Читать книгу Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications - Valeri Obukhovskii - Страница 18

Theorem of maximum, which is called sometimes the principle of continuity of optimal solutions plays an important role in the applications of multivalued maps in the theory of games and mathematical economics (see Chapter 4).

Theorem 1.3.29. Let X, Y be topological spaces, Φ : X → K(Y) a continuous multimap, f : X × Y → a continuous function. Then the function φ : X → ,

is continuous and the multimap F : X → P(Y)

has compact values and is upper semicontinuous.

Remark 1.3.30. The function φ and the multimap F are often called marginal.

The proof of Theorem 1.3.29 will be based on the following two assertions.

Lemma 1.3.31. Let a multimap Φ : X → K(Y) be lower semicontinuous, a function f : X × Y → lower semicontinuous (in the single-valued sense). Then the function ,

is lower semicontinuous.

Proof. Choose a point x ∈ X and assume at first that φ(x) < +∞. Fix ε > 0; then there exists a point y ∈ Φ(x) such that f(x, y) ≥ φ(x) − ε. By the lower semicontinuity of f there exist neighborhoods U₀(x) of x and V(Y) of y such that, for each x′ ∈ U₀(x), y′ ∈ V(Y) we have

By the lower semicontinuity of the multimap Φ there exists a neighborhood U₁(x) of x such that x″ ∈ U₁(x) implies Φ(x″) ∩ V(Y) ≠ . Further, if then there exists and then , hence .

The case φ(x) = +∞ can be considered similarly.

Lemma 1.3.32. Let a multimap Φ : X → K(Y) be upper semicontinuous, a function f : X × Y → be upper semicontinuous (in the single-valued sense). Then the function φ : X → ,

is upper semicontinuous.

Proof. Fix ε > 0. For each pair x ∈ X, y ∈ Φ(x) there exist neighborhoods U_y(x), V(Y) such that x′ ∈ U_y(x), y′ ∈ V(Y) implies f(x′, y′) < f(x, y) + ε. Since the set Φ(x) is compact, there exist a finite number of points y₁, . . . , y_n such that the neighborhoods V(y_i), 1 ≤ i ≤ n form a cover of Φ(x). If now , and then from x″ ∈ U₀(x), y″ ∈ V(Φ(x)) it follows that

Let U₁(x) be a neighborhood of x such that Φ(U₁(x)) ⊂ V(Φ(x)). Then yields and for each we have implying .

Proof of Theorem 1.3.29. For every x ∈ X, the set

is nonempty. From Lemmas 1.3.31 and 1.3.32 it follows that the function φ is continuous but then the multimap Γ : X → C(Y) is closed (see Example 1.2.27). Now, notice that F = Φ ∩ Γ and apply Theorem 1.3.3.