Читать книгу Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications - Valeri Obukhovskii - Страница 17

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

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,

is continuous. We have

and conclusion follows from Theorem 1.3.11.

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),

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.

Proof. The multimap f × F : X → P( × Y) is l.s.c. by Theorem 1.3.15. The map φ : × Y → Y,

is continuous. Then the multimap

is l.s.c. by Theorem 1.3.11.

The following statement can be proved by a similar application of Theorems 1.3.17 and 1.3.11.

Theorem 1.3.24. If a multimap F : X → K(Y) is upper semicontinuous and a function f : X → is continuous then the product f · F : X → K(Y) is upper semicontinuous.

Definition 1.3.25. Let Y be a linear topological space, F : X → P(Y) a multimap. The multimap ,

is called the convex closure of the multimap F.

Theorem 1.3.26. Let Y be a Banach space. If a multimap F : X → K(Y) is u.s.c. (l.s.c.) then the convex closure is u.s.c. (l.s.c.)

Proof. First, we note that the multimap has compact values by Mazur’s theorem (see Chapter 0). Let the multimap F be u.s.c. Consider a point x ∈ X and let ε > 0. Then for every ε₁, 0 < ε₁ < ε there exists a neighborhood U(x) of x such that F(x′) ⊂ F_ε1(x) for all x′ ∈ U(x) (Theorem 1.2.39). But F_ε1(x) ⊂ U_ε1((x)), hence coF(x′) ⊂ U_ε1((x)) since the set U_ε1((x)) is convex. Then

for each x′ ∈ U(x) proving, by Theorem 1.2.39, the upper semicontinuity of the multimap .

The lower semicontinuity of the multimap can be proved in a similar way by applying Theorem 1.2.40.

Remark 1.3.27. The property of closedness of a multimap can be lost under the operation of convex closure, as the following example shows.

Example 1.3.28. The multimap F : → C(),

is closed, but its convex closure : → Cv(),

is not closed.