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

Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

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

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href="#fb3_img_img_8ed18411-0455-5df1-a748-bdd183551e1a.png" alt="figure"/> we have
, i.e., figure, that means the lower semicontinuity of the multimap F0 ∩ F1 at x0.

figure

      As a corollary, we can obtain now a sufficient condition for the lower semicontinuity of the intersection of l.s.c. multimaps.

      Theorem 1.3.10. Let X be a topological space, Y a finite-dimensional linear topological space, F0, F1 : X → Cv(Y) l.s.c. multimaps. Assume that F0(x) ∩ F1(x) ≠

for all x ∈ X and

figure

      for some x0 ∈ X. Then the intersection F0 ∩ F1 : X → Cv(Y) is l.s.c. at x0.

      Proof. From Theorem 1.3.8 it follows that the multimap F1 is quasi-open at x0. Let y ∈ (F0 ∩ F1) be an arbitrary point and figure.

      It is clear that figure and figure. This means that the multimaps F0 and F1 satisfy the assumptions of Theorem 1.3.9.

figure

      It is worth noting that the loss of the lower semicontinuity for the intersection of multimaps in Example 1.3.6 occurs exactly at the points where the above condition is violated.

      Now consider some continuity properties of the composition of multimaps (see Definition 1.2.9).

      Let X, Y, and Z be topological spaces.

      Proof. The assertion follows immediately from Theorems 1.2.15(b), 1.2.19(b) and Lemma 1.2.10.

figure

      Theorem 1.3.12. Let F0 : X → K(Y) be a u.s.c. multimap and F1 : Y → C(Z) a closed multimap. Then the composition F1 ○ F0 : X → C(Z) is a closed multimap.

      Proof. Let z ∈ Z be such that z ∉ F1 ○ F0(x), x ∈ X. Applying Theorem 1.2.24(b) to the closed multimap F1 we can find for each point y ∈ F0(x), neighborhoods Wy(z) of z and V(y) of y such that

figure

      Let figure be a finite cover of the set F0(x). If now U(x) is a neighborhood of x such that

figure

      then

figure

      and the application of Theorem 1.2.24(b) concludes the proof.

figure

      Remark 1.3.13. The condition of upper semicontinuity of the multimap F0 is essential. The following example shows that the composition of closed multimaps is not necessarily a closed multimap.

      Example 1.3.14. The multimaps F0 :

→ K(
),

figure

      and F1 :

→ K(
),

figure

      are closed but not u.s.c. Their composition F1 ○ F0 :

→ K(
),

figure

      is not closed.

      We consider now the Cartesian product of multimaps (see Definition 1.2.11).

      Proof. Notice that the sets V0 × V1, where V0 ⊂ Y, V1 ⊂ Z are open sets form a base for the topology of the space Y × Z and apply Theorem 1.2.19(d) and Lemma 1.2.12(b).

figure

      Theorem 1.3.16. If multimaps F0 : X → C(Y), F1 : X → C(Z)

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