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) ≠
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
(c)for every closed set W ⊂ Y, the set
(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)
4) (d) ⇒ (c): if D is closed then
Example 1.2.16. The multimaps from Examples 1.1.4 (a),