Читать книгу Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications - Valeri Obukhovskii - Страница 14

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), the value

is called the deviation of the set A from the set B. The function ϱ_* : C(Y) × C(Y) → ∪ {∞} possesses the following properties.

Theorem 1.2.41.

(a)ϱ_*(A, B) ≥ 0 for each A, B ∈ C(Y);

(b)ϱ_*(A, B) = 0 implies A ⊂ B;

(c)in a general case ϱ_* (A, B) = ϱ_*(B, A);

(d)if ϱ_*(A, B) < ∞ then ϱ_*(A, B) ≤ ϱ_*(A, C) + ϱ_*(C, B) for every C ∈ C(Y);

(e)if ϱ_*(A, B) < ∞ then ϱ_*(A, B) = inf {ε | A ⊂ U_ε(B)}.

Proof. (a) Follows immediately from the definition.

(b) For each x ∈ A we have ϱ(x, B) = 0. Hence x is a limit point for a certain sequence of points from B. Since B is closed, we get x ∈ B.

(c) Take A = {a} ∈ Y, B = {a} ∪ {b} ∈ Y, a ≠ b. Then ϱ_*(A, B) = 0, ϱ_*(B, A) = ϱ(b, a) ≠ 0.

(d) By the triangle inequality, for each x ∈ A we have

where z is an arbitrary point of C. Then

for each z ∈ C. Whence

Then ϱ_*(A, B) ≤ ϱ_*(A, C) + ϱ_*(C, B).

(e) Let ε > ϱ_*(A, B), then for each point x ∈ A there exists a point y ∈ B such that x ∈ B_ε(y). Therefore A ⊂ Uε(B), i.e., inf{1|A ⊂ U_ε(B)} ≤ ϱ_*(A, B). In case when ε > 0 is such that A ⊂ U_ε(B), for every x ∈ A we have ϱ(x, B) < ε. Then ϱ_*(A, B) ≤ ε, i.e., ϱ_*(A, B) ≤ inf{ε|A ⊂ U_ε(B)}. Comparing the obtained inequalities we get the desired property.

Consider the function h : C(Y) × C(Y) → ∪ [∞],

Applying the previous result one can verify (do it!) that this function has the next properties:

For each A, B ∈ C(Y) the following holds true:

1)h(A, B) ≥ 0;

2)h(A, B) = 0 is equivalent to A = B;

3)h(A, B) = h(B, A);

4)If h(A, B) < ∞ then h(A, B) ≤ h(A, C) + h(C, B) for each C ∈ C(Y).

Definition 1.2.42. The function h is called the extended Hausdorff metric on the set C(Y).

Here the term “extended” means that the function h can take infinite values.

Denote Cb(Y) the collection of all nonempty closed bounded subsets of Y. From the above properties it immediately follows that the function h is a usual metric on this set. It is called the Hausdorff metric.

Notice that from Theorem 1.2.41(e) it follows that for every A, B ∈ Cb(Y) the Hausdorff metric may be defined as

Definition 1.2.43. A multimap F : X → Cb(Y) is called Hausdorff continuous, if it is continuous as a single-valued map into the metric space (Cb(Y), h).

For multimaps with compact values we can obtain now the following useful characterization of the continuity.

Theorem 1.2.44. A multimap F : X → K(Y) is continuous if and only if it is Hausdorff continuous.

Proof. The statement of the theorem directly follows from Theorems 1.2.39 and 1.2.40.

Now, let Y be a separable metric space. The following criteria of lower and upper semicontinuity of multimaps will be useful in the sequel.

Let be a countable dense subset of Y. For a multimap F : X → P(Y) define the functions , φ_n : X → ,

Theorem 1.2.45. For the lower semicontinuity of a multimap F : X → P(Y) it is necessary and sufficient that all the functions φ_n are upper semicontinuous (in the single-valued sense).

Proof. For each a > 0 and n the set

coincides with the set . To verify the assertion it remains to notice that the balls centered at the points r_n form the base of the topology of the space Y and to use Theorem 1.2.19 (d).

For the further reasonings we need the following statement.

Lemma 1.2.46. Let F : X → P(Y) be a multimap, W ⊂ Y a closed subset. Let sets Wm be such that W ⊂ W_m ⊂ U_εm(W) for a certain sequence , ε_m > 0, ε_m → 0. Then:

(a);

(b)if the values of the multimap F are compact then

Proof. The inclusions

are evident.

(a) If then there exists y ∈ F(x) such that y ∉ W.

But then ε_m < ϱ(y, W) yields that proves (a).

(b) If then F(x) ∩ W = and, since the set F(x) is compact there exists ε > 0 such that F_ε(x) ∩ W = . But then ε_m < ε implies and (b) is also proved.

Theorem 1.2.47. For the upper semicontinuity of a multimap F : X → K(Y) it is necessary and, in the case of compactness of the multimap F, also sufficient that all the functions φn are lower semicontinuous (in the single-valued sense).

Proof. 1) Necessity. If the multimap F is upper semicontinuous then for each a > 0 and n the set

is open.

2) Necessity. Since the multimap F is compact, it is sufficient to show that for every compact set K ⊂ Y the set is closed. For a certain sequence , ε_m > 0, ε_m → 0 consider the finite covers of K by closed balls of the radius ε_m centered at points from the set :

For each m, the set

is closed. Applying Lemma 1.2.46 (b) we obtain from where the closedness of the set follows.

In conclusion of this section notice that for metric spaces we have the following refinement of Theorem 1.2.32.

Theorem 1.2.48. Let X and Y be metric spaces and F : X → K(Y) a closed quasicompact multimap Then F is upper semicontinuous.

Proof. Let x ∈ X be a point and V ⊂ Y an open set such that F(x) ⊂ V. If F is not u.s.c. at x there exists a sequence {x_n} ⊂ X, x_n → x such that we can choose a sequence y_n ∈ F(x_n)\V for all n = 1, 2, ... By virtue of the quasicompactness condition we can assume without loss of generality that y_n → y ∉ V, contrary to y ∈ F(x).