Michael selection theorem
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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 φ : X → E a lower hemicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection f : X → E of φ.
- 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
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.
See also
References
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- 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