On derivatives of fuzzy multi-dimensional mappings and applications under generalized differentiability

Abstract

This article identifies and presents the generalized difference (g-difference) of fuzzy numbers, Fréchet and Gâteaux generalized differentiability (g-differentiability) for fuzzy multi-dimensional mapping which consists of a new concept, fuzzy g-(continuous linear) function; Moreover, the relationship between Fréchet and Gâteaux g-differentiability is studied and shown. The concepts of directional and partial g-differentiability are further framed and the relationship of which will the aforementioned concepts are also explored. Furthermore, characterization is pointed out for Fréchet and Gâteaux g-differentiability; based on level-set and through differentiability of endpoints real-valued functions a characterization is also offered and explored for directional and partial g-differentiability. The sufficient condition for Fréchet and Gâteaux g-differentiability, directional and partial g-differentiability based on level-set and through employing level-wise gH-differentiability (LgH-differentiability) is expressed. Finally, to illustrate the ability and reliability of the aforementioned concepts we have solved some application examples.

Keywords

Fuzzy multi-dimensional mappings g-(linear continuous) function g-differentiability Fréchet g-derivative Gâteaux g-derivative Directional g-derivative Partial g-derivative

1 Introduction

Many studies have focused on the theoretical and practical aspects of the fuzzy sets since Zadeh [19] began the basic concepts and definitions of fuzzy theory. Zadeh proposed the concept of fuzzy numbers in [20, 21] so that the fuzzy numbers have been extensively researched by many researchers so far. In mathematical models, fuzzy numbers are used as a tool for modeling uncertainty, processing ambiguous concepts, or dealing with subjective information. Fuzzy numbers have been developed in different ways and are used in many different practical problems such as fuzzy optimization [1], fuzzy transport problems [25, 31], fuzzy differential equations [8 , 32], etc. Subsequently, several researchers have begun to study and use the differentiability and integrability of fuzzy mapping to develop fuzzy numbers theory and its applications. First, Puri and Ralescu [24] began to develop the fuzzy mapping derivative from an open subset of a normed space into the n-dimensional fuzzy numbers, and then they generalized the concept of Hukuhara differentiability to set-valued mappings. In 1987, Kaleva [26] discussed fuzzy differentiability and considered the necessary and sufficient conditions for H-differentiability of fuzzy mappings from [a, b] into E. In 2003, Wang and Wu [9] proposed fuzzy mappings from $ℝ^{n}$ into E by using H-difference, directional derivative, differentiability, and subdifferential. However, the H-difference between two fuzzy numbers occurs only in very restricted conditions [26] and the H-difference between two fuzzy numbers cannot always come into existence [12, 13]. Then the gH-difference between two fuzzy numbers is defined under less restrictive conditions, although the gH-difference is not always existed [11, 12]. As a result, the generalized difference proposed in [3, 13] as well as in [10], by Gomes and Barros, eliminates all the short-comes discussed in the above differences so that the g-difference of two fuzzy numbers always exists [3, 14]. In 2013, Bede and Stefanini studied and introduced the concept of generalized differentiability for fuzzy-valued functions by using the new generalization of gH-difference. We introduce the concept of g-differentiability for fuzzy multi-dimensional mappings based on the concept of g-differentiability [3], so that it exists for a larger class of fuzzy multi-dimensional mappings. We present the concepts of Fréchet and Gâteaux g-differentiability, directional and partial g-differentiability for fuzzy multi-dimensional mappings. In this regard, characterization is appointed for Fréchet and Gâteaux g-differentiability; based on level-set and through differentiability of endpoints real-valued functions a characterization is also offered and explored for directional and partial g-differentiability. The sufficient condition for Fréchet and Gâteaux g-differentiability, directional and partial g-differentiability in terms of level-set and through employing uniformity of level-wise gH-differentiability (L_gH-differentiability) is expressed. Fréchet and Gâteaux gH-differentiability, directional and partial gH-differentiability are introduced for interval-valued functions [7, 18], however, the advantage of our work in this paper is that all the concepts of fuzzy differentiability that were already mentioned, are investigated for fuzzy multi-dimensional mappings using the g-differentiability based on the concept of fuzzy g-(continuous linear) function. The purpose of this paper is to implement the fuzzy g-differentiability proposed by Bede and Stefanini [3] and to extend it to the g-differentiability of fuzzy multi-dimensional mappings. This article is divided into the following:

In Section 3, the generalized differentiability and the concept of Fréchet and Gâteaux g-differentiability based on the new concept of fuzzy g-(continuous linear) function for fuzzy multi-dimensional mappings are presented, as well as the relationship between Fréchet and Gâteaux g-derivative and their properties have been expressed and discussed. we present the directional generalized differentiability and its properties in section 4. Section 5, the partial generalized differentiability and its relation to the Fréchet and directional g-differentiability are stated, and eventually, for better understanding and showing of the strength of the relationships between the above concepts, several practical examples are provided.

2 Preliminaries

The basic definitions and theorems that we need in this article are considered. In addition, some new concepts are proved here.

We denote E, the set of fuzzy numbers, that is, normal, fuzzy convex, upper semi-continuous, and compactly supported fuzzy sets that are defined over the real line. Let u ∈ E be a fuzzy number; for 0 < r ≤ 1, the r-level set (or r-cut) of u is defined by $[u]_{r} = {x \in ℝ^{n} | u (x) \geq r}$ , and for r = 0 is defined by the closure of support $[u]_{0} = cl {x \in ℝ^{n} | u (x) > 0}$ . We denote $[u]_{r} = [u_{r}^{-}, u_{r}^{+}]$ , so the r-level set [u] _r is a closed interval for all r ∈ [0, 1].

If u, v ∈ E and $λ \in ℝ$ , the addition and scalar multiplication are defined as having the r-levels of [u + v] _r = [u] _r + [v] _r and [λu] _r = λ [u] _r, respectively.

A trapezoidal fuzzy number, denoted by u = 〈a, b, c, d〉, where a ≤ b ≤ c ≤ d, has level-set [u] _r = [a + r (b - a) , d - r (d - c)] for 0 ≤ r ≤ 1; if b = c we have a triangular fuzzy number. The support of fuzzy number u is defined as follows: $supp (u) = cl {x \in ℝ^{n} | u (x) > 0},$ where cl is the closure of set ${x \in ℝ^{n} | u (x) > 0}$ . Here, conv (X) denotes the convex hull of set X. We indicate Eⁿ as the set of fuzzy numbers [5 , 27]. It is clear that any $u \in ℝ^{n}$ can be regarded as a fuzzy number $\tilde{u}$ defined by $\tilde{u} (x) = {\begin{matrix} 1, x = u, \\ 0, otherwise . \end{matrix}$ Particularly, the fuzzy number $\tilde{0}$ is expressed as $\tilde{0} (x) = 1$ if x = 0, otherwise $\tilde{0} (x) = 0$ .

Theorem 2.1. [5, 26] If u ∈ Eⁿ, then

[u] _r is a nonempty compact convex subset of $ℝ^{n}$ for any r ∈]0, 1],

[u] _r
₁ ⊆ [u] _r
₂, whenever 0 ≤ r₂ ≤ r₁ ≤ 1,

if r_k > 0 and r_k is a nondecreasing sequences converging to r ∈]0, 1], then $⋂_{k = 1}^{\infty} [u]_{r_{n}} = [u]_{r}$ .

Conversely, if ${[A]_{r} \subseteq ℝ^{n} : r \in [0, 1]}$ verifies the conditions (1)-(3), then there exist a unique u ∈ Eⁿ such that [u] _r = [A] _r for each r ∈]0, 1] and [u] ₀ = cl (⋃ _r∈(0,1) [u] _r) ⊆ A₀.

Let u, v ∈ Eⁿ and $k \in ℝ$ . For any $x \in ℝ^{n}$ , the addition u + v and scalar multiplication ku can be defined, respectively, as: $(u + v) (x) = sup_{s + t = x} min {u (s), v (t)},$ (1) $(ku) (x) = u (\frac{x}{k}), k \neq 0,$ (2)

It is obvious that for any u, v ∈ Eⁿ and $k \in ℝ$ , the addition u + v and the scalar multiplication ku have the level-sets ${[u + v]}_{r} = {[u]}_{r} + {[v]}_{r} = {x + y : x \in {[u]}_{r}, y \in {[v]}_{r}},$ (3) ${[k u]}_{r} = k {[u]}_{r} = {k x : x \in {[u]}_{r}} .$ (4) For x = (x₁, x₂, . . . , x_n), $y = (y_{1}, y_{2}, . . ., y_{n}) \in ℝ^{n}$ , d (x, y) denote the Euclidean metric between x and y, and ∥x∥ denote the Euclidean norm of x. Let $K_{C}^{n}$ be the space of non empty compact and convex sets of $ℝ^{n}$ . The gH-difference of two sets $A, B \in K_{C}^{n}$ (gH-difference for short), which we recall from [3 , 27], is defined as follows: $A \underset{gH}{⊖} B = C \Leftrightarrow {\begin{matrix} (a) A = B + C, \\ or (b) B = A + (- 1) C . \end{matrix}$ Note that the difference of sets and itself is zero, that is $A \underset{gH}{⊖} A = {0}$ . Also, the gH-difference of two intervals always exists and is equal to $\begin{matrix} A \underset{gH}{⊖} B = [min {a^{-} - b^{-}, a^{+} - b^{+}}, \\ max {a^{-} - b^{-}, a^{+} - b^{+}}] . \end{matrix}$

Definition 2.1. [13] Let u, v ∈ E be two fuzzy numbers. Then the gH-difference is the fuzzy number w, if it exists, such that $u \underset{gH}{⊖} v = w \Leftrightarrow {\begin{matrix} (i) u = v + w, \\ or (ii) v = u + (- 1) w . \end{matrix}$

In term of r-levels we have $[u \underset{gH}{⊖} v]_{r} = [min {u_{r}^{-} - v_{r}^{-}, u_{r}^{+} - v_{r}^{+}}, max {u_{r}^{-} - v_{r}^{-}, u_{r}^{+} - v_{r}^{+}}]$ , and if the H-difference exists, then $u ⊖ v = u \underset{gH}{⊖} v$ . The conditions for the existence of $w = u \underset{gH}{⊖} v \in E$ are case (i) ${\begin{matrix} w_{r}^{-} = u_{r}^{-} - v_{r}^{-} and w_{r}^{+} = u_{r}^{+} - v_{r}^{+}, \forall r \in [0, 1], \\ with w_{r}^{-} increasing, w_{r}^{+} decreasing, w_{r}^{-} \leq w_{r}^{+}, \end{matrix}$ case (ii) ${\begin{matrix} w_{r}^{-} = u_{r}^{+} - v_{r}^{+} and w_{r}^{+} = u_{r}^{-} - v_{r}^{-}, \forall r \in [0, 1], \\ with w_{r}^{-} increasing, w_{r}^{+} decreasing, w_{r}^{-} \leq w_{r}^{+} . \end{matrix}$ It is easy to show that (i) and (ii) are both valid if and only if w is a crisp number.

Definition 2.2. [3] Let u, v ∈ E, write u ⪯ v ⇔ $[u]_{r} = [u_{r}^{-}, u_{r}^{+}]$ ≤ $[v]_{r} = [v_{r}^{-}, v_{r}^{+}]$ for any r ∈ [0, 1]. And [u] _r ≤ [v] _r ⇔ $u_{r}^{-} \leq v_{r}^{-}$ and $u_{r}^{+} \leq v_{r}^{+}$ . And u ≺ v ⇔ u ⪯ v and u ≠ v.

Definition 2.3. [3] The generalized difference (or g-difference for short) of two fuzzy number u, v ∈ E is defined by its level sets as $[u \underset{g}{⊖} v]_{r} = cl (conv ⋃_{β \geq r} [u]_{β} \underset{gH}{⊖} [v]_{β}), \forall r \in [0, 1],$ (5) where the gH-difference $\underset{gH}{⊖}$ is with interval operands [u] _β and [v] _β.

Proposition 2.1. [3]The g-difference (5) is given by the expression $[u \underset{g}{⊖} v]_{r} = [inf_{β \geq r} min {u_{β}^{-} - v_{β}^{-}, u_{β}^{+} - v_{β}^{+}}, sup_{β \geq r} max {u_{β}^{-} - v_{β}^{-}, u_{β}^{+} - v_{β}^{+}}] .$

The following proposition gives a simplified notation for $u \underset{g}{⊖} v$ and $v \underset{g}{⊖} u$ .

Proposition 2.2. [3] For any two fuzzy numbers u, v ∈ E the two g-difference $u \underset{g}{⊖} v$ and $v \underset{g}{⊖} u$ exist and, for any r ∈ [0, 1], we have $u \underset{g}{⊖} v = - (v \underset{g}{⊖} u)$ with $[u \underset{g}{⊖} v]_{r} = [d_{r}^{-}, d_{r}^{+}] and [v \underset{g}{⊖} u]_{r} = [- d_{r}^{+}, - d_{r}^{-}],$ where $d_{r}^{-} = inf (D_{r}), d_{r}^{+} = sup (D_{r})$ and the sets D_r are $D_{r} = {u_{β}^{-} - v_{β}^{-} | β \geq r} \cup {u_{β}^{+} - v_{β}^{+} | β \geq r} .$

Remark 2.1. [3] We assume that $u \underset{gH}{⊖} v \in E$ and furthermore $u \underset{gH}{⊖} v = u \underset{g}{⊖} v$ .

Proposition 2.3. [3] Let u, v ∈ E be two fuzzy numbers; then

$u \underset{g}{⊖} v = u \underset{gH}{⊖} v$ , whenever the expression on the right exists; in particular $u \underset{g}{⊖} u = 0$ ,

$(u + v) \underset{g}{⊖} v = u$ ,

$0 \underset{g}{⊖} (u \underset{g}{⊖} v) = v \underset{g}{⊖} u$ ,

$u \underset{g}{⊖} v = v \underset{g}{⊖} u = w$ if and only if w = - w; furthermore, w = 0 if and only if u = v.

Definition 2.4. Let φ : K ⊆ X ⟶ E be a fuzzy mapping. The fuzzy mapping φ is said to be fuzzy g-continuous at x⁰ ∈ K, if for any h ∈ X with x⁰ + h ∈ K, we have $lim_{h \to 0} φ (x^{0} + h) \underset{g}{⊖} φ (x^{0}) = 0 .$

Definition 2.5. Let φ : X ⟶ E be a fuzzy mapping. The fuzzy mapping φ is said to be fuzzy g-linear function if

φ (λx) = λ ⊙ φ (x), for all $λ \in ℝ$ and

$φ (x_{1} + x_{2}) \underset{g}{⊖} φ (x_{1}) + φ (x_{2}) = 0$ , for all x₁, x₂ ∈ X.

Definition 2.6. Let (X, ∥ . ∥ ) be a normed space and L (X, E) = {Λ : X ⟶ E | Λ isafuzzyg - (continuous linear) function}. Where we denote L (X, E) is the set of all fuzzy g-(continuous linear) functions from X in to E.

Note that $L (ℝ^{n}, E) = {Λ : ℝ^{n} ⟶ E | Λ is a fuzzy$ g-(continuous linear) function} = {U = (u₁, u₂, . . . , u_n) |u_i ∈ E, i = 1, 2, . . . , n} = Eⁿ .

Note 2.1. The set of all fuzzy g-(continuous linear) functions from X into E, i.e., L (X, E), in a special case Eⁿ equipped with the norm ∥ . ∥ _E = D (. , 0) is a normed quasi-linear space with respect to the operations ${+, \underset{g}{⊖}, ⊙}$ .

Definition 2.7. [3] The Hausdorff distance on E is defined by $D (u, v) = sup_{r \in [0, 1]} {∥ [u]_{r} \underset{gH}{⊖} [v]_{r} ∥_{*}},$ where, for an interval [a, b], the norm is $∥ [a, b] ∥_{*} = max {| a |, | b |} .$ The metric D is well defined since the gH-difference of intervals, $[u]_{r} \underset{gH}{⊖} [v]_{r}$ always exists. Also, this allows us to deduce that (E, D) is a complete metric space.

Definition 2.8. [18] We denote by $K_{C}$ the collection of all bounded closed intervals in $ℝ$ , i.e., $K_{C} = {[a^{-}, a^{+}] \in ℝ and a^{-} \leq a^{+}} .$

Proposition 2.4. [3] For all u, v ∈ E we have $D (u, v) = sup_{r \in [0, 1]} ∥ [u]_{r} \underset{gH}{⊖} [v]_{r} ∥_{*} = ∥ u \underset{g}{⊖} v ∥,$ where ∥ . ∥ = D (. , 0) .

Remark 2.2. [3] We observe that since $u \underset{g}{⊖} v = u \underset{gH}{⊖} v$ whenever the right side exists we can also conclude $D (u, v) = ∥ u \underset{g}{⊖} v ∥ = ∥ u \underset{gH}{⊖} v ∥,$ whenever $u \underset{gH}{⊖} v$ exists.

Proposition 2.5. [3] A fuzzy number u is completely determined by any pair u = (u^-, u⁺) of functions $u^{-}, u^{+}; [0, 1] ⟶ ℝ$ , defining the end-points of the level-set, satisfying the three conditions:

$u^{-} : r ⟶ u_{r}^{-} \in ℝ$ is a bounded monotonic nondecreasing left continuous function ∀ r ∈]0, 1] and right continuous for r = 0;

$u^{+} : r ⟶ u_{r}^{+} \in ℝ$ is a bounded monotonic nondecreasing left continuous function ∀ r ∈]0, 1] and right continuous for r = 0;

$u_{1}^{-} \leq u_{1}^{+}$ for r = 1, which implies $u_{r}^{-} \leq u_{r}^{+}$ ∀ r ∈]0, 1].

Proposition 2.6. [3] Let {U_r|r ∈]0, 1]} be a family of real intervals such that the following three conditions are satisfied:

U_r is a nonempty compact interval ∀r ∈]0, 1];

if 0 < r < β ≤ 1 then U_β ⊆ U_r;

given any nondecreasing sequence r_n ∈]0, 1] with $lim_{n \to \infty} r_{n} = r > 0$ it is $U_{r} = ⋂_{n = 1}^{\infty} U_{r_{n}}$ .

Then three exists a unique LU-fuzzy quantity u such that [u] _r = U_r, ∀r ∈]0, 1] and [u] ₀ = cl (⋃ _r∈]0,1]U_r).

Lemma 2.1. [3] Let $φ : ℝ ⟶ E$ be a fuzzy number-valued function. Let $x^{0} \in ℝ$ . Then if

$lim_{x \to x^{0}} [φ (x)]_{r} = U_{r} = [u_{r}^{-}, u_{r}^{+}]$ uniformly with respect to r ∈ [0, 1],

$u_{r}^{-}, u_{r}^{+}$ fulfill the conditions in Proposition 2.5 or equivalently U_r fulfill the conditions in Proposition 2.6, then $lim_{x \to x^{0}} φ (x) = u$ , with $[u]_{r} = U_{r} = [u_{r}^{-}, u_{r}^{+}]$ .

Remark 2.3. [3] We observe that there are other possible different expressions for the g-difference as e.g., $[u \underset{g}{⊖} v]_{r} = [min {inf_{β \geq r} (u_{β}^{-} - v_{β}^{-}), inf_{β \geq r} (u_{β}^{+} - v_{β}^{+})}, max {sup_{β \geq r} (u_{β}^{-} - v_{β}^{-}), sup_{β \geq r} (u_{β}^{+} - v_{β}^{+})}] .$ The next proposition shows that the g-difference is well defined for any two fuzzy numbers u, v ∈ E.

Proposition 2.7. [13] For any fuzzy numbers u, v ∈ E the g-difference $u \underset{g}{⊖} v$ exists and it is a fuzzy number.

Definition 2.9. [3] Let x⁰ ∈] a, b [ and h be such that x⁰ + h ∈] a, b [, then the gH-derivative of a function φ :] a, b [→ E at x⁰ is defined as $φ_{gH}^{'} (x^{0}) = lim_{h \to 0} \frac{1}{h} [φ (x^{0} + h) \underset{gH}{⊖} φ (x^{0})] .$ (6) If $φ_{gH}^{'} (x^{0}) \in E$ satisfying (6) exists, we say that φ is generalized Hukuhara differentiable (gH-differentiable for short) at x⁰.

Definition 2.10. [3] Let x⁰ ∈] a, b [ and h be such that x⁰ + h ∈] a, b [, then the level-wise gH-derivative (L_gH-derivative for short) of a function φ :] a, b [⟶ E at x⁰ is defined as the set of interval-valued gH-derivatives, if they exist, $φ_{L_{gH}}^{'} (x^{0})_{r} = lim_{h \to 0} \frac{1}{h} [φ (x^{0} + h)]_{r} \underset{gH}{⊖} [φ (x^{0})]_{r} .$ (7) If $φ_{L_{gH}}^{'} (x^{0})_{r}$ is a compact interval for all r ∈ [0, 1], we say that φ is level-wise generalized Hukuhara differentiable (L_gH-differentiable for short) at x⁰ and the collection of intervals ${φ_{L_{gH}}^{'} (x^{0})_{r} | r \in [0, 1])}$ is the L_gH-derivative of φ at x⁰ denoted by $φ_{L_{gH}}^{'} (x^{0})$ .

Definition 2.11. [3] Let φ : [a, b] ⟶ E and x⁰ ∈ (a, b), with $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ both differentiable at x⁰. Hence we say that

φ is [i - gH]-differentiable at x⁰ if $[φ_{i - gH}^{'} (x^{0})]_{r} = [(φ_{r}^{-})^{'} (x^{0}), (φ_{r}^{+})^{'} (x^{0})], r \in [0, 1],$ (8)

φ is [ii - gH]-differentiable at x⁰ if $[φ_{ii - gH}^{'} (x^{0})]_{r} = [(φ_{r}^{+})^{'} (x^{0}), (φ_{r}^{-})^{'} (x^{0})], r \in [0, 1] .$ (9)

Proposition 2.8. [3] The (i)-gH derivative and (ii)-gH derivative are additive functions, i.e., if φ and g are both (i)-gH-differentiable or both (ii)-gH-differentiable then

$(φ + g)_{(i) - gH}^{'} = φ_{(i) - gH}^{'} + g_{(i) - gH}^{'}$ ,

$(φ + g)_{(ii) - gH}^{'} = φ_{(ii) - gH}^{'} + g_{(ii) - gH}^{'}$ .

Remark 2.4. [3] From Proposition 2.8, it follows that (i)-gH derivative and (ii)-gH derivative are semi-linear functions (i.e., additive and positive homogeneous). They are not linear in general since we have $(k φ_{gH})_{(i) - gH}^{'} = k (φ_{gH})_{(ii) - gH}^{'}$ if k < 0.

Theorem 2.2. [3] Let φ :] a, b [⟶ E be such that $[φ (x)]_{r} = [φ_{r}^{-} (x), φ_{r}^{+} (x)]$ . Suppose that the functions $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ are real-valued functions, differentiable with respect to x, uniformly with respect to r ∈ [0, 1]. Then the function φ (x) is gH-differentiable at a fix x ∈] a, b [ if and only if one of the following two cases holds:

$(φ_{r}^{-})^{'} (x)$ is increasing, $(φ_{r}^{+})^{'} (x)$ is decreasing as function of r, and $(φ_{1}^{-})^{'} (x) \leq (φ_{1}^{+})^{'} (x)$ , or

$(φ_{r}^{-})^{'} (x)$ is decreasing, $(φ_{r}^{+})^{'} (x)$ is increasing as function of r, and $(φ_{1}^{+})^{'} (x) \leq (φ_{1}^{-})^{'} (x)$ .

Also, ∀ r ∈ [0, 1] we have $[φ_{gH}^{'} (x)]_{r} = [min {(φ_{r}^{-})^{'} (x), (φ_{r}^{+})^{'} (x)}, max {(φ_{r}^{-})^{'} (x), (φ_{r}^{+})^{'} (x)}] .$

Definition 2.12. [3] Let x⁰ ∈] a, b [ and h be such that x⁰ + h ∈] a, b [, then the g-derivative of a function φ :] a, b [⟶ E at x⁰ is defined as $φ_{g}^{'} (x^{0}) = lim_{h \to 0} \frac{1}{h} [φ (x^{0} + h) \underset{g}{⊖} φ (x^{0})] .$ (10) If $φ_{g}^{'} (x^{0}) \in E$ satisfying (10) exists, we say that φ is generalized differentiable (g-differentiable for short) at x⁰.

Theorem 2.3. [3] Let φ : [a, b] ⟶ E be such that $[φ (x)]_{r} = [φ_{r}^{-} (x), φ_{r}^{+} (x)]$ . If $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ are differentiable real-valued functions with respect to x, uniformly for r ∈ [0, 1], then φ (x) is g-differentiable and we have $[φ_{g}^{'} (x)]_{r} = [inf_{β \geq r} min {(φ_{β}^{-})^{'} (x), (φ_{β}^{+})^{'} (x), sup_{β \geq r} max {(φ_{β}^{-})^{'} (x), (φ_{β}^{+})^{'} (x)}] .$

Theorem 2.4. [3] Let φ :] a, b [⟶ E be uniformly L_gH-differentiable at x⁰. Then φ is g-differentiable at x⁰ and for any r ∈ [0, 1], $[φ_{g}^{'} (x^{0})]_{r} = cl (conv ⋃_{β \geq r} φ_{LgH}^{'} (x^{0})_{β}) .$

Definition 2.13. A mapping $φ : K \subseteq ℝ^{n} ⟶ E$ is called to be fuzzy mapping. For each r ∈ [0, 1], we associate with φ the collection of interval-valued functions $φ : K ⟶ K_{C}$ given by φ (x) _r = [φ (x)] _r, for all r ∈ [0, 1], we denote $φ (x)_{r} = [φ_{r}^{-} (x), φ_{r}^{+} (x)]$ . Here, the endpoint $φ_{r}^{-}, φ_{r}^{+} : K \subseteq ℝ^{n} ⟶ ℝ$ are called lower and upper mappings of φ.

3 g-differentiability of fuzzy multi-dimensional mappings

In this segment, a concept of the g-differentiability for fuzzy multi-dimensional functions is introduced. From now on, K indicates an open subset of $ℝ^{n}$ . In the following, using the new g-difference the g-differentiability of Fréchet and Gâteaux for fuzzy multi-dimensional mappings on $K \subseteq ℝ^{n}$ are defined and the relation between them is discussed.

Definition 3.1. Let $(ℝ^{n}, ∥ . ∥)$ be a normed space. Let $K \subseteq ℝ^{n}$ and let x⁰ ∈ K, $h \in ℝ^{n}$ be such that x⁰ + h ∈ K for all $h \in ℝ^{n}$ with ∥h ∥ < δ for a given δ > 0. We say that the fuzzy mapping $φ : K \subseteq ℝ^{n} ⟶ E$ is Fréchet generalized differentiable (F_g-differentiable for short) at x⁰, if there exist a fuzzy g-(continuous linear) function U = (u₁, u₂, . . . , u_n) ∈ Eⁿ such that $lim_{h \to 0} \frac{(φ (x^{0} + h) ⊖_{g} φ (x^{0})) ⊖_{g} U ⊙ h}{∥ h ∥} = 0 .$ (11) If U ∈ Eⁿ uniquely determined by (11). In this case, U ∈ Eⁿ is called Fréchet generalized derivative (F_g-derivative for short) of φ at x⁰ and denoted by $φ_{F_{g}}^{'} (x^{0})$ , i.e., $φ_{F_{g}}^{'} (x^{0}) = U$ . Notice that, the above definition is equivalent to the following definition.

Definition 3.2. Let $(ℝ^{n}, ∥ . ∥)$ be a normed space. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping and let x⁰ ∈ K be such that x⁰ + h ∈ K, for any $h \in ℝ^{n}$ with ∥h ∥ < δ for a given δ > 0. We say that φ is F_g-differentiable at x⁰ if and only if there exist a fuzzy g-(continuous linear) U = (u₁, u₂, . . . , u_n) ∈ Eⁿ and fuzzy function ɛ (h) with $lim_{h \to 0} ɛ (h) = 0$ , such that, for any $0 \neq h \in ℝ^{n}$ , $φ (x^{0} + h) \underset{g}{⊖} φ (x^{0}) = U ⊙ h + ∥ h ∥ ⊙ ɛ (h),$ also we get η (h) = ∥ h ∥ ⊙ ɛ (h), we have $φ (x^{0} + h) \underset{g}{⊖} φ (x^{0}) = U ⊙ h + η (h),$ where η (h) →0 when h → 0. The fuzzy g-(continuous linear) function $φ_{F_{g}}^{'} : ℝ^{n} ⟶ E$ defined, for $h = (h_{1}, h_{2}, . . ., h_{n}) \in ℝ^{n}$ , by $φ_{F_{g}}^{'} (x^{0}) ⊙ h = U ⊙ h = \sum_{i = 1 h_{i} \geq 0}^{n} h_{i} ⊙ u_{i} \underset{g}{⊖} \sum_{i = 1 h_{i} < 0}^{n} | h_{i} | ⊙ u_{i},$ is called the F_g-differential of φ at x⁰ and $φ_{F_{g}}^{'} (x^{0}) ⊙ h$ is the fuzzy g-(continuous linear) function F_g-differential of φ at x⁰ with respect to h.

Using the same concept of fuzzy g-(continuous linear) function on Eⁿ, the definition of Gâteaux generalized derivative is as follows:

Definition 3.3. Let $K \subseteq ℝ^{n}$ and x⁰ ∈ K, $y \in ℝ^{n}$ , there exists δ > 0 such that x⁰ + αy ∈ K for any real number α ∈ (0, δ). We say that the fuzzy mapping $φ : K \subseteq ℝ^{n} ⟶ E$ is Gâteaux generalized differentiable (G_g-differentiable for short) at x⁰, if there exist a fuzzy g-(continuous linear) function U = (u₁, u₂, . . . , u_n) ∈ Eⁿ such that for all $y \in ℝ^{n}$ $lim_{α \to 0^{+}} \frac{φ (x^{0} + α y) ⊖_{g} φ (x^{0})}{α} = U ⊙ y .$ (12) If U ∈ Eⁿ is uniquely determined by (12) (by unicity of the limit for all $y \in ℝ^{n}$ ), and U = (u₁, u₂, . . . , u_n) ∈ Eⁿ is called Gâteaux generalized derivative (G_g-derivative for short) of φ at x⁰, and denoted by ∇_gφ (x⁰) = (u₁, u₂, . . . , u_n) ∈ Eⁿ, i.e., ∇_gφ (x⁰) = U.

Example 3.1. Let $φ : ℝ^{2} ⟶ E$ be defined by $φ (x_{1}, x_{2}) = U_{1} ⊙ x_{1} + U_{2} ⊙ (x_{2}^{2} e^{x_{1}}),$ where U₁, U₂ ∈ E. Given x⁰ = (0, 0) and for any $y = (y_{1}, y_{2}) \in ℝ^{2}$ indeed, we have $\nabla_{g} φ (0, 0) ⊙ (y_{1}, y_{2}) = lim_{α \to 0^{+}} \frac{φ ((0, 0) + α (y_{1}, y_{2})) ⊖_{g} φ (0, 0)}{α} = lim_{α \to 0^{+}} \frac{(α ⊙ U_{1} ⊙ y_{1} + α^{2} ⊙ U_{2} ⊙ (y_{2}^{2} e^{α y_{1}}))}{α} = U_{1} ⊙ y_{1} .$ Clearly ∇_gφ (0, 0) on $ℝ^{2}$ is fuzzy g-(continuous linear) in y₁. So, φ is G_g-differentiable at x⁰ = (0, 0).

Theorem 3.1. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping and x⁰ ∈ K. If the F_g-derivative of φ at x⁰ exists, then the G_g-derivative of φ at x⁰ for any $y \in ℝ^{n}$ exists and both the g-derivative values are equal.

Proof. Let $φ_{F_{g}}^{'} (x^{0})$ be the F_g-derivative of φ at x⁰, enough we get h = αy, α > 0 and $y \in ℝ^{n}$ , then we have $lim_{α \to 0^{+}} \frac{(φ (x^{0} + α y) ⊖_{g} φ (x^{0})) ⊖_{g} φ_{F_{g}}^{'} (x^{0}) ⊙ (α y)}{∥ α y ∥} = 0, \forall y \in ℝ^{n},$ or $lim_{α \to 0^{+}} \frac{(φ (x^{0} + α y) ⊖_{g} φ (x^{0})) ⊖_{g} φ_{F_{g}}^{'} (x^{0}) ⊙ (α y)}{α}$ (13) $= 0, \forall y \in ℝ^{n} .$ By the hypothesis since $φ_{F_{g}}^{'} (x^{0})$ with respect to y is g-(continuous linear) function, and then $φ_{F_{g}}^{'} (x^{0}) ⊙ (α y) = α ⊙ φ_{F_{g}}^{'} (x^{0}) ⊙ y$ , from (13), we have $lim_{α \to 0^{+}} \frac{(φ (x^{0} + α y) ⊖_{g} φ (x^{0})) ⊖_{g} α ⊙ φ_{F_{g}}^{'} (x^{0}) ⊙ y}{α} = 0, \forall y \in ℝ^{n},$ or $φ_{F_{g}}^{'} (x^{0}) ⊙ y = lim_{α \to 0^{+}} \frac{φ (x^{0} + α y) ⊖_{g} φ (x^{0})}{α}, \forall y \in ℝ^{n} .$ Hence, φ is G_g-differentiable at x⁰ and $\nabla_{g} φ (x^{0}) = φ_{F_{g}}^{'} (x^{0}) .$ □

For instance, consider the following example.

Example 3.2. There are some the fuzzy mappings which are G_g-differentiable of φ at x⁰ but does not F_g-differentiable of φ at x⁰. For instance, consider the fuzzy mapping $φ : ℝ^{2} ⟶ E$ be defined by φ (x₁, x₂) = ${\begin{matrix} U ⊙ (\frac{x_{1}^{2} x_{2}}{x_{1}^{4} + x_{2}^{2}} \sqrt{x_{1}^{2} + x_{2}^{2}}) if (x_{1}, x_{2}) \neq (0, 0) \\ 0, otherwise . \end{matrix}$ Where U ∈ E. Given $x^{0} = (0, 0) \in ℝ^{2}$ and $y = (y_{1}, y_{2}) \in ℝ^{2}$ , we obtain $\nabla_{g} φ (0, 0) ⊙ (y_{1}, y_{2}) = lim_{α \to 0^{+}} \frac{φ ((0, 0) + α (y_{1}, y_{2})) ⊖_{g} φ (0, 0)}{α} = lim_{α \to 0^{+}} \frac{φ (α y_{1}, α y_{2})}{α} = 0 .$ Since, the function ∇_gφ (0, 0) ⊙ y = 0 is fuzzy g-(continuous linear) in y then φ is G_g-derivative at x⁰ = (0, 0) and ∇_gφ (0, 0) ⊙ y = 0 for all $y = (y_{1}, y_{2}) \in ℝ^{2}$ .

If φ is F_g-differentiable at x⁰ = (0, 0), then by Theorem 3.1, $φ_{F_{g}}^{'} (0, 0) ⊙ h = \nabla_{g} φ (0, 0) ⊙ y = 0$ . However, we have $lim_{h \to 0} \frac{(φ (x^{0} + h) ⊖_{g} φ (x^{0})) ⊖_{g} φ_{F_{g}}^{'} (x^{0}) ⊙ h}{∥ h ∥} = lim_{h \to 0} \frac{(φ (h) ⊖_{g} φ (0, 0)) ⊖_{g} φ_{F_{g}}^{'} (0, 0) ⊙ h}{∥ h ∥} = lim_{h \to 0} \frac{φ (h)}{∥ h ∥} = lim_{h \to 0} \frac{U ⊙ (h_{1}^{2} h_{2} \sqrt{h_{1}^{2} + h_{2}^{2}})}{(h_{1}^{4} + h_{2}^{2}) \sqrt{h_{1}^{2} + h_{2}^{2}}} = lim_{h \to 0} \frac{U ⊙ (h_{1}^{2} h_{2})}{h_{1}^{4} + h_{2}^{2}},$ which does not exist. Hence, φ is not F_g-differentiable at x⁰ = (0, 0).

The above definitions of F_g-differentiability and G_g-differentiability for fuzzy mapping on a subset $K \subseteq ℝ^{n}$ , which is an extension of the g-differentiability for the fuzzy-valued function on $ℝ$ .

The following definition provides a natural extension of L_gH-differentiability of the fuzzy-valued function for the fuzzy mapping, defined on a subset in $ℝ^{n}$ .

Definition 3.4. Let $K \subseteq ℝ^{n}$ and x⁰ ∈ K, $y \in ℝ^{n}$ , there exists δ > 0 such that x⁰ + αy ∈ K for any real number α ∈ (0, δ), the given r ∈ [0, 1], the level-wise generalized Gâteaux derivative (G_{L
_gH}-derivative for short) of a fuzzy mapping φ : K ⟶ E at x⁰ of the corresponding interval-valued functions $φ_{r} : K ⟶ K_{C}$ at x⁰ for all $y \in ℝ^{n}$ is defined as $lim_{α \to 0^{+}} \frac{[φ (x^{0} + α y)]_{r} ⊖_{gH} [φ (x^{0})]_{r}}{α} = \nabla_{L_{gH}} φ (x^{0})_{r} y,$ where $\nabla_{L_{gH}} φ (x^{0})_{r} y = \sum_{i = 1 y_{i} \geq 0}^{n} y_{i} [\nabla_{L_{gH}} φ_{r}^{-} (x^{0}), \nabla_{L_{gH}} φ_{r}^{+} (x^{0})]$ $+ \sum_{i = 1 y_{i} < 0}^{n} y_{i} [\nabla_{L_{gH}} φ_{r}^{+} (x^{0}), \nabla_{L_{gH}} φ_{r}^{-} (x^{0})],$ if $\nabla_{L_{gH}} φ (x^{0})_{r} \in K_{C}$ exists for all r ∈ [0, 1].

Then we say that φ is the level-wise generalized Gâteaux differentiable (G_{L
_gH}-differentiable for short) at x⁰, the collection of intervals {∇ _{L
_gH}φ (x⁰) _r : r ∈ [0, 1] } is the G_{L
_gH}-derivatives of φ at x⁰ and denoted by ∇_{L
_gH}φ (x⁰).

The next theorem indicates the G_g-derivative is well defined for a large class of fuzzy mappings. Furthermore, a characterization and a practical formula for fuzzy mappings on a subset $K \subseteq ℝ^{n}$ are introduced. The symbol ∇_g is used For the G_g-derivative of fuzzy mapping with an associated fuzzy g-(continuous linear) function $Λ : ℝ^{n} ⟶ E$ and the symbol ∇ is used the Gâteaux derivative of real-valued mappings with an associated continuous linear function $Λ : ℝ^{n} ⟶ ℝ$ .

Theorem 3.2. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping and x = (x₁, x₂, . . . , x_n) ∈ K be such that $[φ (x)]_{r} = [φ_{r}^{-} (x), φ_{r}^{+} (x)]$ . If real-valued multi variables functions $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ are real-valued Gâteaux differentiable with respect to x for all $y \in ℝ^{n}$ , uniformly for r ∈ [0, 1], then φ is G_g-differentiable at x for all $y \in ℝ^{n}$ with an associated fuzzy g-(continuous linear) function $Λ : ℝ^{n} ⟶ E$ , and we have

$[\nabla_{g} φ (x)]_{r} . y = [inf_{β \geq r} min {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y}, sup_{β \geq r} max {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y}] .$

Proof. According to Proposition 2.1, we have $\frac{1}{α} [φ (x + α y) \underset{g}{⊖} φ (x)]_{r} = \frac{1}{α} [inf_{β \geq r} min {φ_{β}^{-} (x + α y) - φ_{β}^{-} (x), φ_{β}^{+} (x + α y) - φ_{β}^{+} (x)}, sup_{β \geq r} max {φ_{β}^{-} (x + α y) - φ_{β}^{-} (x), φ_{β}^{+} (x + α y) - φ_{β}^{+} (x)}] .$ Then $\frac{1}{α} [φ (x + α y) \underset{g}{⊖} φ (x)]_{r} = \frac{1}{α} [inf_{β \geq r} min {φ_{β}^{-} (x + α y) - φ_{β}^{-} (x), φ_{β}^{+} (x + α y) - φ_{β}^{+} (x)}, sup_{β \geq r} max {φ_{β}^{-} (x + α y) - φ_{β}^{-} (x), φ_{β}^{+} (x + α y) - φ_{β}^{+} (x)}] .$ Then by the hypothesis the functions $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ are real-valued multi variables Gâteaux differentiable at x, we have $[\nabla_{g} φ (x)]_{r} . y = lim_{α \to 0^{+}} \frac{1}{α} [φ (x + α y) \underset{g}{⊖} φ (x)]_{r}$ i.e., $[inf_{β \geq r} min {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y}, sup_{β \geq r} max {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y}] = [inf_{β \geq r} min {φ_{β}^{' -} (x, y), φ_{β}^{' +} (x, y)}, sup_{β \geq r} max {φ_{β}^{' -} (x, y), φ_{β}^{' +} (x, y)}],$ for each $y \in ℝ^{n}$ and r ∈ [0, 1]. Also, let us consider that if $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ are left continuous with respect to r ∈]0, 1] and right continuous at 0. Taking a sequences α_n ⟶ 0, the functions $\frac{φ_{r}^{-} (x + α_{n} y) - φ_{r}^{-} (x)}{α_{n}}$ and $\frac{φ_{r}^{+} (x + α_{n} y) - φ_{r}^{+} (x)}{α_{n}}$ are left continuous for r ∈]0, 1] and right continuous at 0. Also, for all $y \in ℝ^{n}$ the functions $\begin{matrix} inf_{β \geq r} min {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y} \\ = inf_{β \geq r} min {\frac{φ_{β}^{-} (x + α_{n} y) - φ_{β}^{-} (x)}{α_{n}}, \\ \frac{φ_{β}^{+} (x + α_{n} y) - φ_{β}^{+} (x)}{α_{n}}} . \\ sup_{β \geq r} max {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y} \\ = sup_{β \geq r} max {\frac{φ_{β}^{-} (x + α_{n} y) - φ_{β}^{-} (x)}{α_{n}}, \\ \frac{φ_{β}^{+} (x + α_{n} y) - φ_{β}^{+} (x)}{α_{n}}}, \end{matrix}$ satisfying the same properties. Therefore, it follows that $\begin{matrix} inf_{β \geq r} min {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y}, \\ sup_{β \geq r} max {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y}, \end{matrix}$ are left continuous at r ∈]0, 1] and right continuous at 0. It is easy to see that for all $y \in ℝ^{n}$ $inf_{β \geq r} min {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y}$ is increasing function with respect to r ∈ [0, 1] and $sup_{β \geq r} max {\nabla φ_{β}^{-} (x) . y, \nabla φ_{β}^{+} (x) . y}$ is decreasing function with respect to r ∈ [0, 1], by Proposition 2.5, they define a fuzzy number. Consequently, the level-sets of ∇_gφ (x) _r . y for each $y \in ℝ^{n}$ and r ∈ [0, 1], define a fuzzy number, and hence, by Lemma 2.1, we obtain that φ (x) is G_g-differentiable at x for each $y \in ℝ^{n}$ . □

In the following theorem, the uniform G_{L
_gH}- differentiability is sufficient for the G_g-differentiability.

Theorem 3.3. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be uniformly G_{L
_gH}-differentiable at x⁰. Then the fuzzy mapping φ is G_g-differentiable at x⁰ with an associated fuzzy g-(continuous linear) function $Λ : ℝ^{n} ⟶ E$ , and for every r ∈ [0, 1] $[\nabla_{g} φ (x^{0})]_{r} = cl (conv ⋃_{β \geq r} \nabla_{L_{gH}} φ (x^{0})_{β}) .$

Proof. The proof is similar to Theorem 2.4 in [3]. □

Example 3.3. Consider the fuzzy function $φ : ℝ ⟶ E$ be defined by $φ (x) = {\begin{matrix} 〈 - 1, 0, 1 〉 ⊙ (x^{2} \sin \frac{1}{x}) if x \neq 0, \\ 〈 - 1, 0, 1 〉 if x = 0 . \end{matrix}$ It is easy to see that φ is F_g-differentiable at x⁰ = 0, i.e., $lim_{h \to 0} \frac{(φ (0 + h) ⊖_{g} φ (0)) ⊖_{g} φ_{F_{g}}^{'} (0) ⊙ h}{| h |} = 0 lim_{h \to 0} \frac{(〈 - 1, 0, 1 〉 ⊙ h^{2} \sin \frac{1}{h} ⊖_{g} 〈 - 1, 0, 1 〉) ⊖_{g} φ_{F_{g}}^{'} (0) ⊙ h}{| h |} = 0,$ then $φ_{F_{g}}^{'} (0) = 〈 - 1, 0, 1 〉 \in E$ .

4 Directional g-differentiability

Now, we introduce a definition of directional g-derivative for multi-dimensional fuzzy mapping on a subset $K \subseteq ℝ^{n}$ .

Definition 4.1. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, where K is any open subset of $ℝ^{n}$ . Let x⁰ ∈ K, $y \in ℝ^{n}$ . We say that φ has the one-sided directional generalized derivative (directional g-derivative for short) at x⁰ in the direction of y $φ_{g}^{'} (x^{0}, y) = lim_{α \to 0^{+}} \frac{φ (x^{0} + α y) ⊖_{g} φ (x^{0})}{α} .$ (14) If also $φ_{g}^{'} (x^{0}, y) = lim_{α \to 0^{-}} \frac{φ (x^{0}) ⊖_{g} φ (x^{0} - α y)}{α},$ (15) of the above fuzzy mapping exists and two are equal, we say that φ has two-sided directional g-derivative at x⁰ in the direction of y.

If $φ_{g}^{'} (x^{0}, y) \in E$ satisfying (14) exists, we say that φ is directional generalized differentiable (directional g-differentiable for short) at x⁰ in the direction of y.

Definition 4.2. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping. Let x⁰ ∈ K, $y \in ℝ^{n}$ . We say that φ is directionally g-differentiable at x⁰ if $φ_{g}^{'} (x^{0}, y)$ exist, for any $y \in ℝ^{n}$ .

Theorem 4.1. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping and x⁰ ∈ K. Then $φ_{g}^{'} (x^{0}, .)$ on $ℝ^{n}$ is a fuzzy positively homogeneous.

Proof. Let λ > 0, x⁰ ∈ K be fixed and arbitrary. Consider $φ_{g}^{'} (x^{0}, λ y) = lim_{α \to 0^{+}} \frac{φ (x^{0} + α (λ y)) ⊖_{g} φ (x^{0})}{α} = λ ⊙ lim_{α λ \to 0^{+}} \frac{φ (x^{0} + (α λ) y) ⊖_{g} φ (x^{0})}{α λ} .$ Let s : = αλ, since α → 0⁺ ⇔ αλ → 0⁺, ∀ λ > 0, i.e., s → 0⁺ $= λ ⊙ lim_{s \to 0^{+}} \frac{φ (x^{0} + sy) ⊖_{g} φ (x^{0})}{s} = λ ⊙ φ_{g}^{'} (x^{0}, y), \forall y \in ℝ^{n} .$ Hence, $φ_{g}^{'} (x^{0}, .)$ on $ℝ^{n}$ is fuzzy positively homogeneous. □

Proposition 4.1. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping and x = (x₁, x₂, . . . , x_n) ∈ K be such that $[φ (x)]_{r} = [φ_{r}^{-} (x), φ_{r}^{+} (x)]$ .

If $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ are real-valued multi variables functions directional differentiable at x in the direction of y, uniformly for r ∈ [0, 1], then φ (x) is directional g-differentiable at x in the direction of y, and we have $φ_{g}^{'} (x, y)_{r} = [inf_{β \geq r} min {φ_{β}^{' -} (x, y), φ_{β}^{' +} (x, y)}, sup_{β \geq r} max {φ_{β}^{' -} (x, y), φ_{β}^{' +} (x, y)}] .$

Consequently, the level-sets of $φ_{g}^{'} (x, y)_{r}$ define a fuzzy number, and hence, by Lemma 2.1, we obtain that φ (x) is directional g-differentiable at x in the direction of y. □

Definition 4.3. [18] Let K be a non-empty open subset of $ℝ^{n}$ and $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, with x⁰ ∈ K. If $y \in ℝ^{n}$ is any admissible direction at x⁰ ∈ K, then given r ∈ [0, 1], the directional level-wise generalized derivative (directional L_gH-derivative for short) of corresponding interval-valued function $φ_{r} : K \subseteq ℝ^{n} ⟶ K_{C}$ at x⁰ in the direction of y is defined as $φ_{L_{gH}}^{'} (x^{0}; y)_{r} = lim_{α \to 0^{+}} \frac{[φ (x^{0} + α y)]_{r} ⊖_{gH} [φ (x^{0})]_{r}}{α},$ (16) if it exists.

(1) If $φ_{L_{gH}}^{'} (x^{0}; y)_{r} \in K_{C}$ exists for all r ∈ [0, 1], then φ is said to have the directional L_gH-derivative at x⁰ in the direction of y;

(2) We say that φ is directionally (or weak) level-wise generalized differentiable (directionally or weak L_gH-differentiable) at x⁰ if φ admits directional L_gH-derivatives at x⁰ in any direction $y \in ℝ^{n}$ and for all r ∈ [0, 1]; the collection of intervals ${φ_{L_{gH}}^{'} (x^{0}; y)_{r} : r \in [0, 1]}$ is the directional L_gH-derivative of φ at x⁰ in direction y, denoted as ${φ_{L_{gH}}^{'} (x^{0}; y)_{r}}$ ;

(3) We say that φ is directionally (weak) gH-differentiable at x⁰ if it is directionally (weak) L_gH-differentiable at x⁰ in any direction y and the directional L_gH-derivative ${φ_{L_{gH}}^{'} (x^{0}; y)}$ defines a fuzzy interval (i.e., the intervals $φ_{L_{gH}}^{'} (x^{0}; y)_{r}$ define the level-cuts of a fuzzy interval);

(4) φ is said directionally (weak) L_gH-differentiable on K if it is directionally L_gH-differentiable at each point x⁰ ∈ K and is said directionally (weak) gH-differentiable on K if it is directionally gH-differentiable at each point x⁰ ∈ K.

Proposition 4.2. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping and x = (x₁, x₂, . . . , x_n) ∈ K is directional L_gH-differentiable at x in the direction of $y \in ℝ^{n}$ uniformly for r ∈ [0, 1]. Then φ is directional g-differentiable at x in the direction of y, for any r ∈ [0, 1] $φ_{g}^{'} (x, y)_{r} = cl (conv ⋃_{β \geq r} φ_{L_{gH}}^{'} (x, y)_{β}) .$

Proof. The proof is similar to Theorem 4 in [3]. □

Definition 4.4. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, x ∈ K. If for $y \in ℝ^{n}$ , there exists δ > 0 such that x + αy ∈ K for any real number α ∈ (0, δ), then the quotient fuzzy mapping has the generalized Hukuhara difference as follows: $S_{gH}^{x, y} (α) = \frac{φ (x + α y) ⊖_{gH} φ (x)}{α},$ as a function of α at x in the direction of y, for given r ∈ [0, 1], the quotient of interval-valued function has the level-wise generalized Hukuhara difference (L_gH-quotient for short) at x in the direction of y is defined as $S_{L_{gH}}^{x, y} (α)_{r} = \frac{[φ (x + α y)]_{r} ⊖_{gH} [φ (x)]_{r}}{α} .$ If $S_{L_{gH}}^{x, y} (α)_{r} \in K_{C}$ for all r ∈ [0, 1] and the collection of intervals ${S_{L_{gH}}^{x, y} (α)_{r} : r \in [0, 1]}$ is the L_gH-quotient of φ as a function of α at x in the direction of y, and denoted by $S_{L_{gH}}^{x, y} (α)$ .

Example 4.1. Let $φ : ℝ^{n} ⟶ E$ be defined by $φ (x) = 〈 - 1, 0, 1 〉 ⊙ ∥ x ∥; \forall x \in ℝ^{n},$ its level-set, r ∈ [0, 1], are defined by $[φ (x)]_{r} = [φ_{r}^{-} (x), φ_{r}^{+} (x)] = [(r - 1) ∥ x ∥, (1 - r) ∥ x ∥] .$ For all r ∈ [0, 1], the functions $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ are real-valued Gâteaux differentiable at each point $0 \neq x \in ℝ^{n}$ , then $\nabla φ_{r}^{-} (x) = (r - 1) \frac{x}{∥ x ∥}$ and $\nabla φ_{r}^{+} (x) = (1 - r) \frac{x}{∥ x ∥}$ . Now for all r ∈ [0, 1], the two functions $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ are not real-valued Gâteaux differentiable at x = 0. Also φ (x) is not G_{L
_gH}-differentiable at x = 0.

However, for all r ∈ [0, 1] the functions $φ_{r}^{' -} (x, y) = (r - 1) ∥ y ∥$ and $φ_{r}^{' +} (x, y) = (1 - r) ∥ y ∥$ are real-valued multi variables directional differentiable at $0 = x \in ℝ^{n}$ in any direction $y \in ℝ^{n}$ , and satisfy the conditions in Proposition 2.5, in fact for all $y \in ℝ^{n}$ , r ∈ [0, 1] we have $φ_{g}^{'} (0, y)_{r} = [inf_{β \geq r} min {φ_{β}^{' -} (0, y), φ_{β}^{' +} (0, y)}, sup_{β \geq r} max {φ_{β}^{' -} (0, y), φ_{β}^{' +} (0, y)}] = [inf_{β \geq r} min {(β - 1) ∥ y ∥, (1 - β) ∥ y ∥}, sup_{β \geq r} max {(β - 1) ∥ y ∥, (1 - β) ∥ y ∥}] = [(r - 1) ∥ y ∥, (1 - r) ∥ y ∥] = [r - 1, 1 - r] ∥ y ∥,$ it is follows that $φ_{g}^{'} (0, y)_{r} = [r - 1, 1 - r] ∥ y ∥ .$ Hence φ is directionally g-differentiable at x = 0 in any direction $y \in ℝ^{n}$ .

5 Partial g-derivative

Below, we define the generalized partial derivative a natural extension of g-derivative for multi-dimensional fuzzy mapping $φ : K \subseteq ℝ^{n} ⟶ E$ . Denote e_i = (t₁, t₂, . . . , t_i, . . . , t_n) with $t_{j} = {\begin{matrix} 1, if j = i, \\ 0, if j \neq i . \end{matrix}$ Where e_i is the ith unit vector (all components or 0 accept for the ith component which is 1).

Definition 5.1. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, fix some x = (x₁, x₂, . . . , x_n) ∈ K, and consider the expression $\frac{\partial_{g} φ (x)}{\partial x_{i}} = lim_{α \to 0} \frac{φ (x + α e_{i}) ⊖_{g} φ (x)}{α} .$ (17) If $\frac{\partial_{g} φ (x)}{\partial x_{i}} \in E (i = 1, 2, . . ., n)$ satisfying (17) exists, it is called generalized partial derivative of φ (((g-p))-derivative for short) at x with respect to x_i, (i = 1, 2, . . . , n).

Definition 5.2. Consider $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, where K is any open subset of $ℝ^{n}$ . If y = e_i is any admissible direction at x ∈ K, we say that φ to be right and left generalized partial differentiable (right and left ((g-p))-differentiable for short) at x with respect to x_i, denoted by $\frac{\partial_{g} φ_{+} (x)}{\partial x_{i}} = lim_{α \to 0^{+}} \frac{φ (x + α e_{i}) ⊖_{g} φ (x)}{α},$ and $\frac{\partial_{g} φ_{-} (x)}{\partial x_{i}} = lim_{α \to 0^{+}} \frac{φ (x - α e_{i}) ⊖_{g} φ (x)}{α} .$ If $\frac{\partial_{g} φ_{+} (x)}{\partial x_{i}}$ and $\frac{\partial_{g} φ_{-} (x)}{\partial x_{i}} \in E$ both exist and are equal, we obtain the ((g-p))-derivative of φ at x ∈ K with respect to x_i: $\frac{\partial_{g} φ (x)}{\partial x_{i}} = lim_{α \to 0} \frac{φ (x + α e_{i}) ⊖_{g} φ (x)}{α} .$

Proposition 5.1. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, fix some x = (x₁, x₂, . . . , x_n) ∈ K. If $φ_{r}^{-} (x)$ and $φ_{r}^{+} (x)$ are real-valued multi variables functions partially differentiable at x with respect to x_i, uniformly for r ∈ [0, 1], then φ (x) is ((g-p))-differentiable at x with respect to x_i and we have $[\frac{\partial_{g} φ (x)}{\partial x_{i}}]_{r} = [inf_{β \geq r} min {\frac{\partial φ_{β}^{-} (x)}{\partial x_{i}}, \frac{\partial φ_{β}^{+} (x)}{\partial x_{i}}}, sup_{β \geq r} max {\frac{\partial φ_{β}^{-} (x)}{\partial x_{i}}, \frac{\partial φ_{β}^{+} (x)}{\partial x_{i}}}] .$

Proof. The proof is similar to the Proposition 4.1, in particular, for y = e_i, i = 1, . . . , n. □

Definition 5.3. [18] Let $φ : K \subseteq ℝ^{n} ⟶ E$ with x = (x₁, x₂, . . . , x_n) ∈ K, the given r ∈ [0, 1], the partially level-wise generalized derivative (partially L_gH-derivative for short) at x with respect to x_i of corresponding interval-valued functions $φ_{r} : K \subseteq ℝ^{n} ⟶ K_{C}$ at x with respect to x_i partially gH-derivatives, if they exist $\frac{\partial_{L_{gH}} φ (x)_{r}}{\partial x_{i}} = lim_{α \to 0} \frac{[φ (x + α e_{i})]_{r} ⊖_{gH} [φ (x)]_{r}}{α} .$ If $\frac{\partial_{L_{gH}} φ (x)_{r}}{\partial x_{i}} \in K_{C}$ exists for all r ∈ [0, 1], we say that φ is partially level-wise generalized Hukuhara differentiable (partially L_gH-differentiable for short) at x with respect to x_i and the collection of intervals ${\frac{\partial_{L_{gH}} φ (x)_{r}}{\partial x_{i}} : r \in [0, 1]}$ is the partially L_gH-derivative of φ at x with respect to x_i (i = 1, 2, . . . , n), denoted by $\frac{\partial_{L_{gH}} φ (x)}{\partial x_{i}}$ .

Definition 5.4. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, x ∈ K. If for $e_{i} \in ℝ^{n}$ , there exists δ > 0 such that x + αe_i ∈ K for any real number α ∈ (0, δ), then the quotient fuzzy mapping has the generalized Hukuhara difference as follows $S_{i - gH}^{x} (α) = \frac{φ (x + α e_{i}) ⊖_{gH} φ (x)}{α},$ as a function of α at x in the direction of e_i, if $S_{i - gH}^{x} (α) \in E$ .

Definition 5.5. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, x ∈ K. If for $e_{i} \in ℝ^{n}$ , there exists δ > 0 such that x + αe_i ∈ K for any real number α ∈ (0, δ), then for given r ∈ [0, 1], the quotient of interval-valued function has the level-wise generalized Hukuhara difference (L_gH-quotient for short) at x in the direction of e_i is defined as $S_{i - L_{gH}}^{x} (α)_{r} = \frac{[φ (x + α e_{i})]_{r} ⊖_{gH} [φ (x)]_{r}}{α} .$ If $S_{i - L_{gH}}^{x} (α)_{r} \in K_{C}$ for all r ∈ [0, 1] and the collection of intervals ${S_{i - L_{gH}}^{x} (α)_{r} : r \in [0, 1]}$ is the L_gH-quotient of φ as a function of α at x in the direction of e_i, and denoted by $S_{i - L_{gH}}^{x} (α)$ , (i = 1, 2, . . . , n).

Example 5.1. Let $φ : ℝ^{2} ⟶ E$ be a fuzzy mapping and be define by φ (x₁, x₂) = 〈1, 2, 3〉 ⊙ |x₁ - 2| + x₂ for any $x = (x_{1}, x_{2}) \in ℝ^{2}$ . Its level-sets are, for any r ∈ [0, 1] $φ (x_{1}, x_{2})_{r} = [(r - 1) | x_{1} - 2 | + x_{2}, (3 - r) | x_{1} - 2 | + x_{2}] .$ Then the quotient fuzzy mapping of φ has the (L_gH-quotient) as follows: $S_{i - L_{gH}}^{x} (α)_{r} = \frac{[φ (x + α e_{i})]_{r} ⊖_{gH} [φ (x)]_{r}}{α},$ for all r ∈ [0, 1], as a function of α at x in the direction of e_i with i ∈ {1, 2}. Given x = (2, 0), we have that for any r ∈ [0, 1] and for i = 1, 2, the (L_gH-quotient) of φ as a function of α at x in the direction of e_i is given by $S_{1 - L_{gH}}^{x} (α)_{r} = \frac{[φ ((2, 0) + α e_{1})]_{r} ⊖_{gH} [φ (2, 0)]_{r}}{α}$ $= \frac{[(r - 1) | α |, (3 - r) | α |]}{α}, \forall r \in [0, 1],$ and $S_{2 - L_{gH}}^{x} (α)_{r} = \frac{[φ ((2, 0) + α e_{2})]_{r} ⊖_{gH} [φ (2, 0)]_{r}}{α}$ $= \frac{[(r - 1) + α, (3 - r) + α]}{α}, \forall r \in [0, 1] .$ The collection ${S_{i - L_{gH}}^{x} (α)_{r}, i \in {1, 2}, r \in [0, 1]}$ is the L_gH-quotient of φ as a function of α at x in the direction of e_i, and denoted by $S_{i - L_{gH}}^{x} (α), (i = 1, 2)$ .

Theorem 5.1. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, x ∈ K and $y \in ℝ^{n}$ , if there exist δ > 0 such that x + αy ∈ K for every real-number α ∈ (0, δ), the function φ (x) uniformly partially L_gH-differentiable at x with respect to x_i. Then φ (x) is ((g-p))-differentiable at x with respect to x_i, and $S_{i - g}^{x} (α)_{r} = cl (conv ⋃_{β \geq r} (\frac{[φ (x + α e_{i})]_{β} ⊖_{gH} [φ (x)]_{β}}{α})),$ uniformly with respect to r ∈ [0, 1] converge to $\frac{\partial_{g} φ (x)_{r}}{\partial x_{i}}$ as α → 0, respectively (i = 1, 2, . . . , n).

Proof. Assume that x ∈ K, there exists δ > 0 such that x + αy ∈ K for any α ∈ (0, δ), and consider for r ∈ [0, 1] be fixed, the intervals $S_{i - L_{gH}}^{x} (α)_{r} = \frac{1}{α} [φ (x + α e_{i})]_{r} \underset{gH}{⊖} [φ (x)]_{r},$ $b_{i} (r) = lim_{α \to 0} S_{i - L_{gH}}^{x} (α)_{r} = \frac{\partial_{g} φ_{L_{gH}} (x)_{r}}{\partial x_{i}},$ $S_{i - g}^{x} (α)_{r} = cl (conv ⋃_{β \geq r} S_{i - L_{gH}}^{x} (α)_{β}),$ $= \frac{1}{α} ([φ (x + α e_{i})]_{r} \underset{g}{⊖} [φ (x)]_{r}),$ $B_{i} (r) = cl (conv ⋃_{β \geq r} b_{β}) .$ Suppose that the fuzzy sets $S_{i - g}^{x} (α)$ and B_i be the fuzzy number having the collection of intervals ${S_{i - g}^{x} (α)_{r} : r \in [0, 1]}$ and {B_i (r) : r ∈ [0, 1]} as level-sets respectively.

In fact, we show that the fuzzy numbers $S_{i - g}^{x} (α)$ and B_i in Proposition 5.1 the level-sets ${S_{i - g}^{x} (α)_{r} : r \in [0, 1]}$ and {B_i (r) : r ∈ [0, 1]} satisfying the conditions in Proposition 2.5. We observe that the following limit exists, i.e., $lim_{α \to 0} S_{i - g}^{x} (α) = B_{i} .$ As a conclusion, that the ((g-p))-derivative of φ at x with respect to x_i exist and equal to B_i.

Consider the intervals $S_{i - g}^{x} (α)_{r} = [S_{i - g}^{- x} (α)_{r}, S_{i - g}^{+ x} (α)_{r}],$ and $B_{i} (r) = [B_{i}^{-} (r), B_{i}^{+} (r)],$ such that $S_{i - g}^{- x} (α)_{r} = inf_{β \geq r} S_{i - L_{gH}}^{- x} (α)_{β},$ $S_{i - g}^{+ x} (α)_{r} = sup_{β \geq r} S_{i - L_{gH}}^{+ x} (α)_{β},$ $B_{i}^{-} (r) = inf_{β \geq r} b_{β}^{-}, B_{i}^{+} (r) = sup_{β \geq r} b_{β}^{+},$ then by the hypothesis of φ is uniformly partially L_gH-differentiability at x with respect to x_i (i = 1, 2, . . . , n), for the chosen ɛ > 0 be given for each i = 1, 2, . . . , n there exists δ_i > 0, such that $\begin{matrix} | α | < δ = min {δ_{1}, δ_{2}, . . ., δ_{n}} \\ \Rightarrow b_{i}^{-} (r) - \frac{ɛ}{4} < S_{i - L_{gH}}^{- x} (α)_{r} \\ < b_{i}^{-} (r) + \frac{ɛ}{4}, \forall r \in [0, 1], \\ | α | < δ = min {δ_{1}, δ_{2}, . . ., δ_{n}} \\ \Rightarrow b_{i}^{+} (r) - \frac{ɛ}{4} < S_{i - L_{gH}}^{+ x} (α)_{r} \\ < b_{i}^{+} (r) + \frac{ɛ}{4}, \forall r \in [0, 1] . \end{matrix}$ In the other words, from the definition of infimum and supremum, we have for ɛ > 0 be arbitrary and for any r ∈ [0, 1] and any α, there exist λ₁ ≥ r, λ₂ ≥ r, λ₃ ≥ r, λ₄ ≥ r, such that $\begin{matrix} S_{i - g}^{- x} (α)_{r} > S_{i - L_{gH}}^{- x} (α)_{λ_{1}} - \frac{ɛ}{4} > b_{λ_{1}}^{-} - \frac{ɛ}{4} - \frac{ɛ}{4} \\ \geq B_{i}^{-} (r) - \frac{ɛ}{2}, \\ B_{i}^{-} (r) > b_{λ_{1}}^{-} - \frac{ɛ}{2} > S_{i - L_{gH}}^{- x} (α)_{λ_{2}} - \frac{ɛ}{4} - \frac{ɛ}{4} \\ > S_{i - g}^{- x} (α)_{r} - \frac{ɛ}{2}, \\ S_{i - g}^{+ x} (α)_{r} < S_{i - L_{gH}}^{+ x} (α)_{λ_{3}} + \frac{ɛ}{4} < b_{λ_{3}}^{+} + \frac{ɛ}{4} + \frac{ɛ}{4} \\ \leq B_{i}^{+} (r) + \frac{ɛ}{2}, \\ B_{i}^{+} (r) > b_{λ_{4}}^{+} + \frac{ɛ}{4} < S_{i - L_{gH}}^{+ x} (α)_{λ_{4}} + \frac{ɛ}{4} + \frac{ɛ}{4} \\ < S_{i - g}^{+ x} (α)_{r} + \frac{ɛ}{2}, \end{matrix}$ and finally, for the sufficiently small α, we have $\begin{matrix} ∥ S_{i - g}^{x} (α) \underset{g}{⊖} B_{i} ∥ = sup_{r \in [0, 1]} ∥ S_{i - g}^{x} (α)_{r} - B_{i} (r) ∥_{*} \\ = sup_{r \in [0, 1]} max {| S_{i - g}^{- x} (α)_{r} - B_{i}^{-} (r) |, | S_{i - g}^{+ x} (α)_{r} \\ - B_{i}^{+} (r) |} \leq \frac{ɛ}{2} < ɛ . \end{matrix}$ Passing to limit as α → 0, we obtain $lim_{α \to 0} S_{i - g}^{x} (α) = B_{i} .$ The proof is complete.□

Proposition 5.2. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping. If φ is F_g-differentiable at x⁰ ∈ K, then all ((g-p))-derivative of φ at x⁰ exist, it is $\frac{\partial_{g} φ (x^{0})}{\partial x_{i}} = u_{i}, i = 1, 2, . . . . n .$

Proof. By using the definition of F_g-differentiability of φ at x⁰, taking h = αe_i, α ≠ 0, we have that for all i = 1, 2, . . . , n, $φ (x^{0} + α e_{i}) \underset{g}{⊖} φ (x^{0}) = α ⊙ u_{i} + | α | ⊙ ɛ (α e_{i}),$ with $lim_{α \to 0} ɛ (α e_{i}) = 0$ ; then the following limit exists $\begin{matrix} lim_{α \to 0} \frac{φ (x^{0} + α e_{i}) ⊖_{g} φ (x^{0})}{α} \\ = lim_{α \to 0} u_{i} + \frac{| α |}{α} ⊙ ɛ (α e_{i}) = u_{i} . \end{matrix}$ Hence $\frac{\partial_{g} φ (x^{0})}{\partial x_{i}} = u_{i}, i = 1, 2, . . ., n .$ The proof is complete. □

Proposition 5.3. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping. If φ is F_g-differentiable at x⁰ ∈ K, then for a given i, the ((g-p))-derivative $\frac{\partial_{g} φ (x^{0})}{\partial x_{i}}$ exists, then $\frac{\partial_{g} φ (x^{0})}{\partial x_{i}} = u_{i} .$

Proof. Suppose that φ is F_g-differentiable at x⁰. We get h = αe_i for α > 0, we have $\frac{φ (x^{0} + α e_{i}) ⊖_{g} φ (x^{0})}{α} = u_{i} + ɛ (α e_{i}),$ and, passing the limit for α → 0⁺, we obtain $lim_{α \to 0^{+}} \frac{φ (x^{0} + α e_{i}) ⊖_{g} φ (x^{0})}{α} = u_{i},$ similarly, if we get h = - αe_i, α > 0, we have $lim_{α \to 0^{+}} \frac{φ (x^{0} - α e_{i}) ⊖_{g} φ (x^{0})}{α} = u_{i},$ therefore $lim_{α \to 0^{+}} \frac{φ (x^{0} + α e_{i}) ⊖_{g} φ (x^{0})}{α}$ $= lim_{α \to 0^{+}} \frac{φ (x^{0} - α e_{i}) ⊖_{g} φ (x^{0})}{α} = u_{i} .$ (18) The ((g-p))-derivative of φ at x with respect to x_i, i.e., $\frac{\partial_{g} φ (x^{0})}{\partial x_{i}} = lim_{α \to 0^{+}} \frac{φ (x^{0} + α e_{i}) ⊖_{g} φ (x^{0})}{α},$ exists and, from (18) we obtain $\frac{\partial_{g} φ (x^{0})}{\partial x_{i}} = u_{i}$ .

The proof is complete. □

Definition 5.6. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping, is G_g-differentiable at x⁰ ∈ K. Then n-dimensional vector of fuzzy u_i ∈ E, i = 1, 2, . . . , n is called the g-gradient of φ at x⁰ and denoted by ∇_gφ (x⁰) = (u₁, u₂, . . . , u_n). Note that the g-gradient is well-defined by Definition 3.3 is unique.

Let us consider that the g-gradient of a (g-p)-differentiable and L_gH-gradient of a partial L_gH-differentiable function for each level-sets of fuzzy mapping as follows.

Definition 5.7. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping. The g-gradient of a fuzzy mapping φ at x ∈ K is given $\nabla_{g} φ (x) = (\frac{\partial_{g} φ (x)}{\partial x_{1}}, . . ., \frac{\partial_{g} φ (x)}{\partial x_{n}}),$ where $\frac{\partial_{g} φ (x)}{\partial x_{i}}$ , i = 1, 2, . . . , n the ((g-p))-derivative of φ at x with respect to x_i. In this case, the ith partial L_gH-derivative of φ at x, as follows: $\frac{\partial_{L_{gH}} φ (x)_{r}}{\partial x_{i}} \in K_{C}, \forall r \in [0, 1],$ the collection of intervals ${\frac{\partial_{L_{gH}} φ (x)_{r}}{\partial x_{i}} : r \in [0, 1]}$ is the partial L_gH-derivative of φ at x with respect to x_i and denoted by $\frac{\partial_{L_{gH}} φ (x)}{\partial x_{i}}$ .

Also, the collection of intervals {∇ _{L
_gH}φ (x) _r : r ∈ [0, 1]} is the L_gH-gradient of φ at x and denoted by ∇_{L
_gH}φ (x) as follows: $\nabla_{L_{gH}} φ (x) = (\frac{\partial_{L_{gH}} φ (x)}{\partial x_{1}}, . . ., \frac{\partial_{L_{gH}} φ (x)}{\partial x_{n}}) .$

In the following theorem connection between directional g-derivative and G_g-derivative is given.

Theorem 5.2. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping. If φ is G_g-differentiable at x⁰, i.e., ∇_gφ (x) exist at $x^{0} \in ℝ^{n}$ , then the directional g-derivative of φ at x⁰, i.e., $φ_{g}^{'} (x^{0}, y)$ exists for any fixed direction $y \in ℝ^{n}$ , and satisfies $φ_{g}^{'} (x^{0}, y) = \nabla_{g} φ (x^{0}) ⊙ y, \forall y \in ℝ^{n} .$

Proof. Suppose that φ is G_g-differentiable at $x^{0} \in ℝ^{n}$ , then by Definition 3.3 for all $y \in ℝ^{n}$ there exists the following limit: $lim_{α \to 0^{+}} \frac{φ (x^{0} + α y) ⊖_{g} φ (x^{0})}{α} = \nabla_{g} φ (x^{0}) ⊙ y .$ (19) It follows from (19) that $φ_{g}^{'} (x^{0}, y) = \nabla_{g} φ (x^{0}) ⊙ y$ , $for any y \in ℝ^{n}$ , i.e., φ′ (x⁰, .) on $ℝ^{n}$ is a fuzzy g-(continuous linear) function whenever g-gradient of φ at x⁰, i.e., ∇_gφ (x⁰) exists.

The proof is complete. □

Corollary 5.1. Let $φ : K \subseteq ℝ^{n} ⟶ E$ be a fuzzy mapping. If φ is G_g-differentiable at x⁰, then all the ((g-p))-derivatives of φ at x⁰ exist, and $\frac{\partial_{g} φ (x^{0})}{\partial x_{i}} = φ_{g}^{'} (x^{0}, e_{i}) = u_{i},$ for i = 1, 2, . . . , n. Furthermore, all directional g-derivatives of φ at x⁰ exists, and $φ_{g}^{'} (x^{0}, y) = \sum_{i = 1 y_{i} \geq 0}^{n} y_{i} ⊙ u_{i} \underset{g}{⊖} \sum_{i = 1 y_{i} < 0}^{n} y_{i} ⊙ u_{i} .$

Note 5.1. In Theorem 5.2, we have shown the G_g-differentiability of φ at x⁰, i.e., ∇_gφ (x⁰) exists, then $φ_{g}^{'} (x^{0}, y)$ exists for all $y \in ℝ^{n}$ . But, the converse is not true, i.e., the existence of all directional g-derivatives of φ at x⁰ does not imply the G_g-differentiability of f at x⁰.

Considering the following example:

Example 5.2. Let $φ : ℝ^{2} ⟶ E$ be defined by $φ (x_{1}, x_{2}) = {\begin{matrix} U_{1} ⊙ x_{1} + U_{2} ⊙ (x_{2} e^{\frac{x_{1}}{x_{2}}}), if x_{2} \neq 0, \\ 0, otherwise . \end{matrix}$ Where U₁, U₂ ∈ E. The directional g-derivative of φ at $x^{0} = (0, 0) \in ℝ^{2}$ in the direction of $y = (y_{1}, y_{2}) \in ℝ^{2}$ ; indeed, we have $φ_{g} ((0, 0), (y_{1}, y_{2})) = lim_{α \to 0^{+}} \frac{φ ((0, 0) + α (y_{1}, y_{2})) ⊖_{g} φ (0, 0)}{α} = lim_{α \to 0^{+}} \frac{φ (α y_{1}, α y_{2})}{α} = {\begin{matrix} U_{1} ⊙ y_{1} + c_{2} ⊙ (y_{2} e^{\frac{y_{1}}{y_{2}}}), if y_{2} \neq 0, \\ 0, otherwise . \end{matrix}$ Therefore, φ is directionally g-derivative at x⁰ = (0, 0) in any direction $(y_{1}, y_{2}) \in ℝ^{2}$ . But ∇_gφ (0, 0) is not fuzzy g-continuous, and hence φ does not the G_g-derivative at x⁰ = (0, 0).

In particular, in the one-dimensional case, we can state a rule to calculate the directional g-derivative via the g-derivative as follows.

Proposition 5.4. Let $φ : K \subseteq ℝ ⟶ E$ be a fuzzy-valued function. If φ is g-differentiable on K, then φ is directionally g-differentiable on K, and $φ_{g}^{'} (x^{0}, y^{0}) = φ_{g}^{'} (x^{0}) ⊙ y^{0}, \forall x^{0} \in K, y^{0} \in ℝ .$

Proof. Let x⁰ ∈ K and $y^{0} \in ℝ$ be arbitrary by Definition 4.2, we have $φ_{g}^{'} (x^{0}, y^{0}) = lim_{α \to 0^{+}} \frac{φ (x^{0} + α y^{0}) ⊖_{g} φ (x^{0})}{α},$ (20) exists, since φ is g-differentiable at x⁰ then, by Definition 2.12, there exist the following limit: $φ_{g}^{'} (x^{0}) = lim_{h \to 0^{+}} \frac{φ (x^{0} + h) ⊖_{g} φ (x^{0})}{h},$ where $φ_{g}^{'} (x^{0}) \in E$ . As a conclude, we have the following expressions: $lim_{h \to 0^{+}} \frac{φ (x^{0} + h) ⊖_{g} φ (x^{0})}{h} = lim_{h \to 0^{-}} \frac{φ (x^{0} + h) ⊖_{g} φ (x^{0})}{h}$ $= φ_{g}^{'} (x^{0}) .$ (21) From (20) and (21), it follows: $φ_{g}^{'} (x^{0}, y^{0}) = lim_{α \to 0^{+}} \frac{φ (x^{0} + α y^{0}) ⊖_{g} φ (x^{0})}{α y^{0}} ⊙ y^{0} \underset{h^{'} = α y^{0}}{=} {\begin{matrix} lim_{h^{'} \to 0^{+}} \frac{φ (x^{0} + h^{'}) ⊖_{g} φ (x^{0})}{h^{'}} ⊙ y^{0}, if y^{0} > 0 \\ lim_{h^{'} \to 0^{-}} \frac{φ (x^{0} + h^{'}) ⊖_{g} φ (x^{0})}{h^{'}} ⊙ y^{0}, if y^{0} < 0 \end{matrix} = φ_{g}^{'} (x^{0}) ⊙ y^{0}, \forall x^{0} \in K, y^{0} \in ℝ .$ □ In the next examples, we have shown the description and relations of directional g-derivative, (g-p)-derivative, directional L_gH-differentiability, and partial L_gH-derivative with details.

Example 5.3. Let $φ : ℝ^{2} ⟶ E$ be a fuzzy mapping be define by $φ (x_{1}, x_{2}) = 〈 - 1, 0, 1 〉 ⊙ | x_{1} - x_{2} | - x_{1} - x_{1} x_{2} - x_{2}^{3},$ for every $(x_{1}, x_{2}) \in ℝ^{2}$ . Its level-set, for all r ∈ [0, 1], $φ (x_{1}, x_{2})_{r} = [(r - 1) | x_{1} - x_{2} | - x_{1} - x_{1} x_{2} - x_{2}^{3}, (1 - r) | x_{1} - x_{2} | - x_{1} - x_{1} x_{2} - x_{2}^{3}] .$ Given x⁰ = (0, 0) and $y = (y_{1}, y_{2}) \in ℝ^{2}$ , for all r ∈ [0, 1], we have $φ_{L_{gH}}^{'} ((0, 0), (y_{1}, y_{2}))_{r} = lim_{α \to 0^{+}} \frac{1}{α} [φ ((0, 0) + α (y_{1}, y_{2}))]_{r} \underset{gH}{⊖} [φ (0, 0)]_{r} = lim_{α \to 0^{+}} \frac{1}{α} [φ (α y_{1}, α y_{2})]_{r} = lim_{α \to 0^{+}} \frac{1}{α} [(r - 1) | α y_{1} - α y_{2} | - α y_{1} - α^{2} y_{1} y_{2} - α^{2} y_{2}^{3}, (1 - r) | α y_{1} - α y_{2} | - α y_{1} - α^{2} y_{1} y_{2} - α^{2} y_{2}^{3}] = [(r - 1) | y_{1} - y_{2} | - y_{1}, (1 - r) | y_{1} - y_{2} | - y_{1}] .$ Then, φ is directionally L_gH-differentiable at x⁰ = (0, 0) in any direction $y = (y_{1}, y_{2}) \in ℝ^{2}$ and $φ_{g}^{'} ((0, 0), (y_{1}, y_{2}))_{r} = cl (conv ⋃_{β \geq r} φ_{L_{gH}}^{'} ((0, 0), (y_{1}, y_{2}))_{β}) = cl (conv ⋃_{β \geq r} ((β - 1) | y_{1} - y_{2} | - y_{1}, (1 - β) | y_{1} - y_{2} | - y_{1})) = [(r - 1) | y_{1} - y_{2} | - y_{1}, (1 - r) | y_{1} - y_{2} | - y_{1}], \forall y = (y_{1}, y_{2}) \in ℝ^{2}, r \in [0, 1] .$ Hence, φ is directionally g-differentiable at x⁰ = (0, 0) in any direction $y = (y_{1}, y_{2}) \in ℝ^{2}$ . But, if we get r = 0.25 and define the unit vector e₁ = (1, 0), we have $\frac{\partial_{L_{gH}} φ (0, 0)_{0.25}}{\partial x_{1}} = lim_{α \to 0} \frac{1}{α} [φ ((0, 0) + α e_{1})]_{0.25} \underset{gH}{⊖} [φ (0, 0)]_{0.25} = lim_{α \to 0} \frac{1}{α} [- 0.75 | α | - α, 0.75 | α | - α],$ which does not exist. Finally, φ does not the partial L_gH-differentiable at x⁰ = (0, 0) with respect to x₁.

Example 5.4. Let $φ : ℝ^{2} ⟶ E$ be a fuzzy mapping be define by $φ (x_{1}, x_{2}) = 〈 - 1, 0, 1 〉 ⊙ sin (x_{1} + x_{1}^{2} + x_{1}^{3} - | x_{2} |) + | x_{2} |, \forall (x_{1}, x_{2}) \in ℝ^{2},$ for x⁰ = (0, 0) and $y = (y_{1}, y_{2}) \in ℝ^{2}$ , we have $φ_{g}^{'} ((0, 0), (y_{1}, y_{2})) = lim_{α \to 0^{+}} \frac{φ ((0, 0) + α (y_{1}, y_{2})) ⊖_{g} φ (0, 0)}{α} = lim_{α \to 0^{+}} \frac{〈 - 1, 0, 1 〉 ⊙ sin (α y_{1} + α^{2} y_{1}^{2} + α^{3} y_{1}^{3}}{α} \frac{- | α y_{2} |) + | α y_{2} |}{α} = lim_{α \to 0^{+}} 〈 - 1, 0, 1 〉 ⊙ (y_{1} + 2 α y_{1}^{2} + 3 α^{2} y_{1}^{3} - | y_{2} |) cos (α y_{1} + α^{2} y_{1}^{2} + α^{3} y_{1}^{3} - | α y_{2} |) + | y_{2} | = 〈 - 1, 0, 1 〉 ⊙ y_{1}, \forall y_{1} \in ℝ .$ Therefore φ is directionally g-differentiable at x⁰ = (0, 0) in any direction y₁ i.e., $φ_{g}^{'} (x^{0}, .)$ on $ℝ^{2}$ is a fuzzy g-(continuous linear) function in y₁.

But, for x⁰ = (0, 0), and the unit vector e₁ = (1, 0), we have that $\frac{\partial_{g} φ (0, 0)}{\partial x_{1}} = lim_{α \to 0} \frac{φ ((0, 0) + α (1, 0)) ⊖_{g} φ (0, 0)}{α} = lim_{α \to 0} \frac{φ (α, 0) ⊖_{g} φ (0, 0)}{α} = lim_{α \to 0} \frac{〈 - 1, 0, 1 〉 ⊙ sin (α + α^{2} + α^{3})}{α} = lim_{α \to 0} 〈 - 1, 0, 1 〉 ⊙ (1 + 2 α + 3 α^{2}) cos (α + α^{2} + α^{3}) = 〈 - 1, 0, 1 〉 \in E .$ Therefore, φ is the (g-p)-derivative at x⁰ = (0, 0) with respect to x₁.

Example 5.5. Let $φ : ℝ^{3} ⟶ E$ be a fuzzy mapping be define by the level-set $[φ (x_{1}, x_{2}, x_{3})]_{r} = [φ_{r}^{-} (x_{1}, x_{2}, x_{3}), φ_{r}^{+} (x_{1}, x_{2}, x_{3})] = {\begin{matrix} [min {0, \frac{(1 - r) x_{1} x_{2}^{2} x_{3}}{x_{1}^{4} + x_{2}^{4} + x_{3}^{4}}, max \frac{(1 - r) x_{1} x_{2}^{2} x_{3}}{x_{1}^{4} + x_{2}^{4} + x_{3}^{4}}}], \\ if (x_{1}, x_{2}, x_{3}) \neq (0, 0, 0) \\ [0, 0, 0], if (x_{1}, x_{2}, x_{3}) = (0, 0, 0) . \end{matrix}$ Given $x = (x_{1}, x_{2}, x_{3}) \in ℝ^{3}$ and for all r ∈ [0, 1], take note of it is well defined. Given r ∈ [0, 1] and $x = (x_{1}, x_{2}, x_{3}) \in ℝ^{3}$ , if x₁ ≠ 0, x₂ ≠ 0 and x₃ ≠ 0, then the real-valued functions $φ_{r}^{-}$ and $φ_{r}^{+}$ are differentiable at x. If x₁ = x₂ = x₃ = 0, we have $\frac{\partial_{L_{gH}} φ (0, 0, 0)_{r}}{\partial x_{1}} = lim_{α \to 0} \frac{[φ (0, 0, 0) + α (1, 0, 0)]_{r} ⊖_{gH} [φ (0, 0, 0)]_{r}}{α} = lim_{α \to 0} \frac{[φ (0 + α, 0, 0)]_{r} ⊖_{gH} [φ (0, 0, 0)]_{r}}{α} = lim_{α \to 0} \frac{[0, 0, 0]}{α} = [0, 0, 0] \in K_{C}, \frac{\partial_{L_{gH}} φ (0, 0, 0)_{r}}{\partial x_{2}} = lim_{α \to 0} \frac{[φ (0, 0, 0) + α (0, 1, 0)]_{r} ⊖_{gH} [φ (0, 0, 0)]_{r}}{α} = lim_{α \to 0} \frac{[φ (0, 0 + α, 0, 0)]_{r} ⊖_{gH} [φ (0, 0, 0)]_{r}}{α} = lim_{α \to 0} \frac{[0, 0, 0]}{α} = [0, 0, 0] \in K_{C}, \frac{\partial_{L_{gH}} φ (0, 0, 0)_{r}}{\partial x_{3}} = lim_{α \to 0} \frac{[φ (0, 0, 0) + α (0, 0, 1)]_{r} ⊖_{gH} [φ (0, 0, 0)]_{r}}{α} = lim_{α \to 0} \frac{[φ (0, 0, 0 + α)]_{r} ⊖_{gH} [φ (0, 0, 0)]_{r}}{α} = lim_{α \to 0} \frac{[0, 0, 0]}{α} = [0, 0, 0] \in K_{C} .$ Therefore φ is the partial L_gH-derivative at $x = (0, 0, 0) \in ℝ^{3}$ . However, there does not exist the directional L_gH-derivative of φ at $x = (0, 0, 0) \in ℝ^{3}$ .

Consider the direction of $y = (2, 1, - 1) \in ℝ^{3}$ and r ∈ [0, 1], we have $φ_{L_{gH}}^{'} (x, y)_{r} = lim_{α \to 0^{+}} \frac{[φ (0, 0, 0) + α (2, 1, - 1)]_{r} ⊖_{gH} [φ (0, 0, 0)]_{r}}{α} = lim_{α \to 0^{+}} \frac{[φ (2 α, α, - α)]_{r}}{α} = lim_{α \to 0^{+}} [\frac{- 1 (1 - r)}{9 α^{8}}, 0],$ hence the limit does not exist, consequently, φ is not directional L_gH-derivative at x = (0, 0, 0) in the direction of y = (2, 1, - 1).

6 Conclusion

In this article, g-differentiability to fuzzy multi-dimensional mappings based on a new concept of fuzzy g-(continuous linear) function from $ℝ^{n}$ into E was extended by using the new g-difference. Moreover Fréchet and Gâteaux g-differentiability and the relation of them were defined and analyzed. The directional g-differentiability and partial g-derivatives proposed here and their properties, and so association of them in terms of a level-wise gH-differentiability were discussed. Furthermore, some examples to illustrate the ability and reliability of our concepts were solved. The results of this study could directly be applied to the fuzzy optimization. In line with the content suggested in this article, the next step could be to investigate the g-subdifferential of fuzzy multi-dimensional mappings plus the applications in the convex fuzzy optimization.

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