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

Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

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

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alt="figure"/> from where the asserion follows.

figure figure

      is upper semicontinuous.

      (b) Let multimaps Fj : X → C(Y) be closed and figure. then the intersection figure is closed.

      Proof. (a) At first, let us prove the assertion for the case of two multimaps F0 and F1. For x ∈ X, let V be a neighborhood of the set F(x) = (F0 ∩ F1)(x). If at least one of the sets F0(x) or F1(x) is contained in V then the existence of such a neighborhood U(x) of x that F(U(x)) ⊂ V is evident. Otherwise F0(x) \ V and F1(x) \ V are nonempty disjoint closed sets. By virtue of the normality of the space Y there exist disjoint open sets W0 and W1 such that (Fj(x) \ V) ⊂ Wj, j = 0, 1. Then for every j = 0, 1 we have

figure

      From the upper semicontinuity of the multimaps Fj it follows that for each j = 0, 1 there exists a neighborhood Uj(x) of x such that

figure

      But if U(x) = U0(x) ∩ U1(x) then for every x′ ∈ U(x) we have

figure

      proving the upper semicontinuity of F at x.

      The validity of the statement in the general case now follows from the mathematical induction principle.

      (b) The statement follows from the fact that the graph of the multimap figure is the intersection of the graphs of the multimaps Fj.

figure

      Let us mention also the following assertion.

figure

      Then the intersection F = F0 ∩ F1 : X → K(Y) is upper semicontinuous.

      Proof. For an arbitrary x ∈ X let V ⊂ Y be any open neighborhood of the set (F0 ∩ F1)(x). We will show that there exists an open neighborhood U(x) of x such that (F0 ∩ F1)(U(x)) ⊂ V.

      When F1(x) ⊂ V the existence of such a neighborhood follows from the upper semicontinuity of F1. If K = F1(x) \ V ≠

then the set K is compact and K ∩ F0(x) =
. As F0 is a closed multimap, for each point y ∈ K there exist neighborhoods V(y) ⊂ Y of y and Uy(x) ⊂ X of x such that F0(Uy(x)) ∩ V(y) =
(see Theorem 1.2.24(b)).

      Let {figure be a finite cover of K formed by such neighborhoods V(y), and figure. The open set V ∪ V(K) contains F1(x), hence there exists a neighborhood U1(x) of x such that F1(U1(x)) ⊂ (V ∪ V(K)). Then the neighborhood

figure

      is the required one. In fact, F0(U(x)) ∩ V(K) =

and F1(U(x)) ⊂ (V ∪ V(K)), therefore (F0 ∩ F1)(U(x)) ⊂ V.

figure

      Corollary 1.3.4. Let a multimap F : X → K(Y) be upper semicontinuous, C ⊂ Y a closed set and F(x) ∩ C ≠

, ∀x ∈ X. Then the multimap F : X → K(Y),

figure

      is upper semicontinuous.

      Proof. It is clear that the multimap F0 : X → C(Y),

figure

      is closed. Take F1 = F and apply the previous theorem.

figure

      Corollary 1.3.5. Let Y be a Hausdorff topological space, multimaps figure upper semicontinuous and figure. Then the intersection figure is upper semicontinuous.

      Proof. Let Fj0 be one of the multimaps from the family. Since al the multimaps Fj are closed (see Remark 1.2.30) the multimap

figure

      is also closed (Скачать книгу