On multi-valued maps with images in the space of closed subsets of a metric space
© Zhukovskiy and Panasenko; licensee Springer 2013
Received: 27 August 2012
Accepted: 22 December 2012
Published: 10 January 2013
A new metric in the space of all closed subsets of a metric space X is proposed. This metric, unlike the generalized Hausdorff metric, takes finite values only, and the convergence of a sequence of closed sets , , with respect to this metric is equivalent to the convergence (in the sense of Hausdorff) for any of the unions of with a closed ‘exterior ball’ of radius r. Using this metric allows one to investigate multi-valued maps that have images in and are not continuous in the Hausdorff metric. In the work, the necessary and sufficient conditions for a multi-valued map to be continuous and Lipschitz with respect to the metric presented are studied, a connection of these properties with their analogues in the Hausdorff metric is derived, and a generalization of the Nadler fixed point theorem is obtained.
MSC:47H04, 47H10, 54E35.
the distance between any closed sets is finite;
if a sequence of closed sets is convergent with respect to the Hausdorff metric, then it is convergent with respect to the ‘new’ metric;
the convergence of a sequence means the convergence for any (with respect to the Hausdorff metric) of the sequence of bounded subsets , , such that and as soon as .
For the space of convex closed subsets of , in , a metric (called the Hausdorff-Bebutov metric) satisfying (1)-(3) was suggested. It allowed to investigate differential inclusions with convex unbounded right-hand sides. For studying multi-valued maps with non-convex unbounded images in , the Hausdorff-Bebutov metric turns out to be inefficient since, considered in the space , it does not meet requirements (2) and (3). Moreover, in the structure of the Hausdorff-Bebutov metric, the local compactness of is fundamental, so the definition of this metric itself cannot be used for an arbitrary metric space X. In what follows, a different construction for a metric in the space is offered. It fulfills all the listed requirements and can be used in the case of arbitrary X.
The function takes finite values only and, as it will be shown later, satisfies all the axioms of a metric. This metric first was introduced by the authors in  for the space of all nonempty closed subsets of a finite dimensional space. In , the metric was used for studying the dynamical system of translations in the space of multi-valued maps with images in .
In the work, we point out the benefits the metric gives when it comes to dealing with continuous and Lipschitz multi-valued maps. We emphasize that a crucial role here is played by the construction (2) used in the metric , i.e., associating to every set the set , (representing the union of H with the closed ‘exterior ball’ of radius r centered at θ).a For a map , such an ‘extension’ of the values allows, in particular, to obtain a fixed point theorem (which will be proved in the last section of the paper) for multi-valued maps that in the Hausdorff metric may not be contracting or even continuous and to which the known fixed point principles (for example, Nadler’s theorem) are not applicable. One can also use this idea for a map that is not contracting or continuous. In many cases, after the corresponding extension, the map gets these properties.
2 The space
We start with studying the space .
(the latter occurs only if ).
We will need the following properties of this function: for every , the function is non-decreasing; for every , the function is non-increasing.
Finally, if , , then .
Let us illustrate the definition of the function D on some concrete metric spaces.
Example 1 For any linear normed space X with any choice of , the function is given by , .
If , , then no matter what point is chosen as , for any , , the value of is equal to the number of integers belonging to the interval .
One gets the same values for if , , and . If , the value coincides with the number of integers in the set when or , and is equal to the number of integers in the set when .
- (1)if , then
- (2)if , then
For arbitrary closed and , we determine now . Next, we show that this distance is finite and also give the properties of the function ; these properties will be used in the sequel.
the function is non-decreasing;
- (2)for every , the inequality(6)and the relation(7)
denoted , the inequality holds for each ;
denoted , the inequality holds for each .
for every , we get the inequality . Similarly, it can be checked that . So, (8) is proved.
Thus, the function is non-decreasing.
Now, we prove equality (7). If , then , for , so (7) is true.
Let . First, suppose that . Since the function is non-decreasing and bounded, there exists . Then for each r, the inequality takes place, and it follows that for any , every , and , one has and . Thus, taking into account , one gets , i.e., . As above, for each point , holds. Hence, for any , which means that . The inequality is easily obtained by passing to the limit in (6).
Now, let . If (7) is not true, then there exists . So, arguing as before, one gets the inequality , which contradicts the initial assumption. Thus, equality (7) is proved.
Theorem 1 Equality (5) defines a metric in the space . If X is a complete metric space, then is also complete.
, and if and only if ;
Thus, is a metric space.
Now, let X be complete; show that is also a complete metric space.
From (10), it follows that the number sequence is fundamental, so it is convergent; denote its limit by ; obviously, .
Similarly, and relation (11) is proved.
and therefore . Moreover, (this equality follows from statement (4) of Lemma 1).
For every , the set is not empty for . Prove that for any , holds. Let . Then there exists a sequence , , convergent to x. This means that ; moreover, . Conversely, let , then , , and hence there is a sequence , , . Starting with some index I, all the members of this sequence satisfy the condition , and therefore . Since converges to , it follows that . So, .
From what has been proved, one may conclude that as r increases, the sets ‘expand’ in the following way: if , then (i.e., is a subset of containing elements x such that .) Define now the set and show it is closed. Note that for every r, holds and the set is closed. Consider a sequence convergent to y. This sequence is bounded, so starting with some index i, the inclusions , , where , hold. Thus, since is closed, it follows that ; on the other hand, . So, and F is closed.
According to Lemma 1, the function is non-decreasing. Hence, from (12), it follows that for every . Next, for , one has . Thus, for all , and therefore , i.e., . □
holds, which contradicts relation (13).
Let us now give a criterion for a sequence of closed sets to be convergent with respect to the metric .
Lemma 2 Given , , the convergence implies and for any .
Conversely, let . If there exist and such that for every , the sequence converges in the metric dist to some set , then for any r, such that , holds, and in the space the sequence converges (with respect to ) to the set ; moreover, .
We omit the proof of this statement since it repeats that of the corresponding result for the finite dimensional case .
We conclude this section by proving a theorem on the connection between convergence in the metric and convergence in the metric dist. Note that if a sequence converges in the Hausdorff metric, then by statement (2) of Lemma 1, it converges (to the same limit) in the metric ; moreover, as it will be shown later, in the metric . The converse is not true. For example, in ℝ the sequence of sets , , converges in the metric to the set . On the other hand, for any i, j, one has , i.e., the sequence is divergent in the Hausdorff metric.
if , then ;
if and there is a such that , , then and .
Proof (1) Suppose , . Then, for an arbitrary , there is a number such that for every . So, according to inequality (9), one gets , and from (6) it follows that the inequality holds for every . Thus, by Lemma 2, .
that contradicts the relation . Hence .
hence . □
3 Continuous multi-valued maps
Let T be a topological space. Recall that a map is said to be continuous at a point if for any , one can find a neighborhood of such that for every . This definition corresponds to endowing the space with the Hausdorff metric. In what follows, we call a map satisfying the listed requirements continuous in the Hausdorff metric in order to distinguish it from the one considered in the space .
Definition 1 A map is said to be continuous at a point in the metric if for any , there is a neighborhood of such that for every .
Since the space with the metric is a subspace of , one can also apply the given definition to a map , i.e., to a map having closed bounded images.
Following the standard terminology, the map F is called continuous on a set in the metric dist or metric if it is continuous in the corresponding metric at every point of .
If is continuous at a point in the Hausdorff metric, then it is continuous at in the metric .
If is continuous at a point in the metric and there exist a number and a neighborhood of such that for every , then F is continuous at in the Hausdorff metric.
Proof (1) Suppose F is continuous at some point in the Hausdorff metric and let . Then there is a neighborhood of such that for every . So, for every , according to estimate (9), one gets the inequality and, according to (6), the inequality . Therefore, , .
i.e., . Similarly, .
Thus, the map F is continuous at a point in the Hausdorff metric. □
So, the continuity of in the Hausdorff metric implies its continuity in the metric . The following example shows that the converse is not true.
So, due to relations (15) and (17), F is continuous in the metric .
4 Lipschitz multi-valued maps
We consider now multi-valued maps defined on a metric space.
Naturally, this definition is applicable to a map . In what follows, a map satisfying inequality (18) will be called q-Lipschitz in the Hausdorff metric.
To give the definition of a Lipschitz map in the case when is endowed with the metric , note that this metric is given by a pair of functions defined by (3), (4) (the second of which is a metric itself (see Remark 1)).
In the case of and , a map F satisfying condition (18) is called contracting or q-contracting (in the metric dist). Similarly, a map will be called contracting in the metric implying it satisfies inequality (19) with a constant , and contracting in the metric if it satisfies inequalities (19), (20).
In what follows, the conditions for a map F to be Lipschitz in the metrics dist, , and , as well as connections between these properties are derived.
We begin with the criteria of the q-Lipschitzness of F in the metrics dist and .
A map is q-Lipschitz in the metric dist if and only if the map is q-Lipschitz in the metric dist for each .
- (2)A map is q-Lipschitz in the metric if and only if for every and any such that , the following holds:(21)
Proof Statement (1) is a direct consequence of the relations between and obtained in paragraph (2) of Lemma 1: the necessity follows from estimate (6) and sufficiency from equality (7).
Therefore, inequality (22) takes place for all , such that . Since, according to paragraph (1) of Lemma 1, the function does not decrease for any , , inequality (22) remains valid for every .
and thus . □
Remark 2 A multi-valued map is said to be -uniformly locally Lipschitz (see, e.g., ) if for any such that , the inequality holds. So, according to Theorem 4, a map is q-Lipschitz in the metric if and only if for any , the map is -uniformly locally Lipschitz for all .
The following definition allows to consider an important class of metric spaces often appearing in applications and to which the use of Theorem 4 (and the statements proved further on) is of particular interest.
Definition 3 
This property of convex metric spaces follows immediately from [, Theorems 14.1, 15.1].
If is complete and convex, then paragraph (2) of Theorem 4 can be specified.
Corollary 1 Let Ω be a complete, convex metric space. Then a map is q-Lipschitz in the metric if and only if inequality (21) holds for every and any (i.e., if and only if the map is q-Lipschitz in the metric dist for every ).
Proof Necessity. For and any , one has , so and (21) holds for all , .
The sufficiency part of the theorem follows directly from Theorem 4. □
Note that the hypotheses of completeness and convexity of Ω in Corollary 1 are crucial. Consider some examples.
Estimate for arbitrary .
Evidently, if , , then .
In all the other cases, i.e., when , , , , or when , , , , doing the analogous calculations one also gets estimation (25).
So, the map F is contracting in the metric with a constant . Moreover, since for any , , the map F is contracting in the metric .
Note that the map considered does not have fixed points in X. This means that, for example, in the Nadler theorem, one cannot replace the classical assumption of F be contracting in the metric dist with the one of F be contracting in the metric . Nevertheless, using the map instead of the map F allows to weaken the known conditions of fixed points existence. The corresponding result will be given in the next section.
Example 4 Let , , , where , , , . This space is convex, but not complete. Define the map by (24).
The next theorem establishes a connection between the concepts of Lipschitzness considered in different metrics of the space .
If a map is q-Lipschitz in the Hausdorff metric, then it is q-Lipschitz in the metric and all the more in the metric .
In the case when Ω is a complete and convex metric space, the concepts of q-Lipschitzness of in the metrics , , and dist are equivalent.
Here, a solid arrow means that the Lipschitzness of F in the first metric implies its Lipschitzness in the second metric; a dashed arrow corresponds to a relation between these properties that takes place only if the metric space Ω is complete and convex.
Proof (1) The correctness of the implication follows from statements (2) and (4) of Lemma 1. Further, by Definition 2, the q-Lipschitzness of the map F in the metric implies its q-Lipschitzness in the metric .
(2) Now, let Ω be complete, convex and be q-Lipschitz in the metric . Then, according to Corollary 1, the map is q-Lipschitz for every and hence, by statement (1) of Theorem 4, the map F is q-Lipschitz in the metric dist. Therefore, F is q-Lipschitz in the metric (see paragraph (1) of the theorem). □
Let us show that the convexity of a metric space is essential for equivalence of the properties of q-Lipschitzness in the metrics dist, , .
Note that the maps considered in Examples 3 and 4 are contracting (with a constant ) in the metrics and . With respect to the metric dist, they are only Lipschitz with a constant . Give an example of a map that is contracting in the metric and not Lipschitz in the metric dist.
and therefore, there is no constant q for F to be q-Lipschitz in the Hausdorff metric.
Next, consider a map that is q-Lipschitz in the metric and not q-Lipschitz in the metric .
and thus estimate (20) does not hold.
5 Fixed point theorems
In this section, we prove a generalization of the Nadler fixed point theorem for a multi-valued map . The idea of the generalization consists in replacing the image by the set . As it has been mentioned already, this transformation allows to reduce the distance between the images of a map and, in a number of cases, to turn a multi-valued map that is not even continuous into a contracting one. And if it can be proved that a fixed point of the ‘new’ map does not belong to , then this point becomes a fixed one also for the initial map F.
- (1)the map is q-contracting (in the metric dist), i.e.,(26)
- (2)there exists such that(27)
Then there is a point satisfying the inclusion and the estimate .
So, and , and it follows that is a fixed point of the map F. □
In the classical theorem of Nadler, if a map is contracting (in the Hausdorff metric) with a constant and with a fixed point satisfying the condition , then this map has at least one more fixed point different from (see ). For maps meeting the requirements of Theorem 6, the analogous statement may not be true. In fact, it is easy to explain: arguing as in [, Theorem 2.1.3], take , , as an initial point and, using the iteration procedure, find a fixed point of the map . Since , it is easy to show that . It may turn out, however, that and , i.e., the point is not a fixed point of the map F. This situation does not occur if the element satisfies condition (2) of Theorem 6. Thus, the following statement holds.
then F has at least one more fixed point .
Example 7 Consider the map given by (assume , ). We show first that this map is not q-Lipschitz (for no q) and, moreover, at any point is neither upper semicontinuous nor lower semicontinuous. More precisely, we prove that for any close enough, , hold.
Consider two situations: and .
So, inequality (31) is correct.
If , then , and inequality (31) holds as well.
Despite the fact that for any close enough , , Theorem 6 can be applied to the map F. Put , , and prove that relations (26) and (27) are valid. To determine the set , we will use equality (2′) since in the space ℝ definitions (2) and (2′) lead to the same values of , (see the footnote on page 2), but the calculations by means of (2′) are less ponderous.
i.e., the map is contracting with a constant .
so condition (2) of Theorem 6 is also satisfied. Thus, there exists a fixed point of the map F such that (it is easy to check that is the minimal solution of the equation ).
This very definition was used in  for . Using (2′) for an arbitrary metric space does not seem possible since, first of all, a sphere may result in an empty set and as a consequence, the set will be empty when .
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