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

Скачать книгу
× Z) is closed.

      Proof. Consider nets {xα} ⊂ X, {υα} ⊂ Y × Z such that xα → x, υα ∈ (F0 × F1)(xα), υα → υ. Then υα = yα × zα, yα ∈ F0(F1(xa). from the definition of the topology in Y × Z, the convergence υα → υ = (y, z) implies the convergences yα → y and zα → z. From the closedness of the multimaps F0 and F1 it follows that y ∈ F0(x), z ∈ F1(x) (Theorem 1.2.24(c)) but it means that υ ∈ (F0 × F1)(x) concluding the proof.

figure

      To consider the upper semicontinuity of the Cartesian product of multimaps we need the compactness of their values.

      Proof. The Tychonoff theorem (see Chapter 0) implies that the multimap F0 × F1 has compact values. For an arbitrary point x ∈ X, let G ⊃ (F0 × F1)(x) be an open subset of Y × Z. From the definition of the product topology in Y × Z it follows that for every (y, z)(F0 × F1)(x) there exist open sets G0(y, z) ⊂ Y and G1(y, z) ⊂ Z such that (y, z) ∈ G0(y, z) × G1(y, z) ⊂ G. For each y ∈ F0(x) consider the cover figure of the set F1(x). Since the set F1(x) is compact we can select a finite subcover figure. The set figure is a neighborhood of y in Y and the set figure is a neighborhood of F1(x) in Z and moreover (V(Y) × Wy) ⊂ G. The sets V(Y), y ∈ F0(x) form an open cover of the compact set F0(x). Choose a finite subcover figure and set figure and figure. Then V is a neighborhood of F0(x) in Y whereas W is a neighborhood of F1(x) in Z and V × W ⊂ G.

      Then from Lemma 1.2.12(a) and the upper semicontinuity of the multimaps F0 and F1 it follows that there exists a neighborhood U(x) of x such that (F0 × F1)(U(x)) ⊂ V × W ⊂ G proving the upper semicontinuity of F0 × F1.

figure

      Let X be a topological space, Y a linear topological space.

      Definition 1.3.18. Let F0, F1 : X → P(Y) be multimaps. The multimap F0 + F1 : X → P(Y) defined as

figure

      is called the sum of the multimaps F0 and F1.

      Theorem 1.3.19. If multimaps F0, F1 : X → P(Y) are lower semicontinuous then their sum F0 + F1 : X → P(Y) is lower semicontinuous.

      Proof. The multimap F0 × F1 : X → P(Y × Y) is l.s.c. by Theorem 1.3.15. The single-valued map f : Y × Y → Y,

figure

      is continuous. We have

figure

      and conclusion follows from Theorem 1.3.11.

figure

      Similar application of Theorems 1.3.17 and 1.3.11 yields the following result.

      Theorem 1.3.20. If multimaps F0, F1 : X → K(Y) are upper semicontinuous then their sum F0 + F1 : X → K(Y) is upper semicontinuous.

      Remark 1.3.21. Notice that the assumption of compactness of the values of the multimaps F0 and F1 is essential. In fact, it was mentioned already that the multimap F in Example 1.1.8 is not u.s.c. But it may be represented as the sum of the identity map F0(x) = {x} and the constant multimap F1(x) = {(z1, z2) | (z1, z2) ∈

2, z1z2 = 1, z1 > 0, z2 > 0}.

      Definition 1.3.22. Let F : X → P(Y) be a multimap, f : X →

a function. The multimap f · F : X → P(Y),

figure

      is called the product of f and F.

      Theorem 1.3.23. If a multimap F : X → P(Y) is lower semicontinuous and a function f : X →

is continuous then the product f · F : X → P(Y) is lower semicontinuous.

Скачать книгу