Hyers-Ulam stability of Hermite fuzzy differential equations and fuzzy Mellin transform

Abstract

In this paper, we introduce a fuzzy Mellin transform method for solving Hermite fuzzy differential equations. The fuzzy Mellin transform reduce the problem of solving a Hermite fuzzy differential equation to a problem of solving a difference equation, whose inverse transform gives the solution of the fuzzy differential equation at hands. Under some conditions, we also give some Hyers-Ulam stability result of Hermite fuzzy differential equations.

Keywords

Mellin transform method Hermite fuzzy differential equation H-difference

1 Introduction

The Ulam stability of functional equations was originally raised by Ulam in 1940 in a talk given at Wisconsin University [1]. Hyers firstly gave the results on the Ulam stablity in the case of Banach space [2]. Aoki generalized Hyers’ theorem for approximately additive mappings [3]. Th.M. Rassias provided a generalized version of Hyers’ result which allows the Cauchy difference to be unbounded [4]. J.M. Rassias and Xu generalized the Hyers stability result by introducing two weaker conditions controlled by a product of different powers of norms and a mixed product-sum of powers of norms, respectively [5 –8]. Furthermore, Jung proved the Ulam–Hyers stability of linear functional equations [9 –11]. By applying a fixed point theorem in a generalized complete metric space, Wang et al. presented the Hyers–Ulam–Rassias stability, Hyers–Ulam stability and four types of Mittag–Leffler–Ulam stability for fractional differential equations [12].

The study on fuzzy differential equations has been rapidly advancing in recent years. Fuzzy differential equations have exited considerable interests in both mathematics and engineering areas. Many research papers have been published to consider solutions of fuzzy differential equations [13 –16].

Agarwal et al. proposed the concept of solutions for fractional differential equations with uncertainty [17]. They considered the Riemann-Liouville differentiability to solve the equation which is a combination of Hukuhara difference and Riemann-Liouville derivative. In a few papers by Bede and Gal, the shortcomings of applications of Hukuhara difference were discussed [18]. The existence and uniqueness of the solution were later established in [19, 20], and in [21 –24], authors derived the explicit solution of the fuzzy fractional differentiable equation using the Riemann-Liouville H-derivative. Allahviranloo et al. introduced the fuzzy Caputo fractional differential equation (FCFDE) under the Generalized Hukuhara differentiability and obtained the analytical solution of the Cauchy problem [25]. By Adomian decomposition method, Abbasbandy et al. solved the fuzzy system of Fredholm integral equation of the second kind [26]. In the paper [27], the authors proposed an improved predictor-corrector (IPC) method and gave some examples to show that the proposed method is more accurate. Salahshour et al. considered the fuzzy Volterra integral with separable kernel by using fuzzy differential transform method [28].

Various methods for solving the fuzzy differential equation are available in literature. Salahshour et al. applied fuzzy Laplace transform to solve fuzzy differential equations [29, 30]. Butera er al. proposed a Mellin transform method for solving multi-order differential equations [31]. Butzer et al. presented a new approach to fractional integration and differentiation in terms of Mellin analysis [32]. Motivated by their work, we introduce the notion of fuzzy Mellin transform (See [33]), and use the Mellin transform method to solve Hermite fuzzy differential equations. The fuzzy Mellin transform reduces the problem of solving a fuzzy differential equation to a problem of solving a fuzzy difference equation, whose inverse transform gives the solution of the fuzzy differential equation at hands. Furthermore, we consider the Hyers-Ulam stability of the Hermite fuzzy differential equation associated with the inhomogeneous Hermite fuzzy differential equation.

This paper is organized as follows. In Section 2, we review relating basic definitions and theoretical backgrounds needed in the paper. In Section 3, we solve the Hermite fuzzy differential equations by Mellin transform method and give the main results on its Hyers-Ulam stability.

2 Preliminaries

Let $(X, d)$ be a metric space. We denote the set of convex compact subsets of $X$ by $P (X)$ . For $A, B \in P (X)$ , the Hausdorff metric $D_{1}$ is defined as

$D_{1} (A, B) = inf {A \subset N (B, \in), N (A, \in)},$

where $N (A, \in) = {a \in A | d (a, b) < \in for some b \in A}$ .

Denote the space of fuzzy sets by $X_{F}$ in which the fuzzy set $μ : X \to [0, 1]$ satisfies the following conditions:

μ is normal, i.e., there exists an $a \in X$ such that μ (a) =1;

μ is convex, i.e., μ (ta + (1 - t) b) ≥ min {μ (a) , μ (b)} , for $t \in [0, 1], a, b \in X$ ;

μ is semi-continuous on $X$ ;

the closure of the set $\bar{{a \in X | μ (a) > 0}}$ is compact.

For 0 < γ ≤ 1, $[μ]_{γ} : = {μ_{0} \in X | μ (μ_{0}) \geq γ}$ is called a γ-level set of μ. And let $[μ]_{0} : = \bar{{μ_{0} \in X | μ (μ_{0}) > 0}}$ .

Let $X = R$ . The γ-level set [μ] _γ can be explicitly represented as a closed and bounded interval, i.e., [μ] _γ : = μ (γ) = [μ_l (γ) , μ_r (γ)] for each γ ∈ [0, 1].

For $μ, ν \in ℝ_{F}, λ \in ℝ$ , the addition μ + ν and scalar multiplication λ · ν can be defined, levelwise, by

$[μ + ν]_{γ} = [μ]_{γ} + [ν]_{γ}$

and

$[λ \cdot μ]_{γ} = λ \cdot [μ]_{γ}$

for all γ ∈ [0, 1].

A mapping $F : ℝ \to ℝ_{F}$ is called a fuzzy-valued function.

The supremum metric between μ and ν is defined by $D : ℝ_{F} \times ℝ_{F} \to ℝ^{+} \cup {0}$ ,

$D (μ, ν) = sup_{γ \in [0, 1]} D_{1} ([μ]_{γ}, [ν]_{γ}),$ (1) it can be show that $(ℝ_{F}, D)$ is a complete metric space.

Definition 1. [22] Let $F : [a, \infty] \to ℝ_{F}$ be a fuzzy-valued function, and it can be represented by [F_l (t; r), F_r(t; r)]. For any fixed r ∈ [0, 1], assume that the endpoint functions F_l(t; r) and F_r(t; r)) are both Riemann-integrable on [a, b] for b ≥ a. If there exist two positive M_l (r) and M_r (r) such that $\int_{a}^{b} | F_{l} (t; r) | d t \leq M_{l} (r)$ and $\int_{a}^{b} | F_{r} (t; r) | d t \leq M_{r} (r)$ for every b ≥ a, then F (t) is improper fuzzy Riemann-integrable on [a, ∞).

Remark 1. The improper fuzzy Riemann-integral is a fuzzy number and it can be represented as the following

$\begin{matrix} I F (t; r) = \int_{a}^{\infty} F (t; r) d t \\ = [\int_{a}^{\infty} F_{l} (t; r) d t, \int_{a}^{\infty} F_{r} (t; r) d t] . \end{matrix}$ (2.2)

We denote the space of Riemann-integrable complex fuzzy-valued functions on the bounded interval $[a, b] \subset ℝ$ by $L^{F} [a, b]$ .

Remark 2. There are some different definitions such as the Hukuhara difference, the generalized Hukuhara difference, the generalized difference, and so on. Some authors have considered the connections between these different differences and defined various notions of derivative based on these differences. In this paper, we adopt the follow H-difference.

Let $x, y \in ℝ_{F}$ . If there exists $z \in ℝ_{F}$ such that x = y + z, then z is called the H-difference of x and y and it is denoted by x ⊖ y.

Definition 2. We call a mapping $F : [a, b] \to ℝ_{F}$ strongly generalized differentiable at t₀ if there exists some F^′(t₀) such that

(i) there exist differences F (t₀ + h) - F(t₀), F(t₀)-F (t₀-h) and F^′(t₀) =

$lim_{h \to 0^{+}} \frac{F (t_{0} + h) - F (t_{0})}{h} = lim_{h \to 0^{+}} \frac{F (t_{0}) - F (t_{0} - h)}{h},$

(ii) there exist differences F t₀) - F (t₀ + h), F(t₀ - h) - F (t₀) and

$lim_{h \to 0^{+}} \frac{F (t_{0}) - F (t_{0} + h)}{- h} = lim_{h \to 0^{+}} \frac{F (t_{0} - h) - F (t_{0})}{- h},$

(iii) there exist differences F (t₀ + h) - F (t₀) , F (t₀ - h) - F (t₀) and F^′ (t₀) =

$lim_{h \to 0^{+}} \frac{F (t_{0} + h) - F (t_{0})}{h} = lim_{h \to 0^{+}} \frac{F (t_{0} - h) - F (t_{0})}{- h},$

(iv) there exist differences F (t₀) - F (t₀ + h) , F (t₀) - F (t₀ - h) and F^′ (t₀) =

$lim_{h \to 0^{+}} \frac{F (t_{0}) - F (t_{0} + h)}{- h} = lim_{h \to 0^{+}} \frac{F (t_{0}) - F (t_{0} - h)}{h} .$

Moreover, if F^′ (t₀) is (i)-differentiable, then we have

where r ∈ [0, 1] . And in this paper, we only consider the (i)-differentiable.

The Laplace transform and Fourier transform of fuzzy-valued functions have been studied by many authors (See [29 , 35–37]). By Fourier or Laplace transform, the image function of a fuzzy-valued function can be represented by the endpoint function model such as a interval form [f_l (x r) , f_r (x r)].

We then continue to define the fuzzy Mellin transform for a fuzzy-valued function.

Definition 3. Let $F \in L^{F} (ℝ)$ be a fuzzy-valued function. The fuzzy Mellin transform of the fuzzy-valued function F is defined as follows:

$M [F] (s) \equiv [F]_{M} (s) = \int_{0}^{\infty} τ^{s - 1} \cdot F (τ) d τ$ (2.3)

where $s = c + i t, t \in ℝ .$

Remark 3. For 0 ≤ r ≤ 1, if F (t) is a fuzzy-valued function, then we have

$\begin{matrix} M [F] (s; r) \\ = [\int_{0}^{\infty} t^{s - 1} F_{l} (t; r) d t, \int_{0}^{\infty} t^{s - 1} F_{r} (t; r) d t] \\ = [M [F_{l}] (s; r), M [F_{r}] (s; r)] . \end{matrix}$

Example 1. Let F (t) = e^-λ·t with [λ] _r = [1 + r, 3 - r]. We have

$M [e^{- λ \cdot t}] (s; r) = [(3 - r)^{- s} Γ (s), (1 + r)^{- s} Γ (s)] .$ (2.4)

Definition 4. The inversion formula for the fuzzy Mellin transform is defined as

$F (t) = M^{- 1} [\tilde{F} (s)] (t) = \frac{1}{2 π i} \cdot pv \int_{c - i \infty}^{c + i \infty} t^{- s} \cdot \tilde{F} (s) d s .$ (2.5)

Example 2. Let [λ] _r = [1 + r, 3 - r] , r ∈ [0, 1] be a triangular fuzzy number.

$\begin{matrix} M^{- 1} [λ^{- s} \cdot Γ (s)] (t) \\ = \frac{1}{2 π i} \cdot pv \int_{c - i \infty}^{c + i \infty} t^{- s} \cdot λ^{- s} \cdot Γ (s) d s = e^{- λ \cdot t} . \end{matrix}$ (2.6)

We can establish the connection with the fuzzy Laplace transform defined by the authors in papers [30 , 35]. We then have the following theorem:

Theorem 1.

$M [F (t)] (s) = L [F (e^{- t})] (s) .$ (2.7)

Proof. By a change of variables x = exp(- t), we have

$\begin{matrix} L [F (e^{- t})] (s) = \int_{- \infty}^{+ \infty} e^{- st} \cdot F (e^{- t}) d t \\ = \int_{0}^{\infty} x^{s - 1} \cdot F (x) d x = M [F (t)] (s) . \end{matrix}$ (2.8)

□

By using Theorem 1, we have the following example.

Example 3. Let $F \in L^{F} (ℝ)$ . Assume that a ≠ 0 .

$\begin{matrix} M [F (t^{a})] (s) & = L [F (e^{- a τ})] (s) = \frac{1}{| a |} \cdot L [F (e^{- τ})] (\frac{s}{a}) \\ = \frac{1}{| a |} \cdot M [F (τ)] (\frac{s}{a}) . \end{matrix}$ (2.9)

Definition 5. The fuzzy Mellin convolution product, denoted by F * G, of two functions $G : ℝ^{+} \to ℂ$ and $F \in L^{F} [a, b]$ , is defined by

$\begin{matrix} (F * G) (t) : = \int_{0}^{\infty} G (\frac{t}{τ}) \cdot F (τ) \cdot \frac{d τ}{τ} . \end{matrix}$ (2.10)

Moreover, for 0 ≤ r ≤ 1, if F is a fuzzy-valued function, then we can obtain

$\begin{matrix} (F * G) (t; r) : = [\int_{0}^{\infty} G (\frac{t}{τ}) F_{l} (τ; r) \frac{1}{τ} d τ, \int_{0}^{\infty} G (\frac{t}{τ}) F_{r} (τ; r) \frac{1}{τ} d τ] \\ = [(F_{l} * G) (t; r), (F_{r} * G) (t; r)] . \end{matrix}$ (2.11)

Theorem 2. If $G : ℝ^{+} \to ℂ$ , $F \in L^{F} [a, b]$ and s = c + it, then

$M [F * G] (s) = M [F] (s) \cdot M [G] (s) .$

Proof.

$\begin{matrix} M [F (t) * G (t)] (s) \\ = \int_{0}^{\infty} t^{s - 1} \cdot \int_{0}^{\infty} F (\frac{t}{τ}) \cdot G (τ) d τ d t \\ = \int_{0}^{\infty} t^{s - 1} \cdot F (\frac{t}{τ}) d t \cdot \int_{0}^{\infty} \frac{1}{τ} G (τ) d τ \\ = M [F (t)] (s) \cdot M [G (t)] (s) . \end{matrix}$ (2.12)

□

In fact, by using fuzzy Laplace transform, we can obtain another proof as follows:

$\begin{matrix} M [F (t) * G (t)] (s) = L [F (e^{- t}) ★ G (e^{- t})] \\ = L [F (e^{- t})] \cdot L [G (e^{- t})] \\ = M [F (t)] (s) \cdot M [G (t)] (s) . \end{matrix}$ (2.13)

In order to get a series solution of the fuzzy differential equation, we need the following theorem related with the level convergence of a sequence of fuzzy numbers.

Theorem 3. [38] Let μ_k be a sequence of fuzzy numbers such that $lim_{k \to \infty} μ_{k, l} (λ) = α (λ)$ and $lim_{k \to \infty} μ_{k, r} (λ) = β (λ)$ for each λ ∈ [0, 1]. Then the pair of functions α and β determines a fuzzy number if and only if the sequences of functions {μ_k,l (λ)} and {μ_k,r (λ)} are eventually equi-left-continuous at each λ ∈ [0, 1] and eventually equi-right-continuous at λ = 0.

Thus, it is deduced that the series $\sum_{k = 0}^{\infty} μ_{k, l} (λ) = μ_{l} (λ)$ and $\sum_{k = 0}^{\infty} μ_{k, r} (λ) = μ_{r} (λ)$ define a fuzzy number if the sequences ${S_{n, l} (λ)}_{n} = {\sum_{k = 0}^{n} μ_{k, l} (λ)}$ and ${S_{n, r} (λ)}_{n} = {\sum_{k = 0}^{n} μ_{k, r} (λ)}$ satisfy the conditions of above theorem (for more details see [38]).

3 Hyers-Ulam stability of Hermite fuzzy differential equation

The fuzzy Mellin transform method can be used in solving the fuzzy ordinary differential equation with polynomial coefficients, such as Hermite differential equation, Euler differential equation and Legendre differential equation. We here consider the following Hermite fuzzy differential equation,

${H^{″}}_{v} (t) + 2 v H_{v} (t) = 2 t {H^{'}}_{v} (t),$ (3.1)

where ν > 0 and H_ν (t) is a fuzzy-valued function.

Lemma 1. Let $H (t) \in L^{F} (ℝ)$ . Assume that H^″ (t), H^′ (t) and H_ν (t) exist. If the fuzzy Mellin transforms of H^″ (t), H^′ (t) and H_ν (t) exist, then we have

$M [H ″ (t)] (s) = (s - 1) (s - 2) M [H] (s - 2),$ (3.2)

and

$M [H^{'} (t)] (s) = - s M [H] (s),$ (3.3)

where $s \in ℂ$ .

Proof. By the classic Mellin transform, we can get the following properties:

$M [f^{″} (t)] (s) = (s - 1) (s - 2) M [f] (s - 2),$ (3.4)

and

$M [tf' (t)] (s) = - s M [f] (s),$ (3.5)

where f (t) is a real-valued function. Then for any r ∈ [0, 1], we have

$\begin{matrix} M [H^{″} (t)] (s; r) \\ \begin{matrix} = [\int_{0}^{\infty} t^{s - 1} {H^{″}}_{l} (t; r), \int_{0}^{\infty} t^{s - 1} {H^{″}}_{r} (t; r)] \\ = [(s - 1) (s - 2) M [H_{l}] (s - 2; r)] \\ = (s - 1) (s - 2) M [H] (s - 2; r) . \end{matrix} \end{matrix}$ (3.6)

Similarly, one can obtain

$\begin{matrix} M [t H^{'} (t)] (s; r) \\ = [\int_{0}^{\infty} t^{s} {H^{'}}_{l} (t; r), \int_{0}^{\infty} t^{s} {H^{'}}_{r} (t; r)] \\ = [- s M [H_{r}] (s; r), - s M [H_{l}] (s; r)] \\ = - s M [H] (s; r) . \end{matrix}$ (3.7)

□

Lemma 2. Assume that the fuzzy Mellin transforms of H^″(t), H^′(t) and H_ν (t) exist. Then the Hermite Equation (3.1) can be transformed as the following:

$\begin{matrix} (s - 1) (s - 2) M [H_{ν}] (s - 2) + 2 ν M [H_{ν}] (s) \\ = - 2 s M [H_{ν}] (s) . \end{matrix}$ (3.8)

Proof. By Lemma 1, we have

$M [{H^{″}}_{v}] (t; r) + 2 v M [H_{v}] (t; r) = M [2 t {H^{'}}_{v}] (t; r) .$ (3.9)

Then, we can obtain

$\begin{matrix} (s - 1) (s - 2) M [H_{ν, l}] (s - 2; r) + 2 ν M [H_{ν, l}] (s; r) \\ = - 2 s M [H_{ν, r}] (s; r), \end{matrix}$ (3.10)

and

$\begin{matrix} (s - 1) (s - 2) M [H_{ν, r}] (s - 2; r) + 2 ν M [H_{ν, r}] (s; r) \\ = - 2 s M [H_{ν, l}] (s; r) . \end{matrix}$ (3.11)

Then, the Equation (3.8) is obvious, and the proof is complete.

□

Theorem 4. Assume that the fuzzy Mellin transforms of H^″(t), H^′(t) and H_ν (t) exist. The solution of (3.1) can be expressed as a power series,

$\begin{matrix} H_{ν} (t) = K \cdot (\sum_{n = 0}^{\infty} \frac{(- 2)^{n}}{(2 n)!} t^{2 n} \prod_{j = 0}^{n - 1} (ν - 2 j) \\ + \sum_{n = 0}^{\infty} \frac{(- 2)^{n}}{(2 n + 1)!} t^{2 n + 1} \prod_{j = 0}^{n - 1} [ν - (2 j + 1)]), \end{matrix}$ (3.12)

where $K \in ℝ_{F}$ are arbitrary constants.

Proof. By Lemma 2, we have

$\begin{matrix} (s - 1) (s - 2) M [H_{ν}] (s - 2) + 2 ν M [H_{ν}] (s) \\ = - 2 s M [H_{ν}] (s) . \end{matrix}$ (3.13)

Therefore, we further have

$\begin{matrix} (s - 1) (s - 2) M [H_{ν}] (s - 2) \\ = - 2 s M [H_{ν}] (s) ⊖ 2 ν M [H_{ν}] (s) . \end{matrix}$ (3.14)

It is apparent that the Mellin transform does not give $M [H_{ν}] (s)$ directly; rather we must solve a fuzzy difference Equation (3.14).

Setting s = -2x and $M [H_{ν}] (s) = T (x)$ , we can reduce the difference Equation (3.14) to standard form. Thus, we only need to solve

$2 (2 x + 1) (x + 1) T (x + 1) = 4 xT (x) ⊖ 2 ν T (x) .$ (3.15)

A particular solution of (3.15) is given by

$T (x) = K (x - ν / 2 - 1)! (- x - 1 / 2)! (- x - 1)!,$ (3.16)

where $K \in ℝ_{F}$ is an arbitrary fuzzy constant. It is easy to see that T (x) is not the only solution of the difference equation, since we can multiply T (x) by any function that satisfies

$Y (x + 1) = Y (x) .$ (3.17)

At this point, we appeal to the fact that $M [H_{ν}] (x)$ is a Mellin transform, defined only in some strip α < Re (x) < β. Therefore, (3.15) is valid only in the overlap of the strips

$α < Re (x) < β, α < Re (x - 2) < β,$

and there is no such overlap unless β > α + 2. Thus, Y (x) cannot have poles, since they would give rise to a row of poles in S (x) separated by exactly two units. Also, Y (x) cannot grow faster that |x| as Im (x)→ ∞ in the inversion strip, otherwise the inversion integral would diverge. Therefore, by (3.17), Y (x) is a bounded entire function, and thus equal to a constant by Liouville’s theorem. Hence, (3.16) is the only acceptable solution, and even then only if Re (ν) < -2, and we have

$\begin{matrix} H_{ν} (t) = \frac{K}{2 π i} \\ \int_{c - i \infty}^{c + i \infty} (x - ν / 2 - 1)! (- x - 1 / 2)! (- x - 1)! t^{2 x} d x . \end{matrix}$ (3.18)

The poles of (x - ν/2 -1) ! lie at x = (ν/2 - k) , k = 0, 1, 2, . . . , with residues (-1) ^k/k !. Thus,

$\begin{matrix} H_{ν} (t) = K \times \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k!} (- ν / 2 + k - 1 / 2)! \\ (- ν / 2 + k - 1)! t^{ν - 2 k} . \end{matrix}$ (3.19)

Closing the contour to the right leads to the expansion

where $K \in ℝ_{F}$ is an arbitrary constant. □

Remark 4. Let

$\begin{matrix} H (t) : = \sum_{n = 0}^{\infty} \frac{(- 2)^{n}}{(2 n)!} t^{2 n} \prod_{j = 0}^{n - 1} (ν - 2 j) \\ + \sum_{n = 0}^{\infty} \frac{(- 2)^{n}}{(2 n + 1)!} t^{2 n + 1} \prod_{j = 0}^{n - 1} [ν - (2 j + 1)] . \end{matrix}$ (3.21)

In fact, the solution of (3.1) can also be expressed by

$\begin{matrix} H_{ν} (t; r) = [H_{ν, l} (t; r), H_{ν, r} (t; r)] \\ = [μ_{l} (r) \cdot H (t), μ_{r} (r) \cdot H (t)], \end{matrix}$ (3.22)

where μ_l (r) and μ_r (r) are arbitrary real endpoint functions such that $H_{ν} (t) \in ℝ_{F}$ .

We then consider the general solution of the following inhomogeneous Hermite fuzzy differential equation

${H^{″}}_{v} (t) + 2 v H_{V} (t) = 2 t {H^{'}}_{v} (t) + \sum_{m = 0}^{\infty} a_{m} t^{m}$ (3.23)

for all t ∈ (- ρ, ρ), where ν is given real number and the coefficients $a_{m} \in ℝ_{F}$ of the power series are given such the radius of convergence is ρ > 0.

(H1) Assume that for each $m \in ℕ \cup {0}$ there exists the fuzzy number sequence ${c_{m}}_{m = 0}^{\infty}$ such that the following recurrence formula

$(m + 2) (m + 1) c_{m + 2} = a_{m} + 2 (m - ν) c_{m}$ (3.24)

holds.

(H2) For t ∈ (- ρ, ρ) , r ∈ [0, 1], the following two series

$\sum_{n = 1}^{\infty} \frac{t^{2 n - 2}}{(2 n - 2)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 2)! a_{2 i - 2} B$

and

$\sum_{n = 1}^{\infty} \frac{t^{2 n - 1}}{(2 n - 1)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 1)! a_{2 i - 1} A$

are both converged in the setting of Theorem 3, where $\prod_{j = i}^{n - 1} [ν - (2 j + 1)] = A$ , $\prod_{j = i}^{n - 1} (ν - 2 j) = B$ .

Theorem 5. Assume that hypotheses (H1)-(H2) are satisfied. And assume that the radius of convergence of the power series $\sum_{m = 0}^{\infty} a_{m} t^{m}$ is ρ, where ρ is either a positive real number or the infinity. If the solution $H : (- ρ, ρ) \to ℝ_{F}$ of the inhomogeneous Hermite fuzzy differential equation (3.23) can be represented by a power series, then the solution can be expressed by

$\begin{matrix} H (t) = H_{ν} (t) + \sum_{n = 1}^{\infty} \frac{t^{2 n}}{(2 n)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 2)! \\ a_{2 i - 2} \prod_{j = i}^{n - 2} (λ - 2 j) + \sum_{n = 1}^{\infty} \frac{t^{2 n + 1}}{(2 n + 1)!} \sum_{i = 1}^{n} (- 2)^{n - i} \\ (2 i - 1)! a_{2 i - 1} \prod_{j = i}^{n - 1} [λ - (2 j + 1)] \end{matrix}$ (3.25)

for all t ∈ (- ρ, ρ), where H_ν (t) is a solution of the Hermite Equation (3.1).

Proof. See that the solution of the inhomogeneous Hermite fuzzy differential Equation (3.23) can be represented by a power series. Assume that $H : (- ρ, ρ) \to ℝ_{F}$ can be defined by (3.25), where H_ν (t) is a solution of the Hermite differential fuzzy Equation (3.1). Set

$H (t) = H_{p} (t) + H_{ν} (t) .$

We shall show that H_p (t) is a solution of inhomogeneous Hermite fuzzy differential Equation (3.23). For simplicity, let $\prod_{j = i}^{n - 1} [ν - (2 j + 1)] = A$ , $\prod_{j = i}^{n - 1} (ν - 2 j) = B$ and a_i (r) = [a_i,l (r) , a_i,r (r)] for r ∈ [0, 1]. By (H1) and (H2), we have

$\begin{matrix} {H^{″}}_{p} (t; r) - 2 t {H^{'}}_{p} (t; r) + 2 v H_{p} (t; r) \\ = [{H^{″}}_{p, l} (t; r) - 2 t {H^{'}}_{p, l} (t; r) + 2 v H_{p, l} (t; r), {H^{″}}_{p, r} (t; r) - 2 t {H^{'}}_{p, r} (t; r) + 2 v H_{p, r} (t; r)], \end{matrix}$ (39)

where

$\begin{matrix} {H^{″}}_{p, l} (t; r) = \\ \sum_{n = 1}^{\infty} \frac{x^{2 n - 2}}{(2 n - 2)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 2)! a_{2 i - 2, l} (r) B + \sum_{n = 1}^{\infty} \frac{t^{2 n - 1}}{(2 n - 1)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 1)! a_{2 i - 1, l} (r) A, \end{matrix}$ $\begin{matrix} 2 t {H^{'}}_{p, l} (t; r) = \\ \sum_{n = 1}^{\infty} \frac{2 t^{2 n}}{(2 n - 1)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 2)! a_{2 i - 2, l} (r) B + \sum_{n = 1}^{\infty} \frac{2 t^{2 n + 1}}{(2 n)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 1)! a_{2 i - 1, l} (r) A, \end{matrix}$ $\begin{matrix} 2 v H_{p, l} (t; r) = \\ \sum_{n = 1}^{\infty} \frac{2 v t^{2 n}}{(2 n)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 2)! a_{2 i - 2, l} (r) B + \sum_{n = 1}^{\infty} \frac{2 v t^{2 n + 1}}{(2 n + 1)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 1)! a_{2 i - 1, l} (r) A, \end{matrix}$ $\begin{matrix} {H^{″}}_{p, r} (t; r) = \\ \sum_{n = 1}^{\infty} \frac{x^{2 n - 2}}{(2 n - 2)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 2)! a_{2 i - 2, r} (r) B + \sum_{n = 1}^{\infty} \frac{t^{2 n - 1}}{(2 n - 1)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 1)! a_{2 i - 1, r} (r) A, \end{matrix}$ $\begin{matrix} 2 t {H^{'}}_{p, r} (t; r) = \\ \sum_{n = 1}^{\infty} \frac{2 t^{2 n}}{(2 n - 1)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 2)! a_{2 i - 2, r} (r) B + \sum_{n = 1}^{\infty} \frac{2 t^{2 n + 1}}{(2 n)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 1)! a_{2 i - 1, r} (r) A, \end{matrix}$ $\begin{matrix} 2 v H_{p, r} (t; r) = \\ \sum_{n = 1}^{\infty} \frac{2 v t^{2 n}}{(2 n)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 2)! a_{2 i - 2, r} (r) B + \sum_{n = 1}^{\infty} \frac{2 v t^{2 n + 1}}{(2 n + 1)!} \sum_{i = 1}^{n} {(- 2)}^{n - i} (2 i - 1)! a_{2 i - 1, r} (r) A . \end{matrix}$

We can also obtain that

$\begin{matrix} \sum_{n = 1}^{\infty} \frac{t^{2 n - 2}}{(2 n - 2)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 2)! a_{2 i - 2, l} (r) B \\ = \sum_{n = 0}^{\infty} \frac{t^{2 n}}{(2 n)!} \sum_{i = 1}^{n} (- 2)^{n + 1 - i} (2 i - 2)! a_{2 i - 2, l} (r) B \\ = \sum_{n = 0}^{\infty} a_{2 n, l} (r) t^{2 n} + \sum_{n = 1}^{\infty} \frac{2 t^{2 n}}{(2 n - 1)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 2)! \\ a_{2 i - 2, l} (r) B - \sum_{n = 1}^{\infty} \frac{2 ν t^{2 n}}{(2 n)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 2)! \\ a_{2 i - 2, l} (r) B, \end{matrix}$

$\begin{matrix} \sum_{n = 1}^{\infty} \frac{t^{2 n - 2}}{(2 n - 2)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 2)! a_{2 i - 2, r} (r) B \\ = \sum_{n = 0}^{\infty} \frac{t^{2 n}}{(2 n)!} \sum_{i = 1}^{n} (- 2)^{n + 1 - i} (2 i - 2)! a_{2 i - 2, r} (r) B \\ = \sum_{n = 0}^{\infty} a_{2 n, r} (r) t^{2 n} + \sum_{n = 1}^{\infty} \frac{2 t^{2 n}}{(2 n - 1)!} \\ \times \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 2)! \\ a_{2 i - 2, r} (r) B - \sum_{n = 1}^{\infty} \frac{2 ν t^{2 n}}{(2 n)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 2)! \\ a_{2 i - 2, r} (r) B . \end{matrix}$

We can similarly get the following:

$\begin{matrix} \sum_{n = 1}^{\infty} \frac{t^{2 n - 1}}{(2 n - 1)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 1)! a_{2 i - 1, l} (r) A \\ = \sum_{n = 0}^{\infty} a_{2 n + 1, l} (r) t^{2 n + 1} + \sum_{n = 1}^{\infty} \frac{2 t^{2 n + 1}}{(2 n)!} \sum_{i = 1}^{n} (- 2)^{n - i} \\ (2 i - 1)! a_{2 i - 1, l} (r) A - \sum_{n = 1}^{\infty} \frac{2 ν t^{2 n + 1}}{(2 n + 1)!} \\ \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 1)! a_{2 i - 1, l} (r) A, \end{matrix}$ (3.27) $\begin{matrix} \sum_{n = 1}^{\infty} \frac{t^{2 n - 1}}{(2 n - 1)!} \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 1)! a_{2 i - 1, r} (r) A \\ = \sum_{n = 0}^{\infty} a_{2 n + 1, l} (r) t^{2 n + 1} + \sum_{n = 1}^{\infty} \frac{2 t^{2 n + 1}}{(2 n)!} \sum_{i = 1}^{n} (- 2)^{n - i} \\ (2 i - 1)! a_{2 i - 1, r} (r) A - \sum_{n = 1}^{\infty} \frac{2 ν t^{2 n + 1}}{(2 n + 1)!} \\ \sum_{i = 1}^{n} (- 2)^{n - i} (2 i - 1)! a_{2 i - 1, r} (r) A . \end{matrix}$ (3.28)

Finally, it follows from (3.26) that

${H^{″}}_{p} (t; r) + 2 v H_{p} (t; r) = 2 t {H^{'}}_{p} (t; r) + \sum_{m = 0}^{\infty} a_{m} (r) t^{m} .$

The above equation implies that

${H^{″}}_{p} (t) + 2 v H_{p} (t) = 2 t {H^{'}}_{p} (t) + \sum_{m = 0}^{\infty} a_{m} t^{m},$

which proves that H_p (t) is a particular solution of the inhomogeneous Equation (3.23). Every solution of (3.23) can be expressed as a sum of a solution H_ν (t) of the homogeneous equation and a particular solution H_p (t) of the inhomogeneous equation. Hence, the solution can be expressed by (3.25). □

Theorem 6. Assume that hypotheses (H1)-(H2) are satisfied. Let ν ≥ 1, and $H : (- ρ, ρ) \to ℝ_{F}$ be an analysis function which can be represented by a power-series expansion centered at t = 0. Assume that the radius of convergence of the power series $\sum_{m = 0}^{\infty} a_{m} t^{m}$ is ρ, where ρ is either a positive real number or the infinity. Suppose that there exists a constant ɛ > 0 such that for t ∈ (- ρ, ρ)

$D (H^{″} (t) + 2 v H (t), 2 t H^{'} (t)) \leq ε .$ (3.29)

Also, suppose that for any t ∈ (- ρ, ρ)

$H^{″} (t) + 2 v H (t) = 2 t H^{'} (t) + \sum_{m = 0}^{\infty} a_{m} t^{m},$ (3.30)

and

$\sum_{m = 0}^{\infty} D (a_{m} t^{m}, 0) \leq K \cdot D (\sum_{m = 0}^{\infty} a_{m} t^{m}, 0),$ (3.31)

where K is a constant. Then there exists a solution of (3.1), $z : (- ρ, ρ) \to ℝ_{F}$ such that

$D (H (t), Z (t)) \leq C ɛ t^{2} e^{t^{2}}$ (3.32)

for all t ∈ [- ρ, ρ], where C is some constant which depends on ρ, ν and H.

Moreover, the Hermite fuzzy differential Equation (3.1) is Hyers-Ulam stable.

Proof. Since the power series $\sum_{m = 0}^{\infty} a_{m} t^{m}$ is absolutely convergent on its interval of convergence, which includes the interval [- ρ, ρ], and the power series $\sum_{m = 0}^{\infty} a_{m} t^{m}$ is continuous on [- ρ, ρ], there exists a constant K > 0 with

$\sum_{m = 0}^{\infty} D (a_{m} t^{m}, 0) \leq K$ (3.33)

for all t ∈ (- ρ, ρ).

On the other hand, we have

$\begin{matrix} \nabla : = D (\sum_{n = 1}^{\infty} \frac{t^{2 n}}{(2 n)!} \sum_{i = 1}^{n} (- 2)^{n - j} (2 i - 2)! a_{2 i - 2} \\ \prod_{j = i}^{n - 1} (ν - 2 j), 0) \\ \leq \sum_{n = 1}^{\infty} t^{2 n} \sum_{i = 1}^{n} \frac{(2 i - 2)!}{(2 n)!} D (a_{2 i - 2}, 0) \prod_{j = i}^{n - 1} | 4 j - 2 ν | \\ = \sum_{n = 1}^{\infty} t^{2 n} \sum_{i = 1}^{n} \frac{(2 i - 2)!}{(2 n)!} D (a_{2 i - 2}, 0) 4^{n - i} \end{matrix}$

$\begin{matrix} \frac{(n - 1)}{(i - 1)!} \prod_{j = i}^{n - 1} | 1 - \frac{ν}{2 j} | \\ = \sum_{n = 1}^{\infty} t^{2 n} \sum_{i = 1}^{n} \frac{(i - 1)!}{n! 2 (2 i - 1)} D (a_{2 i - 2}, 0) \prod_{j = i}^{n - 1} | 1 - \frac{ν}{2 j} | . \end{matrix}$ (3.34)

Then, we have

$\begin{matrix} \nabla & = C_{0} \sum_{n = 1}^{\infty} \frac{t^{2 n}}{n!} \sum_{i = 1}^{n} \frac{(i - 1)!}{2 (2 i - 1)} D (a_{2 i - 2}, 0) \\ \leq C_{0} \sum_{i = 1}^{\infty} \frac{(i - 1)!}{2 (2 i - 1)} D (a_{2 i - 2}, 0) \sum_{n = i}^{\infty} \frac{t^{2 n}}{n!} \\ = C_{0} \sum_{i = 1}^{\infty} \frac{t^{2}}{2 i (2 i - 1)} D (a_{2 i - 2}, 0) t^{2 i - 2} \\ \sum_{n = i}^{\infty} \frac{(t^{2})^{n - i}}{(n - i)!} \cdot \frac{i!}{n (n - 1) . . . (n - i + 1)} \\ \leq C_{0} t^{2} e^{t^{2}} \sum_{i = 0}^{\infty} \frac{1}{2 (i + 1) (2 i + 1)} D (a_{2 i - 2}, 0) t^{2 i} \\ \leq \frac{C_{0}}{2} t^{2} e^{t^{2}} \sum_{m = 0}^{\infty} D (a_{2 m}, 0) | t |^{2 m}, \end{matrix}$ (3.35)

where C₀ is a positive constant satisfying

$\prod_{j = i}^{n - 1} | 1 - \frac{ν}{2 j} | \leq C_{0}$

for all $i, n \in ℕ$ . (The above product diverges to 0 as n→ ∞ because of ν ≥ 1, and C₀ depends on ν.)

Since the power series $\sum_{m = 0}^{\infty} a_{m} t^{m}$ absolutely converges on its convergence interval, and so

$C_{0} \frac{(i - 1)!}{2 (2 i - 1)} D (a_{2 i - 2}, 0) | t |^{2 m}$

is absolutely convergent.

Analogously, we obtain

$\begin{matrix} D (\sum_{n = 1}^{\infty} \frac{t^{2 n + 1}}{(2 n + 1)!} \sum_{i = 1}^{n} (- 2)^{n - j} (2 i - 1)! a_{2 i - 1} \\ \prod_{j = i}^{n - 1} [ν - (2 j + 1)], 0) \end{matrix}$

$\begin{matrix} \leq \frac{C_{1}}{6} t^{2} e^{t^{2}} \sum_{m = 0}^{\infty} D (a_{2 m + 1}, 0) | t |^{2 m + 1}, \end{matrix}$ (3.36)

where C₁ is a positive constant satisfying

$\prod_{j = i}^{n - 1} | 1 - \frac{ν - 1}{2 j} | \leq C_{1}$

for all $i, n \in ℕ$ . (The above product diverges to 0 as n→ ∞ because of ν ≥ 1, and C₀ depends on ν.)

In view of, H (t) is a solution of the inhomogeneous Hermite equation. Set C₂ = 2K · max {C₀/2, C₁/6}. Then there exists a solution z (t) of Hermite equation such that

$D (H (t), z (t)) \leq C_{2} ɛ t^{2} e^{t^{2}} .$ (3.37)

Then, the Hyers-Ulam stability result of (3.1) is easy to show. □

Example 4. Consider the fuzzy-valued function $H (t) = λ \cdot exp (t^{3}), (λ \in ℝ_{F})$ which can be expressed by the power series

$H (t; r) = [\sum_{m = 1}^{\infty} \frac{x^{3 m}}{m!} λ_{l} (r), \sum_{m = 1}^{\infty} \frac{x^{3 m}}{m!} λ_{r} (r)],$ (3.38)

where r ∈ [0, 1]. And set for r ∈ [0, 1], $a_{3 k} (r) = [\frac{- 6 k + 2}{k!} λ_{l} (r), \frac{- 6 k + 2}{k!} λ_{r} (r)]$ , $a_{3 k + 1} (r) = [\frac{9 k + 6}{k!} λ_{l} (r), \frac{9 k + 6}{k!} λ_{r} (r)]$ , and a_3k+2 = 0.

Thus, we have

$\begin{matrix} \sum_{m = 0}^{\infty} D (a_{m} t^{m}, 0) \leq \sum_{m = 0}^{\infty} D (a_{3 m} t^{3 m}, 0) \\ \sum_{m = 0}^{\infty} D (a_{3 m + 1} t^{3 m + 1}, 0) + \sum_{m = 0}^{\infty} D (a_{3 m + 2} t^{3 m + 2}, 0) \\ \leq \sum_{k = 0}^{\infty} | \frac{- 6 k + 2}{k!} t^{3 k} | D (λ, 0) + \sum_{k = 0}^{\infty} | \frac{9 k + 6}{k!} t^{3 k + 1} | D (λ, 0) \\ \leq (9 ρ^{4} + 6 ρ^{3} + 6 ρ + 2) exp (ρ^{3}) D (λ, 0) \end{matrix}$ (3.39)

for all x ∈ (ρ, ρ). Then, Theorem 6 implies that there exists an Hermite function H_ν : (- ρ, ρ) satisfying

$D (exp (t^{3}), H_{ν} (t)) \leq {Ct}^{2} exp (t^{2})$

for all x ∈ (ρ, ρ).

Competing interests

The authors declare that they have no competing interests.

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