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∈X, all neighborhoods V of 0 there exists a neighborhood U of x such that ${\displaystyle \cap _{u\in U}\{F(u)+V\}\neq \emptyset .}$ Precisely, Deutsch and Kenderov theorem states that if X is paracompact, E a normed vector space and F(x) is nonempty convex for each x∈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:X → E such that for each x ∈ X, f(x) ∈ F(X) + V.^[1]

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

See also

• Zero-dimensional Michael selection theorem
• List of selection theorems

References

1. 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.
2. 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, Ernest (1956), "Continuous selections. I", Annals of Mathematics. Second Series, Annals of Mathematics, 63 (2): 361–382, doi:10.2307/1969615, JSTOR 1969615, MR 0077107
• Dušan Repovš; Pavel V. Semenov (2014). "Continuous Selections of Multivalued Mappings". In Hart, K. P.; van Mill, J.; Simon, P. Recent progress in general topology III. Berlin: Springer. pp. 711–749. ISBN 978-94-6239-023-2.
• Jean-Pierre Aubin, Arrigo Cellina Differential Inclusions, Set-Valued Maps And Viability Theory, Grundl. der Math. Wiss., vol. 264, Springer - Verlag, Berlin, 1984
• J.-P. Aubin and H. Frankowska Set-Valued Analysis, Birkh¨auser, Basel, 1990
• Klaus Deimling Multivalued Differential Equations, Walter de Gruyter, 1992
• D.Repovs and P. V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht 1998.
• D.Repovs and P. V. Semenov, Ernest Michael and theory of continuous selections, Topol. Appl. 155:8 (2008), 755-763.
• Aliprantis, Kim C. Border Infinite dimensional analysis. Hitchhiker's guide Springer
• S.Hu, N.Papageorgiou Handbook of multivalued analysis. Vol. I Kluwer