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

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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.

      (a)the multimap F is l.s.c.;

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

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

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

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

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

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

figure

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

      (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 figure, xn → x0 and each y0 ∈ F(x0) there exists a sequence figure, yn ∈ F(xn) such that yn → 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 figure, xn → x0 and figure, yn ∈ F(xn) the following relation holds: ϱY(yn, 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 yn ∈ F(xn) which converges to it.

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

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