An Extension of the Carathéodory Differentiability to Set-Valued Maps

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.

1. Introduction

Given and nonempty sets, if for each element belonging to is assigned, due to a certain law denoted here by , a set contained in , we say that 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 ([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 as ” being an arbitrary function, a multifunction with and 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 and and consider the family of affine multifunctions (see [11] for details of affine multifunctions and the Fréchet differentiability of multifunctions):

where is the collection of all bounded closed convex subsets of , provided with a suitable metric. Then, we associate the differentiability (Carathéodory) of the multifunctions to the existence of continuous functions

This paper is organized as follows. In Section 2, we present some definitions and basic results of generalized Hukuhara difference and affine multifunction. Section 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

Let and be finite-dimensional normed vector spaces over the set of real numbers

A multifunction from into assigns to each a (possibly empty) subset

The setsare called the effective domain and the graph of the multifunction , respectively.

By , we denote the collection of all bounded closed convex subsets of . The collection equipped with the Minkowsky addition:and multiplication by nonnegative real numbers:is a semilinear space. The functionwhere is the unit ball in , is called the Hausdorff distance, and determines a structure of a metric space on

Remark 1. It is well known that the Minkowsky addition is associative and commutative with neutral element If , the scalar multiplication gives the opposite but, in general,i.e., the opposite of is not the inverse of unless is a singleton. The first implication of this fact is that, in general, the additive simplification is not valid, that is or

Remark 2. The partial solution for (7) is the Hukuhara difference, this difference has been introduced as a set for which , and the important property of is and , The Hukuhara difference is unique, but it does not always exist (a necessary condition for exist is that contains a translation of .)

A generalization of the Hukuhara difference proposed in [17] aims to overcome this situation.

Definition 1 (see [9]). The generalized Hukuhara difference of two sets (gH-difference for short) is defined as follows:

Proposition 1 (see [18]). If exists, it is unique and if also exists then

Remark 3 (see [18]). A necessary condition for to exists is that either A contains a translation of (as for ) or contains a translation of In fact, for any given we get from (i) or from (ii).

Remark 4 (see [18]). If exists, then exist and

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]). Let then(1)If the gH-difference exists, it is unique and it is a generalization of the usual Hukuhara difference, since whenever exists(2),(3)if exists in the sense , then exists in the sense and vice versa(4)(5)(6)We have if and only if and From now on, in this paper, we will consider the generalized Hukuhara difference,

2.1. Affine Multifunction and Fréchet Differentiability

Let and be finite-dimensional normed vector spaces over the set of real numbers

A multifunction is said to be

iffor all and

iffor all and

Proposition 3 (see [11]). An affine multifunction is uniformly continuous on in the Hausdorff sense.

Definition 2. A multifunction is said to be Fréchet differentiable at a point , if there exists an affine multifunction such that and

This definition is equivalent to the following: for any there exists such that for all , we havewhere and are unit balls in and , respectively. So immediately, we have that .

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.

Let and be finite-dimensional normed vector space over real , define the space:

Example 1. Let be a linear function in , such that . Choose the function defined by , for . Then, .
In effect, .

Proposition 4. For ,is a metric on , where is the Hausdorff metric.

Proof. (1)If , then for all such that , we have , thus for such that On the other hand, since is affine and for , we haveTherefore, (2)If and are multifunctions, then for each such that we haveTherefore, ☐

Definition 3. Let be finite-dimensional normed vector spaces over the set of real numbers . A function is said to be continuous at , if given , there exists such that, if , then

Example 2. Let be a fixed point and in . By the Hahn-Banach theorem (see [19], page 3.), there is a linear function in , such that

Let be fixed . By Example 1, the functions defined by are in . Now, the functionis continuous at .

In effect, and

Theorem 1. If are continuous at the point Then, is continuous at

Proof. Since and are continuous at , given and there are and such that, if then and . Thus, for and the definition of affine multifunction, we haveNow, if , from Definition 1, there is and , such that ; thus, from (1) and (7), we havewhere and SoThen, this is, and since and is convex. Therefore, . In consequence, . Analogously, . Thus, is continuous.☐

Now, motivated by the Carathéodory’s Definition in [14], Definition 1, and Proposition 2, we define the following:

Definition 4. A multifunction is CH-differentiable (Carathéodory and Hukuhara differentiable) at a point , if there are functions continuous at a point , where is a neighborhood of the point , such that and such that

Remark 6. Given . It is natural to ask the following question. Are the functions 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 function defined by , for is CH-differentiable at X. In effect, , where .

Example 4. Let be a differentiable function at and . The function defined by , is CH-differentiable at . In effect, where exists because is differentiable, thus, Carathéodory differentiable. Now, consider defined by . Then,

Theorem 2. Let be two Banach spaces and be a CH-differentiable at . Suppose that, there are functions and satisfying the CH-differentiability conditions. Then, .

Proof. Since and , for all . Now define the functionby . Observe that . ThusNow, since and are continuous at . We have is continuous at . Whence, given , there is such that if , thenNow, for each with there is , such that for , there exists , with and (just take the sequence on the line segment joining to ). Hence, we haveThen,Therefore,
and ☐

ؙ

Lemma 1. Let be two multifunctions and if and Then

Proof. In effect, if and , then given and , there exists , such that if . Then, and . Hence, we obtain thatIt follows that if then , with and , thus, and , with , , and , so . As , we have . Thus, . Using and similarly, . Therefore☐

Theorem 3. If is a CH-differentiable multifunction at . Then, is Hausdorff continuous at

Proof. Since we have, from the above proposition,☐

Theorem 4. Let be two CH-differentiable multifunctions at Then, , with , is CH-differentiable at .

Proof. We have, and , thusHence, since is continuous, the result follows.☐

Theorem 5. If is a differentiable function at and is a CH-differentiable multifunction at , then , is CH-differentiable at .

Proof. Observe that, and , thus☐

Theorem 6. If , then is a CH-differentiable multifunction at

Proof. Let and . Choose small, thusNow, choose , defined by ☐

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. Letbe inand
Consider defined by

Then, is an affine multifunction.

Proof. Let and thenwith On the other hand, since We have☐

Theorem 7. If is a CH-differentiable multifunction at , then is Fréchet differentiable at

Proof. Using the CH-differentiability of at , there exist continuous functions at , where is a neighborhood at , such that and such that On the other hand, since is an affine multifunction, using Lemma 4.1,is an affine multifunction. We affirm that is Fréchet differentiable. In effect, given , there exists such that if , thenfor all , such that In particular, for we haveThis is equivalent toNow, since we haveAnalogously,☐

The reciprocal of the above theorem is not always true, because the Hukuhara difference does not exist (see Remark 2). But when this difference exists, we have a relationship between Fréchet’s differentiability and CH-differentiability.

Theorem 8. If the Hukuhara difference belongs to and is a Fréchet differentiable multifunction at , then is CH-differentiable at

Proof. Let be the Fréchet differential of at and . By the Hahn-Banach theorem, there exists a lineal function in , such thatThen, we can define the function byThis function is continuous at (see Example 2). AndAnalogously, forwe have that is continuous and ☐

Now, we remember the generalized Hukuhara differentiability.

Definition 5: (see [9]). Let and be a multifunction, then the gH-derivative of the multifunction at is defined as

If satisfying (52) exists, we say that is generalized Hukuhara differentiable (gH-differentiable for short) at

Theorem 9. Let be a multifunction, if is CH-differentiable at . Then, is generalized Hukuhara differentiable at

Proof. Using the CH-differentiability of , there exist multifunctions continuous at (neighborhood at ), such that andfor all such that Hence,Now, from the continuity of , given there exists such that, if then in particularthus☐

Remark 7. Throughout this article, we confine ourselves to multifunctions as the generalized Hukuhara difference not always exist, to study the inverse of the Theorem (4.5), consider the case (where denote the set of closed bounded interval of the real line), and the gH-difference exist for any element in These multifunctions are called interval-valued functions and are studied in detail in [9].

In the case , the Hausdorff distance has the following properties:

Proposition 5 (see [9]). For , we have

Theorem 10. Let be a multifunction, if is generalized Hukuhara differentiable at , then is CH-differentiable at

Proof. Using the generalized Hukuhara differentiability of , there is satisfying (52). We definegiven by By Lemma 4.1, the multifunction (58) is affine, moreover On the other hand,Thus, by Proposition 5, we havesince Therefore, is Fréchet differentiable, and by Theorem 8, is CH-differentiable.☐

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication.