Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications. Valeri Obukhovskii
y0.
(b)(i) Let a multimap F be u.s.c. at a point x0. For an arbitrary ε > 0, consider an ε-neighborhood of the set F(x0):
Since Uε(F(x0)) is the open set it follows that there exists a number nε such that for all n ≥ nε we get F(xn) ⊂ Uε(F(x0)) implying ϱY(yn, F(x0)) ≤ ε.
(b)(ii) Let V ⊂ Y be an arbitrary open set containing F(x0). Then, by virtue of compactness of the set F(x0) there exists (see Chapter 0) ε > 0 such that Uε(F(x0)) ⊂ V. But then condition (**) yields the existence of such a neighborhood U(x0) of the point x0 such that F(U(x0)) ⊂ Uε(F(x0)) ⊂ V.
Definition 1.2.21. If a multimap F is upper and lower semicontinuous it is called continuous.
It is clear that in the case of a single-valued map, both upper, as well as lower semicontinuity mean usual continuity. Notice also that the constant multimap F(x) ≡ Y1 ⊂ Y is obviously continuous.
Example 1.2.22. (a) The multimaps from Examples 1.1.4 (a), (c); 1.1.5; 1.1.7; 1.1.8 are u.s.c. It can be verified by application of Definition 1.2.14 (do it!). Whence the multimaps from Examples 1.1.4 (a)(; 1.1.5; 1.1.7 are continuous. The multimap from Example 1.1.4 (b) is u.s.c., but not l.s.c., whereas the multimaps from Examples 1.1.4 (c)); 1.1.8 are l.s.c., but not u.s.c. In particular, for the multimap F from Example 1.1.8 we have
(b) Let T be a compact space; X a metric space, C(T; X) denote the space of continuous functions endowed with the usual sup-norm. For an arbitrary nonempty subset Ω ⊂ C(T, X), the multimap Q : T → P(x) defined as
is l.s.c. It can be checked up by using Theorem 1.2.20. Verify that if the set Ω is compact then the multimap Q is u.s.c. and hence continuous.
One more important class consists of closed multimaps.
Definition 1.2.23. A multimap F is called closed if its graph ΓF (see Definition 1.1.2) is a closed subset of the space X × Y.
Consider some tantamount formulations.
Theorem 1.2.24. The following conditions are equivalent:
(a)the multimap F is closed;
(b)for each pair x ∈ X, y ∈ Y such that y ∉ F(x) there exist neighborhoods U(x) of x and V(y) of y such that F(U(x)) ∩ V(y) =
(c)for every nets {xa} ⊂ X, {yα} ⊂ Y such that xα → x, yα ∈ F(xα), yα → y, we have y ∈ F(x).
Proof. 1) (a) ⇔ (b): condition (b) means that a point (x, y) ∈ X × Y belongs to the complement of the graph ΓF with a certain neighborhood;
2) (a) ⇔ (c): condition (c) means that if a net {(xα, yα)} ⊂ ΓF converges to a point (x, y) ∈ X × Y then (x, y) ∈ ΓF.
Notice that in the case when X and Y are metric spaces, it is sufficient to consider in condition (c) usual sequences.
Example 1.2.25. The multimaps from Examples 1.1.4 (a), (b); 1.1.5 - 1.1.8 are closed.
Example 1.2.26. Consider Example 1.1.9. If X, Y are topological spaces, the space Y is Hausdorff and f : X → Y is a continuous surjective map, then the inverse multimap F = f−1 : Y → P(x) is closed.
Example 1.2.27. Consider Example 1.1.10. If X, Y, Z are topological spaces and the maps f and g are continuous then the implicit multimap F is closed.
Example 1.2.28. Consider Example1.1.11. If X, Y are topological spaces and function f is continuous then the multimap Fr is closed.
The validity of assertions in Examples 1.2.26 – 1.2.28 may be verified by applying Theorem 1.2.24(c) (do it!).
Introduce some notation which we will use in the sequel.
Let Y be a topological space.
Denote by C(Y), K(Y) the collections consisting of all nonempty closed, or respectively, compact subsets of Y. If the topological space Y