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

• Topological fixed point theory of multivalued mappings, Lech Górniewicz, Springer, 1999, ISBN 978-0-7923-6001-8
• Topological degree theory and applications, Donal O'Regan, Yeol Je Cho, Yu Qing Chen, CRC Press, 2006, ISBN 978-1-58488-648-8
• Mapping Degree Theory, Enrique Outerelo, Jesus M. Ruiz, AMS Bookstore, 2009, ISBN 978-0-8218-4915-6