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

Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

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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.

Example 1.3.6. Consider the multimaps F₀, F₁ : [0, π] → Kv(

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

whereas the multimap F₁ is defined as

(See fig. 8).

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

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),

has the open graph Γ_intF ⊂ X × Y.

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

Theorem 1.3.8. Let Y be a finite-dimensional linear topological space. A multimap F : X → Cv(Y) is quasi-open at a point x ∈ X if and only if intF(x) ≠

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 B_δ(y) ⊂ F(x) for some δ > 0. Take δ₁, 0 < δ₁ < δ. Since the space Y is finite-dimensional, the ball B_δ(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 B_δ(y) ⊂ F_η(x′), where η = δ → δ₁.

Let now y′ ∈ B_δ1(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 B_η(y′) ⊂ B_δ(y) ⊂ F_η(x′). Therefore, B_δ1(y) ⊂ F(x′) for all x′ ∈ U(x).

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

Theorem 1.3.9. Let X, Y be topological spaces; a multimap F₀ : X → P(Y) be lower semicontinuous at x₀ ∈ X and a multimap F₁ : X → P(Y) be quasi-open at x₀, and

for all x ∈ X. If

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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