Michael selection theorem

Last updated January 02, 2023

In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following:^[1]

Examples

A function that satisfies all requirements

The function: $F(x)=[1-x/2,~1-x/4]$ , shown by the grey area in the figure at the right, is a set-valued function from the real interval [0,1] to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example: $f(x)=1-x/2$ or $f(x)=1-3x/8$ .

A function that does not satisfy lower hemicontinuity

The function

$F(x)={\begin{cases}3/4&0\leq x<0.5\\\left[0,1\right]&x=0.5\\1/4&0.5<x\leq 1\end{cases}}$

is a set-valued function from the real interval [0,1] to itself. It has nonempty convex closed values. However, it is not lower hemicontinuous at 0.5. Indeed, Michael's theorem does not apply and the function does not have a continuous selection: any selection at 0.5 is necessarily discontinuous.^[2]

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 an 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, $Y$ 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 $Y$ there is a continuous function $f\colon X\mapsto Y$ such that for each $x\in X$ , $f(x)\in F(X)+V$ .^[3]

In a note Xu proved that Deutsch–Kenderov theorem is also valid if $Y$ is a locally convex topological vector space.^[4]

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References

↑ Michael, Ernest (1956). "Continuous selections. I". Annals of Mathematics . Second Series. 63 (2): 361–382. doi:10.2307/1969615. hdl: 10338.dmlcz/119700 . JSTOR 1969615. MR 0077107.
↑ "proof verification - Reducing Kakutani's fixed-point theorem to Brouwer's using a selection theorem". Mathematics Stack Exchange. Retrieved 2019-10-29.
↑ 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 .