Topological degree theory

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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.

[edit] Further reading

• Lech Górniewicz (1999). Topological Fixed Point Theory of Multivalued Mappings. Springer. ISBN 978-0792360018. 
• Donal O'Regan, Yeol Je Cho, Yu-Qing Chen (2006). Topological Degree Theory and Applications. Chapman & Hall/CRC. ISBN 978-1584886488.