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 a lower hemicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection 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

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 is said to be almost lower hemicontinuous if at each , all neighborhoods of there exists a neighborhood of such that Precisely, Deutsch–Kenderov theorem states that if is paracompact, a normed vector space and is nonempty convex for each , then is almost lower hemicontinuous if and only if has continuous approximate selections, that is, for each neighborhood of in there is a continuous function such that for each , .[1]

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

See also

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.

Further reading

  • Michael, Ernest (1956). "Continuous selections. I". Annals of Mathematics. Second Series. 63 (2): 361–382. doi:10.2307/1969615. JSTOR 1969615. MR 0077107.
  • Repovš, Dušan; Semenov, Pavel V. (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. arXiv:1401.2257. Bibcode:2014arXiv1401.2257R. ISBN 978-94-6239-023-2.
  • Aubin, Jean-Pierre; Cellina, Arrigo (1984). Differential Inclusions, Set-Valued Maps And Viability Theory. Grundl. der Math. Wiss. 264. Berlin: Springer-Verlag. ISBN 3-540-13105-1.
  • Aubin, Jean-Pierre; Frankowska, H. (1990). Set-Valued Analysis. Basel: Birkhäuser. ISBN 3-7643-3478-9.
  • Deimling, Klaus (1992). Multivalued Differential Equations. Walter de Gruyter. ISBN 3-11-013212-5.
  • Repovš, Dušan; Semenov, Pavel V. (1998). Continuous Selections of Multivalued Mappings. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-5277-7.
  • Repovš, Dušan; Semenov, Pavel V. (2008). "Ernest Michael and Theory of Continuous Selections". Topology and its Applications. 155 (8): 755–763. doi:10.1016/j.topol.2006.06.011.
  • Aliprantis, Charalambos D.; Border, Kim C. (2007). Infinite Dimensional Analysis : Hitchhiker's Guide (3rd ed.). Springer. ISBN 978-3-540-32696-0.
  • Hu, S.; Papageorgiou, N. Handbook of Multivalued Analysis. Vol. I. Kluwer. ISBN 0-7923-4682-3.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.