On the Ulam stability of fuzzy differential equations

1.   Introduction

Stability is one of the most important characteristics of dynamic systems. One of the methods to study stability is introduced by Ulam ^[1], in which the problem of functional equations concerning the stability of group homomorphisms was first discussed. Ulam method-based stability analysis was then investigated by many researchers such as Hyers ^[2], Rassias ^[3], and so on. Then, Obloza ^[4] explored the Ulam stability of differential equations. Afterwards, Alsina and Ger ^[5] proved the differential equation $y' = y$ satisfied Hyers-Ulam stability. Miura et al. ^[6,7] and Takahasi et al. ^[8] discussed the Ulam stability of the differential equation $y' = \lambda y$ in some abstract spaces.

As is known to all, fuzzy differential equations play an important role in practical applications, such as power systems and artificial intelligence. Kaleva ^[9] proved the existence and uniqueness of a solution to a fuzzy differential equation $x'(t) = f(t, x(t))$ provided $f$ satisfied a Lipschitz condition. After that, Kaleva ^[10] and Kloeden ^[11] discussed the peano theorem of fuzzy differential equations one after another. In the following year, Tomasiello and Macías-Díaz ^[12] extended a Picard-like approach to fuzzy fractional differential equations. Rashid et al. ^[13] established a relation between the solutions and approximate solutions of complex fuzzy differential equations with complex membership grades. Nowadays, the study of fuzzy differential equations is still active, See ^{[14,15,16,17,18,19,20,21,22]}.

It is worth noting that the stability of fuzzy differential equations has got scholars' much attentions in recent years. Diamond ^[23] studied the Lyapunov stability. Song et al. ^[24] studied the stability properties of the trivial fuzzy solution of fuzzy differential equations. After that, Yakar et al. ^[25] considered the Lagrange stability. Other results about the stability of fuzzy differential equations can be found in Refs. ^[26,27,28]. Recently, the Ulam stability of fuzzy differential equations is developed. For example, Shen and Wang ^[29] discussed the Ulam stability of fuzzy differential equations under generalized differentiability by a fixed point approach. Shen ^[30] explored the Ulam stability of the following three linear fuzzy differential equations in the real number space:

where $u$ and $\sigma$ are fuzzy number-valued functions (mappings), $\delta$ is a real valued function and equality symbol means identity of membership functions on each side. Ren et al. ^[31] introduced a fuzzy Mellin transform method for solving Hermite fuzzy differential equations and give some Hyers-Ulam stability results of Hermite fuzzy differential equations. Wang and Sun ^[32] investigated the existence and uniqueness of solution to Cauchy problems for a class of nonlinear fuzzy fractional differential equations with the Riemann-Liouville H-derivative.

In this paper, we further the discussions of Ulam stability for Eqs (1.1)–(1.3) under the different differentiability and domain space from those in Ref. ^[30]. Here, the concept of differentiability is the generalization of that introduced in ^[33]. Meanwhile, the domain space is a Banach space instead of the real number space. We first give some preliminaries in section 2, and then, investigate the Ulam stability of Eqs (1.1)–(1.3) in sections 3–5, respectively.

In this paper, $R$ denotes the set of all real numbers, ${R_ + } = \left({0, \infty } \right)$, $I = [0, 1]$, ${I_0} = (0, 1)$, $T = [a, b]$ and ${T_0} = (a, b)$, where $a, b \in R$ with $a < b$. $X$ is a Banach space, ${P_{kc}}(X)$ denotes the set of all non-empty compact convex subsets of $X$.

2.   Preliminaries

In this section, we review some necessary notions and fundamental results which are useful in this paper.

If a function $u:X \to [0, 1]$ satisfies the following conditions:

(ⅰ) $u$ is normal, i.e., there exists $x_0\in X$ such that $u(x_0) = 1$;

(ⅱ) ${[u]^\alpha } = \{ x \in X:u(x) \ge \alpha \} \in {P_{kc}}(X)$, $\forall \alpha \in (0, 1]$;

(ⅲ) $u$ is upper-semicontinuous; ;

(ⅲ) the support set of $u:{[u]^0} = {\mathop{\rm supp}\nolimits} (u) = \rm cl\{ x:u(x) > 0\}$ is a compact set, where "cl" means the closure operator.

Then $u$ is called a fuzzy number on $X$. The set of all fuzzy numbers on $X$ is denoted by $X_F$.

A mapping $D:{X_F} \times {X_F} \to {R_ + } \cup \{ 0\}$ is defined by $D(u, v) = \mathop {\sup }\limits_{\alpha \in I} {d_H}\left({{{[u]}^\alpha }, {{[v]}^\alpha }} \right)$, where $d_H$ is Hausdorff metric, then $(X_F, D)$ is a complete metric space. For $u, v\in X_F$, $\lambda\in R$, the addition $u+v$ and scalar multiplication $\lambda\cdot u$ are introduced by Zadeh extension principle. If there exists $w\in X_F$ such that $u = v+w$, then $w$ is called the H-difference of $u$ and $v$, and denoted by $u-v$. Obviously, $u = v + (u - v)$.

The details about the fuzzy number space can be found in ^[9,22,34] and so on. On these bases, we obtain the following result easily:

Theorem 2.1. Let $u, v, w, \gamma\in {X_F}$, and the H-difference has the following properties:

(H1) For $u, v, w\in {X_F}$, if $u-w$ exists, then $(u+v)-w$ and $(u + v) - w = (u - w) + v$ exist;

(H2) For $u, v, w, \gamma\in {X_F}$, if $u-v$ exists, then $(\gamma + u) - (\gamma + v)$ and $(\gamma + u) - (\gamma + v) = u - v$ exist.

The following definition extends the corresponding concept in ^[33] from the real number space to a Banach space.

Definition 2.2. Let $F:{T_0} \to {X_F}$ be a fuzzy number-valued mapping, ${t_0} \in {T_0}$, $F'({t_0}) \in {X_F}$. If for all $h > 0$ sufficiently small,

(ⅰ) the H-differences $F({t_0} + h) - F({t_0})$ and $F({t_0}) - F({t_0} - h)$ exist, and

or

(ⅱ) the H-differences $F({t_0}) - F({t_0} + h)$ and $F({t_0}-h) - F({t_0})$ exist, and

or

(ⅲ) the H-differences $F({t_0} + h) - F({t_0})$ and $F({t_0}-h) - F({t_0})$ exist, and

or

(ⅳ) the H-differences $F({t_0}) - F({t_0} + h)$ and $F({t_0}) - F({t_0} - h)$ and exist, and

($h$ and $-h$ at denominators denote $\frac{1}{h}$ and $-\frac{1}{h}$, respectively), then $F$ is said to be (ⅰ)-differentiable or (ⅱ)-differentiable or (ⅲ)-differentiable or (ⅳ)-differentiable at $t_0$, respectively. If $F$ satisfies all of (ⅰ)-(ⅳ), then $F$ is said to be differentiable at $t_0$. If $F$ is (ⅰ)-differentiable ((ⅱ)-differentiable, (ⅲ)-differentiable, (ⅳ)-differentiable, differentiable) at every point of $T_0$, then $F$ is said to be (ⅰ)-differentiable ((ⅱ)-differentiable, (ⅲ)-differentiable, (ⅳ)-differentiable, differentiable) on $T_0$.

In this paper, we develop our discussion in the cases of (ⅲ) and (ⅳ) differentiabilities.

Theorem 2.3. Let $F, G:{T_0} \to {X_F}$ be two fuzzy mappings. Suppose the H-difference $F(t) - G(t)$ exists for $t\in T_0$.

(1) If $F$ is (ⅲ)-differentiable and $G$ is (ⅳ)-differentiable on $T_0$, then $F-G$ is (ⅲ)-differentiable on $T_0$;

(2) If $F$ is (ⅳ)-differentiable and $G$ is (ⅲ)-differentiable on $T_0$, then $F-G$ is (ⅳ)-differentiable on $T_0$.

Moreover, in the case (1) or (2), for all $t\in T_0$, we have $\left({F - G} \right)'(t) = F'(t) + (- 1) \cdot G'(t)$.

Proof. We only prove (1), the method used here is similar to Theorem 9 in ^[33].

Since $F$ is (ⅲ)-differentiable and $G$ is (ⅳ)-differentiable, for any $t\in T_0$ and $h > 0$, there exist $u_1(t, h)$, $u_2(t, h)$, $v_1(t, h)$, $v_2(t, h)\in X_F$ such that

and

From (2.1) and (2.3), we get

Since the H-differences $F(t) - G(t)$ and $F(t + h) - G(t + h)$ exist, from (2.7) and Theorem 2.1 we have

which implies that $\left({F(t + h) - G(t + h)} \right) - \left({ F(t)-G(t)} \right) = {u_1}\left({t, h} \right) + {v_1}\left({t, h} \right)$.

From (2.5) and (2.6), we get

Similarly, from (2.2), (2.4)–(2.6) and Theorem 2.1, we obtain

Hence, $F-G$ is (ⅲ)-differentiable on $T_0$ and $\left({F - G} \right)'(t) = F'(t) + (- 1) \cdot G'(t)$. The proof ends.

For convenience, we introduce the following conditions:

(C1) For a given $t\in T_0$, if $F(t + h) - F(t)$ and $F(t - h) - F(t)$ exist for sufficiently small $h > 0$;

(C2) For a given $t\in T_0$, if $F(t) - F(t+h)$ and $F(t) - F(t-h)$ exist for sufficiently small $h > 0$;

(C3) For a given $t\in T_0$, if $F(t + h) - F(t)$ and $F(t) - F(t-h)$ exist for sufficiently small $h > 0$;

(C4) For a given $t\in T_0$, if $F(t) - F(t+h)$ and $F(t - h) - F(t)$ exist for sufficiently small $h > 0$.

Based on Theorem 2.5 in ^[30], we can verify the following theorem.

Theorem 2.4. Let $f:{T_0} \to R$ be differentiable, and $G:{T_0} \to {X_F}$ be (ⅲ)-differentiable or (ⅳ)-differentiable on $T_0$,

(ⅰ) if $f(t) \cdot f'(t) > 0$ and $G$ is (ⅲ)-differentiable on $T_0$, then $f \cdot G$ is (ⅲ)-differentiable on $T_0$ and

(ⅱ) if $f(t) \cdot f'(t) < 0$ and $G$ is (ⅲ)-differentiable on $T_0$, then $f \cdot G$ is (ⅲ)-differentiable on $T_0$ and

(ⅲ) if $f(t) \cdot f'(t) < 0$ and $G$ is (ⅲ)-differentiable on $T_0$ and $f \cdot G$ satisfies condition (C2), then $f \cdot G$ is (ⅳ)-differentiable on $T_0$ and

(ⅳ) if $f(t) \cdot f'(t) > 0$ and $G$ is (ⅳ)-differentiable on $T_0$ and $f \cdot G$ satisfies condition (C1), then $f \cdot G$ is (ⅲ)-differentiable on $T_0$ and

(ⅴ) if $f(t) \cdot f'(t) > 0$ and $G$ is (ⅳ)-differentiable on $T_0$, then $f \cdot G$ is (ⅳ)-differentiable on $T_0$ and

(ⅵ) if $f(t) \cdot f'(t) < 0$ and $G$ is (ⅳ)-differentiable on $T_0$, then $f \cdot G$ is (ⅳ)-differentiable on $T_0$ and

Proof. The proofs of them is similar, we only show (ⅵ). Suppose that $f(t) < 0$, $f'(t) > 0$ and $G$ is (ⅳ)-differentiable on $T_0$, then for any $t\in T_0$ and any sufficiently small $h > 0$, there exists $u\left({t, h} \right)\in X_F$ such that

We also have $f(t) = f(t + h) + v\left({t, h} \right)$, where $\mathop {\lim }\limits_{h \to 0} \frac{{v\left({t, h} \right)}}{{(- h)}}{\rm{ = }}f'(t)$. Then

so

Hence,

Similarly, we have

The proof ends.

Theorem 2.5. Let $F:T \to {X_F}$ be (ⅲ)-differentiable or (ⅳ)-differentiable and ${t_1}, {t_2} \in T$, then

(1) $F({t_2}) - F({t_1})$ exists;

(2) there exists a positive real number $M$ such that $D\left({F({t_2}), F({t_1})} \right) \le M\left| {{t_2} - {t_1}} \right|$.

Proof. We only show the case that $F$ is (ⅲ)-differentiable.

Case 1: If $t_1 = t_2$, we have $F({t_2}) - F({t_1}) = \theta$ and $D\left({F({t_2}), F({t_1})} \right) = 0$ directly.

Case 2: If $t_1 < t_2$. For all $s \in [{t_1}, {t_2}]$, by the assumption there exists $\delta (s) > 0$ such that $F(s+h)-F(s)$ exists and $D\left({\frac{{F(s + h) - F(s)}}{h}, F'(s)} \right) < {\rm{1}}$ or $D\left({F(s + h) - F(s), hF'(s)} \right) < h$ for $h \in (0, \delta (s))$, hence

where ${D_{F'}}(s) = D\left({F'(s), \theta } \right)$.

So we can find a finite sequence: ${t_1} = {s_1} < {s_2} \cdot \cdot \cdot < {s_n} = {t_2}$ such that the interval set $\{I_i: {I_i} = ({s_i} - \delta ({s_i}), {s_i} + \delta ({s_i})), i = 1, 2, \cdot \cdot \cdot, n\}$ cover $[{t_1}, {t_2}]$ and ${I_i} \cap {I_{i + 1}} \ne \phi$ $(i = 1, 2, 3 \cdot \cdot \cdot, n-1)$. Let ${v_i} \in {I_i} \cap {I_{i + 1}}$, such that ${s_i} < {v_i} < {s_{i + 1}}$ $(i = 1, 2, 3 \cdot \cdot \cdot, n-1)$, then there exist ${\gamma _1}^{(i)}, {\gamma _2}^{(i)} \in {X_F}$, such that $F({s_{i + 1}}) = F({v_i}) + {\gamma _1}^{(i)} = F({s_i}) + {\gamma _2}^{(i)} + {\gamma _1}^{(i)} = F({s_i}) + {\eta _i}$ and

where ${D_{F'}} = \mathop {\max }\limits_{1 \le i \le n - 1} \left\{ {{D_{F'}}({s_i}), {D_{F'}}({v_i})} \right\}$, ${\eta _i} = {\gamma _2}^{(i)} + {\gamma _1}^{(i)} \in {X_F}$ $(i = 1, 2 \cdot \cdot \cdot, n - 1)$.

Let $\sum\limits_{i = 1}^{n - 1} {{\eta _i}} = u \in {X_F}$, then $F({t_2}) = F({t_1}) + u$ and

Let ${M_{\rm{1}}}{\rm{ = }}(1 + {D_{F'}}) > 0$, then $D\left({F({t_2}), F({t_1})} \right) < M_1({t_2} - {t_1})$.

Case 3: $t_1 > t_2$. By the similar discussion in Case 2, we can prove there exists $M_2 > 0$ such that $D\left({F({t_2}), F({t_1})} \right) < M{}_{\rm{2}}({t_{\rm{1}}} - {t_{\rm{2}}})$. The proof ends..

The following contents about the integral of a fuzzy mapping are generalizations of these in ^[9].

Definition 2.6. A mapping $F:T \to {X_F}$ is measurable if for all $\alpha \in \left({0, 1} \right]$ the set-valued mapping ${F_\alpha }:T \to {P_{kc}}(X)$ defined by

is measurable. The set of all measurable selections of ${F_\alpha }(t)$ is denoted by ${S_{{F_\alpha }}}$.

Definition 2.7. Let $F:T \to {X_F}$ be measurable. If there exists $u \in {X_F}$ such that

for all $\alpha \in \left({0, 1} \right]$, then $F$ is said to be integrable over $T$, $u$ is the integral of $F$ over $T$ and denoted by $\int_T {F(t)dt}$ or $\int_a^b {F(t)dt}$, where

Definition 2.8. A mapping $F:T \to {X_F}$ is called integrably bounded if there exists an integrable mapping $h:\left[{a, b} \right] \to X$ such that $|x| \le h\left(t \right)$ for all $x \in {\left[{F\left(t \right)} \right]^0}$.

It is well known that if a measurable mapping $F:T \to {X_F}$ is integrably bounded, then it is integrable.

Theorem 2.9. If $F:[a, b] \to {X_F}$ is integrable and $c \in [a, b]$, then

Remark. Let $\int_a^b {F(t)dt} = \left({ - \int_b^a {F(t)dt} } \right)$ when $a > b$, we can prove that (2.8) still holds for $a > c > b$.

The set of all continuous mapping $F:T \to \left({{X_F}, D} \right)$ is denoted by $C[T, {X_F}]$.

The following theorem is the generalization of Corollary 4.1 in ^[9] from $R^n$ to a Banach space.

Theorem 2.10. If $F:T \to {X_F}$ be a continuous fuzzy mapping, then it is integrable.

Simillar to Theorems 2.3 and 2.4 in ^[30], we can show the following theorem.

Theorem 2.11. Let $F:T \to {X_F}$ be a continuous fuzzy mapping. Then, for any $x\in T$,

(1) the mapping $H(x) = \int_a^x {F(t)dt}$ is (ⅰ)-differentiable, and $H(x)$ is Lipschitz continuous. Moreover, $H'(x) = F(x)$;

(2) $H(x) = \gamma - \int_a^x {\left({(- 1) \cdot F(t)} \right)dt}$ is (ⅱ)-differentiable, and $H'(x) = F(x)$.

3.   Ulam stability for Eq (1.1)

In this section, we mainly discuss the Ulam stability for the linear fuzzy differential Eq (1.1) in two cases: (1) $u$ is (ⅲ)-differentiable and $\delta (t) > 0$; (2) $u$ is (ⅳ)-differentiable and $\delta (t) < 0$.

Theorem 3.1. Let $\sigma \in C[T, {X_F}]$ and $\delta :T \to {R_ + }$ be a continuous function. Suppose that $u:{T_0} \to {X_F}$ is continuous and (ⅲ)-differentiable, and the H-difference $\exp \left({\int_a^t {\delta (\tau)d\tau } } \right)u(t) - \int_a^t {\exp \left({\int_a^v {\delta (\tau)d\tau } } \right)\sigma (v)dv}$ exists for each $t\in T_0$.

(1) there exists a ${u_0} \in {X_F}$ such that the mapping: $\forall t \in {T_0}$,

satisfies

where $M, M' > 0$.

(2) If $\tilde u(t)$ satisfies the condition (C3), then $\tilde u(t)$ is the (ⅰ)-differentiable solution of Eq (1.1).

Proof. (1) For convenience, we set

for each $t\in T_0$. Then for arbitrary $s, t\in T_0$ with $t < s$, we have

From (ⅰ) of Theorem 2.4, we know that $G(t)$ is (ⅲ)-differentiable on $T_0$, which together with Theorem 2.5 implies that there exists $M > 0$ such that

Since $\sigma$ and $\delta$ are continuous functions and by Theorem 2.10, we know that $\exp \left({\int_a^t {\delta (\tau)d\tau } } \right)\sigma (t)$ is integrable on $T$. Then from (1) of Theorem 2.11, we obtain

where $M' > 0$. Thus

which means that ${\{ w(s)\} _{s \in {T_0}}}$ is a Cauchy net in $X_F$. The completeness of the metric space $(X_F, D)$ implies that there exists a $u_0\in X_F$ such that $w(s)$ converges to $u_0$ as $s \to {b^ - }$. Based on the above arguments, we obtain

It follows from (3.3) that the expression (3.4) tends to $\exp \left({ - \int_a^t {\delta (\tau)d\tau } } \right)\left({M + M'} \right)(b - t)$ as $s \to {b^ - }$. That is, the inequality (6) holds.

Now we prove the uniqueness of $u_0$. Assume that there exists $u_1\in X_F$ such that inequality (3.2) holds where $u_0$ in (3.1) is replaced by $u_1$. It follows from (3.2) that

for each $t\in T_0$. So

for each $t\in T_0$. Since $2\left({M{\rm{ + }}M'} \right)(b - t)$ tends to $0$ as $t \to {b^ - }$, we get ${u_0} \equiv {u_1}$. This completes the proof.

(2) Next, we shall consider the differentiability of $\tilde u$ given by (3.2). Setting

From (1) of Theorem 2.11, it is easy to see that $H(t)$ is (ⅰ)-differentiable on $T_0$. If $f(t)\cdot H(t)$ satisfies the condition (C3) on $T_0$, by (ⅱ) of Theorem 5 in ^[30], we obtain that $\tilde u(t) = f(t)\cdot H(t)$ is (ⅰ)-differentiable on $T_0$, since $f(t) \cdot f'(t) < 0$ holds for $t\in T_0$. Moreover, we have

Thus $\tilde u'(t) + \delta (t)\tilde u(t) = \sigma (t)$. That is to say, $\tilde u(t)$ is a (ⅰ)-differentiable solution of the fuzzy differential Eq (1.1).

Theorem 3.2. Let $\sigma \in C[T, {X_F}]$ and $\delta :T \to {R_ - }$ be a continuous function. Suppose $u:{T_0} \to {X_F}$ is continuous and (ⅳ)-differentiable.

(1) Then there exists a ${u_0} \in {X_F}$ such that

as $t \to {b^ - }$.

(2) If the H-difference ${u_0} - \int_a^t { - \exp \left({\int_a^v {\delta (\tau)d\tau } } \right)\sigma (v)dv}$ exists on $T_0$, then the mapping: $\forall t\in T_0$,

satisfies

where $M, M' > 0$.

(3) If $\tilde u(t)$ satisfies the condition (C4), then $\tilde u(t)$ is the (ⅱ)-differentiable solution of Eq (1.1).

Proof. (1) For simplicity, let

for each $t\in T_0$. By (ⅵ) of Theorem 2.4, we obtain $G(t)$ is (ⅳ)-differentiable. Then, for any $s, t\in T_0$ with $t < s$, we can infer from Theorem 2.5 that there exists $M > 0$ such that

Since $\sigma$ and $\delta$ are continuous functions and by Theorem 2.10, we get $\exp \left({\int_a^t {\delta (\tau)d\tau } \sigma (t)} \right)$ is integrable on $T$, which together with (1) of Theorem 2.11, we have

where $M' > 0$. Then

From the proof of Theorem 3.1, we obtain that $w(s)$ converges to $u_0$ as $s \to {b^ - }$. That is to say, the expression (3.5) holds.

(2) If the H-difference ${u_0} - \int_a^t { - \exp \left({\int_a^v {\delta (\tau)d\tau } } \right)\sigma (v)dv} = {u_{0H}}(t)$ exists on $T_0$, then

for any $t\in T_0$. It is easy to see that the expression (3.8) tends to $\exp \left({ - \int_a^t {\delta (\tau)d\tau } } \right)\left({M{\rm{ + }}M'} \right)(b - t)$ as $s \to {b^ - }$. Thus, the inequality (3.7) holds.

Now we prove the uniqueness of $u_0$. Assume that there exists $u_1\in X_F$ such that the H-difference

exists on $T_0$ and inequality (3.7) holds if $u_0$ in (3.6) is replaced by $u_1$. Then, we have

for each $t\in T_0$. Therefore for each $D({u_0}, {u_1}) \le 2\left({M{\rm{ + }}M'} \right)(b - t)$. Since $2\left({M{\rm{ + }}M'} \right)(b - t)$ tends to $0$ as $t \to {b^ - }$, we get ${u_0} \equiv {u_1}$. This completes the proof.

(3) Next, we shall consider the differentiability of the fuzzy number-valued mapping $\tilde u(t)$ given by (3.6). Setting

By (2) of Theorem 2.11, it is easy to see that $H(t)$ is (ⅱ)-differentiable and $H'(t) = \exp \left({\int_a^t {\delta (\tau)d\tau } } \right)\sigma (t)$. Moreover, $f(t) \cdot f'(t) > 0$. From (ⅴ) of Theorem 5 in ^[30] and the assumption that $\tilde u(t) = f(t) \cdot H(t)$ satisfies the condition (C4) on $T_0$, we know that $\tilde u(t)$ is (ⅱ)-differentiable on $T_0$ and fulfills

which implies that $\tilde u'(t) + \delta (t)\tilde u(t) = \sigma (t)$, i.e., $\tilde u(t)$ is a (ⅱ)-differentiable solution of the fuzzy differential Eq (1.1).

4.   Ulam stability for Eqs (1.2) and (1.3)

In this section, with the similar proof method of Theorems 3.1 and 3.2, we can prove the Ulam stability for the linear fuzzy differential Eqs (1.2) and (1.3) in two cases: (1) $u$ is (ⅲ)-differentiable and $\delta (t) > 0$; (2) $u$ is (ⅳ)-differentiable and $\delta (t) < 0$.

Theorem 4.1. Let $\sigma \in C[T, {X_F}]$ and $\delta :T \to {R_ + }$ be a continuous function. Suppose $u:{T_0} \to {X_F}$ is continuous and (ⅲ)-differentiable, $exp \left({ - \int_a^t {\delta (\tau)d\tau } } \right)u(t)$ satisfies the condition (C1) on $T_0$. If the H-difference $\exp \left({ - \int_a^t {\delta (\tau)d\tau } } \right)u(t)- \int_a^t {\exp \left({ - \int_a^v {\delta (\tau)d\tau } } \right)\sigma (v)dv}$ exists for any $t \in {T_0}$, then

(1) there exists a ${u_0} \in {X_F}$ such that the mapping: $\forall t \in {T_0}$,

satisfies

where $M, M' > 0$.

(2) $\tilde u(t)$ is the (ⅰ)-differentiable solution of Eq (1.2).

Theorem 4.2. Let $\sigma \in C[T, {X_F}]$ and $\delta :T \to {R_ - }$ be a continuous function. Suppose $u:{T_0} \to {X_F}$ is continuous and (ⅳ)-differentiable, $\exp \left({ - \int_a^t {\delta (\tau)d\tau } } \right)u(t)$ satisfies the condition (C2) on $T_0$.

(1) Then there exists a ${u_0} \in {X_F}$ such that the mapping

as $t \to {b^ - }$.

(2) If the H-difference ${u_0} - \int_a^t { - \exp \left({ - \int_a^v {\delta (\tau)d\tau } } \right)\sigma (v)dv}$ exists on $T_0$, then ${u_0}$ is unique such that the mapping: $\forall t \in {T_0}$,

satisfies

where $M, M' > 0$.

(3) $\tilde u(t)$ is the (ⅱ)-differentiable solution of Eq (1.2).

Theorem 4.3. Let $\sigma \in C[T, {X_F}]$ and $\delta :T \to {R_ + }$ be a continuous function. Suppose $u:{T_0} \to {X_F}$ is continuous and (ⅲ)-differentiable, $\exp \left({ - \int_a^t {\delta (\tau)d\tau } } \right)u(t)$ satisfies the condition (C1) on $T_0$.

(1) Then there exists a ${u_0} \in {X_F}$ such that the mapping

as $t \to {b^ - }$.

(2) If the H-difference ${u_0} - \int_a^t {\exp \left({ - \int_a^v {\delta (\tau)d\tau } } \right)\sigma (v)dv}$ exists for each $t \in {T_0}$, then the mapping: $\forall t \in {T_0}$,

satisfies

where $M, M' > 0$.

(3) If $\tilde u(t)$ satisfies the condition (C3), then $\tilde u(t)$ is the (ⅰ)-differentiable solution of Eq (1.3).

Theorem 4.4. Let $\sigma \in C[T, {X_F}]$ and $\delta :T \to {R_ - }$ be a continuous function. Suppose $u:{T_0} \to {X_F}$ is continuous and (ⅳ)-differentiable, $\exp \left({ - \int_a^t {\delta (\tau)d\tau } } \right)u(t)$ satisfies the condition (C2) on $T_0$ and the H-difference $\exp \left({ - \int_a^t {\delta (\tau)d\tau } } \right)u(t) - \int_a^t { - \exp \left({ - \int_a^v {\delta (\tau)d\tau } } \right)\sigma (v)dv}$ exists for each $t\in T_0$.

(1) Then there exists a ${u_0} \in {X_F}$ such that the mapping: $\forall t\in T_0$,

satisfies

where $M, M' > 0$.

(2) If $\tilde u(t)$ satisfies the condition (C4), then $\tilde u(t)$ is the (ⅱ)-differentiable solution of Eq (1.3).

5.   Conclusion

The paper introduces some new definitions of differentiabilities for fuzzy number-valued mappings in Banach spaces and gives their important properties. Under suitable conditions, the paper shows the solutions of some fuzzy differential equations are stable in the sense of Ulam. By the techniques introduced in this paper, more results about Ulam stability theorems for linear fuzzy differential equations are prospective. They will play important roles in the applications of fuzzy differential equations.

Acknowledgments

The research work is supported by National Natural Science Foundation of China under Grant No. 11971343. The authors are grateful to the editor and referees for their valuable comments which led to the improvement of this paper.

Conflict of interest

The authors declare that there is no conflicts of interest in this paper.