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

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


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=
.

      Let us mention also the following property.

      Theorem 1.2.37. Let X and Y be topological spaces; AX a connected set and F : XP(Y) a multimap. If one of the following conditions holds true:

      (i)F is upper or lower semicontinuous and the values F(x) are connected for each xA;

      (ii)F is continuous and a value F(x0) is connected for some x0A

      then F(A) is a connected subset of Y.

      Proof. (i) Consider the case of a upper semicontinuous multimap F. Suppose the contrary, then there exist open sets V0 and V1 in the space Y such that:

      a)F(A) ⊂ (V0V1);

      b)F(A) ∩ Vi

, i = 0, 1;

      c)(F(A) ∩ V0) ∩ (F(A) ∩ V1) =

.

      Consider the sets

for each i = 0, 1 and (AU0) ∩ (AU1) =
that contradicts to the connectedness of the set A.

      In the case when the multimap F is lower semicontinuous, it is sufficient to note that open sets arising in the definition of a connected set may be replaced with closed ones and to carry out the same reasonings as above, by using Theorem 1.2.19 (c).

      (ii) Also suppose the contrary. Then, by virtue of its connectedness, the set F(x0) must lie either in V0 or V1. Suppose for determinacy that F(x0) ⊂ V0 and hence figure. Then we get

figure

      and moreover, by the continuity of the multimap F, both last sets are non-empty, disjoint and open. But this contradicts to the connectedness of A.

figure

      In the case when a multimap acts into a metric space we can obtain a few convenient characterizations for the above considered types of continuity.

      Everywhere in this section, (Y, ϱ) is a metric space.

      Definition 1.2.38. Let F : XP(Y) be a multimap. The multimap Fε : XP(Y),

figure

      is called an ε-enlargement of the multimap F.

      Proof. 1) Necessity. Notice that

figure

      is an open set containing F(x) and apply Definition 1.2.13.

      2) Sufficiency. Let F(x) ⊂ V, where V is an open set. Then (see Ch. 0) there exists ε > 0 such that Fε(x) ⊂ V. But then there exists a neighborhood U(x) of x such that F(U(x)) ⊂ Fε(x) ⊂ V.

figure

      Proof. 1) Necessity. Take ε > 0 and let y1, . . . , yn be points of the set F(x) such that the collection of balls figure, 1 ≤ in forms an open cover of F(x). Since F is l.s.c., for every i, 1 ≤ in, there exists an open neighborhood Ui(x) of the point x such that from x′ ∈ Ui(x) it follows that figure. But then, figure implies figure for all i, 1 ≤ in and hence the neighborhood U(x) is the desired one.

      2) Sufficiency. Let V be an open set in Y and F(x) ∩ V

. Take an arbitrary point yF(x) ∩ V and let ε > 0 be such that Bε(y) ⊂ V. Let U(x) be a neighborhood of x such that x′ ∈ U(x) implies F(x) ⊂ Fε(x′). Then F(x′) ∩ Bε(y) ≠
for all x′ ∈ U(x) proving that F is l,s.c. at x.

figure

      It is worth noting that in the necessary part of Theorem 1.2.39 and in the sufficient part of Theorem 1.2.40 the compactness of the values of the multimap F is not used.

      As earlier, let C(Y) denote the collection of all nonempty closed subsets of Y. For A, BC(Y),


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