# An Extension of the Carathéodory Differentiability to Set-Valued Maps

Authors: Pedro Hurtado; Alexander Leones; M. Martelo; J. B. Moreno
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

This paper uses the generalization of the Hukuhara difference for compact convex set to extend the classical notions of Carathéodory differentiability to multifunctions (set-valued maps). Using the Hukuhara difference and affine multifunctions as a local approximation, we introduce the notion of CH-differentiability for multifunctions. Finally, we tackle the study of the relation among the Fréchet differentiability, Hukuhara differentiability, and CH-differentiability.


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## Body

## 1. Introduction

GivenX and Y nonempty sets, if for each element x belonging to X is assigned, due to a certain law denoted here by F:X⇉Y, a set Fx contained in Y, we say that F is a multifunction (we also use the term multivalued function or set-valued map). Among the many and various examples and uses of multifunctions, we can find from those simpler provided by elementary algebra and trigonometry to those appearing, for example, in applications of control theory involving optimal control problems and structural properties of control systems such as stability, controllability/reachability ([1–4]), and perturbation theory for ordinary differential equations in the real Euclidean space ℝn ([5, 6]); of particular interest is also studying the differentiability of multifunctions applied to perturbed optimization problems as observed in [7], where it is considered the problem “minimize fw,x as x∈Aw” being f:W×X⟶ℝ∪+∞ an arbitrary function, A:W⇉X a multifunction with 0∈domA and W,X Banach spaces. The concept of differentiability for multifunction has been discussed by many authors([8–13]). In [11], it extends the classical notion of Fréchet differentiability to multifunctions, using affine multifunctions and Hausdorff distance.Other concepts of differentiability can be found in the literature of classical analysis, as the Carathéodory derivative proposed in [14], the strength of Carathéodory’s formulation relies on the concept of continuity, and its proof uses strongly the properties of continuous functions. Another advantage about this formulation is that it does not require the difference quotient present in the Fréchet formulation, which is the key to generalize it to a function of several variables. In [15], it explored some of the advantages of Carathéodory’s characterization over Fréchet’s characterization proposed in [16].In this work, motivated by the Carathéodory derivative proposed in [14] and generalization of Hukuhara difference proposed ([9, 17, 18]), an extension of the classical notion of Carathéodory differentiability to multifunctions defined on finite-dimensional normed spaces is approached. For this purpose, we assume finite-dimensional normed vector spaces X and Y and consider the family of affine multifunctions (see [11] for details of affine multifunctions and the Fréchet differentiability of multifunctions):(1)MAFF0X,Y=A:X⟶bccY∣AisaffineandA0=0,wherebccY is the collection of all bounded closed convex subsets of Y, provided with a suitable metric. Then, we associate the differentiability (Carathéodory) of the multifunctions to the existence of continuous functions(2)ϕ1,ϕ2:X⟶MAFF0X,Y.This paper is organized as follows. In Section2, we present some definitions and basic results of generalized Hukuhara difference and affine multifunction. Section 3 presents our extension of Carathéodory differentiability to multifunctions (see Definition 4), named here as CH-differentiability (Carathéodory Hukuhara differentiability), which is based on the set in (1) and the generalized Hukuhara difference, [9]. Also, we show an example of CH-differentiable multifunctions (see Example 4) and study some properties of the CH-differentiability such as uniqueness, continuity, sum, and composition. Finally, in Section 4, we provided a relation involving the Fréchet differentiability, generalized Hukuhara differentiability, and CH-differentiability (Theorems 7, 8, 9, 10).


## 2. Preliminaries

LetX and Y be finite-dimensional normed vector spaces over the set of real numbers ℝ.A multifunctionF:X⇉Y from X into Y assigns to each x∈X a (possibly empty) subset Fx⊂Y.The sets(3)domF=x∈X:Fx≠∅,grF=x,y∈X×Y:y∈Fx,are called the effective domain and the graph of the multifunction F, respectively.BybccY, we denote the collection of all bounded closed convex subsets of Y. The collection bccY equipped with the Minkowsky addition:(4)A+B=a+b:a∈A,b∈BA,B∈bccYand multiplication by nonnegative real numbers:(5)αA=αa:a∈AA∈bccY,α≥0is a semilinear space. The function(6)dHA,B=infα≥0:A⊂B+αBY,B⊂A+αBYwhere BY is the unit ball in Y, is called the Hausdorff distance, and determines a structure of a metric space on bccY.Remark 1.
It is well known that the Minkowsky addition is associative and commutative with neutral element0. If α=−1, the scalar multiplication gives the opposite −A=−1A=−a∣a∈A but, in general,(7)A+−A≠0,i.e., the opposite of A is not the inverse of A unless A=a is a singleton. The first implication of this fact is that, in general, the additive simplification is not valid, that is A+C=B+C¿A=B or A+B−B≠A.Remark 2.
The partial solution for (7) is the Hukuhara difference, this difference has been introduced as a set C for which A⊖B=C⇔A=B+C, and the important property of ⊖ is A⊖A=0 and A+B⊖B=A, ∀A,B∈bccX. The Hukuhara difference is unique, but it does not always exist (a necessary condition for A⊝B exist is that A contains a translation c+B of B.)A generalization of the Hukuhara difference proposed in [17] aims to overcome this situation.Definition 1 (see [9]).
The generalized Hukuhara difference of two setsA,B∈bccY (gH-difference for short) is defined as follows:(8)A⊝gB=C⇔aA=B+C;orbB=A+−1C.Proposition 1 (see [18]).
IfC=A⊝gB exists, it is unique and if also A⊝B exists then A⊝gB=A⊝B.Remark 3 (see [18]).
A necessary condition forA⊝gB to exists is that either A contains a translation of B (as for A⊝B) or B contains a translation of A. In fact, for any given c∈C, we get B+c⊆A from (i) or A+−c⊆B from (ii).Remark 4 (see [18]).
IfA⊝gB exists, then B⊝gA exist and B⊝gA=−A⊝gB.Remark 5 (see Proposition 4 in [9]).
The generalized Hukuhara difference exists for any two compact intervals.The generalized Hukuhara difference has the following properties:Proposition 2 (see [18]).
LetA,B∈bccY then(1)
If the gH-difference exists, it is unique and it is a generalization of the usual Hukuhara difference, sinceA⊝gB=A⊝B whenever A⊝B exists(2)
A⊝gA=0,(3)
ifA⊝gB exists in the sense a, then B!gA exists in the sense b and vice versa(4)
A+B⊝gB=A(5)
0⊝gA⊝gB=−B⊝g−A,(6)
We haveA⊝gB=B⊝gA=C if and only if C=0 and A=B
From now on, in this paper, we will consider the generalized Hukuhara difference,⊝g.


### 2.1. Affine Multifunction and Fréchet Differentiability

LetX and Y be finite-dimensional normed vector spaces over the set of real numbers ℝ.A multifunctionH:X⟶bccY is said to beconvex if(9)αHx1+1−αHx2⊂Hαx1+1−αx2,for all x1,x2∈domH and α∈0,1;affine if(10)αHx1+1−αHx2=Hαx1+1−αx2,for all x1,x2∈domH and α∈0,1.Proposition 3 (see [11]).
An affine multifunctionF:X⟶bccY is uniformly continuous on domF in the Hausdorff sense.Definition 2.
A multifunctionF:X⟶bccY is said to be Fréchet differentiable at a point x0∈intdomF, if there exists an affine multifunction Ax0∣·:X⟶bccY such that 0∈intdomAx0∣· and(11)limh⟶0dHFx0+h,Ax0∣hh=0.This definition is equivalent to the following: for anyε>0 there exists δ>0 such that for all h∈δBX, we have(12)Fx0+h⊂Ax0∣h+εhBY,Ax0∣h⊂Fx0+h+εhBY,where BX and BY are unit balls in X and Y, respectively. So immediately, we have that Ax0∣0=Fx0.


## 2.1. Affine Multifunction and Fréchet Differentiability

LetX and Y be finite-dimensional normed vector spaces over the set of real numbers ℝ.A multifunctionH:X⟶bccY is said to beconvex if(9)αHx1+1−αHx2⊂Hαx1+1−αx2,for all x1,x2∈domH and α∈0,1;affine if(10)αHx1+1−αHx2=Hαx1+1−αx2,for all x1,x2∈domH and α∈0,1.Proposition 3 (see [11]).
An affine multifunctionF:X⟶bccY is uniformly continuous on domF in the Hausdorff sense.Definition 2.
A multifunctionF:X⟶bccY is said to be Fréchet differentiable at a point x0∈intdomF, if there exists an affine multifunction Ax0∣·:X⟶bccY such that 0∈intdomAx0∣· and(11)limh⟶0dHFx0+h,Ax0∣hh=0.This definition is equivalent to the following: for anyε>0 there exists δ>0 such that for all h∈δBX, we have(12)Fx0+h⊂Ax0∣h+εhBY,Ax0∣h⊂Fx0+h+εhBY,where BX and BY are unit balls in X and Y, respectively. So immediately, we have that Ax0∣0=Fx0.


## 3. CH-Differentiability and Some Properties

In this section, we use the Hukuhara difference and the affine multifunction to build the concept of differentiability of a multifunction.LetX and Y be finite-dimensional normed vector space over real ℝ, define the space:(13)MAFF0X,Y=A:X⟶bccY∣AisaffineandA0=0Example 1.
Letρ:X⟶ℝ be a linear function in X∗, such that ρ0=0. Choose the function F:X⇉bccY defined by Fx=ρxB, for B∈bccY. Then, F∈MAFF0X,bccY.
In effect,Fαx+1−αy=αρx+1−αρyB=αFx+1−αFy.Proposition 4.
ForF,G∈MAFF0X,Y,(14)d~F,G=sup∥x∥≤1dHFx,Gx,is a metric on MAFF0X,Y, where dH is the Hausdorff metric.Proof.
(15)d~F,F=sup∥x∥≤1dHFx,Fx=sup∥x∥≤10=0.(1)
Ifd~F,G=0, then for all x∈X such that x≤1, we have dHFx,Gx=0, thus Fx=Gx for x∈X such that x≤1. On the other hand, since F is affine and F0=0, for x>1, we have(16)Fx=x·1xFx=x·1xFx+1−1x0=x·F1x·x+1−1x·0=x·Fxx=x·Gxx=Gx.
Therefore,F=G.(17)d~F,G=supx≤1dHFx,Gx=supx≤1dHGx,Fx=d~G,F.(2)
IfF,G, and H are multifunctions, then for each x∈X such that x≤1, we have(18)dHFx,Hx≤dHFx,Gx+dHGx,Hx≤d~F,G+d~G,H.
Therefore,d~F,H≤d~F,G+d~G,H.☐Definition 3.
LetX,Y be finite-dimensional normed vector spaces over the set of real numbers ℝ. A function ϕ:X⟶MAFF0X,bccY is said to be continuous at x0, if given ε>0, there exists δ>0 such that, if x−x0<δ, then d~ϕx,ϕx0<ε.Example 2.
Letx0 be a fixed point and x in X. By the Hahn-Banach theorem (see [19], page 3.), there is a linear function ρx:X⟶ℝ in X∗, such that(19)ρx0=0,ρxx−x0=x−x0X2,andρxX∗=x−x0X.Let be fixedB∈bccY. By Example 1, the functions Fx:X⇉bccY defined by Fxh=ρxhB are in MAFF0X,bccY. Now, the function(20)ϕ:X⟶MAFF0X,bccY,x⟶Fx,is continuous at x0.In effect,ϕx0=0 and(21)d~ϕx,ϕx0=suph≤1dHFxh,Fx0h=suph≤1dHρxhB,0≤ρxX∗=x−x0X.Theorem 1.
Ifϕ,φ:X⟶MAFF0X,Y are continuous at the point x0. Then, ηx=ϕx⊝gφx is continuous at x0.Proof.
Sinceϕ and φ are continuous at x0, given ε>0 and α∈0,1 there are δ1>0 and δ2>0 such that, if x−x0<δ=minδ1,δ2 then d~ϕx,ϕx0<αε and d~φx,φx0<1−αε. Thus, for h<δ and the definition of affine multifunction, we have(22)ϕx0+hh⊆ϕx0h+αεhBY,φx0+hh⊆φx0h+1−αεhBY,ϕx0h⊆ϕx0+hh+αεhBY,φx0h⊆φx0+hh+1−αεhBY.
Now, ifx∈ηx0+hh, from Definition 1, there is a∈ϕx0+hh and b∈φx0+hh, such that x=a−b; thus, from (1) and (7), we have(23)a=a1+αεha2,b=b1+1−αεhb2.where a1∈ϕx0h,b1∈φx0h, and a2,b2∈BY. So(24)=a−b=a1+αεha2−b1−1−αεhb2=a1−b1+εhαa2+1−α−b2=x1+εhx2.
Then,x1=a1−b1∈ϕx0h⊝gφx0h, this is, x1∈ηx0h and x2=αa2+1−α−b2∈BY, since a2,−b2∈BY and BY is convex. Therefore, x∈ηx0h+εhBY. In consequence, ηx0+h⊆ηx0h+εhBY. Analogously, ηx0⊆ηx0+hh+εhBY. Thus, η is continuous.☐Now, motivated by the Carathéodory’s Definition in [14], Definition 1, and Proposition 2, we define the following:Definition 4.
A multifunctionF:X⟶bccY is CH-differentiable (Carathéodory and Hukuhara differentiable) at a point x0∈intDomF, if there are functions ϕ1,ϕ2:Vx0⟶MAFF0X,bccY continuous at a point x0∈Vx0, where Vx0 is a neighborhood of the point x0, such that ϕ1x0=ϕ2x0 and(25)Fx0+h⊝gFx0=ϕ1x0+hh,Fx0⊝gFx0+h=ϕ2x0+hh,
∀h∈X such that x0+h∈Vx0.Remark 6.
GivenF:X⇉bccY. It is natural to ask the following question. Are the functions ϕ′s in Definition 4 unique, the answer is no, even it does not fit for functions (see [15]). Nevertheless, we have the uniqueness of the differential.Example 3.
The constant functionF:X⇉bccY defined by Fx=B, for B∈bccY is CH-differentiable at X. In effect, Fx0+h⊝gFx0=B⊝gB=0=ϕx0+hh, where ϕx0+hh=0∀h∈X.Example 4.
Letf:ℝn⟶ℝ be a differentiable function at x0∈ℝn and b∈ℝm. The function F:ℝn⇉bccℝm defined by Fx=fxb, is CH-differentiable at x0. In effect, Fx0+h⊝gFx0=fx+hb⊝gfx0b=fx0+h−fx0b=ψx0+hhb where ψ exists because f is differentiable, thus, Carathéodory differentiable. Now, consider ϕ:Vx0⟶MAFF0X,bccY defined by ϕxz=ψxzb. Then, Fx0+h⊝gFx0=ϕx0+hh.Theorem 2.
LetX,Y be two Banach spaces and F:X⇉bccY be a CH-differentiable at x0. Suppose that, there are functions ϕ and φ satisfying the CH-differentiability conditions. Then, ϕx0=φx0.Proof.
SinceFx0+h⊝gFx0=ϕx0+hh and Fx0+h⊝gFx0=φx0+hh, for all h∈Vx0. Now define the function(26)η:Vx0⟶MAFF0X,bccY,by ηx=ϕx⊝gφx. Observe that ηx0+hh=0. Thus(27)dHηx0h,0=dHηx0h,ηx0+hh.
Now, sinceϕ and φ are continuous at x0. We have η is continuous at x0. Whence, given ε>0, there is δ>0 such that if h<δ, then(28)dHηx0hh,ηx0+hhh≤d~ηx0,ηx0+h<ε.
Now, for eachz∈X with z=1 there is N∈ℕ, such that for n>N, there exists hn∈X, with hn<δ and hn/hn⟶z (just take the sequence on the line segment joining 0 to z). Hence, we have(29)limn⟶+∞dHηx0hnh,0=0.
Then,(30)ηx0limn⟶+∞hnh=limn⟶+∞ηx0hnh=0.
Therefore,
ηx0z=0 and ϕx0=φx0.☐ؙLemma 1.
LetF,G:X⟶bccY be two multifunctions and x0∈X, if limx⟶x0Fx=Fx0 and limx⟶x0Gx=Gx0. Then(31)limx⟶x0Fx+Gx=Fx0+Gx0.Proof.
In effect, iflimx⟶x0Fx=Fx0 and limx⟶x0Gx=Gx0, then given ε>0 and α∈0,1, there exists δ>0, such that if dx,x0<δ. Then, dFx,Fx0<αε and dGx,Gx0<1−αε. Hence, we obtain that(32)Fx⊆Fx0+αεBY,Gx⊆Gx0+1−αεBY.
It follows that ifz∈Fx+Gx then z=a+b, with a∈Fx and b∈Gx, thus, a=a1+αεa2 and b=b1+1−αεb2, with a1∈Fx0, b1∈Gx0, and a2,b2∈BY, so z=a1+b2+εαa2+1−αb2. As αa2+1−αb2∈BY, we have z∈Fx0+Gx0+εBY. Thus, Fx+Gx⊆Fx0+Gx0+εBY. Using Fx0⊆Fx+αεBY and Gx0⊆Gx+1−αεBY, similarly, Fx0+Gx0⊆Fx+Gx+εBY. Therefore(33)dHFx+Gx,Fx0+Gx0<ε.☐Theorem 3.
IfF:X⟶bccY is a CH-differentiable multifunction at x0∈X. Then, F is Hausdorff continuous at x0.Proof.
SinceFx0+h⊝gFx0=ϕx0+hh, we have, from the above proposition,(34)limh⟶0Fx0+h=limh⟶0Fx0+ϕx0+hh=Fx0+ϕx00=Fx0.☐Theorem 4.
LetF,G:X⟶bccY be two CH-differentiable multifunctions at x0. Then, αFx+βGx, with α,β∈ℝ, is CH-differentiable at x0.Proof.
We have,Fx0+h⊝gFx0=ϕx0+hh and Gx0+h⊝Gx0=φx0+hh, thus(35)αFx0+h+βGx0+h=αFx0+ϕx0+hh+βGx0=+φx0+hh=αFx0+βGx0+αϕ+βφx0+hh.
Hence, sinceαϕ+βφ is continuous, the result follows.☐Theorem 5.
Iff:X⟶X is a differentiable function at x0 and F:X⟶bccY is a CH-differentiable multifunction at fx0, then F∘f:X⟶bccY, is CH-differentiable at x0.Proof.
Observe that,Ffx0+h⊝gFfx0=ϕfx0+hh and fx0+h−fx0=ψx0+hh, thus(36)F∘fx0+h⊝gF∘fx0=Ffx0+h⊝gFfx0=Ffx0+ψx0+hh⊝gFfx0=ϕfx0+ψx0+hhψx0+hh=ϕfx0+ψx0+hh∘ψx0+hh.☐Theorem 6.
IfF∈MAFF0X,bccY, then F is a CH-differentiable multifunction at x0.Proof.
LetF∈MAFF0X,bccY and α∈0,1. Choose h=1−α/αw small, thus(37)Fx0+h=Fx0+1−ααw=F1ααx0+1−αw=1ααFx0+1−αFw=Fx0+1−ααFw=Fx0+F1−ααw=Fx0+Fh.
Now, chooseϕ:Vx0⟶MAFF0X,bccY, defined by ϕh=F.☐


## 4. CH-Differentiability vs. Fréchet Differentiability and Generalized Hukuhara Differentiability

The following result is useful for studying the relationship between CH-differentiability and Fréchet differentiability.Lemma 4.
LetAbe inbccYandϕ∈MAFF0X,bccY.
Considerφ:X⟶bccY defined by(38)φx=A+ϕx.Then,φ is an affine multifunction.Proof.
Letx1,x2∈X and α∈0,1, then(39)ϕαx1+1−αx2=αϕx1+1−αϕx2with ϕ0=0. On the other hand, since A=αA+1−αA. We have(40)φαx11−αx2=A+ϕαx1+1−αx2=αA+1−αA+αϕx1+1−αϕx2=αA+ϕx1+1−αA+ϕx2=αφx1+1−αφx2.☐Theorem 7.
IfF:X⟶bccY is a CH-differentiable multifunction at x0, then F is Fréchet differentiable at x0.Proof.
Using the CH-differentiability ofF at x0, there exist ϕ1,ϕ2:Vx0⟶MAFF0X,bccY continuous functions at x0∈Vx0, where Vx0 is a neighborhood at x0, such that ϕ1x0=ϕ2x0 and(41)Fx0+h⊝gFx0=ϕ1x0+hh,Fx0⊝gFx0+h=ϕ2x0+hh,
∀h∈X such that x0+h∈Vx0. On the other hand, since ϕ1x0 is an affine multifunction, using Lemma 4.1,(42)Ah=Fx0+ϕ1x0h,is an affine multifunction. We affirm that A is Fréchet differentiable. In effect, given ε>0, there exists δ>0 such that if h<δ, then(43)dHϕ1x0+hz,ϕ1x0z≤d~ϕ1x0+h,ϕ1x0<ε,for all z∈X, such that z≤1. In particular, for z=h/h, we have(44)dH1hϕ1x0+hh,1∥h∥ϕ1x0h=dHϕ1x0+hhh,ϕ1x0hh<ε.
This is equivalent to(45)ϕ1x0+hh⊆ϕ1x0h+εhBY,ϕ1x0h⊆ϕ1x0+hh+εhBY.
Now, sinceFx0+h⊝Fx0=ϕ1x0+hh, we have(46)Fx0+h=Fx0+ϕ1x0+hh⊆Fx0+ϕ1x0h+εhBY⊆Ah+εhBY.
Analogously,(47)Ah=Fx0+ϕ1x0h⊆Fx0+ϕ1x0+hh+εhBY⊆Fx0+h+εhBY.☐The reciprocal of the above theorem is not always true, because the Hukuhara difference does not exist (see Remark2). But when this difference exists, we have a relationship between Fréchet’s differentiability and CH-differentiability.Theorem 8.
If the Hukuhara difference belongs tobccY and F:X⟶bccY is a Fréchet differentiable multifunction at x0, then F is CH-differentiable at x0.Proof.
LetA be the Fréchet differential of F at x0 and x∈X. By the Hahn-Banach theorem, there exists a lineal function ρx:X⟶ℝ in X∗, such that(48)ρx0=0,ρxx−x0=x−x0X2,andρxX∗=x−x0X.
Then, we can define the functionϕ:X⟶MAFF0X,bccY by(49)ϕ1x=1x−x0X2ρx·fx⊝gfx0!gAx−x0+A·,ifx≠x0,A·,ifx=x0.
This function is continuous atx0 (see Example 2). And(50)ϕ1xx−x0=1x−x0X2ρxx−x0fx⊝gfx0⊝gAx−x0+Ax−x0=1x−x0X2x−x0X2fx⊝gfx0⊝gAx−x0+Ax−x0=fx⊝gfx0⊝gAx−x0+Ax−x0=fx⊝gfx0.
Analogously, for(51)ϕ2x=1∥x−x0∥X2ρx·fx0⊝gfx⊝gAx−x0+A·,ifx≠x0,A·,ifx=x0.,we have that ϕ2 is continuous and ϕ2xx−x0=fx0⊝gfx.☐Now, we remember the generalized Hukuhara differentiability.Definition 5: (see [9]).
Letx0∈ℝ and F:ℝ⟶bccY be a multifunction, then the gH-derivative of the multifunction F at x0 is defined as(52)∂Fx0=limh⟶0Fx0+h⊝gFx0h.If∂Fx0∈bccY satisfying (52) exists, we say that F is generalized Hukuhara differentiable (gH-differentiable for short) at x0.Theorem 9.
LetF:ℝ⟶ccY be a multifunction, if F is CH-differentiable at x0∈ℝ. Then, F is generalized Hukuhara differentiable at x0∈ℝ.Proof.
Using the CH-differentiability ofF, there exist multifunctions ϕ1,ϕ2:Vx0⟶MAFF0ℝ,bccY continuous at x∈Vx0 (neighborhood at x0), such that ϕ1x0=ϕ2x0 and(53)Fx0+h⊝gFx0=ϕ1x0+hh,Fx0⊝gFx0+h=ϕ2x0+hh,for all h∈ℝ such that x0+h∈Vx0. Hence,(54)limh⟶0Fx0+h⊝gFx0h=limh⟶0ϕ1x0+hhh=limh⟶0ϕ1x0+h1.
Now, from the continuity ofϕ1, given ε>0 there exists δ>0 such that, if ∣h∣<δ then d~ϕx0+h,ϕx0<ε, in particular(55)dHϕx0+h1,ϕx01<ε,thus(56)limh⟶0Fx0+h⊝gFx0h=ϕ1x01.☐Remark 7.
Throughout this article, we confine ourselves to multifunctionsF:X⟶bccY, as the generalized Hukuhara difference not always exist, to study the inverse of the Theorem (4.5), consider the case bccℝ=I (where I denote the set of closed bounded interval of the real line), and the gH-difference exist for any element in I. These multifunctions are called interval-valued functions and are studied in detail in [9].In the caseI, the Hausdorff distance has the following properties:Proposition 5 (see [9]).
ForA,B∈I, we have(57)dHkA,kB=kdHA,B,∀k∈ℝ,dHA,B=dHA⊝gB,0.Theorem 10.
LetF:ℝ⟶I be a multifunction, if F is generalized Hukuhara differentiable at x0∈ℝ, then F is CH-differentiable at x0∈ℝ.Proof.
Using the generalized Hukuhara differentiability ofF, there is ∂Fx0∈I satisfying (52). We define(58)Ax0:ℝ⟶I,given by Ax0h=Fx0+h∂Fx0. By Lemma 4.1, the multifunction (58) is affine, moreover Ax00=Fx0. On the other hand,(59)limh⟶0dHFx0+h,Ax0hh=limh⟶0dHFx0+h,Fx0+h∂Fx0h.
Thus, by Proposition5, we have(60)limh⟶0dHFx0+h,Ax0hh=limh⟶0dHFx0+h⊝gFx0h,∂Fx0=0,since limh⟶0Fx0+h⊝gFx0/h=∂Fx0. Therefore, F is Fréchet differentiable, and by Theorem 8, is CH-differentiable.☐



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