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

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

∪ {∞} possesses the following properties.

      (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 ⊂ Uε(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

figure

      where z is an arbitrary point of C. Then

figure

      for each z ∈ C. Whence

figure

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

figure

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

∪ [∞],

figure

      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

figure

      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.

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      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

,

figure

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

      Proof. For each a > 0 and n the set

figure

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

figure

      For the further reasonings we need the following statement.

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