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

rel="nofollow" href="#ulink_fb40e556-4739-5b8a-9b9f-075727ad5b12">1.1.5; 1.1.7 are u.s.c. The subdifferential multimap from Example 1.1.17 is also u.s.c. (see, e.g., [104]).

      Definition 1.2.17. A multimap F is called lower semicontinuous at a point x ∈ X if for every open set V ⊂ Y such that F(x) ∩ V ≠

there exists a neighborhood U(x) of x such that F(x′) ∩ V ≠

for all x′ ∈ U(x).

      Definition 1.2.18. A multimap F is said to be lower semicontinuous (l.s.c.) if it is lower semicontinuous at every point x ∈ X.

      The lower semicontinuity also admits tantamount definitions.

Theorem 1.2.19. The following conditions are equivalent:

      (a)the multimap F is l.s.c.;

      (b)for every open set V ⊂ Y, the set is open in X;

      (c)for every closed set W ⊂ Y, the set is closed in X;

      (d)if a system of open sets {Vj}_j∈J forms a base for the topology of the space Y then for each Vj, the set is open in X;

      (e)if D ⊂ Y is an arbitrary set then ;

      (f)if A ⊂ X is an arbitrary set then .

      Proof. 1) the equivalence (a) ⇔ (b) is evident;

      2) the equivalences (b) ⇔ (c) and (c) ⇔ (e) can be proved similarly to the corresponding statements of Theorem 1.2.15;

      3) the equivalence (b) ⇔ (d) follows from the fact that each set Vj is open and from Lemma 1.2.4 (d);

      4) , but by virtue of Lemma 1.2.3(a) , hence . From Lemma 1.2.3(b) it follows: , therefore ;

      5) but by virtue of Lemma 1.2.3 (b): , yielding . Applying to both sides of the last inclusion and using Lemma 1.2.3 (a) we get .

      In the case of metric spaces we may obtain the following convenient sequential characterization of the lower and upper semicontinuity.

Theorem 1.2.20. Let X and Y be metric spaces.

      (a)For the lower semicontinuity of a multimap F : X → P(Y) at a point x₀ ∈ X it is necessary and sufficient that:

      (*) for every sequence , x_n → x₀ and each y₀ ∈ F(x₀) there exists a sequence , y_n ∈ F(x_n) such that y_n → y₀.

      (b)For the upper semicontinuity of a multimap F : X → P(Y) at a point x₀ ∈ X it is necessary, and in the case of the compactness of the set F(x₀) it is also sufficient that:

      (**) for every sequences , x_n → x₀ and , y_n ∈ F(x_n) the following relation holds: ϱY(y_n, F(x₀)) → 0.

      Proof. (a)(i) Let condition (*) holds. If the multimap F is not l.s.c. at the point x₀ then there exist an open set V ⊂ Y such that F(x₀) ∩ V ≠

and a sequence

for all n = 1, 2, ... But these relations are in contradiction to the fact that we can, choosing a point y₀ ∈ F(x₀) ∩ V, to find a sequence y_n ∈ F(x_n) which converges to it.

      (a)(ii) Let a multimap F be l.s.c. at a point x₀ and a certain sequence , x_n → x₀ and a point y₀ ∈ F(x₀) be given. Consider the sequence of open balls , m = 1, 2, ... centered at the point y₀. Let a number n₁ be such that for all n ≥ n₁. For every n < n₁ choose y_n ∈ F(x_n) arbitrarily. Further, let us find a number n₂ ≥ n₁ such that for all n ≥ n₂. For every n, n₁ ≤ n < n₂ choose y_n ∈ F(x_n) ∩ B₁(y₀).