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

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      The continuity properties of the intersection of lower semicontinuous multimaps are more complicated. The following example demonstrates that in general case such an intersection is not lower semicontinuous.

2) defined in the following way. The multimap Fo is constant:

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      whereas the multimap F1 is defined as

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      (See fig. 8).

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      Fig. 8

      The multimaps F0 and F1 are l.s.c. (they are even continuous) but their intersection F0 ∩ F1 defined on the whole interval [0, π] loses the lower semicontinuity property at the points 0 and π (why?)

      To clarify the conditions under which we can guarantee the lower semi-continuity of the intersection of multimaps the following notion is useful.

      Definition 1.3.7. A multimap F : X → P(Y) is called quasi-open at a point x ∈ X if

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      and for every y ∈ intF (x) there exist neighborhoods V(y) ⊂ Y of y and U(x) ⊂ X of x such that V(y) ⊂ F(x′) for all x′ ∈ U(x). A multimap F is said to be quasi-open provided it is quasi-open at every point x ∈ X.

      It is easy to see that a multimap F : X → P(Y) such that intF(x) ≠

for all x ∈ X is quasi-open if and only if the multimap intF : X → P(Y),

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      has the open graph ΓintF ⊂ X × Y.

      We have the following important characterization of a quasi-open multimap.

and F is lower semicontinuous at x.

      Proof. 1) Let F be quasi-open at x ∈ X then intF(x) ≠

. If V ⊂ Y is an open set such that V ∩ F(x) ≠
then it is easy to see that V ∩ intF(x) ≠
. For an arbitrary y ∈ Y ∩ intF(x) let V(y) ⊂ Y and U(x) ⊂ X be neighborhoods such that V(y) ⊂ F(x′) for all x′ ∈ U(x). But V(y) ∩ V ≠
implies that V ∩ F(x′) ≠
for all x′ ∈ U(x), giving the lower semicontinuity of F at x.

      2) Conversely, let intF(x) ≠

and F be l.s.c. at x ∈ X. Let y ∈ intF(x) and Bδ(y) ⊂ F(x) for some δ > 0. Take δ1, 0 < δ1 < δ. Since the space Y is finite-dimensional, the ball Bδ(y) is relatively compact. By applying the reasonings similar to those that were used while proving the necessity part of Theorem 1.2.40 we get that there exists a neighborhood U(x) of x such that for each point x′ ∈ U(x) we have Bδ(y) ⊂ Fη(x′), where η = δ → δ1.

      Let now y′ ∈ Bδ1(y) but y′ ∉ F(x′) for some x′ ∈ U(x). Then from the convexity of the set F(x′) it follows that the ball (y′) will contain points whose distance from F(x′) is greater than η. But this contradicts to the fact that Bη(y′) ⊂ Bδ(y) ⊂ Fη(x′). Therefore, Bδ1(y) ⊂ F(x′) for all x′ ∈ U(x).

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      We now formulate a condition that guarantees the lower semicontinuity of the intersection of multimaps.

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      for all x ∈ X. If

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      then the intersection F0 ∩ f1 is lower semicontinuous at x0.

      Proof. Let V ⊂ Y be an open set such that V ∩ (F0 ∩ F1)(x0) ≠

. From the assumptions it follows that there exists a point y ∈ V ∩ (F0 ∩ F1)(x0) which is an interior point of the set F1(x0). Let V(y) be a neighborhood of y such that V(y) ⊂ (V ∩ F1(x0)). By using the quasi-openness of the multimap F1 we can assume, without loss of generality, that there exists a neighborhood U1(x0) of x0 such that V(y) ⊂ F1(x′) for all x′ ∈ U1(x0).

      Since y ∈ F0(x0) and the multimap F0 is l.s.c. there exists a neighborhood U0(x0) of x0 such that F0(x″) ∩ V(y) ≠

for all x″ ∈ U0(x0).

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