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:

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

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.

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

...
Wikipedia