Quasilinear approximation for interval-valued functions via generalized Hukuhara differentiability

In this paper, a new generalized Hukuhara differentiability concept for interval-valued functions defined on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n}$$\end{document} is proposed, which extends the classical Fréchet differentiability notion and provides an interval quasilinear approximation for an interval-valued function in a neighborhood of a point at which such function is gH-differentiable. Moreover, it overcomes the shortcomings generated by the use of the gH-differentiability concept previously presented in the literature, and this presents a good perspective on interval and fuzzy environments. Several properties of this new concept are investigated and compared with the previous concept properties. Furthermore, the gH-differentiability concept is extended for a fuzzy function, and its introduction is argued and illustrated with examples.


Introduction
It is known that real-valued function differentiability is one of the most important concepts of real analysis, which has been allowing the rigorous formalizations of important physical Communicated by Zhong-Zhi Bai.
The research has been supported by Ministry of Economy, Knowledge, Business and University (Junta de Andalucía) and ERDF co-financing 80% FEDER Andalucía 2014-2020 (UPO-1381297). theories as well as many phenomena of real-world description through mathematical models. Moreover, the differential calculus development has provided valuable mathematical tools to several areas as computer science, biology, engineering, economics, among others.
The interval analysis is a mathematics area introduced in the works of Moore (1959), Sunaga (1958), and Warmus (1956), which have been received great highlight for providing mathematical tools that allow modelling and dealing with problems under interval type uncertainties. However, there exist some shortcomings in the algebraic operation "difference" given in these works, which do not allow developing a consistent differential theory in interval spaces, and it has motived the introduction of some different kinds of algebraic arithmetic in interval space (Lodwick 2015;Markov 1979;Plotnikova 2005 and its references).
In Markov (1979), a "difference" on interval spaces is defined, with which was introduced an interval version of the Gâteaux differentiability and was demonstrated that calculus for interval-valued functions of a real variable could be developed. Also in this paper, it was stated that such interval version of the Gâteaux differentiability is equivalent to a specific interval version of the Frechet differentiability. That statement does not hold as it is shown in that paper. Basically, the shortcoming in such statement arises from the fact of that interval spaces do not have a vector structure.
Years later, Stefanini (2008) proposed a generalization of the Hukuhara difference called the generalized Hukuhara difference (g H-difference for short) that coincides with the difference defined in Markov (1979). Using the g H-difference, Stefanini and Bede (2009) introduced a generalization of the Hukuhara differentiability for interval-valued functions of a real variable called the generalized Hukuhara differentiability (g H-differentiability for short), which is also an interval version of the Gâteaux differentiability and which coincides with the differentiability concept given in Markov (1979) (see Chalco-Cano et al. 2011 for details). Also, in Stefanini and Bede (2009), the authors compared differentiability concepts for interval-valued functions previously proposed in the literature and showed that the g H-differentiability is more general than these others. Since then, the g H-differentiability concept has been applied to different fields (Armand et al. 2016;Bede and Stefanini 2013;Chalco-Cano et al. 2012, 2013aLong et al. 2015;Majumder et al. 2016;Villamizar-Roa et al. 2015;Wang et al. 2019), and has been proved to be a powerful tool with many applications in interval and fuzzy-valued functions spaces.
In Ahmad et al. (2016), Chalco-Cano et al. (2013a, b), and Luhandjula and Rangoaga (2014), the g H-differentiability was extended for interval-valued functions of several real variables. However, we understand that these definitions have some important drawbacks which are exposed herein.
Recently, Stefanini and Arana-Jiménez (2019) introduced a new g H-differentiability concept for interval-valued functions of several real variables and they showed that such concept is equivalent to an interval version of the Gâteaux differentiability.
From the approximation theory viewpoint, it is very interesting to obtain a g Hdifferentiability definition for interval-valued functions of several real variables that extend the classical Fréchet differentiability preserving a linear approximation for an interval-valued function in a neighborhood of a point at which such function is g H-differentiable. Nevertheless, due to the nonexistent vector structure on interval spaces, such linear approximation, in general, it is not possible. In this work, we present a new g H-differentiability definition for interval-valued functions of several real variables which extends the classical Fréchet differentiability and which provides an interval quasilinear approximation for an interval-valued function in a neighborhood of a point at which such function is g H-differentiable. Also, it is shown that this definition is equivalent to the one given (Stefanini and Arana-Jiménez 2019), and for the one-dimensional case, it also coincides with definitions given by Markov (1979) and Stefanini in Stefanini (2008), and gives a correct meaning for the Markov's statement above cited.
This work is organized as follows: Sect. 2 recalls some known results about interval analysis. Section 3 shows that the condition for g H-differentiability given by Markov (1979) is only sufficient, and then, a necessary and sufficient condition for such g H-differentiability, which is the starting point for the introduction of g H-differentiability of interval-valued functions of several real variables, is introduced. Section 4 shows that some of the g Hdifferentiability previous concepts for interval-valued functions of several real variables introduced in the literature have shortcomings, and it provides a new g H-differentiability definition for interval-valued functions of several real variables that extends the classical version of Fréchet differentiability, generating a g H-differential that is a quasilinear interval-valued function. Section 4 also shows the g H-differentiability concept for interval-valued functions of several real variables given by Stefanini and Arana-Jiménez (2019) which is equivalent to the g H-differentiability one herein introduced. The extension of g H-differentiability to fuzzy environment is presented in Sect. 5. Finally, Sect. 6 presents our last considerations.

Preliminaries
Consider the space I (R) of all closed and bounded intervals of real numbers, that is, I (R) = a, a / a, a ∈ R and a ≤ a . Given A, B ∈ I (R) and λ ∈ R, the interval arithmetic operations are defined by and where μ(A) denotes the length of an interval A = [a, a], i.e., μ(A) = a − a.
We write A = [α ∨ β] if α and β are the end-points of the interval A ∈ I (R), but α ≤ β it is not necessarily satisfied. The g H-difference of two intervals always exists and it is equal (Stefanini and Bede 2009) to The space (I (R), H ) is a complete and separable metric space (Diamond and Kloeden 1994), where H (A, B) = max{|a − b|, |a − b|} is the distance between A, B ∈ I (R). The absolute value of A ∈ I (R) is the real number A given by H (A, [0, 0]) = A . Next, some known properties related to the interval arithmetic operations and to intervals length are recalled.
functions, such that f (x) ≤ f (x), for all x ∈ S, and f and f are its end-point functions. If n = 1, then F is called an interval-valued function of a real variable.
Based on the limit concept of set-valued function (Aubin and Cellina 1984), and on the g H-difference given by Stefanini (2008), the following differentiability concept for intervalvalued functions was introduced.
Definition 2.1 (Stefanini and Bede 2009) Let S ⊆ R be an open and nonempty set and let F : S → I (R) be an interval-valued function, and then, the generalized Hukuhara derivative (g H-derivative, for short) of F at x * ∈ S is defined by If F gH (x * ) ∈ I (R) satisfying (4) exists, we say that F is generalized Hukuhara differentiable (g H-differentiable, for short) at x * .

The relationships between the g H-differentiability of an interval-valued function
and the differentiability of its end-point functions, f (x) and f (x), have been completely studied in Qiu (2020). Markov (1979) states that a necessary and sufficient condition for the gH-differentiability of F at x * is the existence of an interval A ∈ I (R) and an interval-valued function P :

New necessary and sufficient conditions for gH-differentiability on R
However, this condition, in general, it is not necessary as it is shown in the following example.
. Then, f and f are differentiable functions, and consequently (see Chalco- for all x ∈ (−1, 1) and h ∈ R, such that x + h ∈ (−1, 1), then given x < 0, there exists However, does not exist any interval-valued function P : Indeed, let us suppose that there exists an interval-valued function P : for all h ∈ (− , ) such that (6) holds, then setting˜ = min{ , 1 } and given h ∈ (0,˜ ), it follows that p(h) = −h and p(h) = −2h, that is, p(h) < p(h), which contradicts the fact that P is an interval-valued function. Therefore, F is g H-differentiable, but does not exist an interval A ∈ I (R) and an interval-valued function P : (− , ) → I (R), such that (5) holds.
Although the condition given by Markov is not necessary for g H-differentiability of F at x * ∈ S ⊆ R, the next result shows that the condition is sufficient. Moreover, such result presents a relation between the left and right sides of (5) when F it is g H-differentiable at x * ∈ S ⊆ R . Before to present such result, a technical proposition is necessary. Stefanini and Bede (2009) , and from definition of gH , it follows that: Next results follow directly from Proposition 3.1.
The following result provides a necessary and sufficient g H-differentiability condition for interval-valued functions of a real variable providing a correct sense for the Markov's statement. This important result is the starting point to define the g H-differentiability for interval-valued functions of several variables.

Theorem 3.2 Let S ⊆ R be an open and nonempty set. An interval-valued function F : S → I (R) is g H -differentiable at x * ∈ S if and only if there exist an interval A ∈ I (R) and an interval-valued function
. Given > 0, let P : (− , ) → I (R) be the interval-valued function given by Then, lim h→0 P(h) = [0, 0]. Moreover, given h ∈ (− , )\{0}, it follows that: Thus, (8) holds with A = F gH (x * ). On the other hand, it is easy to see that (8) Reciprocally, if there exist an interval A ∈ I (R) and an interval-valued function P : (− , ) → I (R) with lim h→0 P(h) = [0, 0], such that (8) holds, then from Proposition 2.1 and Proposition 7 in Stefanini and Bede (2009), it follows that: Since the g H-differentiability condition given in Theorem 3.2 is an interval version of the Fréchet differentiability, this result shows that the g H-differentiability concept given in Markov (1979) and Stefanini and Bede (2009), which is an interval version of Gatêax differentiability, is equivalent to an interval version of the Fréchet differentiability for intervalvalued functions of a real variable.

gH-differentiability for interval-valued functions on R n
This section recalls the g H-differentiability concepts for interval-valued functions of several variables that were introduced in the literature, and it presents a short discussion about the shortcomings generated by such concepts. Moreover, this section presents a new g Hdifferentiability concept which overcomes the shortcomings above mentioned and it shows that this concept is equivalent to the g H-differentiability one for interval-valued functions of several variables given by Stefanini i , then we say that F has the ith partial g H -derivative at x * that is defined and denoted by of intervals is called the gradient of F and it is denoted by ∇ gH F(x * ).  and F gH is continuous at x 0 . Therefore, Definition 4.2 does not coincide with Definition 2.1. Moreover, if F : S ⊆ R → I (R) is a real-valued function, then Definition 4.2 is more restrictive than the classical real-valued differentiability concept.
Besides Theorem 3.2, our proposal of g H-differentiability concept for interval-valued functions of several variables is presented based on the following remark.

Remark 4.2
It is known that in the classical differential calculus theory, where S is an open set and V is a normed vector space, a function f : S ⊆ R → V is differentiable at x * ∈ S if and only if there exist a linear function L x * : R → V given by L x * (h) = f (x * )h and a function p : From Theorem 3.2, we have a similar situation for the g H-differentiability on R. Indeed, given F :

However, the interval-valued function T x * (h), satisfying (8), may not be linear, since I (R)
is not a vector space. For example, let us consider F : . Considering x * = 0 and given P : it follows that lim h→0 P(h) = [0, 0] and (F(0 + h) gH F(0)) gH h[−1, 0] = |h|P(h). Therefore, A = [−1, 0] = F gH (0). However, given h 1 = − 1 2 and h 2 = 1 2 , it follows that: On the other hand, it is easy to prove that if an interval-valued function F is gHdifferentiable at a point x * ∈ S ⊆ R, then the interval-valued function T x * : R → I (R) given by T x * (h) = h F gH (x * ) is a quasilinear interval-valued function in the following sense.
Clearly, the condition (C3) holds true for every x, y ∈ R n . Thus, an interval-valued function : our g H-differentiability definition for interval-valued functions of several real variables is given as follows. 0 = (0, . . . , 0) ∈ R n and S ⊆ R n an open and nonempty set, let F : S → I (R) be an interval-valued function. We say that F is g H-differentiable at x * ∈ S if there exist a continuous and quasilinear interval-valued function T x * : R n → I (R) and an interval-valued function P :

Definition 4.4 Given
where T x * : R n → I (R) is a continuous and quasilinear interval-valued function that satisfies (9).
Proof If F is g H-differentiable at x * ∈ S, then directly from Definition 4.4 and Proposition 2.1, it follows that (10) holds. Reciprocally, if (10) holds, then setting P : it follows that (9) holds with lim v→0 P(v) = [0, 0], and consequently, F is g H-differentiable at x * .
When n = 1, Definition 4.4 coincides with Definition 2.1 (see Stefanini and Bede 2009). On the other hand, recently, Stefanini and Arana-Jiménez (2019) presented the following definition and result, respectively.

Definition 4.5 Let
and let x * ∈ S, such that (x * + v) ∈ S for all v ∈ R n with v < δ for some given δ > 0. We say that F is g H-differentiable at x * ∈ S if and only if there exist two vectorsŵ = (ŵ 1 , . . . ,ŵ n ) and w = (w 1 , . . . ,w n ) in R n and two functionsˆ (v), The interval-valued function D gH F(x * ) : is called the g H-differential (or total g H-derivative) of F at x * and D gH F(x * )(v) is the interval-valued differential of F at x * with respect to v.
and let x * ∈ S such that (x * + v) ∈ S for all v ∈ R n with v < δ for some given δ > 0. Then, F is g H -differentiable at x * ∈ S if and only if there exist two vectorsŵ ∈ R n andw ∈ R n , such that the following limit condition is true In this case, for the differential function, we have that The following result allows us to know exactly the continuous and quasilinear intervalvalued function T x * given in Definition 4.4 and allows us to see that the g H-differentiability definition for interval-valued functions of several real variables given in Stefanini and Arana-Jiménez (2019) is consistent with Theorem 3.2.

Theorem 4.2 Let
Then, F is g H -differentiable at x * ∈ S in the sense of Definition 4.4 if and only if F is g H -differentiable at x * ∈ S in the sense of Definition 4.5 Proof From the unicity of the limit, Theorem 4.1, and from Proposition 4.1, it follows that for proving this result is sufficient to prove that D gH F(x * ) : R n → I (R) is quasilinear, since from its definition, it is easy to see that it is continuous.
Given v ∈ R n and λ ∈ R with λ < 0, from definition of D gH F(x * ) and from (1), it follows that: Now, given λ ∈ R with λ > 0, from definition of D gH F(x * ) and from (1), it follows that: Given u, v ∈ R n , it follows that: and Since (13) and (14), it follows that: Therefore, D gH F(x * ) : R n → I (R) is quasilinear and continuous, and consequently,

An application of gH-differentiability for interval-valued functions: extension of gH-differentiability notion to fuzzy functions with several real variables
Let us recall some basic notions about the fuzzy environment to understand the extension to fuzzy context. A fuzzy set on R n is a mapping defined as u : R n → [0, 1]. The α-level set of a fuzzy set, 0 ≤ α ≤ 1, is defined as where cl(supp u) denotes the closure of the support of u, supp(u) = {x ∈ R n | u(x) > 0}.
Definition 5.1 A fuzzy number is a fuzzy set u on R with the following properties: 1. u is normal, that is, there exists x 0 ∈ R, such that u(x 0 ) = 1; 2. u is an upper semi-continuous function; [u] 0 is compact.
Let F C be the set of all fuzzy numbers on R.
Obviously, if u ∈ F C , then [u] α ∈ K C for all α ∈ [0, 1], and thus, the α-level sets of a fuzzy number are given by [u] For fuzzy numbers, u, v ∈ F C , represented by u α , u α and v α , v α respectively, and for any real number θ , we define the following operations between fuzzy numbers: It is known that for every α ∈ [0, 1] and if u gH v exists, then, in terms of α-level sets, we can deduce that (see Stefanini and Bede (2009); Stefanini (2010)) Given u, v ∈ F C , we define the distance between u and v as Therefore, (F C , D) is a complete metric space. We recall the usual order relations between fuzzy numbers (Osuna-Gómez et al. 2016): (2) u v if u v and u = v, i. e.
[u] α [v] α for every α ∈ [0, 1], and ∃α 0 ∈ [0, 1], such that Note that is a partial order relation on F C . Hence, v u can be written instead of u v. We observe that if u ≺ v, then u v and, therefore, u v.
Let us considerf : S ⊆ R n → F C a fuzzy function or fuzzy mapping where S is an open and nonempty subset of R n . We associate withf the family of interval-valued functions In Bede and Stefanini (2013), we find the g H-differentiability and level-wise g Hdifferentiability notions for fuzzy functions of a single variable. The g H-differentiability for fuzzy functions on R is a more general concept than the H-differentiability and Gdifferentiability, but less general than level-wise g H-differentiability, that is based on the g H-differentiability of every interval-valued function f α .
Let us consider the definition of level-wise g H-differentiable fuzzy function when n ≥ 1.

Definition 5.3 Let us considerf
for each α ∈ [0, 1] and let x * ∈ S such that x * + v ∈ S for all v ∈ R n with ||v|| < ( > 0). We say thatf is level-wise g H-differentiable at x * ∈ S if and only if for each α ∈ [0, 1], there exist two interval-valued functions, T α x * continuous and quasilinear and P α with lim v→0 P α (v) = [0, 0], such that When S ⊆ R (n = 1), from Theorem 3.2, this definition coincides with Definition 23 in Bede and Stefanini (2013). In the general case, S ⊆ R n with n > 1, from Theorem 4.2, it coincides with Definition 9-A in Stefanini and Arana-Jiménez (2019).

Conclusions
In this paper, we present and study a g H-differentiability definition for interval-valued functions of several real variables, which extends the classical Fréchet differentiability notion.
This concept herein introduced overcomes some drawbacks of the g H-differentiability definition extended for interval-valued functions of several real variables in the previous literature, generating a g H-differential that is a quasilinear interval-valued function.
The g H-differentiability concept given by Markov (1979), Stefanini and Bede (2009), and Stefanini and Arana-Jiménez (2019) is extension of the classical Gâteaux differentiability to interval-valued functions. We prove that these concepts and the new definition, which is equivalent to an interval version of the Fréchet differentiability, are equivalents.
The extension of the new concept of differentiability to the fuzzy environment is presented.
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