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

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⊂ Uεm(W) for a certain sequence figure, εm > 0, εm → 0. Then:

      (a)figure;

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

figure

      Proof. The inclusions

figure

      are evident.

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

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

      (b) If

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

figure

      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

figure

      is open.

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

figure

      For each m, the set

figure

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

figure

      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 {xn} ⊂ X, xn → x such that we can choose a sequence yn ∈ F(xn)\V for all n = 1, 2, ... By virtue of the quasicompactness condition we can assume without loss of generality that yn → y ∉ V, contrary to y ∈ F(x).

figure

      In mathematics there are no symbols for obscure thoughts.

      —Henri Poincaré

      The variety of operations that can be defined on multimaps is intrinsically richer than for single-valued maps: such operations as union, intersection of multimaps and some others have no “single-valued” analogs. In this section we investigate the preserving of continuity properties of multimaps with respect to various operations on them.

      Let X, Y be topological spaces; {Fj}j∈J, Fj : X → P(Y) a family of multimaps.

      Theorem 1.3.1. (a) Let multimaps Fj be upper semicontinuous. If the set of indices J is finite then the union of multimaps figure

figure

       is upper semicontinuous;

      (b) Let the multimaps Fj be lower semicontinuous. Then their union figure is lower semicontinuous;

      (c) Let multimaps Fj : X → C(Y) be closed. If the set of indices J is finite then the union figure is closed.

      Proof. (a) Let V ⊂ Y be open, then in accordance with Lemma 1.2.7 (a)

figure

      and hence this set is open and by Theorem 1.2.15 (b) the multimap figure is u.s.c.

      (b) The assertion similarly follows from Lemma 1.2.8 (a) and Theorem 1.2.19 (b).

      (c) It is easy to verify (do it!) that the graph figure of the multimap figure is the union of the graphs

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