Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications. Valeri Obukhovskii

      is called the deviation of the set A from the set B. The function ϱ_* : C(Y) × C(Y) →

∪ {∞} possesses the following properties.

Theorem 1.2.41.

      (a)ϱ_*(A, B) ≥ 0 for each A, B ∈ C(Y);

      (b)ϱ_*(A, B) = 0 implies A ⊂ B;

      (c)in a general case ϱ_* (A, B) = ϱ_*(B, A);

      (d)if ϱ_*(A, B) < ∞ then ϱ_*(A, B) ≤ ϱ_*(A, C) + ϱ_*(C, B) for every C ∈ C(Y);

      (e)if ϱ_*(A, B) < ∞ then ϱ_*(A, B) = inf {ε | A ⊂ U_ε(B)}.

      Proof. (a) Follows immediately from the definition.

      (b) For each x ∈ A we have ϱ(x, B) = 0. Hence x is a limit point for a certain sequence of points from B. Since B is closed, we get x ∈ B.

      (c) Take A = {a} ∈ Y, B = {a} ∪ {b} ∈ Y, a ≠ b. Then ϱ_*(A, B) = 0, ϱ_*(B, A) = ϱ(b, a) ≠ 0.

      (d) By the triangle inequality, for each x ∈ A we have

      where z is an arbitrary point of C. Then

      for each z ∈ C. Whence

      Then ϱ_*(A, B) ≤ ϱ_*(A, C) + ϱ_*(C, B).

      (e) Let ε > ϱ_*(A, B), then for each point x ∈ A there exists a point y ∈ B such that x ∈ B_ε(y). Therefore A ⊂ Uε(B), i.e., inf{1|A ⊂ U_ε(B)} ≤ ϱ_*(A, B). In case when ε > 0 is such that A ⊂ U_ε(B), for every x ∈ A we have ϱ(x, B) < ε. Then ϱ_*(A, B) ≤ ε, i.e., ϱ_*(A, B) ≤ inf{ε|A ⊂ U_ε(B)}. Comparing the obtained inequalities we get the desired property.

      Consider the function h : C(Y) × C(Y) →

∪ [∞],

      Applying the previous result one can verify (do it!) that this function has the next properties:

      For each A, B ∈ C(Y) the following holds true:

      1)h(A, B) ≥ 0;

      2)h(A, B) = 0 is equivalent to A = B;

      3)h(A, B) = h(B, A);

      4)If h(A, B) < ∞ then h(A, B) ≤ h(A, C) + h(C, B) for each C ∈ C(Y).

      Definition 1.2.42. The function h is called the extended Hausdorff metric on the set C(Y).

      Here the term “extended” means that the function h can take infinite values.

      Denote Cb(Y) the collection of all nonempty closed bounded subsets of Y. From the above properties it immediately follows that the function h is a usual metric on this set. It is called the Hausdorff metric.

      Notice that from Theorem 1.2.41(e) it follows that for every A, B ∈ Cb(Y) the Hausdorff metric may be defined as

      Definition 1.2.43. A multimap F : X → Cb(Y) is called Hausdorff continuous, if it is continuous as a single-valued map into the metric space (Cb(Y), h).

      For multimaps with compact values we can obtain now the following useful characterization of the continuity.

      Theorem 1.2.44. A multimap F : X → K(Y) is continuous if and only if it is Hausdorff continuous.

      Proof. The statement of the theorem directly follows from Theorems 1.2.39 and 1.2.40.

      Now, let Y be a separable metric space. The following criteria of lower and upper semicontinuity of multimaps will be useful in the sequel.

      Let

,

      Theorem 1.2.45. For the lower semicontinuity of a multimap F : X → P(Y) it is necessary and sufficient that all the functions φ_n are upper semicontinuous (in the single-valued sense).

      Proof. For each a > 0 and n the set

      coincides with the set . To verify the assertion it remains to notice that the balls centered at the points r_n form the base of the topology of the space Y and to use Theorem 1.2.19 (d).

      For the further reasonings we need the following statement.

Lemma 1.2.46. Let F : X → P(Y) be a multimap, W ⊂