Existence and uniqueness solutions of fuzzy integration-differential mathematical problem by using the concept of generalized differentiability

1.   Introduction

The theory of calculus, which deals with the investigation and applications of derivatives and integrals of arbitrary order has a long history. The theory of calculus developed mainly as a pure theoretical field of mathematics, in the last decades it has been used in various fields as rheology, viscoelasticity, electrochemistry, diffusion processes, etc ^[33,34]. Calculus have undergone expanded study in recent years as a considerable interest both in mathematics and in applications. One of the recently influential works on the subject of calculus is the monograph of Podlubny ^[50] and the other is the monograph of Kilbas et al. ^[34]. The differential equations have great application potential in modeling a variety of real world physical problems, which deserves further investigations. Among these we might include the modeling of earthquakes, the fluid dynamic traffic model with derivatives, the measurement of viscoelastic material properties, etc. Consequently, several research papers were done to investigate the theory and solutions of differential equations (see ^{[18,21,36,38]} and references therein).

The concept of solution for differential equations with uncertainty was introduced by Agarwal, Lakshmikantham and Nieto ^[1]. They considered Riemann-Liouville differentiability concept based on the Hukuhara differentiability to solve fuzzy differential equations. Arshad and Lupulescu in ^[12] proved some results on the existence and uniqueness of solution to fuzzy differential equation under Hukuhara Riemann-Liouville differentiability. Some existence results for nonlinear fuzzy differential equations of order involving the Riemann-Liouville derivative have been proposed in ^[30,31]. The solutions of fuzzy differential equations are investigated by using the fuzzy Laplace transforms in ^[52]. Recently, the concepts of derivatives for a fuzzy function are either based on the notion of Hukuhara derivative ^[25] or on the notion of strongly generalized derivative. The concept of Hukuhara derivative is old and well known, but the concept of strongly generalized derivative was recently introduced by Bede and Gal ^[13]. Using this new concept of derivative, the classes of fuzzy differential equations have been extend and studied in some papers such as: Ahmad et al. ^[4], Allahviranloo et al. ^[9,10,11,49], Bede et al. ^{[14,15,16,17]}, Gasilov ^[20], Khastan et al. ^[27,28,29], Malinowski ^[42,43,44] and Nieto ^[46]. Furthermore, by using this new concept of derivative, Allahviranloo et al. in ^[7,8] have studied the concepts about generalized Hukuhara Riemann-Liouville and Caputo differentiability of fuzzy valued functions. Later, authors have proved the existence and uniqueness of solution for fuzzy differential equation by using different methods. Alikhani et al. in ^[6] have proved the existence and uniqueness results for nonlinear fuzzy integral and integration and differential equations by using the method of upper and lower solutions. Mazandarani et al. ^[45] studied the solution to fuzzy initial value problem under Caputo-type fuzzy derivatives by a modified Euler method. Besides, authors studied some results on the existence and uniqueness of solution to fuzzy differential equation under Caputo type-2 fuzzy derivative and the definition of Laplace transform of type-2 fuzzy number-valued functions ^[46,47,48]. Salahshour et al. ^[48,51] proposed some new results toward existence and uniqueness of solution of fuzzy differential equation. According to the concept of Caputo-type fuzzy derivative in the sense of the generalized fuzzy differentiability, Fard et al. ^[19] extended and established some definitions on fuzzy calculus of variation and provide some necessary conditions to obtain the fuzzy Euler-Lagrange equation for both constrained and unconstrained fuzzy variational problems. Ahmad et al. ^[5] proposed a new interpretation of fuzzy differential equations and present their solutions analytically and numerically. The proposed idea is a generalization of the interpretation given in ^[3,4], where the authors used Zadeh’s extension principle to interpret fuzzy differential equations.

In real world systems, delays can be recognized everywhere and there has been widespread interest in the study of delay differential equations for many years. Therefore, delay differential equations (or, as they are called, functional differential equations) play an important role in an increasing number of system models in biology, engineering, physics and other sciences. There exists an extensive amount of literature dealing with delay differential equations and their applications; the reader is referred to the monographs ^[22,35], and the references therein. The study of fuzzy delay differential equations is expanding as a new branch of fuzzy mathematics. Both theory and applications have been actively discussed over the last few years. In the literature, the study of fuzzy delay differential equations has several interpretations. The first one is based on the notion of Hukuhara derivative. Under this interpretation, Lupulescu established the local and global existence and uniqueness results for fuzzy delay differential equations. The second interpretation was suggested by Khastan et al. ^[29] and Hoa et al. ^[24].

In this setting, Khastan et al. proved the existence of two fuzzy solutions for fuzzy delay differential equations using the concept of generalized differentiability. Hoa et al. established the global existence and uniqueness results for fuzzy delay differential equations using the concept of generalized differentiability. Moreover, authors have extended and generalized some comparison theorems and stability theorem for fuzzy delays differential equations with definition a new Lyapunov-like function. Besides that, some very important extensions of the fuzzy delay differential equations In ^{[2,21,28,36,54]}, the authors considered the fuzzy differential equation with initial value

where f : [0, ∞) × E^d → E^d and the symbol ' denotes the first type Hukuhara derivative (classic Hukuhara derivative). O. Kaleva also discussed the properties of differentiable fuzzy mappings in ^[28] and showed that if f is continuous and f (t, x) satisfies the Lipschitz condition with respect to x, then there exists a unique local solution for the fuzzy initial value problem (1.1). V. Lupulescu proved several theorems stating the existence, uniqueness and boundedness of solutions to fuzzy differential equations with the concept of inner product on the fuzzy space under classic Hukuhara derivative in ^[36].

In ^[35], V. Lupulescu considered the fuzzy functional differential equation

Where f : [0, ∞) × C_σ → E^d and the symbol ' denotes the first type Hukuhara derivative (classic Hukuhara derivative). Author studied the local and global existence and uniqueness results for (1.2) by using the method of successive approximations and contraction principle.

In this paper, we consider fuzzy functional integration and differential equations under form

We establish the local and global existence and uniqueness results for (1.3) by using the method of successive approximations and contraction principle. This direction of research is motivated by the results of B. Bede and S. G. Gal ^[17], Chalco-Cano and Roman-Flores ^[23], Marek T. Malinowski ^{[38,39,40,41]}, Bashir Ahmad, S. Sivasundaram ^[1], T. Allahviranloo et al. ^[5,6,7].

The paper is organized as follows. In Section 2, we collect the fundamental notions and facts about fuzzy set space, fuzzy differentiation and integration. In Section 3, we discuss the FFIDEs with a two kinds of fuzzy derivative. Some examples of this class having two different solutions were presented in Section 4.

2.   Preliminaries and notation

In this section, we give some notations and properties related to fuzzy set space, and summarize the major results for integration and differentiation of fuzzy set-valued mappings. We recall some notations and concepts presented in detail in recent series works of Professor V. Lakshmikantham, et al. (see ^[33,34]).

Let K_c(R^d) denote the collection of all nonempty compact and convex subsets of R^d and scalar multiplication in K_c(R^d) as usual, i.e. for A, B ∈ K_c(R^d) and λ ∈ R.

The Hausdorff distance d_H in K_c (R^d) is defined as follows

where A, B ∈ (K_c, R^d), ${{\left\|. \right\|}_{{{R}^{n}}}}$ denotes the Euclidean norm in R^d. It is known that (K_c, R^d), d_H is a complete metric space. Denote E^d = {ω: R^d → [0, 1] such that ω(z) satisfies (ⅰ)-(ⅳ) stated below}

ⅰ. ω is normal, that is, there exists z₀ ∈ R^d such that ω(z0) = 1;

ⅱ. ω is fuzzy convex, that is, for 0 ≤ λ ≤ 1

ω(λz1 + (1 - λ)z2) ≥ min{ω(z1), ω(z2)},

for any z1, z2 ∈ R^d;

ⅲ. ω is upper semi continuous;

ⅳ. [ω]⁰ = cl{z ∈ R^d : ω(z) > 0} is compact, where cl denotes the closure in (R^d, $\left\| \cdot \right\|$).

Although elements of E^d are often called the fuzzy numbers ^[57], we shall just call them the fuzzy sets.

For α ∈ (0, 1], denote [ω]^α = {z ∈ R^d | ω(z) ≥ α}. We will call this set an α-cut (α- level set) of the fuzzy set ω. For ω ∈ E^d one has that [ω]^α ∈ Kc(Rd) for every α ∈ [0, 1]. For two fuzzys ω1, ω2 ∈ E^d, we denote ω1 ≤ ω2 if and only if [ω1]^α ⊂ [ω2]^α.

If g : R^d × R^d → R^d is a function then, according to Zadeh’s extension principle^[47,57], one can extend g to E^d × E^d → E^d by the formula g(ω1, ω2)(z) = sup z = g(z1, z2) min {ω1(z1), ω2(z2)} . It is well known that

if g is continuous then [g (ω1, ω2)]^α = g([ω1]^α, [ω2]^α) for all ω1, ω2 ∈ E^d, α ∈ [0, 1]. Especially, for addition and scalar multiplication in fuzzy set space Ed, we have [ω1 + ω2]^α = [ω1]^α + [ω2]^α, [λω1]^α = λ[ω1]^α. The notation [ω]^α = [ω(α), ω(α)]. We refer to ω and ω as the lower and upper branches of ω, respectively.

For ω∈E^d, we define the length of ω as len (ω) = ω(α) - ω(α) In the case d = 1, we have len(ω) = ω(α) - ω(α). Let us denote D₀ [ω1, ω2] = sup {d_H ([ω1]^α, [ω2]^α) : 0 ≤ α ≤ 1} the distance between ω1 and ω2 in E^d, where d_H ([ω1]^α, [ω2]^α) is Hausdorff distance between two set[ω1]^α, [ω2]^α of (K_c, R^d). Then (E^d, d_H) is a completespace. Some properties of metric D are as follows.

D₀ [ω1 + ω3, ω2 + ω3] = D₀ [ω1, ω2], D₀ [λω1, λω2] = |λ| D₀ [ω1, ω2], D₀ [ω1, ω2] ≤ D₀ [ω1, ω3] + D₀ [ω3, ω2], for all ω1, ω2, ω3 ∈ E^d and λ ∈ R. Let ω1, ω2 ∈ E^d. If there exists ω3 ∈ E^d such that ω1 = ω2 + ω3 then ω3 is called the difference of ω1, ω2 and it is denoted ω1 Ө ω2. Let us remark that ω1 Ө ω2 ≠ ω1 + (-1) ω2.

Remark 2.1. If for fuzzy numbers ω1, ω2, ω3 ∈ E^d there exist Hukuhara difference ω1 Ө ω2, ω1 Ө ω3 then D₀ [ω1 Ө ω2, 0] = D₀ [ω1, ω2] and D₀ [ω1 ω2, ω1 Ө ω3] = D₀ [ω2, ω3].

The strongly generalized differentiability was introduced in ^[17] and studied in ^{[18,23,26,32,37,42,55,56]}.

Definition 2.1. (See ^[17,48,49]) Let x : (a, b) → E^d and t ∈(a, b). We say that x is strongly generalized differentiable at t, if there exists D_H^g x(t) ∈ E^d, such that either

(ⅰ) for all h > 0 sufficiently small, the differences x (t + h) ⊖ x(t), x(t) ⊖x(t - h) exist and the limits (in the metric D₀)

or

(ⅱ) for all h > 0 sufficiently small, the difference x (t) ⊖ x(t + h), x(t - h)⊖ x(t) exist and the limits

or

(ⅲ) for all h > 0 sufficiently small, the difference x (t + h) ⊖ x(t), ∃x(t - h) ⊖ x(t) exist and the limits

(ⅳ) for all h > 0 sufficiently small, the difference x (t) ⊖ x(t + h), ∃x(t) ⊖ x(t - h) exist and the limits

In this definition, case (ⅰ) ((ⅰ)-differentiability for short) corresponds to the classic derivative, so this differentiability concept is a generalization of the Hukuhara derivative. In Ref. ^[17], B. Bede and S.G. Gal consider four cases for derivative. In this paper we consider only the two first of Definition.

In the other cases, the derivative is trivial because it is reduced to a crisp element

Lemma 2.1. (B, Bede and S. G. Gal ^[17]) If x(t) = (z1(t), z2(t), z3(t)) is triangular number valued function, then

(i) if x is (i)-differentiable (i.e. Hukuhara differentiable) then DH g x(t) = (z' 1(t), z '2(t), z '3(t));

(ii) if x is (ii)-differentiable then DH g x(t) = (z' 3(t), z' 2(t), z' 1(t)).

Lemma 2.2. (see ^[23]) Let x ∈ E¹ and put [x(t)]α = [x(t, α), x(t, α)] for each α ∈ [0, 1].

(i) If x is (i)-differentiable then x(t, α), x(t, α) are differentiable functions and we have

(ii) If x is (ii)-differentiable then x(t, α), x(t, α) are differentiable functions and we have:

Definition 2.2. ^[49,53] We say that a point t ∈ (a, b), is a switching for the differentiability of x, if in any neighborhood V of t there exist points t1 < t < t2 such that

(type Ⅰ) at t1 (2.1) holds while (2.2) does not hold and at t2 (2.2) holds and (2.1) not hold, or

(type Ⅱ) at t1 (2.2) holds while (2.1) does not hold and at t2 (2.1) holds and (2.2) not hold.

Lemma 2.3. Let a(t), b(t) and c(t) be real valued nonnegative continuous functions defined on R+, d ≥ 0 is a constant for which the inequality

hold for all t ∈ R+. Then

3.   Main results

For σ > 0 let Cσ = C([-σ, 0], Ed) denote the space of continuous mappings from [-σ, 0] to Ed. Define a metric Dσ in Cσ by

Let p > 0. Denote I = [t₀, t₀ + p], J = [t₀- σ, t₀] ∪ I = [t₀- σ, t₀ + p]. For any t ∈ I denote by the element of Cσ defined by x_t(s) = x (t + s) for s ∈ [-σ, 0].

Let us consider the fuzzy functional integration and differential equations (FFIDEs) with generalized Hukuhara derivative under form

Where f : I × Cσ → E^d, g : I × I × Cσ → E^d, x∈Cσ and the symbol D_H^g denotes the generalized Hukuhara derivative from Definition (2.1). By a solution to equation (3.1) we mean a fuzzy mapping x ∈C (J, E^d), that satisfies:

Lemma 3.1. Assume that f ∈ C (I × Cσ, E^d), g ∈C (I × I × Cσ , E^d) and x ∈ C(J, E^d). Then the fuzzy mapping

Belongs to C(I, E^d).

Remark 3.1. Under assumptions of the lemma above we have the mapping

Is integrable over the interval I.

Remark 3.2. If f : I × Cσ → E^d, g : I × I × Cσ →E^d are jointly continuous functions and x ∈ C(J, Ed), then the mapping

Is bounded on each compact interval I. Also, the function

is bounded on I.

Lemma 3.2. A fuzzy mapping x : J → E^d is called to be a local solution to the problem (3.1) on J if and only

if x is a continuous fuzzy mapping and it satisfies to one of the following fuzzy integral equations

if x is (i)-differentiable or (iii)-differentiable.

if x is (ii)-differentiable or (iv)-differentiable. Let us remark that in (3.3) it is hidden the following statement: there exists Hukuhara difference

Definition 3.1. Let x : J → E^d be a fuzzy function such that (ⅰ)-differentiable. If x and its derivative satisfy problem (3.1), we say that x is a (ⅰ)-solution of problem (3.1).

Definition 3.2. Let x : J → E^d be a fuzzy function such that (ⅱ)-differentiable. If x and its derivative satisfy problem (3.1), we say that x is a (ⅱ)-solution of problem (3.1).

Definition 3.3. A solution x : J → E^d is unique if it holds D [x(t), y(t)] = 0, for any y : J → E^d which is a solution of (3.1).

Theorem 3.1. Let φ(t - t₀) ∈ Cσ and suppose that f ∈C (I × Cσ, E^d), g ∈ C (I × I × Cσ, E^d) satisfy the condition: there exists a constant L > 0 such that for every ξ, ψ ∈ Cσ it holds

Moreover, there exists a M > 0 such that max}D₀[f(t, ξ), 0], D₀[g(t, s, ξ), 0} < = M

Assume that the sequence {xn}^∞ n = 0, xⁿ: J → E^d given by

and for n = 1, 2, ...

is well defined, i.e. the foregoing Hukuhara difference do exist. Then the FFIDE (3.1) has a unique for each case ((i)-differentiable or (ii)-differentiable).

Proof. From assumptions of this Theorem we have

for t ∈ I. Further for every n ≥ 2 and t ∈ I we get D₀ [xⁿ⁺¹(t), xⁿ(t)]

In particular, from (3.4), we get

Therefore, by mathematical induction, for every n ∈ N and t ∈ I

In the inequality (3.5), λ₁, ..., λn are balancing constants. We observe that for every n ∈ {0, 1, 2, ...}, the function xⁿ (·) : J → E^d are continuous. Indeed, since $\varphi$ ∈ Cσ, x⁰(t) is continuous on

t ∈ [-σ, t₀ + p]. We see that

Thus, by mathematical induction, for every n ≥ 2, we deduce that

D₀ [xⁿ (t + h), xⁿ(t)] → 0

as h → 0⁺. A similar inequality is obtained for D₀ [xⁿ(t -h), xⁿ(t)] → 0 as h → 0⁺. In the sequel we shall show that for the {xⁿ(t)} the Cauchy convergence condition is satisfied uniformly in t, and as a consequence{xⁿ(·)} is uniformly convergent. For n > m > 0, from (3.5) we obtain

The convergence of this series implies that for any ε > 0 we find n₀ ∈ N large enough such that for n, m > n₀

Since (E^d, D₀) is a complete metric space and (3.6) holds, the sequence {xⁿ(·)} is uniformly convergent to a mapping x ∈ C(J, E^d). We shall that x is a solution to (3.1). Since xⁿ(t) = φ(t - t₀) for every n = 0, 1, 2, ... and every t ∈ [t₀- σ, t₀], we easily have x(t) = φ (t -t₀). For s ∈ I and n ∈ N

And

As n → ∞ for any t ∈ I. Consequently, we have

We infer that

for every t ∈ I. Therefore x is the solution of (3.3), due to Lemma (3.2) we have that x is a (ⅱ)-solution of (3.1). For the uniqueness of the solution x let us assume that x, y ∈ C (J, E^d) are two solutions of (3.3). By definition of the solution we have x(t) = y(t) if t ∈ [t₀- σ, t₀].Note that for t ∈ I

If we let a(s) = sup _{r∈[s-σ, s]} D0 [x(r), y(r)] , s ∈ [t0, t] ⊂[t₀, t₀ + p], then we have

and by Lemma 2.3 we obtain that a(t) = 0 on I. This prove the uniqueness of the solution for (3.1).

Remark 3.3. The existence and uniqueness theorem for the problem (3.1) can be obtained using the contraction principle.

Now, we shall prove existence and uniqueness results for (3.1) by using the contraction principle, which studied in ^[34]. In the following, for a given k > 0, we consider the set S_k of all continuous fuzzy functions

x ∈ C ([t₀- σ, ∞), E^d) such that x(t) = φ(t - t₀) = x₀ on [t₀- σ, t₀] and

sup _t≥t0-σ {D0[x(t, ω), 0] exp(ˆ -kt) < ∞.

On S_k we can define the following metric

Where k > 0 is chosen suitably later. We easily prove that the space [Sk, Dk] of continuous fuzzy functions

x : [t₀, ∞) → E^d is a complete metric space with distance (3.7).

Theorem 3.2. Assume that

(i) f ∈ C([t₀, ∞) × Cσ, E^d), g ∈ C([t₀, ∞) ×[t₀, ∞) × Cσ, E^d) and there exists a constant L > 0 such that

(ii) $max\{ {D_0}\left[{f\left({t.\xi } \right).f\left({t.\psi } \right)} \right].\{ {D_0}\left[{g\left({t.s.\xi } \right).g\left({t.s.\psi } \right)} \right] \le L{D_\sigma }\left[{\xi.\psi } \right]$ for all ξ, ψ ∈ Cσ and t, s ≥ t0;

(iii) there exists constants M > 0 and b > 0 such that

for all t ≥ t₀, where b < k. Then the FFIDE (3.1) has a unique solution for each case on [t₀, ∞).

Proof. Since the way of the proof is similar for all four cases, we only consider case (ⅱ)-differential for x. In this case, we consider the complete metric space (S_k, D_k), and define an operator

T : Sk → Sk

x → Tx

given by

We can choose a big enough value for k such that T is a contraction, so the Banach fixed point theorem provides the existence of a unique fixed point for T, that is, a unique solution for (3.1).

Step 1: We shall prove that T(S_k) ⊂ S_k with assumption k > b. Indeed, let x ∈ S_k. For each t ≥ t₀, we get

Since x ∈ S_k, there exists ρ such that sup _t≥t0-σ {D₀ [x(t), 0] exp (-kt)} < ρ < ∞. Therefore, for all t ≥ t₀, we obtain ${{D}_{k}}~\left[~\left(\mathbb{T}x \right)\left(t \right).~0 \right]$

We infer that Tx ⊂ S_k.

Step 2: The following steps, we shall prove that T is a contraction by metric D_k. The first, we consider Let x, y ∈ S_k. Then for -σ ≤ s ≤ 0, D₀ [(Tx) (t₀ +s), (Ty)(t₀ + s)] = 0. For each t ≥ t₀, we have ${{D}_{k}}~\left[\left(\mathbb{T}x \right)\left(t \right).\left(\mathbb{T}y \right)\left(t \right) \right]$

Choosing k > b and (1 + k) L/k2 < 1, we have the operator T on S_k is a contraction by using Banach fixed point theorem provides the existence of a unique fixed point for T and the unique fixed of T is in the space S_k, that is a unique solution for (3.1) in case (ⅱ)-differentiable and for each case.

4.   Illustrations

In this section, we shall present some examples being simple illustrations of the theory of FFIDE. We will consider the FFIDE (3.1) with (ⅰ) and (ⅱ) derivative, respectively. Let us start the illustrations with considering the following fuzzy functional integration and differential equation:

Where f : I × E¹ → E¹, k(t, s) : I × I → R. Let [x(t)]^α = [x(t, α), x(t, α)]. By using Zadeh’s extension principle, we obtain [f (t, xt)]^α = [f (t, α, xt(α), xt(α)), f (t, α, xt(α), xt(α))], for α ∈ [0, 1]. By using Lemma 2.2, we have the following two cases. If x(t) is (ⅰ)-differentiable, then [D_H^g x(t)]^α = [x'(t, α), x'(t, α)] and (4.1) is translated into the following delay integration and differential system:

If x(t) is (ⅱ)-differentiable, then [D_H^g x(t)]^α = [x'(t, α), x'(t, α)] and (4.1) is translated into the following delay integration and differential system:

Example 4.1. Let us consider the linear fuzzy functional integration and differential equation under two kinds Hukuhara derivative

Where k (t, s) = λ e^(s-t), φ(t) = (1 - t, 2 - t, 3 -t), λ ∈ R\{0}. In this example we shall solve (4.4) on [0, 1/2].

Case 1: (λ > 0 or k (t, s) > 0) From (4.2), we get

Where α ∈ [0, 1]. By solving delay integration and differential systems (4.5), we obtain (ⅰ)-solution

t ∈ [0, 1/2]. The (ⅰ)-solution of (4.4) on [-1/2, 1/2] are illustrated in Figure 1. From (4.3), we obtain

By solving delay integration and differential systems (4.6), we obtain (ⅱ)-solution

t ∈ [0, 1/2]. The (ⅱ)-solution of (4.4) on [-1/2, 1/2] are illustrated in Figure 2.

Case 2: (λ < 0 or k (t, s) < 0) From (4.2), we get

By solving delay integration and differential systems (4.7), we obtain (ⅰ)-solution

t ∈ [0, 1/2]. The (ⅰ)-solution of (4.4) on [-1/2, 1/2] are illustrated in Figure 3.

From (4.3), we obtain

By solving delay integration and differential systems (4.7), we obtain (ⅱ)-solution

t ∈ [0, 1/2]. The (ⅱ)-solution of (4.4) on [-1/2, 1/2] are illustrated in Figure 4. From Example 4.1, we notice that, the solutions under classic Hukuhara derivative ((ⅰ)-differentiable) have increasing length of its values. Indeed, we can see the Figures 1 and 3.

However, if we consider the second type Hukuhara derivative ((ⅱ)- differentiable) the length of solutions change. Under the second type Hukuhara differentiable solutions have non-increasing length of its values (see Figures 2 and 4).

5.   Conclusions

In this paper, we have obtained a global existence and uniqueness result for a solution to fuzzy functional integration and differential equations. Also, we have proved a local existence and uniqueness results using the method of successive approximation. Results here might be used in further research on fuzzy functional integration and differential equations. Other possible directions of research could be an approach for fuzzy differential equations using other concepts of calculus for fuzzy functions and derivative for fuzzy functions (see ^[3,8]).

Acknowledgments

The authors would like to express his gratitude to the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper.

Conflict of interest

The authors declare no conflict of interests.