Leray–Schauder degree explained

In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres

(S ^{n,} *) \to (S ^{n} , *) or equivalently to a boundary sphere preserving continuous maps between balls (B ^{n,} S ^{n-1} ) \to (B ^{n,} S ^{n-1} ) to boundary sphere preserving maps between balls in a Banach space f: (B(V), S(V)) \to (B(V), S(V)) , assuming that the map is of the form f = id - C where id is the identity map and C 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]}

Notes and References

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