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

Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

Let us mention also the following property.

Theorem 1.2.37. Let X and Y be topological spaces; A ⊂ X a connected set and F : X → P(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 x ∈ A;

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

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) ⊂ (V₀ ∪ V₁);

b)F(A) ∩ V_i ≠

, i = 0, 1;

c)(F(A) ∩ V₀) ∩ (F(A) ∩ V₁) =

Consider the sets

for each i = 0, 1 and (A ∩ U₀) ∩ (A ∩ U₁) =

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 . Then we get

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.

1.2.3Multivalued maps into a metric space

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 : X → P(Y) be a multimap. The multimap F_ε : X → P(Y),

is called an ε-enlargement of the multimap F.

Theorem 1.2.39. For the upper semicontinuity of a multimap F : X → K(Y) at a point x ∈ X, it is necessary and sufficient that for every ε > 0 there exists a neighborhood U(x) of x such that F(x′) ⊂ F_ε(x) for all x′ ∈ U(x).

Proof. 1) Necessity. Notice that

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 F_ε(x) ⊂ V. But then there exists a neighborhood U(x) of x such that F(U(x)) ⊂ F_ε(x) ⊂ V.

Theorem 1.2.40. For the lower semicontinuity of a multimap F : X → K(Y) at a point x ∈ X, it is necessary and sufficient that for every ε > 0 there exists a neighborhood U(x) of x that F(x) ⊂ F_ε(x′) for all x′ ∈ U(x).

Proof. 1) Necessity. Take ε > 0 and let y₁, . . . , y_n be points of the set F(x) such that the collection of balls , 1 ≤ i ≤ n forms an open cover of F(x). Since F is l.s.c., for every i, 1 ≤ i ≤ n, there exists an open neighborhood U_i(x) of the point x such that from x′ ∈ U_i(x) it follows that . But then, implies for all i, 1 ≤ i ≤ n 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 y ∈ F(x) ∩ V and let ε > 0 be such that B_ε(y) ⊂ V. Let U(x) be a neighborhood of x such that x′ ∈ U(x) implies F(x) ⊂ F_ε(x′). Then F(x′) ∩ B_ε(y) ≠

for all x′ ∈ U(x) proving that F is l,s.c. at x.

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, B ∈ C(Y),

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