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

× 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.

1.3.3Theorem of maximum

      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