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

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


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      It is clear that figure.

      Let AX; DY; {Dj}j∈J a family of subsets of Y, J a set of indices. The next properties of small and complete preimages follow immediately from the definitions (verify!).

figure figure

      Let us observe the properties of small and complete preimages while passing to various set-theoretic operations on multimaps.

      Definition 1.2.5. Let F0, F1 : XP(Y) be multimaps. The multimap F0F1 : XP(Y),

figure

      is called the union of the multimaps F0 and F1.

      Definition 1.2.6. Let F0, F1 : XP(Y) be multimaps such that F0(x) ∩ F1(x) ≠

for all xX. The multimap F0F1 : XP(Y),

figure

      is called the intersection of the multimaps F0 and F1.

      The following properties can be easily verified (do it!).

      (a) figure;

      (b) figure.

      (a) figure;

      (b) figure.

figure

      is called the composition of the multimaps F0 and F1.

      (a) figure;

      (b) figure.

       Verify these relations!

figure

      is called the Cartesian product of the multimaps F0 and F1.

      (a) figure;

      (b) figure.

       Verify these relations!

      Let X, Y be topological spaces, F : XP(Y) multimaps.

figure

      Consider some tantamount formulations.

      (a)the multimap F is u.s.c.;

      (b)for every open set VY, the set figure is open in X;

      (c)for every closed set WY, the set figure is closed in X;

      (d)if DY then figure.

      Proof. 1) The equivalence (a) ⇔ (b) is evident;

      2) the equivalence (b) ⇔ (c) follows from Lemma 1.2.3(c) and Lemma 1.2.4(c);

      3) figure is a closed set which contains figure;

      4) (d) ⇒ (c): if D is closed then figure, i.e., figure is closed.

figure

      Example 1.2.16. The multimaps from Examples 1.1.4 (a), Скачать книгу