Topological degree theory
In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can often be used to prove existence of a second, nontrivial, solution. There are different types of degree for different types of maps: e.g. for maps between Banach spaces there is the Brouwer degree in Rn, the Leray-Schauder degree for compact mappings in normed spaces, the coincidence degree and various other types. There is also a degree for continuous maps between manifolds.
Topological degree theory has applications in complementarity problems, differential equations, differential inclusions and dynamical systems.
Further reading
- [https://books.google.com/books?id=qqAOpxsim9EC Topological fixed point theory of multivalued mappings], Lech Górniewicz, Springer, 1999, {{isbn|978-0-7923-6001-8}}
- [https://books.google.com/books?id=6R6kpqHV_bAC Topological degree theory and applications], Donal O'Regan, Yeol Je Cho, Yu Qing Chen, CRC Press, 2006, {{isbn|978-1-58488-648-8}}
- [https://books.google.com/books?id=K94tfITNWucC Mapping Degree Theory], Enrique Outerelo, Jesus M. Ruiz, AMS Bookstore, 2009, {{isbn|978-0-8218-4915-6}}
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