On the stability for the fuzzy initial value problem

Abstract

In this paper, we present the Hyers–Ulam stability and Hyers–Ulam-Rassias stability (HU-stability and HUR-stability for short) for fuzzy initial value problem (FIVP) by using fixed point theorem. We improve and extend some known results on the stability for FDEs by dropping some assumptions. Some examples illustrate the theoretical results.

Keywords

HU stability HUR stability fuzzy differential equation uncertainly fixed point theory 34A12 45J05

1 Introduction

The problem of stability is the subject prominent in the studying of the differential equation (DE) and integrodifferential equation (IDE). The study of the stability of DE has developed by various concepts such as Lyapunov stability, asymptotic stability, HU-stability, and so on. Ulam (1940) introduced the stability of functional equation (for more detail see [25]), and Hyers [12] answered the question of Ulam in the case the Banach spaces in 1941. The HU-stability of linear differential equations was proved by Oblaza, who seems to be the first author to introduce this problem in [20]. Afterward, the HU-stability for DEs was discussed by the various authors (see Jung [13, 14], Zada et al. [28], Huang et al. [11], Sousa et al. [6, 7], Castro et al. [5], Eghbali et al. [9], Başcı et al. [2])

Very recently, by using the fixed point theorem, Shen and Wang [24] established the HU-stability of fuzzy differential equations (FDEs). Shen [22] investigated the HU-stability of FDEs in the linear case. Liu et al. [17] established the HU-stability for the FDEs with impulsive. In [1], Ahmadian et al. introduced the concept E_α-stability for IDE with fractional-order under uncertainly. Using the proposed new concept, the authors proved the existence and uniqueness of the solutions of the above problem. Phung et al. [21] discussed the HU stability and HURS stability for IDE under uncertainly via the Picard’s successive approximation method and the fixed point method. In [27], Vu et al. considered the HU-stability for the fuzzy integral equations with fractional-order by using Banach’s fixed point theorem. Besides, some results related to HU-stability for partial differential equations with uncertainty have also been introduced (see [18 , 23]).

Motivated by the above mentioned ideas, we will improve and extend these results by proving the stability for FIVP in the sense of HU and HUR. Moreover, we also provide some examples to illustrate the results of the paper.

The paper is structured as follows: In Section 2, we recall some notations, definitions, theorem, lemmas, which are used throughout the paper. In Section 3, we prove the HU-stability and HU-stability for FIVP by employing a fixed point technique. In Section 4, some examples illustrate the theoretical results.

2 Preliminaries

Definition 2.1. ([8]) Let X≠ ∅. The mapping d : X × X → [0, + ∞) is called the generalized metric on X, iff it satisfies the following properties: (H1) d (ρ₁, ρ₂) =0 iff ρ₁ = ρ₂; (H2) d (ρ₁, ρ₂) = d (ρ₂, ρ₁), ∀ρ₁, ρ₂ ∈ X; (H3) d (ρ₁, ρ₃) ≤ d (ρ₁, ρ₂) + d (ρ₂, ρ₃), ∀ρ₁, ρ₂, ρ₃ ∈ X.

Theorem 2.1. ([8]) Let (X, d) be a generalized complete metric space (GCM-space for short). Assume that ϒ : X → X is the strictly contractive operator with L < 1. If there is a positive integer l such that d (ϒ^l+1ρ, ϒ^lρ)< ∞ for some ρ ∈ X, the following phrases are true:

The sequence {ϒⁿρ} converges to fixed point ρ^* of X;

ρ^* is the unique the fixed point of ϒ in $\begin{matrix} X^{*} = {ρ_{0} \in X : d (ϒ^{l} ρ, ρ_{0}) < \infty}; \end{matrix}$

If ρ₀ ∈ X^*, then $\begin{matrix} d (ρ_{0}, ρ^{*}) \leq \frac{1}{1 - L} d (ϒ ρ_{0}, ρ_{0}) . \end{matrix}$

A fuzzy number (FN for short) is a fuzzy sets $ω : R \to [0, 1]$ satisfying the following four conditions:

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

ω is fuzzy convex, that is, ω (λρ₁ + (1 - λ) ρ₂) ≥ min {ω (ρ₁) , ω (ρ₂)} for any $ρ_{1}, ρ_{2} \in R$ , $ω \in R_{F}$ and λ ∈ [0, 1];

ω is an upper semi-continuous function;

$[ω]^{0} = \bar{{ρ \in R : ω (ρ) > 0}}$ is compact.

We denote by

R_{F}

the space of all FNs on

R

. The set

[ω]^{α} = [ω_{l}^{α}, ω_{r}^{α}]

is called the α-cut set of FN ω for any α ∈ (0, 1].

Let α ∈ [0, 1] and $λ \in R$ . The operators: addition "+" and multiplication "." of two FNs ω and $\hat{ω}$ are defined by

$[ω + \hat{ω}]^{α} = [ω]^{α} + [\hat{ω}]^{α}$ and $[λ ω] = λ [\hat{ω}]^{α}$ ,

$[ω \cdot \hat{ω}]^{α} = [min {ω_{l}^{α} {\hat{ω}}_{l}^{α}, ω_{l}^{α} {\hat{ω}}_{r}^{α}, ω_{r}^{α} {\hat{ω}}_{l}^{α}, ω_{r}^{α} {\hat{ω}}_{r}^{α}}, max {ω_{l}^{α} {\hat{ω}}_{l}^{α}, ω_{l}^{α} {\hat{ω}}_{r}^{α}, ω_{r}^{α} {\hat{ω}}_{l}^{α}, ω_{r}^{α} {\hat{ω}}_{r}^{α}}]$ ,

Definition 2.2. ([4]) Let $ω, \hat{ω} \in R_{F}$ . If there exists $w \in R_{F}$ such that $ω ⊖ \hat{ω} = w$ , then w is called the Hukuhara difference (H-difference for short) of ω and $\hat{ω}$ , denoted by $ω ⊖ \hat{ω}$ , In general, $ω ⊖ \hat{ω} \neq ω + (- 1) \hat{ω}$ .

Definition 2.3. ([4]) Let $ω, \hat{ω} \in R_{F}$ . The generalized H-difference (gH-difference for short) of ω and $\hat{ω}$ is $w \in R_{F}$ , if w exists, such that $\begin{matrix} ω \underset{gH}{⊖} \hat{ω} = w \Leftrightarrow {\begin{matrix} (i) ω = \hat{ω} + w, \\ or (ii) \hat{ω} = ω + (- 1) w . \end{matrix} \end{matrix}$ Let $ω, \hat{ω} \in R_{F}$ . The distance between of ω and $\hat{ω}$ is defined by $\begin{matrix} D (ω, \hat{ω}) = sup_{α \in [0, 1]} d_{H} ([ω]^{α}, [\hat{ω}]^{α}), \end{matrix}$ where $d_{H} ([ω]^{α}, [\hat{ω}]^{α}) = sup_{α \in [0, 1]} max {| ω_{l}^{α} - {\hat{ω}}_{l}^{α} |, | ω_{r}^{α} - {\hat{ω}}_{r}^{α} |}$ is the Hausdorff-Pompeiu distance between [ω] ^α and $[\hat{ω}]^{α}$ .

Lemma 2.1. Let $Ω_{1}, Ω_{2}, Ω_{3}, Ω_{4} \in R_{F}$ and $λ \in R$ . We have:

D (Ω₁ + Ω₃, Ω₂ + Ω₃) = D (Ω₁, Ω₂);

D (λΩ₁, λΩ₂) = λD (Ω₁, Ω₂);

D (Ω₁ + Ω₂, Ω₃ + Ω₄) ≤ D (Ω₁, Ω₂) + D (Ω₃, Ω₄);

$(R_{F}, D)$ is a complete metric space.

Definition 2.4. ([10]) Let $x_{0} \in (a, b) \subset R_{F}$ . We say that $ω : (a, b) \to R_{F}$ is a continuous fuzzy function at x₀ if for every ε > 0, ∃δ (ε) >0 such that |x - x₀| < δ (ε), ∀x ∈ (a, b) implies that D (ω (x) , ω (x₀)) < ε.

A fuzzy function $ω : (a, b) \to R_{F}$ is continuous on (a, b), if ω is a continuous fuzzy function at every point in its domain.

Let $ω \in R_{F}$ and α ∈ [0, 1]. We define the diameter of $ω \in R_{F}$ as $\begin{matrix} diam ([ω (x_{0})]^{α}) = ω_{r}^{α} (x_{0}) - ω_{l}^{α} (x_{0}), \forall x_{0} \in (a, b) . \end{matrix}$

Definition 2.5. ([26]) A mapping $ω : (a, b) \to R_{F}$ is called d-decreasing (d-increasing) on (a, b) if for every α ∈ [0, 1] the mapping t ↦ diam [ω (x₀)] ^α is non-increasing (non-decreasing) on (a, b). If f is d-decreasing or d-increasing on (a, b), then we say that ω is d-monotone on (a, b).

Definition 2.6. ([4]) Let $ω : (a, b) \to R_{F}$ . The generalized Hukuhara derivative (gH-derivative for short) of ω at x₀ ∈ (a, b) is defined by $D_{H}^{g} ω (x_{0}) = lim_{h \to 0} \frac{ω (x_{0} + h) ⊖_{gH} ω (x_{0})}{h},$ (2.1) for ω₀, x₀ + h ∈ (a, b), h > 0 enough small.

If $D_{H}^{g} ω (x_{0}) \in R_{F}$ satisfies (2.1), we say that ω is a generalized differentiable at x₀ (gH-differentiable for short).

Theorem 2.2. [3, 15] Let the d-monotone fuzzy function $ω : (a, b) \to R_{F}$ be differentiable on (a, b) and assume that $D_{H}^{g} ω$ is integrable over (a, b). Then, we have $\begin{matrix} ω (x) \underset{gH}{⊖} ω (a) = \int_{a}^{x} D_{H}^{g} ω (s) ds, \end{matrix}$ for each x ∈ (a, b).

For convenience, we denote $T = [t_{0}, t_{0} + τ]$ , τ > 0; $C (T, R_{F})$ is the set of all fuzzy continuous functions on $T$ . Consider the following fuzzy initial value problem (FIVP for short)

$D_{H}^{g} ω (t) = f (t, ω (t)), ω (t_{0}) = ω_{0} \in R_{F} .$ (2.2) where $D_{H}^{g}$ is given by Definition 2 and $f \in C (T \times R_{F}, R_{F})$ .

Lemma 2.2. ([15, 16]) Let $f \in C (T \times R_{F}, R_{F})$ and $ω \in C (T, R_{F})$ . The d-monotone fuzzy function ω is called a solution of Eq.(2.2) if and only if ω satisfies the fuzzy integral equation:

$ω (t) \underset{gH}{⊖} ω_{0} = \int_{t_{0}}^{t} f (s, ω (s)) ds, \forall t \in T .$ (2.3)

Remark 2.1. We observe that

If $ω \in C (T, R_{F})$ and ω is d-increasing on $T$ , then Eq.(2.3) can be written as $\begin{matrix} ω (t) = ω_{0} + \int_{t_{0}}^{t} f (s, ω (s)) ds . \end{matrix}$

If $ω \in C (T, R_{F})$ and ω is d-decreasing on $T$ , then Eq.(2.3) can be written as $\begin{matrix} ω (t) = ω_{0} ⊖ (- 1) \int_{t_{0}}^{t} f (s, ω (s)) ds . \end{matrix}$

In next section, we will consider the HU-stability and HUR-stability for the problem (2.2).

3 Main results

Firstly, we will present the necessary result in our further discussion.

In what follows, we set $X : = C (T, R_{F})$ .

Lemma 3.1. Let M > 0 and $ψ \in C (T, [0, + \infty))$ . The mapping $H : X \times X \to [0, + \infty) \cup {+ \infty}$ is defined by $\begin{matrix} H (ω_{1}, ω_{2}) : = inf {C \in [0, + \infty) \cup {+ \infty} : \\ D (ω_{1} (t), ω_{2} (t)) e^{- M (t - t_{0})} \leq C ψ (t), \forall t \in T} . \end{matrix}$ (3.4) Then, $(X, H)$ is a GCM space.

Proof. We easily check that the properties (H1) and (H2) are held. Hence, we will prove that the property (H3) also holds. Assume that the inequality H (ω₁, ω₂) < H (ω₁, ω₃) + H (ω₃, ω₂) does not hold for any $ω_{1}, ω_{2}, ω_{3} \in R_{F}$ . Therefore, by (3.4), there exists a $t_{1} \in T$ such that $\begin{matrix} D (ω_{1} (t_{1}), ω_{2} (t_{1})) e^{- M (t_{1} - t_{0})} \\ > (H (ω_{1}, ω_{3}) + H (ω_{3}, ω_{2})) ψ (t_{1}) \\ = H (ω_{1}, ω_{3}) ψ (t_{1}) + H (ω_{3}, ω_{2}) ψ (t_{1}) \\ \geq D (ω_{1} (t_{1}), ω_{3} (t_{1})) e^{- M (t_{1} - t_{0})} \\ + D (ω_{3} (t_{1}), ω_{2} (t_{1})) e^{- M (t_{1} - t_{0})}, \end{matrix}$ which leads to a contraction. So, H is a generalized metric on $X$ . Let {ω_n} be a Cauchy sequence on $(X, H)$ . Then, for any ε > 0, ∃N = N (ε) >0 such that H (ω_n, ω_m) ≤ ε for all n, m ≥ N (ε). Thus, it follows from the definition (3.4) that $D (ω_{n}, ω_{m}) e^{- M (t - t_{0})} \leq ε ψ (t),$ (3.5) for all $t \in T$ and n, m ≥ N (ε). For any fixed $t \in T$ , by the inequality (3.5), we infer that {ω_n (t)} is a Cauchy sequence in $R_{F}$ . Moreover, $(X, H)$ is complete; hence there exists a fuzzy function $ω : T \to R_{F}$ such that {ω_n (t)} converges to ω (t) for any $t \in T$ . Next, we will prove that ω belongs to $X$ . In the inequality (3.5), letting m→ ∞, we get for arbitrary ε > 0, ∃N (ε) >0 such that $D (ω, ω_{n}) e^{- Mt} \leq ε ψ (t), \forall n \geq N (ε), t \in T .$ (3.6) Since $ψ \in C (T, [0, + \infty))$ , there exists a $\hat{C} > 0$ such that $ψ (t) \leq \hat{C}$ for any $t \in T$ . That is, for arbitrary ε > 0, ∃N (ε) >0 such that $D (ω, ω_{n}) e^{- M (t - t_{0})} \leq ε \hat{C}, \forall n \geq N (ε), t \in T .$ (3.7) This leads to {ω_n} converges uniformly to ω in $(X, H)$ . Hence, we have ω belongs to $X$ . Combining the definition of H and the inequality (3.5), we infer that for any ε > 0, ∃N (ε) >0 such that H (ω, ω_n) < ε, ∀n > N (ε), that is, the sequences {ω_n} converges to ω in $(X, H)$ . This completes the proof.

Now, we will discuss HU stability and HUR stability for the problem (2.2) via the fixed point theorem. Note that in the proofs, we use less assumption than the proofs in Shen et al.[24].

3.1 HU-stability for FIVP (2.2)

Theorem 3.1. Assume that $f \in C (T \times R_{F}, R_{F})$ and it satisfies the conditions as follows: there exists a constant L > 0 such that $D (f (t, ω_{1}), f (t, ω_{2})) \leq LD (ω_{1}, ω_{2})$ (3.8) for each $ω_{1}, ω_{2} \in R_{F}$ and any $t \in T$ . If a continuously d-increasing fuzzy function $ω : T \to R_{F}$ satisfies $D (D_{H}^{g} ω (t), f (t, ω (t))) \leq ε,$ (3.9) for each $ω \in R_{F}$ and some ε > 0, then there exists a unique solution $\tilde{ω} : T \to R_{F}$ of Eq.(2.2) satisfying $\tilde{ω} (t) = ω_{0} + \int_{t_{0}}^{t} f (s, \tilde{ω} (s)) ds$ (3.10) and $D (ω (t), \tilde{ω} (t)) \leq (1 + L) τ ε,$ (3.11) for any $t \in T$ .

Proof. Consider the set as Lemma 3 and the mapping $\hat{H} : X \times X \to [0, + \infty) \cup {+ \infty}$ is defined by $\begin{matrix} \hat{H} (ω_{1}, ω_{2}) : = inf {K \in [0, + \infty) \cup {+ \infty} : \\ D (ω_{1} (t), ω_{2} (t)) e^{- (1 + L) (t - t_{0})} \leq \hat{K}, \forall t \in T} . \end{matrix}$ (3.12) In the view of Lemma 3, then $(X, \hat{H})$ is a GCM space. Now, we define the operator $Λ : X \to X$ by $Λ ω (t) = ω_{0} + \int_{t_{0}}^{t} f (s, ω (s)) ds,$ (3.13) for $ω \in X$ and any $t \in T$ . Since the function f is continuous on $T$ , it leads to the operator Λω is a continuously d-increasing on $T$ ; hence, $Λ ω \in X$ , and thus we infer that for each $z_{0} \in X$ and any $t \in T$ , we have $\begin{matrix} D ((Λ z_{0}) (t), z_{0} (t)) e^{- (1 + L) (t - t_{0})} < \infty, \end{matrix}$ which means $\hat{H} (Λ z_{0}, z_{0}) < \infty$ for each $z_{0} \in X$ . Similarly, we also have $\begin{matrix} D (z_{0} (t), z (t)) e^{- (1 + L) (t - t_{0})} < \infty, \end{matrix}$ for each $z \in X$ and any $t \in T$ , which means $\hat{H} (z_{0}, z) < \infty$ for each $z \in X$ , i.e., $\begin{matrix} {z \in X : \hat{H} (z_{0}, z) < \infty} = X . \end{matrix}$ Now we shall prove that the operator Λ is strictly contrative on $(X, \hat{H})$ . For all $ω_{1}, ω_{2} \in X$ and by (3.8), we have $\begin{matrix} D ((Λ ω_{1}) (t), (Λ ω_{2}) (t)) \\ \leq \int_{t_{0}}^{t} D (f (s, ω_{1} (s)), f (s, ω_{2} (s))) ds \\ \leq L \int_{t_{0}}^{t} D (ω_{1} (s), ω_{2} (s)) e^{- (1 + L) (s - t_{0})} e^{(1 + L) (s - t_{0})} ds \\ \leq LH (ω_{1}, ω_{2}) \int_{t_{0}}^{t} e^{(1 + L) (s - t_{0})} ds \\ \leq \frac{L}{1 + L} H (ω_{1}, ω_{2}) e^{(1 + L) (t - t_{0})}, \forall t \in T . \end{matrix}$

From the inequality above and for all $ω_{1}, ω_{2} \in X$ and $t \in T$ , we have $\begin{matrix} D ((Λ ω_{1}) (t), (Λ ω_{2}) (t)) e^{- (1 + L) (t - t_{0})} \leq \frac{L}{1 + L} H (ω_{1}, ω_{2}) . \end{matrix}$

Therefore, for all $ω_{1}, ω_{2} \in X$ , we obtain $\hat{H} (Λ ω_{1}, Λ ω_{2}) \leq \frac{L}{1 + L} H (ω_{1}, ω_{2}),$ (3.14) which means that Λ is strictly contrative on $(X, H)$ . We note that the Hukuhara difference ω (t) ⊖ ω (t₀) exists and ω is d-increasing on $T$ , then by the inequality (3.9) and Remark 2, we have

$\begin{matrix} D (ω (t), (Λ ω) (t)) \\ = D (ω (t), ω_{0} + \int_{t_{0}}^{t} f (s, ω (s)) ds) \\ = D (ω (t) ⊖ ω_{0}, \int_{t_{0}}^{t} f (s, ω (s)) ds) \\ = D (\int_{t_{0}}^{t} D_{H}^{g} ω (s) ds, \int_{t_{0}}^{t} f (s, ω (s)) ds) \\ \leq \int_{t_{0}}^{t} D (D_{H}^{g} ω (s), f (s, ω (s))) ds \leq (t - t_{0}) ε, \forall t \in T . \end{matrix}$ Multiplying the inequality above with e^{-(1+L)(t-t₀)} we get $\begin{matrix} D (ω (t), (Λ ω) (t)) e^{- (1 + L) (t - t_{0})} \\ \leq (t - t_{0}) ε e^{- (1 + L) (t - t_{0})}, \forall t \in T . \end{matrix}$ By the definition of the metric $\hat{H}$ , one have $\begin{matrix} \hat{H} (ω, Λ ω) \leq ε τ e^{- (1 + L) (t - t_{0})}, \forall t \in T . \end{matrix}$

According to (iii) of Theorem 2, there exists a unique solution $\tilde{ω} : T \times R_{F}$ of Eq.(2.2) satisfying $\begin{matrix} \hat{H} (ω, \tilde{ω}) & \leq \frac{1}{1 - \frac{L}{1 + L}} \hat{H} (Λ ω, ω) \\ \leq (1 + L) ε τ e^{- (1 + L) (t - t_{0})}, \forall t \in T . \end{matrix}$ In view from the definition of $\hat{H} (ω, \tilde{ω})$ , then we have $\begin{matrix} D (ω (t), \tilde{ω} (t)) \leq (1 + L) ε τ, \forall t \in T . \end{matrix}$ The proof is completed.

Corollary 3.1. Assume that $f \in C (T \times R_{F}, R_{F})$ and it satisfies the conditions as follows: there exists a constant L > 0 such that $D (f (t, ω_{1}), f (t, ω_{2})) \leq LD (ω_{1}, ω_{2})$ (3.15) for each $ω_{1}, ω_{2} \in R_{F}$ and any $t \in T$ . If a continuously d-nonincreasing fuzzy function $ω : T \to R_{F}$ satisfies $D (D_{H}^{g} ω (t), f (t, ω (t))) \leq ε,$ (3.16) for each $ω \in R_{F}$ and some ε > 0, then there exists a unique solution $\tilde{ω} : T \to R_{F}$ of Eq.(2.2) satisfying $\tilde{ω} (t) = ω_{0} ⊖ (- 1) \int_{t_{0}}^{t} f (s, \tilde{ω} (s)) ds$ (3.17) and $D (ω (t), \tilde{ω} (t)) \leq (1 + L) τ ε,$ (3.18) for any $t \in T$ .

Remark 3.1. Note that in Theorem 3.1 and Corollary (3.1), we do not use the assumption Lτ < 1, while it is required in Corollary 3.3 and 3.5 in the paper of Shen et al. [24].

3.2 HUR-stability for FIVP (2.2)

Theorem 3.2. Assume that $f \in C (T \times R_{F}, R_{F})$ and it satisfies the following assumptions: there exists a constant L > 0 $D (f (t, ω_{1}), f (t, ω_{2})) \leq LD (ω_{1}, ω_{2})$ (3.19) for each $ω_{1}, ω_{2} \in R_{F}$ and any $t \in T$ . If a continuously d-nonincreasing function $ω : T \to R_{F}$ satisfies $D (D_{H}^{g} ω (t), f (t, ω (t))) \leq γ (t),$ (3.20) for each $ω \in R_{F}$ , where the mapping $γ : T \to [0, + \infty)$ is continuous and nondecreasing which satisfies $\int_{t_{0}}^{t} γ (s) ds \leq C γ (t)$ (3.21) with the constant C > 0, then there exists a unique solution $\tilde{ω} : T \to R_{F}$ of Eq.(2.2) satisfies $\tilde{ω} (t) = ω_{0} ⊖ (- 1) \int_{t_{0}}^{t} f (s, \tilde{ω} (s)) ds$ (3.22) and $D (ω (t), \tilde{ω} (t)) \leq (1 + L) C γ (t),$ (3.23) for any $t \in T$ .

Proof. We consider the set $X$ as in Lemma 3 and introduce the metric $\tilde{H} : X \times X \to [0, + \infty) \cup {+ \infty}$ with $\begin{matrix} \tilde{H} (ω_{1}, ω_{2}) : = inf {K \in [0, + \infty) \cup {+ \infty} : \\ D (ω_{1} (t), ω (t)) e^{- (1 + L) (t - t_{0})} \leq K φ (t), \forall t \in T} . \end{matrix}$ (3.24) Based on Lemma 3, we infer that $(X, \tilde{H})$ is a GCM space. The operator $ϒ : X \to X$ is defined by $ϒ ω (t) = ω_{0} ⊖ (- 1) \int_{t_{0}}^{t} f (s, ω (s)) ds,$ (3.25) for all $ω \in X$ and any $t \in T$ . Similarly as the proof Theorem 3.1, the operator $ϒ ω \in X$ and we can show that $\tilde{H} (ϒ z_{0}, z_{0}) < \infty$ for any $z_{0} \in X$ and ${z \in X : \tilde{H} (z_{0}, z) < \infty} : = X$ . Now we confirm that ϒ is also a strictly contractive on $(X, \tilde{H})$ . Using assumption γ is a nondecreasing continuous function and for any $t \in X$ , we have $\begin{matrix} \int_{t_{0}}^{t} γ (s) e^{(1 + L) (s - t_{0})} ds \\ \leq \frac{1}{1 + L} (γ (t) e^{(1 + L) (t - t_{0})} - \int_{t_{0}}^{t} γ^{'} (s) e^{(1 + L) (s - t_{0})} ds) \\ \leq \frac{γ (t)}{1 + L} e^{(1 + L) (t - t_{0})} \end{matrix}$ (3.26) For any $ω_{1}, ω_{2} \in X$ and let the constant $\tilde{C} \in [0, + \infty) \cup {+ \infty}$ such that $\tilde{H} (ω_{1}, ω_{2}) < \tilde{C}$ , that is $\begin{matrix} D (ω_{1} (t), ω_{2} (t)) e^{- (1 + L) (t - t_{0})} \leq \tilde{C} γ (t), \end{matrix}$ for any $t \in T$ . Then, we obtain $\begin{matrix} D ((ϒ ω_{1}) (t), (ϒ ω_{2}) (t)) \\ = D (ω_{0} ⊖ (- 1) \int_{t_{0}}^{t} f (s, ω_{1} (s)) ds, ω_{0} ⊖ (- 1) \\ \int_{t_{0}}^{t} f (s, ω_{2} (s)) ds) \\ \leq \int_{t_{0}}^{t} D (f (s, ω_{1} (s)), f (s, ω_{2} (s))) ds \\ \leq L \int_{t_{0}}^{t} D (ω_{1} (s), ω_{2} (s)) e^{- (1 + L) (s - t_{0})} e^{(1 + L) (s - t_{0})} ds \\ \leq LH (ω_{1}, ω_{2}) \int_{t_{0}}^{t} γ (s) e^{(1 + L) (s - t_{0})} ds . \end{matrix}$ Combining the inequality above with estimate (3.26), we obtain $\begin{matrix} D ((ϒ ω_{1}) (t), (ϒ ω_{2}) (t)) \\ \leq \frac{L \tilde{C}}{1 + L} γ (t) e^{(1 + L) (t - t_{0})}, \forall t \in T . \end{matrix}$ Thus, for any $ω_{1}, ω_{2} \in X$ and any $t \in T$ . Then we have $\begin{matrix} D ((ϒ ω_{1}) (t), (ϒ ω_{2}) (t)) e^{- (1 + L) (t - t_{0})} \leq \frac{L \tilde{C}}{1 + L} φ (t) \end{matrix}$ By the definition of $\tilde{H}$ , we obtain $\begin{matrix} \tilde{H} (ϒ ω_{1}, ϒ ω_{2}) \leq \frac{L}{1 + L} \tilde{H} (ω_{1}, ω_{2}), \end{matrix}$ for any $ω_{1}, ω_{2} \in X$ . Since $0 < \frac{L}{1 + L} < 1$ with L > 0, we infer that ϒ is a strictly contractive on $(X, \tilde{H})$ . Moreover, in case l = 1 and $X^{*} = X$ , we can see that all conditions of Theorem 2 are holds. We also note that the Hukuhara difference ω (t₀) ⊖ ω (t) exists, since u is d-nondecreasing on $T$ . Therefore, it follows from Remark 2, and the inequalities (3.20) and (3.20), we have $\begin{matrix} D (ω (t), ω (t_{0}) ⊖ (- 1) \int_{t_{0}}^{t} f (s, ω (s)) ds) \\ = D (ω (t_{0}) ⊖ ω (t), (- 1) \int_{t_{0}}^{t} f (s, ω (s)) ds) \\ = D ((- 1) \int_{t_{0}}^{t} D_{H}^{g} ω (s) ds, (- 1) \int_{t_{0}}^{t} f (s, ω (s)) ds) \\ \leq \int_{t_{0}}^{t} D (D_{H}^{g} ω (s), f (s, ω (s))) ds \leq \int_{t_{0}}^{t} φ (s) ds \\ \leq | \int_{t_{0}}^{t} γ (s) ds | \leq K γ (t), \forall t \in T . \end{matrix}$ (3.27) Multiplying this inequality with e^{-(1+L)(t-t₀)}, we get $\begin{matrix} D (ω (t), (ϒ ω) (t)) e^{- (1 + L) (t - t_{0})} \\ \leq K γ (t) e^{- (1 + L) (t - t_{0})}, \forall t \in T, \end{matrix}$ which means that $\begin{matrix} \tilde{H} (ω, ϒ ω) \leq K γ (t) e^{- (1 + L) (t - t_{0})}, \forall t \in T . \end{matrix}$ Finally, from (iii) of Theorem 2 and the inequality above, we infer that $\begin{matrix} \tilde{H} (ω, \tilde{ω}) & \leq \frac{1}{1 - \frac{L}{1 + L}} \tilde{H} (Λ ω, ω) \\ \leq (1 + L) K γ (t) e^{- (1 + L) (t - t_{0})}, \forall t \in T, \end{matrix}$ where $\tilde{ω} : T \to R_{F}$ is unique solution of the problem (2.2) in view of Theorem 2. By the definition of $\tilde{H}$ , we have $\begin{matrix} D (ω (t), \tilde{ω} (t)) e^{- (1 + L) (t - t_{0})} \\ \leq (1 + L) K γ (t) e^{- (1 + L) (t - t_{0})}, \forall t \in T . \end{matrix}$ Hence, we have $\begin{matrix} D (ω (t), \tilde{ω} (t)) \leq (1 + L) K γ (t), \forall t \in T . \end{matrix}$ The proof is complete.

Corollary 3.2. Assume that $f \in C (T \times R_{F}, R_{F})$ and it satisfies the following assumptions: there exists a constant L > 0 $D (f (t, ω_{1}), f (t, ω_{2})) \leq LD (ω_{1}, ω_{2})$ (3.28) for each $ω_{1}, ω_{2} \in R_{F}$ and any $t \in T$ . If a continuously d-increasing function $ω : T \to R_{F}$ satisfies $D (D_{H}^{g} ω (t), f (t, ω (t))) \leq γ (t),$ (3.29) for each $ω \in R_{F}$ , where the mapping $γ : T \to [0, + \infty)$ is continuous and nondecreasing which satisfies $\int_{t_{0}}^{t} γ (s) ds \leq C γ (t)$ (3.30) with the constant C > 0, then there exists a unique solution $\tilde{ω} : T \to R_{F}$ of Eq.(2.2) satisfies $\tilde{ω} (t) = ω_{0} + \int_{t_{0}}^{t} f (s, \tilde{ω} (s)) ds$ (3.31) and $D (ω (t), \tilde{ω} (t)) \leq (1 + L) C γ (t),$ (3.32) for any $t \in T$ .

Remark 3.2. Note that in Theorem 3.2 and Corollary (3.2), we do not use the assumption Lτ < 1, while it is required in Theorem 3.2 and Theorem 3.4 in the paper of Shen et al. [24].

4 Examples

Example 4.1. Let $T = [0, 3 / L]$ , L > 0 and let $A : T \to [0, + \infty)$ be a continuous function such that |A (t) | ≤ L, and let $B : T \to R_{F}$ be continuous on $T$ . Let us consider the FIVP $D_{H}^{g} ω (t) = A (t) ω (t) + B (t), ω (0) = ω_{0} \in R_{F} .$ (4.33) Assume that a continuously d-increasing function $ω : T \to R_{F}$ satisfies $\begin{matrix} D (D_{H}^{g} ω (t), A (t) ω (t) + B (t)) \leq ε, \forall t \in T . \end{matrix}$ If we set f (t, ω (t)) = A (t) ω (t) + B (t) for any $t \in T$ , then it is easy to see that f satisfies Lipschitz condition, i.e.,

$\begin{matrix} D (f (t, ω_{1} (t)), f (t, ω_{2} (t))) \\ = D (A (t) ω_{1} (t) + B (t), A (t) ω_{2} (t) + B (t)) \\ = D (A (t) ω_{1} (t), A (t) ω_{2} (t)) \\ = | A (t) | D (ω_{1} (t), ω_{2} (t)) \leq LD (ω_{1} (t), ω_{2} (t)), \forall t \in T . \end{matrix}$

All the requirements of Theorem 3.1 are fulfilled, and hence there exists a unique continuous fuzzy function $\tilde{ω} : T \to R_{F}$ of the problem (4.33) satisfying $\begin{matrix} \tilde{ω} (t) = ω_{0} + \int_{0}^{t} f (s, \tilde{ω} (s)) ds \end{matrix}$ and $D (ω (t), \tilde{ω} (t)) \leq (1 + L) \frac{3}{2 L} ε,$ (4.34) for any $t \in T$ , which implies the problem (4.33) is HU stable on $T$ . Obverse that $\begin{matrix} T = [t_{0} - τ, t_{0} + τ] = [0, \frac{3}{L}] . \end{matrix}$ Therefore, we imply $t_{0} = τ = \frac{3}{2 L}$ . Since $L τ = L \frac{3}{2 L} = \frac{3}{2} > 1$ , the problem 4.33 is not HU stable on $T$ in the view of Corollary 3.3 in Shen et al. [24].

Example 4.2. Let $T = [0, τ]$ and let A, B be as in Example 4. Consider the FIVP (4.33) on $T$ . In Example 4, we have proved that the right side of the problem (4.33) satisfies the condition (3.19) with L > 0. Now, assume that a continuously d-nondecreasing fuzzy function $u : T \to R_{F}$ satisfies $\begin{matrix} D (D_{H}^{g} ω (t), A (t) ω (t) + B (t)) \leq e^{L - 1} t, \forall t \in T . \end{matrix}$ If we define the function $γ : T \to [0, + \infty)$ by γ (t) = e^L-1t, L > 1, then we have the following estimate $\begin{matrix} \int_{0}^{t} γ (s) ds & \leq \frac{1}{L - 1} e^{(L - 1) t} = \frac{1}{L - 1} γ (t), \forall t \in T, \end{matrix}$ that is, the condition (3.21) in Theorem 3.2 satisfied with $K = \frac{1}{L - 1}$ .

It is easy to see that all the requirements of Theorem 3.2 are fulfilled, and hence there exists a unique continuous fuzzy function $\tilde{ω} : T \to R_{F}$ of the problem (4.33) satisfying $\begin{matrix} \tilde{ω} (t) = ω_{0} ⊖ (- 1) \int_{0}^{t} f (s, \tilde{ω} (s)) ds \end{matrix}$ and

$D (ω (t), \tilde{ω} (t)) \leq \frac{1 + L}{L - 1} e^{(L - 1) t},$ (4.35) for any $t \in T$ , which implies the problem (4.33) is HUR stable on $T$ .

Note that the problem 4.33 is not HUR stable on $T$ in the view of Theorem 3.2 in Shen et al. [24] since $\begin{matrix} KL = \frac{L}{L - 1} > 1 . \end{matrix}$

Example 4.3. Let $T = [0, 1]$ and r > 0. We consider the FIVP of form $D_{H}^{g} ω (t) = r ω (t) (1 \underset{gH}{⊖} ω (t)), ω (0) = ω_{0} \in R_{F} .$ (4.36) In this example, we must assume that the generalized Hukuhara difference $1 \underset{gH}{⊖} ω (t)$ exists for any t ∈ T, i.e., $\begin{matrix} {\begin{matrix} 0 = diam ([1]^{α}) \leq diam ([ω (t)]^{α}), \forall α \in [0, 1], \\ 1 - ω_{r}^{α} (t) is increasing with respect to α, \\ 1 - ω_{l}^{α} (t) is descreasing with respect to α, \end{matrix} \end{matrix}$ and note that we only consider the solutions belong to the interval [χ_{0}, χ_{1}], that is, $ω (t) \in R_{F}$ such that $0 < ω_{l}^{α} (t) < ω_{r}^{α} (t) < 1$ for any α ∈ [0, 1] and $t \in T$ .

Set $f (t, ω) : = r ω (1 \underset{gH}{⊖} ω), \forall t \in T$ . It is easy to check that f is continuous on $T$ . Next, we assume that a continuously d-increasing fuzzy function $ω : T \to R_{F}$ satisfies $\begin{matrix} D (D_{H}^{g} ω (t), f (t, ω)) \leq ε, \forall t \in T . \end{matrix}$ For $ω, \hat{ω} \in R_{F}$ we have $\begin{matrix} min {ω_{l}^{α} (1 - ω_{l}^{α}), ω_{r}^{α} (1 - ω_{l}^{α}), \\ ω_{l}^{α} (1 - ω_{r}^{α}), ω_{r}^{α} (1 - ω_{r}^{α})} = ω_{l}^{α} (1 - ω_{r}^{α}), \end{matrix}$ (4.37) $\begin{matrix} max {ω_{l}^{α} (1 - ω_{l}^{α}), ω_{r}^{α} (1 - ω_{l}^{α}), \\ ω_{l}^{α} (1 - ω_{r}^{α}), ω_{r}^{α} (1 - ω_{r}^{α})} = ω_{r}^{α} (1 - ω_{l}^{α}) \end{matrix}$ (4.38) and

$\begin{matrix} | {\hat{ω}}_{l}^{α} {\hat{ω}}_{r}^{α} - ω_{l}^{α} ω_{r}^{α} | & = | {\hat{ω}}_{l}^{α} ({\hat{ω}}_{r}^{α} - ω_{r}^{α}) + (ω_{l}^{α} - {\hat{ω}}^{α}) ω_{r}^{α} | \\ \leq (| {\hat{ω}}_{l}^{α} | + | ω_{r}^{α} |) D (ω, \hat{ω}) . \end{matrix}$ (4.39)

For $ω, \hat{ω} \in R_{F}$ and the estimations (4.37)-(4.39), we have the following estimate: $\begin{matrix} D (f (t, ω), f (t, \hat{ω})) \\ = D (r ω (1 - ω)), r \hat{ω} (1 - \hat{ω})) \\ = sup_{α \in [0, 1]} d_{H} ([r ω (1 - ω)]^{α}, [r \hat{ω} (1 - \hat{ω})]^{α}) \\ = r sup_{α \in [0, 1]} max {| ω_{l}^{α} (1 - ω_{r}^{α}) - {\hat{ω}}_{l}^{α} (1 - {\hat{ω}}_{r}^{α}) |, \\ | ω_{r}^{α} (1 - ω_{l}^{α}) - {\hat{ω}}_{r}^{α} (1 - {\hat{ω}}_{l}^{α}) |} \\ \leq r sup_{α \in [0, 1]} max {| ω_{l}^{α} - {\hat{ω}}_{l}^{α} | + | {\hat{ω}}_{l}^{α} {\hat{ω}}_{r}^{α} - ω_{l}^{α} ω_{r}^{α} |, \\ | ω_{r}^{α} - {\hat{ω}}_{r}^{α} | + | {\hat{ω}}_{l}^{α} {\hat{ω}}_{r}^{α} - ω_{l}^{α} ω_{r}^{α} |} \\ \leq r (1 + sup_{α \in [0, 1]} (| {\hat{ω}}_{l}^{α} | + | ω_{r}^{α} |)) D (ω, \hat{ω}) \\ \leq r (1 + D (ω, 0) + D (\hat{ω}, 0)) D (ω, \hat{ω}) . \end{matrix}$ According to Theorem 3.1, there exists a unique continuous fuzzy function $\tilde{ω} : T \to R_{F}$ of the problem (4.36) satisfying $\begin{matrix} \tilde{ω} (t) = ω_{0} + \int_{0}^{t} f (s, \tilde{ω} (s)) ds \end{matrix}$ and

$D (ω (t), \tilde{ω} (t)) \leq (1 + r (1 + D (ω, 0) + D (\hat{ω}, 0))) ε,$ (4.40) for any $t \in T$ , which implies the problem (4.36) is HU stable on $T$ .

5 Conclusion

In this paper, we study the HU-stability and the HUR-stability for the FIVP under generalized Hukuhara differentiability with some suitable assumptions. It is a tool useful in studying the problems relevant to the non-linear dynamic system in a fuzzy environment (such as numerical analysis, optimization, etc.), where to determining the exact solution is complicated. Indeed, if we study the HU-stable (or the HUR-stable) system, then one does not need to find the exact solution. All the requirements are to find a fuzzy function which satisfies a suitable approximation inequality. That says, if the system is HU-stable or HUR-stable then there always exists a close exact solution. So, this study provides a method to solving FDEs.

References

Ahmadian

, Salahshour

, Senu

and Ismail

, Some new results on the stability of fractional integro-differential equations under uncertainty. In Rozaida Ghazali, Mustafa Mat Deris, Nazri Mohd Nawi, and Jemal H. Abawajy, editors, Recent Advances on Soft Computing and Data Mining, pp. 53–63, Cham, 2018. Springer International Publishing.

Başcı

, Mısır

and Öğrekçi

, On the stability problem of differential equations in the sense of ulam, Results in Mathematics 75(1) (2019), 6.

Bede

and Gal

S.G.

, Generalizations of the differentiability of fuzzy-numbervalued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems 151(3) (2005), 581–599.

Bede

and Stefanini

, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems 230 (2013), 119–141. Differential Equations Over Fuzzy Spaces - Theory, Applications, and Algorithms.

Castro

L.P.

and Ramos

, Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations, Banach Journal of Mathematical Analysis [electronic only] 3(1) (2008), 36–43.

Vanterler

, Sousa

da C.

and Capelas de Oliveira

, Hyers-Ulam stability of a nonlinear fractional volterra integro-differential equation, Applied Mathematics Letters 81 (2018), 50–56.

Vanterler

, Sousa

da C.

, Kucche

Kishor D.

and Capelas de Oliveira

, Stability of ψ-Hilfer impulsive fractional differential equations, Applied Mathematics Letters 88 (2019), 73–80.

Diaz

J.B.

and Margolis

, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull Amer Math Soc 74(2) (1968), 305–309, 03.

Eghbali

, Kalvandi

and Rassias

J.M.

, A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation, Open Mathematics 14(1) (2016), 237–246.

10.

Gouyandeh

, Allahviranloo

, Abbasbandy

and Armand

, A fuzzy solution of heat equation under generalized hukuhara differentiability by fuzzy fourier transform, Fuzzy Sets and Systems 309 (2017), 81–97. Theme: Analysis.

11.

Huang

and Li

, Hyers-Ulam- stability of linear functional differential equations, Journal of Mathematical Analysis and Applications 426(2) (2015), 1192–1200.

12.

Hyers

D.H.

, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America 27(4) (1941), 222–224.

13.

Jung

S.-M.

, A fixed point approach to the stability of a volterra integral equation, Fixed Point Theory and Applications 2007(1) (2007), 057064.

14.

Jung

S.-M.

and Brzdek

, Hyers-Ulam stability of the delay equation y′ (t) = λy (t - τ), Abstr Appl Anal 2010 (2010), 281 pp. 10.

15.

Kaleva

, Fuzzy differential equations, Fuzzy Sets and Systems 24(3) (1987), 301–317. Fuzzy Numbers.

16.

Khastan

, Nieto

J.J.

and Rodríguez-López

, Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets and Systems 177(1) (2011), 20–33. Theme: Fuzzy Interval Analysis.

17.

Liu

, Wang

J.R.

and O’Regan

, Ulam type stability of first-order linear impulsive fuzzy differential equations, Fuzzy Sets and Systems, 2019.

18.

Long

H.V.

, Kim Son

N.T.

, Thanh Tam

H.T.

and Yao

J.-C.

, Ulam stability for fractional partial integro-differential equation with uncertainty, Acta Mathematica Vietnamica 42(4) (2017), 675–700.

19.

Long

H.V.

and Phuong Thao

H.T.

, Hyers-Ulam stability for nonlocal fractional partial integro-differential equation with uncertainty, Journal of Intelligent & Fuzzy Systems 34(1) (2018), 233–244.

20.

Oblaza

, Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt Prace-Matematyczne 13 (1993), 259–270.

21.

Ngoc Phung

, Bao Quoc

and Vu

, Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy integrodifferential equation, Complexity 2019(Article ID 8275979) (2019), 01–10.

22.

Shen

, On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability, Fuzzy Sets and Systems 280 (2015), 27–57.

23.

Shen

, Hyers-Ulam-Rassias stability of first order linear partial fuzzy differential equations under generalized differentiability, Advances in Difference Equations 315 (2015), 1–18.

24.

Shen

and Wang

, A fixed point approach to the Ulam stability of fuzzy differential equations under generalized differentiability, Journal of Intelligent & Fuzzy Systems 30(6) (2016), 3253–3260.

25.

Ulam

S.M.

, A Collection of the Mathematical Problems. Interscience, New York, 1940.

26.

Van

H.N.

and Ho

, A survey on the initial value problems of fuzzy implicit fractional differential equations, Fuzzy Sets and Systems 2019.

27.

, Rassias

J.M.

and Hoa

N.V.

, Ulam-Hyers-Rassias stability for fuzzy fractional integral equations, Iranian Journal of Fuzzy Systems 17(2) (2020), 17–27.

28.

Zada

, Ali

and Park

, Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of grönwall-bellman-bihari’s type, Applied Mathematics and Computation 350 (2019), 60–65.