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

href="#fb3_img_img_8ed18411-0455-5df1-a748-bdd183551e1a.png" alt="figure"/> we have

, i.e., , that means the lower semicontinuity of the multimap F₀ ∩ F₁ at x₀.

      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

      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 .

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

      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.

Theorem 1.3.11. If the multimaps F₀ : X → P(Y) and F₁ : Y → P(Z) are u.s.c. (l.s.c.) then their composition F₁ ○ F₀ : X → P(Z) is u.s.c. (respectively, l.s.c.).

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

      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 W_y(z) of z and V(y) of y such that

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

      then

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

      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(

),

      and F₁ :

→ K(

),

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

→ K(

),

      is not closed.

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

Theorem 1.3.15. If multimaps F₀ : X → P(Y), F₁ : X → P(Z) are lower semicontinuous then their Cartesian product F₀ × F₁ : X → P(Y × Z) is lower semicontinuous.

      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).

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