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

. 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.

1.4Continuous selections and approximations of multivalued maps

      Beyond each corner, new directions lie in wait.

      —Stanislaw Jerzy Lec

      Let X, Y be topological spaces, F : X → P(Y) a multimap.