Michael selection theorem

In functional analysis, a branch of mathematics, the most popular version of the Michael selection theorem, named after Ernest Michael, states the following:

Let E be a Banach space, X a paracompact space and F : X → E a lower hemicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection f : X → E of F.

Conversely, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values admits continuous selection, then X is paracompact. This provides another characterization for paracompactness.

Applications

Michael selection theorem can be applied to show that the differential inclusion

{\frac {dx}{dt}}(t)\in F(t,x(t)),\quad x(t_{0})=x_{0}

has a C¹ solution when F is lower semi-continuous and F(t, x) is a nonempty closed and convex set for all (t, x). When F is single valued, this is the classic Peano existence theorem.

Generalizations

A theorem due to Deutsch and Kenderov generalizes Michel selection theorem to a equivalence relating approximate selections to almost lower hemicontinuity, where $F$ is said to be almost lower hemicontinuous if at each $x\in X$ , all neighborhoods $V$ of $0$ there exists a neighborhood $U$ of $x$ such that $\cap _{u\in U}\{F(u)+V\}\neq \emptyset .$ Precisely, Deutsch–Kenderov theorem states that if $X$ is paracompact, $E$ a normed vector space and $F(x)$ is nonempty convex for each $x\in X$ , then $F$ is almost lower hemicontinuous if and only if $F$ has continuous approximate selections, that is, for each neighborhood $V$ of $0$ in $E$ there is a continuous function $f\colon X\mapsto E$ such that for each $x\in X$ , $f(x)\in F(X)+V$ .^[1]

In a note Xu proved that Deutsch–Kenderov theorem is also valid if $E$ is a locally convex topological vector space.^[2]

References

↑ Deutsch, Frank; Kenderov, Petar (January 1983). "Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections". SIAM Journal on Mathematical Analysis. 14 (1): 185–194. doi:10.1137/0514015.
↑ Xu, Yuguang (December 2001). "A Note on a Continuous Approximate Selection Theorem". Journal of Approximation Theory. 113 (2): 324–325. doi:10.1006/jath.2001.3622.

Michael selection theorem

Applications

Generalizations

See also

References

Further reading