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

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. Further, if figure then there exists figure and then figure, hence figure.

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

figure
be upper semicontinuous (in the single-valued sense). Then the function φ : X →
,

figure

      is upper semicontinuous.

      Proof. Fix ε > 0. For each pair x ∈ X, y ∈ Φ(x) there exist neighborhoods Uy(x), V(Y) such that x′ ∈ Uy(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, . . . , yn such that the neighborhoods V(yi), 1 ≤ i ≤ n form a cover of Φ(x). If now figure, and figure then from x″ ∈ U0(x), y″ ∈ V(Φ(x)) it follows that

figure

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

figure

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

figure

      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.

figure

      Beyond each corner, new directions lie in wait.

      —Stanislaw Jerzy Lec

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

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