the Lipschitz condition (4) with Lipchitz constantL. For a givenɛ > 0, assume that a continuously (i)-differentiable function $x : T \to ℝ_{F}$ satisfies $D (x^{'} (t), F (t, x (t))) \leq ɛ$ (10)for allt ∈ T. Then there exists a unique continuous fuzzy-number valued function $\tilde{x} : T \to ℝ_{F}$ such that $D (x (t), \tilde{x} (t)) \leq \frac{(b - a) ɛ}{1 - (b - a) L}$ (11)for allt ∈ T, where $\tilde{x} (t)$ is defined as in Theorem 3.2.

3.2 Stability of the equation x′ (t) = F (t, x (t)) under (ii)-differentiability

Theorem 3.4.LetK, L, Fandφbe given as in Theorem 3.2. Assume that a continuously (ii)-differentiable function $x : T \to ℝ_{F}$ satisfies the inequality (5) for allt ∈ T. For eacht ∈ Tand each continuous function $f : T \to ℝ_{F}$ , if the H-difference $x (a) ⊖ \int_{a}^{t} - F (τ, f (τ)) d τ$ exists, then there exists a unique continuous fuzzy-number valued function $\tilde{x} : T \to ℝ_{F}$ such that the inequality (7) holds for allt ∈ T, where $\tilde{x} (t) = x (a) ⊖ \int_{a}^{t} - F (τ, \tilde{x} (τ)) d τ$ .

Proof. Consider the same set X and introduce the same metric d on X as the proof of Theorem 3.2.

Define the operator J : X → X as follows: $(Jf) (t) = x (a) ⊖ \int_{a}^{t} - F (τ, f (τ)) d τ, t \in T,$ for all f ∈ X. By Theorem 2.3, we know that Jf is (ii)-differentiable. Using the same arguments as in the proof of Theorem 5.4 of [14], one can show that Jf is continuous on T.

Now, we assert that J is a strictly contractive mapping on X. For any f, g ∈ X with d (f, g) ≤ η, η ∈ [0, + ∞], we have D (f (t) , g (t)) ≤ ηφ (t) for all t ∈ T. Then we can obtain that

allowdisplaybreaks $\begin{matrix} D ((Jf) (t), (Jg) (t)) \\ = D (x (a) ⊖ \int_{a}^{t} - F (τ, f (τ)) d τ, x (a) \\ ⊖ \int_{a}^{t} - F (τ, g (τ)) d τ) \\ = D (\int_{a}^{t} - F (τ, f (τ)) d τ, \int_{a}^{t} - F (τ, g (τ)) d τ) \\ \leq \int_{a}^{t} D (- F (τ, f (τ)), - F (τ, g (τ))) d τ \\ = \int_{a}^{t} D (F (τ, f (τ)), F (τ, g (τ))) d τ \\ \leq L \int_{a}^{t} D (f (τ), g (τ)) d τ \\ \leq η L \int_{a}^{t} φ (τ) d τ \leq η LK φ (t) \end{matrix}$ for all t ∈ T. This means that d (Jf, Jg) ≤ ηLK. From the analysis above, we conclude that d (Jf, Jg) ≤ LKd (f, g). Immediately, J is a strictly contractive mapping on X since 0 < LK < 1.

Analogous to the proof of Theorem 3.2, one can further show that d (Jf, f)< + ∞ for each f ∈ X, and {g ∈ X : d (f, g) < + ∞} = X. According toTheorem 2.4, there exists a unique fixed point $\tilde{x}$ of J in X such that $J^{n} f \to \tilde{x}$ as n→ ∞.

Notice that the H-difference x (a) ⊖ x (t) exists for every t ∈ T, since $x : T \to ℝ_{F}$ is (ii)-differentiable. Therefore, it follows from Theorem 2.1 and the inequality (5) that $\begin{matrix} D (x (t), x (a) ⊖ \int_{a}^{t} - F (τ, x (τ)) d τ) \\ = D (x (t) \oplus \int_{a}^{t} - F (τ, x (τ)) d τ, x (a)) \\ = D (x (a) ⊖ x (t), \int_{a}^{t} - F (τ, x (τ)) d τ) \\ = D (\int_{a}^{t} - x^{'} (τ) d τ, \int_{a}^{t} - F (τ, x (τ)) d τ) \\ \leq \int_{a}^{t} D (x^{'} (τ), F (τ, x (τ))) d τ \\ \leq \int_{a}^{t} φ (τ) d τ \leq K φ (t) \end{matrix}$ for each t ∈ T. This shows that D ((Jx) (t) , x (t)) ≤ Kφ (t) for each t ∈ T. Obviously, we getd (Jx, x) ≤ K. Thus, by (iv) of Theorem 2.4, we can infer that the inequality (9) holds, and hence the inequality (7) is true for any t ∈ T, which completes the proof of the theorem. □

Remark 3. By Theorem 2.3, it is easy to check that $\tilde{x} (t)$ defined in Theorem 3.4 is the unique continuously (ii)-differentiable solution to the fuzzy differential equation x′ (t) = F (t, x (t)).

The Hyers-Ulam stability of the fuzzy differential equation x′ (t) = F (t, x (t)), as a special case of Theorem 3.4, can be obtained in the sense of (ii)-differentiability.

Corollary 3.5.LetLbe a positive number such that 0 < (b - a) L < 1. Assume that $F : T \times ℝ_{F} \to ℝ_{F}$ is a continuous fuzzy-number valued function which satisfies the Lipschitz condition (4) with Lipchitz constantL. For a givenɛ > 0, assume that a continuously (ii)-differentiable function $x : T \to ℝ_{F}$ satisfies the inequality (10) for allt ∈ T. For eacht ∈ Tand each continuous function $f : T \to ℝ_{F}$ , if the H-difference $x (a) ⊖ \int_{a}^{t} - F (τ, f (τ)) d τ$ exists, then there exists a unique continuous fuzzy-number valued function $\tilde{x} : T \to ℝ_{F}$ such that the inequality (11) holds for allt ∈ T, where $\tilde{x} (t)$ is given by Theorem 3.4.

Remark 4. From Theorems 3.2 and 3.4, as well as Corollaries 3.3 and 3.5, one can see that the corresponding theorem requires a stronger condition, i.e., the existence of H-difference, in order to ensure the same Ulam stability results of fuzzy differential equations under (ii)-differentiability.

4 An example

In this section, we will apply the preceding theorems to derive the Ulam stability results of first order linear fuzzy differential equations.

Example 1. Let T = [0, b] and let ɛ > 0 be an arbitrary positive number such that $\frac{b^{2}}{2 ɛ} - b > 0$ . We choose a positive number K such that $K \geq \frac{b^{2}}{2 ɛ} - b$ . Let $δ : T \to ℝ$ be continuous with |δ (t) | ≤ L and KL < 1, and let $σ : T \to ℝ_{F}$ be continuous on T. Assume that a continuously (i)-differentiable fuzzy-number valued function $x : T \to ℝ_{F}$ satisfies $D (x^{'} (t), δ (t) ⊙ x (t) \oplus σ (t)) \leq t + ɛ$ (12) for all t ∈ T. Then there exists a unique continuous fuzzy-number valued function $\tilde{x} : T \to ℝ_{F}$ such that $D (x (t), \tilde{x} (t)) \leq \frac{K (t + ɛ)}{1 - KL}$ (13) for all t ∈ T.

Let F (t, x (t)) = δ (t) ⊙ x (t) ⊕ σ (t) and φ (t) = t + ɛ. Apparently, F (t, x (t)) is continuous for each function $x : T \to ℝ_{F}$ . Since |δ (t) | ≤ L, for any $x (t), y (t) \in ℝ_{F}$ , we have $\begin{matrix} D (δ (t) ⊙ x (t) \oplus σ (t), δ (t) ⊙ y (t) \oplus σ (t)) \\ = D (δ (t) ⊙ x (t), δ (t) ⊙ y (t)) \\ = | δ (t) | D (x (t), y (t)) \leq LD (x (t), y (t)) \end{matrix}$ for all t ∈ T. Moreover, we can infer that $\begin{matrix} \int_{0}^{t} φ (τ) d τ & = \int_{0}^{t} (τ + ɛ) d τ = \frac{t^{2}}{2} + ɛ t \\ \leq K (t + ɛ) = K φ (t), \end{matrix}$ since $K \geq \frac{b^{2}}{2 ɛ} - b$ . All conditions of Theorem 3.2 are satisfied and so there exists a unique continuous fuzzy-number valued function $\tilde{x} : T \to ℝ_{F}$ such that the inequality (13) holds for all t ∈ T.

In particular, if we let φ (t) = ɛ, then the inequality (12) reduces to $D (x^{'} (t), δ (t) ⊙ x (t) \oplus σ (t)) \leq ɛ$ for all t ∈ T. In this case, the inequality (13) becomes the following form $D (x (t), \tilde{x} (t)) \leq \frac{K ɛ}{1 - KL},$ which implies the Hyers-Ulam stability of the firstlinear fuzzy differential equation x′ (t) = δ (t) ⊙ x (t) ⊕ σ (t) in the sense of (i)-differentiability.

Remark 5. For the first order linear fuzzy differential equation x′ (t) = δ (t) ⊙ x (t) ⊕ σ (t), it remains open to determine the appropriate conditions which can be applied to ensure the Ulam stability of this equation under (ii)-differentiability.

5 Conclusion

At present, the Ulam stability of differential equations has become an important branch in the theory of differential equations. It not only establishes an important foundation for the existence (and even uniqueness) of the solution of differential equations, but also provides a reliable theoretical basis for approximately solving differential equations. However, the research on this area has barely been carried out in the theory of fuzzy differential equations. The main reason is that the space of fuzzy numbers has a special structure such that many similar studies of differential equations are difficult to be extended. In this paper, we studied the Ulam stability of fuzzy differential equation under generalized differentiability by using the fixed point technique. The main results show that the Ulam stability of fuzzy differential equations requires various prerequisites under different types of differentiability. In particular, it should be noted that a stronger condition, i.e., the existence of H-difference, must be imposed under (ii)-differentiability to obtain the Ulam stability of fuzzy differential equations. Essentially, it is not easy to determine the existence of H-difference in Theorem 3.4 and Corollary 3.5. Therefore, it would be interesting to consider the existence of H-difference mentioned above in future studies.

Footnotes

Acknowledgments

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

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