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

all nonempty convex subsets of Y. Introduce also the following symbols:

      When a multimap F maps into the collections C(Y), K(Y) or Pv(Y) we will say that F has closed, compact or convex values respectively.

      From the definition of a closed multimap it follows that it has closed values.

      The consideration of examples shows that closed and upper semicontinuous multimaps are a short distance apart. The relation between them is clarified by the following assertions.

      Theorem 1.2.29. Let X and Y be topological spaces. If the space Y is regular and a multimap F : X → C(Y) is u.s.c. then F is closed.

      Proof. Let y ∈ Y, y ∉ F(x). Since Y is regular there exist an open neighborhood V(y) of the point y and an open set W ⊃ F(x) such that V(y) ∩ W =

. Let U(x) be a neighborhood of x such that F(U(x)) ⊂ W. Then F(U(x)) ∩ V(y) =

and the statement follows from Theorem1.2.24(b).

Remark 1.2.30. It is clear from the proof that when F has compact values the condition of regularity of Y can be replaced with the weaker condition that Y is a Hausdorff space.

      To formulate a sufficient condition for a closed multimap to be u.s.c. we need the following definitions.

      Definition 1.2.31. A multimap F : X → P(Y) is called:

      (a)compact if its range F(x) is relatively compact in Y, i.e., is compact in Y;

      (b)locally compact if every point x ∈ X has a neighborhood U(x) such that the restriction F to U(x) is

      (c)quasicompact if the restriction of F to each compact subset A ⊂ X is compact.

      It is clear that (a) ⇒ (b) ⇒ (c).

Theorem 1.2.32. Let F : X → K(Y) be a closed multimap. If it is locally compact then it is u.s.c.

      Proof. Let x ∈ X, V an open set in Y such that F(x) ⊂ V. Let U(x) be a neighborhood of x such that the restriction of F to it is compact and let

. By the compactness of W we can extract a finite subcover V(y₁), . . . , V(y_n). Consider the following open neighborhood of x:

      Notice now that

for all j = 1, 2, . . . , n and hence F(x′) ∩ W =

. From the other side, . Therefore, .

      The difference between closed and u.s.c. multimaps is illustrated by Examples 1.1.6–1.1.8. As it was mentioned already, the multimaps in these examples are closed, but they are not u.s.c. Notice that the multimap from Example 1.1.6 has compact values and the condition of its upper semicontinuity is violated at the same point x = π/2 in which the condition of the local compactness is not satisfied.

      Let us consider some properties of closed and u.s.c. multimaps.

      Theorem 1.2.33. Let F : X → C(Y) be a closed multimap. If A ⊂ X is a compact set then its image F(A) is a closed subset of Y.

      Proof. The case F(A) = Y is trivial. Let y ∈ Y \ F(A). For any x ∈ A, let U(x) and V_x(y) be neighborhoods of x and y such that

      If U(x₁), . . . , U(x_n) are neighborhoods forming a finite cover of A then

.

      Remark 1.2.34. The condition of compactness of the set A is essential: the image of a closed set under the action of a closed multimap can be a non-closed set. In fact, in Example 1.1.7: .

      In the sequel an important role will be played by the following property of u.s.c. multimaps.

      Theorem 1.2.35. Let F : X → K(Y) be a u.s.c. multimap. If A ⊂ X is a compact set then its image F(A) is a compact subset of A.

      Proof. Let {Vj}_j∈J be an open cover of the set F(A). For each point x ∈ A, the value F(x) can be covered by a finite collection of sets V_j1, ...., V_jn(x). We denote . The sets , x ∈ A form an open cover of A. If we select a finite subcover

      then the sets V_x1, ..., V_xm form an open cover of the set F(A).

      Remark 1.2.36. The condition of a upper semicontinuity is essential in this theorem. In fact, for a closed multimap F with compact