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

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      It is clear that .

      Let A ⊂ X; D ⊂ Y; {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!).

Lemma 1.2.3.

Lemma 1.2.4.

      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 : X → P(Y) be multimaps. The multimap F₀ ∪ F₁ : X → P(Y),

      is called the union of the multimaps F₀ and F₁.

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

for all x ∈ X. The multimap F₀ ∩ F₁ : X → P(Y),

      is called the intersection of the multimaps F₀ and F₁.

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

Lemma 1.2.7. If D ⊂ Y then

      (a) ;

      (b) .

Lemma 1.2.8. If D ⊂ Y then

      (a) ;

      (b) .

Definition 1.2.9. Let X, Y, and Z be sets, F₀ : X → P(Y), F₁ : Y → P(Z) multimaps. The multimap F₁ ○ F₀ : X → P(Z),

      is called the composition of the multimaps F₀ and F₁.

Lemma 1.2.10. Let D ⊂ Z then

      (a) ;

      (b) .

       Verify these relations!

Definition 1.2.11. Let X, Y₀, and Y₁ be sets, F₀ : X → P(Y₀), F₁ : X → P(Y₁) multimaps. The multimap F₀ × F₁ : X → P(Y₀ × Y₁),

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

Lemma 1.2.12. Let D₀ ⊂ Y₀, D₁ ⊂ Y₁, then

      (a) ;

      (b) .

       Verify these relations!

1.2.2Upper and lover semicontinuity, continuity, closedness of multimaps

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

Definition 1.2.13. A multimap F is said to be upper semicontinuous at a point x ∈ X if for every open set V ⊂ Y such that F(x) ⊂ V there exists a neighborhood U(x) of x such that

Definition 1.2.14. A multimap F is called upper semicontinuous (u.s.c.) if it is upper semicontinuous at every point x ∈ X.

      Consider some tantamount formulations.

Theorem 1.2.15. The following conditions are equivalent:

      (a)the multimap F is u.s.c.;

      (b)for every open set V ⊂ Y, the set is open in X;

      (c)for every closed set W ⊂ Y, the set is closed in X;

      (d)if D ⊂ Y then .

      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) is a closed set which contains ;

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

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