Quasilinear approximation for interval-valued functions via generalized Hukuhara differentiability

Computational and Applied Mathematics (2022) 41:149 https://doi.org/10.1007/s40314-021-01746-6 Quasilinear approximation for interval-valued functions via generalized Hukuhara differentiability R. Osuna-Gómez1 ·T. M. Costa2 ·Y. Chalco-Cano3 ·B. Hernández-Jiménez4 Received: 21 April 2021 / Revised: 3 September 2021 / Accepted: 12 December 2021 / Published online: 20 April 2022 © The Author(s) 2022 Abstract In this paper, a new generalized Hukuhara differentiability concept for interval-valued functions defined on Rnis 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. Keywords Interval-valued functions ·Quasilinear functions ·Generalized Hukuhara differentiability Mathematics Subject Classification 90C30 ·90C70 ·90C99 1 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). BR. Osuna-Gómez [email protected] 1Dpto. de Estadística e Investigación Operativa, Universidad de Sevilla, Seville, Spain 2Dpto. de Matemática e Matemática Aplicada-ICET, Universidade Federal de Lavras, Lavras, MG, Brazil 3Dpto. de Matemática, Universidad de Tarapacá, Casilla 7D, Arica, Chile 4Dpto. de Economía, M.Cuantitativos e H. Económica, Universidad Pablo de Olavide, Seville, Spain 123 149 Page 2 of 16 R. Osuna-Gómez et al. 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 (gH-difference for short) that coincides with the difference defined in Markov (1979). Using the gH-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 (gH-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 gH-differentiability is more general than these others. Since then, the gH-differentiability concept has been applied to different fields (Armand et al. 2016; Bede and Stefanini 2013; Chalco-Cano et al. 2012,2013a,b; Long et al. 2015; Majumder et al. 2016; Villamizar- Roa et al. 2015;Wangetal.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 gH-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 gH-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 gH- differentiability 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 gH-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 gH-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 gH-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) 123 Quasilinear approximation for interval-valued… Page 3 of 16 149 and Stefanini in Stefanini (2008), and gives a correct meaning for the Markov’s statement above cited. This work is organized as follows: Sect. 2recalls some known results about interval analysis. Section 3shows that the condition for gH-differentiability given by Markov (1979)is only sufficient, and then, a necessary and sufficient condition for such gH-differentiability, which is the starting point for the introduction of gH-differentiability of interval-valued functions of several real variables, is introduced. Section 4shows that some of the gH- differentiability previous concepts for interval-valued functions of several real variables introduced in the literature have shortcomings, and it provides a new gH-differentiability definition for interval-valued functions of several real variables that extends the classical version of Fréchet differentiability, generating a gH-differential that is a quasilinear interval-valued function. Section 4also shows the gH-differentiability concept for interval-valued functions of several real variables given by Stefanini and Arana-Jiménez (2019) which is equivalent to the gH-differentiability one herein introduced. The extension of gH-differentiability to fuzzy environment is presented in Sect. 5. Finally, Sect. 6presents our last considerations. 2 Preliminaries Consider the space I(R)of all closed and bounded intervals of real numbers, that is, I(R)= a,a/a,a∈Rand a≤a.GivenA,B∈I(R)and λ∈R, the interval arithmetic operations are defined by A+B=a,a+b,b=a+b,a+b,λA=λa,λaif λ≥0 λa,λaif λ<0,(1) and AgH B=C⇔(a)A=B+Cif μ(B)≤μ(A) (b)B=A+(−1)Cif μ(B)>μ(A), (2) where μ(A)denotes the length of an interval A=[a,a], i.e., μ(A)=a−a. We write A=[α∨β]if αand βare the end-points of the interval A∈I(R),butα≤βit is not necessarily satisfied. The gH-difference of two intervals always exists and it is equal (Stefanini and Bede 2009)to AgH B=[min{a−b,a−b},max{a−b,a−b}] = [a−b∨a−b].(3) 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. Proposition 2.1 (Markov 1979; Stefanini and Bede 2009)Given A,B,C∈I(R)and λ∈R, it follows that: 1.μ(A+B)=μ(A)+μ(B), 6.[0,0] gH A=−A, 2.μ(λA)=|λ|μ(A), 7.AgH A=[0,0], 3.μ(AgH B)=|μ(A)−μ(B)|,8.AgH B=[0,0]⇔A=B, 4.λ(AgH B)=λAgH λB,9.(A+B)gH B=A, 5.AgH B=−(BgH A), 10.AgH (A+B)=−B. Given n∈N, an application F:S⊆Rn→I(R), such that F(x)=[f(x), f(x)]is called an interval-valued function of several variables, where f,f:S→Rare real-valued 123 149 Page 4 of 16 R. Osuna-Gómez et al. functions, such that f(x)≤f(x),forallx∈S,and fand fare its end-point functions. If n=1, then Fis 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 gH-difference given by Stefanini (2008), the following differentiability concept for intervalvalued functions was introduced. Definition 2.1 (Stefanini and Bede 2009)LetS⊆Rbe an open and nonempty set and let F:S→I(R)be an interval-valued function, and then, the generalized Hukuhara derivative (gH-derivative, for short) of Fat x∗∈Sis defined by F gH(x∗)=lim h→0 F(x∗+h)gH F(x∗) h.(4) If F gH(x∗)∈I(R)satisfying (4) exists, we say that Fis generalized Hukuhara differentiable (gH-differentiable, for short) at x∗. The relationships between the gH-differentiability of an interval-valued function F(x)= [f(x), f(x)]and the differentiability of its end-point functions, f(x)and f(x), have been completely studied in Qiu (2020). 3 New necessary and sufficient conditions for gH-differentiability on R Markov (1979) states that a necessary and sufficient condition for the gH-differentiability of Fat x∗is the existence of an interval A∈I(R)and an interval-valued function P: (−, ) →I(R)with limh→0P(h)=[0,0], such that F(x∗+h)gH F(x∗)=hA+hP(h). (5) However, this condition, in general, it is not necessary as it is shown in the following example. Example 3.1 Let F:(−1,1)→I(R)be the interval-valued function given by F(x)= (1−x2)[1,2]. Then, fand fare differentiable functions, and consequently (see Chalco- Cano et al. 2011), Fis gH-differentiable with F gH(x)=−2x[1,2].Since F(x+h)gH F(x)=min −h2−2xh,2(−h2−2xh),max −h2−2xh,2(−h2−2xh) for all x∈(−1,1)and h∈R, such that x+h∈(−1,1),thengivenx<0, there exists 1>0, such that h+2x<0forallh∈(0, 1), and consequently F(x+h)gH F(x)=[−h2−2xh,2(−h2−2xh)]. However, does not exist any interval-valued function P:B(−, ) →I(R),P(h)= [p(h), p(h)]for all h∈(−, ), such that [−h2−2xh,−2(−h2−2xh)]=h(−2x)[1,2]+h[p(h), p(h)].(6) Indeed, let us suppose that there exists an interval-valued function P:(−, ) →I(R), P(h)=[p(h), p(h)]for all h∈(−, ) such that (6) holds, then setting ˜=min{, 1}and given h∈(0,˜), it follows that p(h)=−hand p(h)=−2h,thatis,p(h)<p(h),which contradicts the fact that Pis an interval-valued function. Therefore, Fis gH-differentiable, but does not exist an interval A∈I(R)and an interval-valued function P:(−, ) →I(R), such that (5) holds. 123 Quasilinear approximation for interval-valued… Page 5 of 16 149 Although the condition given by Markov is not necessary for gH-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)whenFit is gH-differentiable at x∗∈S⊆R. Before to present such result, a technical proposition is necessary. Proposition 3.1 Let F,P:S⊆R→I(R)be interval-valued functions, such that limx→x0P(x)=[0,0].Given x0∈S, it follows that: (i) if F(x)=A+P(x),thenlimx→x0F(x)=A; (ii) if limx→x0F(x)=A, then F(x)⊆A+P(x). Proof (i) This result follows directly from the metric Hproperties. (ii) If limx→x0F(x)=A, from Proposition 7 in Stefanini and Bede (2009), it follows that limx→x0(F(x)gH A)=[0,0]. Let P:S→I(R)be the interval-valued function given by P(x)=F(x)gH Afor all x∈S. Thus, limx→x0P(x)=[0,0],and from definition of gH, it follows that: (F(x)gH A)=P(x)⇔(a)F(x)=A+P(x), or (b)A=F(x)+(−1)P(x) ⇔(a)f(x)−a=p(x)and f(x)−a=p(x), or (b)f(x)−a=p(x)and f(x)−a=p(x). If (a)holds, then F(x)=A+P(x). On the other hand, if (b)holds, then f(x)−a= p(x)≥p(x)and f(x)−a=p(x)≤p(x), that is, a+p(x)≤f(x)and f(x)≤a+p(x). Therefore, from (a)and (b), it follows that F(x)⊆A+P(x). Next results follow directly from Proposition 3.1. Theorem 3.1 Let S ⊆Rbe an open and nonempty set and let F :S⊆R→I(R)be an interval-valued function. Given x∗∈S, it follows that: (i) If there exist an interval A ∈I(R)and an interval-valued function P :(−, ) →I(R) with limh→0P(h)=[0,0], such that (5)holds, then F is gH-differentiable at x∗. (ii) If F is gH-differentiable at x∗, then there exists an interval-valued function P : (−, ) →I(R)with limh→0P(h)=[0,0], such that F(x∗+h)gH F(x∗)⊆hF gH(x∗)+hP(h). (7) The following result provides a necessary and sufficient gH-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 gH-differentiability for interval-valued functions of several variables. Theorem 3.2 Let S ⊆Rbe an open and nonempty set. An interval-valued function F :S→ I(R)is gH-differentiable at x∗∈S if and only if there exist an interval A ∈I(R)and an interval-valued function P :(−, ) →I(R)with limh→0P(h)=[0,0], such that (F(x∗+h)gH F(x∗)) gH hA =hP(h). (8) Proof If Fis gH-differentiable at x∗, then there exists F gH(x∗)∈I(R), such that limh→0F(x∗+h)gH F(x∗) hgH F gH(x∗)=[0,0].Given >0, let P:(−, ) →I(R) be the interval-valued function given by P(h)=F(x∗+h)gH F(x∗) hgH F gH(x∗)if h= 0, 0ifh=0. 123 149 Page 6 of 16 R. Osuna-Gómez et al. Then, limh→0P(h)=[0,0]. Moreover, given h∈(−, )\{0}, it follows that: hP(h)=(F(x∗+h)gH F(x∗)) gH hF gH(x∗). Thus, (8) holds with A=F gH(x∗). On the other hand, it is easy to see that (8) holds with A=F gH(x∗)whenever h=0. Reciprocally, if there exist an interval A∈I(R)and an interval-valued function P: (−, ) →I(R)with limh→0P(h)=[0,0], such that (8) holds, then from Proposition 2.1 and Proposition 7 in Stefanini and Bede (2009), it follows that: lim h→0F(x∗+h)gH F(x∗) h=A. Therefore, Fis gH-differentiable at x∗. Since the gH-differentiability condition given in Theorem 3.2 is an interval version of the Fréchet differentiability, this result shows that the gH-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. 4 gH-differentiability for interval-valued functions on Rn This section recalls the gH-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 gH- differentiability concept which overcomes the shortcomings above mentioned and it shows that this concept is equivalent to the gH-differentiability one for interval-valued functions of several variables given by Stefanini and Arana-Jiménez (2019). In Ahmad et al. (2016), Chalco-Cano et al. (2013a), and Luhandjula and Rangoaga (2014), the following interval-valued gH-differentiability definitions for interval-valued-functions of several variables are considered. Definition 4.1 Let Fbe an interval-valued function defined on an open and non-empty set S⊆Rnand let x∗=x∗ 1,...,x∗ n∈Sbe fixed. Given the interval-valued function Hi:R→ I(R)defined by Hi(xi)=F(x∗ 1,...,x∗ i−1,xi,x∗ i+1,...,x∗ n),ifHiis gH-differentiable at x∗ i, then we say that Fhas the ith partial gH-derivative at x∗that is defined and denoted by ∂F ∂xi (x∗)=H ix∗ i∈I(R). If there exists ∂F ∂xi(x∗)for all i∈{1,...,n}, then the n-tuple ∂F ∂x1(x∗),... ∂F ∂xn(x∗)of intervals is called the gradient of Fand it is denoted by ∇gH F(x∗). Definition 4.2 Given S⊆Rnan open and nonempty set, let F:S→I(R)be an intervalvalued function and let x∗∈Sbe fixed. Fis said to be gH-differentiable at x∗if there exist all the partial gH-derivatives ∂F ∂x1(x),..., ∂F ∂xn(x)on some neighborhood of x∗and are continuous at x∗. Remark 4.1 Definition 4.2 says that, for n=1, an interval-valued function F:S⊆R→ I(R)is gH-differentiable at x0∈Sif and only if there exist F gH(x)in a neighborhood of x0 123 Quasilinear approximation for interval-valued… Page 7 of 16 149 and F gH is continuous at x0. 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 gH-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 Sis an open set and Vis a normed vector space, a function f:S⊆R→Vis differentiable at x∗∈S if and only if there exist a linear function Lx∗:R→Vgiven by Lx∗(h)=f(x∗)hand a function p:(−, ) →Vwith limh→0p(h)=0, such that f(x∗+h)−f(x∗)=Lx∗(h)+hp(h)for all h∈Rwith x∗+h∈S. From Theorem 3.2, we have a similar situation for the gH-differentiability on R. Indeed, given F:S→I(R), from Theorem 3.2,Fis gH-differentiable at x∗∈Sif there exist the interval-valued function Tx∗:R→I(R)given by Tx∗(h)=hF gH(x∗)and an intervalvalued function P:(−, ) →I(R)with limh→0P(h)=[0,0], such that (F(x0+h)gH F(x0)) gH Tx∗(h)=hP(h). However, the interval-valued function Tx∗(h), satisfying (8), may not be linear, since I(R) is not a vector space. For example, let us consider F:(−1,1)→I(R)given by F(x)= [−1+x,x2]. Considering x∗=0 and given P:(−, ) →I(R)defined by P(h)=(F(h)gH F(0))ghh[−1,0] |h|if h= 0 0ifh=0, it follows that limh→0P(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 h1=− 1 2and h2=1 2, it follows that: Tx∗(h1+h2)=Tx∗(0)=0[−1,0]=[0,0] = −1 2,1 2=0,1 2+−1 2,0 =Tx∗−1 2+Tx∗1 2=Tx∗(h1)+Tx∗(h2). On the other hand, it is easy to prove that if an interval-valued function Fis gH- differentiable at a point x∗∈S⊆R, then the interval-valued function Tx∗:R→I(R) given by Tx∗(h)=hF gH(x∗)is a quasilinear interval-valued function in the following sense. Definition 4.3 (Assev 1986; Rojas-Medar et al. 2005) Considering the inclusion order relation ⊆, an interval-valued function :Rn→(I(R), ⊆)is quasilinear if it satisfies the following conditions: (C1) (λx)=λ(x)for all λ∈R,x∈Rn, (C2) (x+y)⊆(x)+(y)for all x,y∈Rn. (C3) {x}≤{y}⇒(x)⊆(y). Clearly, the condition (C3) holds true for every x,y∈Rn. Thus, an interval-valued function :Rn→(I(R), ⊆)is quasilinear if and only if (C1) and (C2) hold. Based on Theorem 3.2, Remark 4.2, and on the fact that a real-valued function f:S→R, defined on an open set S⊆Rn, is differentiable at x∗∈Sif there exists a continuous 123 149 Page 8 of 16 R. Osuna-Gómez et al. and linear real-valued function Lx∗:Rn→Rand a function p:B(0,) →R,where 0=(0,...,0)∈Rnand B(0,):= {x∈Rn:x<}, with limv→0p(v) =0, such that f(x∗+v) −f(v) =Lx∗(v) +vp(v), our gH-differentiability definition for interval-valued functions of several real variables is given as follows. Definition 4.4 Given 0=(0,...,0)∈Rnand S⊆Rnan open and nonempty set, let F:S→I(R)be an interval-valued function. We say that Fis gH-differentiable at x∗∈S if there exist a continuous and quasilinear interval-valued function Tx∗:Rn→I(R)and an interval-valued function P:B(0,)⊆Rn→I(R)with limv→0P(v) =[0,0], such that (F(x∗+v) gH F(x∗)) gH Tx∗(v) =||v||P(v), (9) for all v∈B(0,) := {x∈Rn:x<}with (x∗+v) ∈S, and in this case, Tx∗(v) is called the gH-derivative of Fat x∗.IfFis gH-differentiable at all x∈S,thenFis said to be gH-differentiable. Theorem 4.1 Given 0=(0,...,0)∈Rnand S ⊆Rnan open and nonempty set, let F:S→I(R)be an interval-valued function. Then, F is gH-differentiable at x∗∈S if and only if lim v→0F(x∗+v) gH F(x∗)gH Tx∗(v) v=[0,0],(10) where Tx∗:Rn→I(R)is a continuous and quasilinear interval-valued function that satisfies (9). Proof If Fis gH-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:B(0,) → I(R)by P(v) =(F(x∗+v)gH F(x∗))gHTx∗(v) vif v= 0 [0,0]if v=0, it follows that (9) holds with limv→0P(v) =[0,0], and consequently, Fis gH-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 F:S⊆Rn→I(R),F(x)=[f(x), f(x)]and let x∗∈S, such that (x∗+v) ∈Sfor all v∈Rnwith v<δfor some given δ>0. We say that Fis gH-differentiable at x∗∈Sif and only if there exist two vectors ˆw=(ˆw1,..., ˆwn)and ˜w=(˜w1,..., ˜wn)in Rnand two functions ˆ(v),˜(v)with limv→0ˆ(v) =limv→0˜(v) =0, such that for all v= 0, f+f 2(x∗+v) −f+f 2(x∗)= n  i=1 viˆwi+vˆ(v), (11) f−f 2(x∗+v) −f−f 2(x∗) = n  i=1 vi˜wi+v˜(v) .(12) 123 Quasilinear approximation for interval-valued… Page 9 of 16 149 The interval-valued function DgH F(x∗):Rn→I(R)defined, for v=(v1,...,v n)∈Rn, by DgH F(x∗)(v) =n  i=1 viˆwi− n  i=1 vi˜wi , n  i=1 viˆwi+ n  i=1 vi˜wi is called the gH-differential (or total gH-derivative) of Fat x∗and DgH F(x∗)(v) is the interval-valued differential of Fat x∗with respect to v. Proposition 4.1 Let F :S⊆Rn→I(R),F(x)=[f(x), f(x)]and let x∗∈S such that (x∗+v) ∈S for all v∈Rnwith v<δfor some given δ>0. Then, F is gH-differentiable at x∗∈S if and only if there exist two vectors ˆw∈Rnand ˜w∈Rn, such that the following limit condition is true lim v→0F(x∗+v) gH F(x∗)gH W(v) v=[0,0], where W(v) is the interval-valued function defined by W(v) =n  i=1 viˆwi− n  i=1 vi˜wi , n  i=1 viˆwi+ n  i=1 vi˜wi. In this case, for the differential function, we have that DgH F(x∗)(v) =W(v). The following result allows us to know exactly the continuous and quasilinear intervalvalued function Tx∗given in Definition 4.4 and allows us to see that the gH-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 F :S→I(R),F(x)=[f(x), f(x)]. Then, F is gH-differentiable at x∗∈S in the sense of Definition 4.4 if and only if F is gH-differentiable at x∗∈Sinthe 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 DgH F(x∗):Rn→I(R)is quasilinear, since from its definition, it is easy to see that it is continuous. Given v∈Rnand λ∈Rwith λ<0, from definition of DgH F(x∗)and from (1), it follows that: DgH F(x∗)(λv) =n  i=1 (λvi)ˆwi− n  i=1 (λvi)˜wi , n  i=1 (λvi)ˆwi+ n  i=1 (λvi)˜wi =λn  i=1 viˆwi+ n  i=1 vi˜wi ,λ n  i=1 viˆwi− n  i=1 vi˜wi =λn  i=1 viˆwi− n  i=1 vi˜wi , n  i=1 viˆwi+ n  i=1 vi˜wi =λDgH F(x∗)(v). 123 149 Page 16 of 16 R. Osuna-Gómez et al. Luhandjula MK, Rangoaga MJ (2014) An approach for solving a fuzzy multiobjective programming problem. 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