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Leray–Schauder degree

From Wikipedia, the free encyclopedia

In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres or equivalently to boundary-sphere-preserving continuous maps between balls to boundary-sphere-preserving maps between balls in a Banach space , assuming that the map is of the form where is the identity map and is some compact map (i.e. mapping bounded sets to sets whose closure is compact).[1]

The degree was invented by Jean Leray and Juliusz Schauder to prove existence results for partial differential equations.[2][3]

References

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  1. ^ Leray, Jean; Schauder, Jules (1934). "Topologie et équations fonctionnelles". Annales scientifiques de l'École normale supérieure. 51: 45–78. doi:10.24033/asens.836. ISSN 0012-9593.
  2. ^ Mawhin, Jean (1999). "Leray-Schauder degree: a half century of extensions and applications". Topological Methods in Nonlinear Analysis. 14: 195–228. Retrieved 2022-04-19.
  3. ^ Mawhin, J. (2018). A tribute to Juliusz Schauder. Antiquitates Mathematicae, 12.