Topological degree theory
Encyclopedia
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 system
s.
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
of a curve in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
. 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
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
in normed spaces, the coincidence degree and various other types. There is also a degree for continuous maps between manifolds
Degree of a continuous mapping
In topology, the degree is a numerical invariant that describes a continuous mapping between two compact oriented manifolds of the same dimension. Intuitively, the degree represents the number of times that the domain manifold wraps around the range manifold under the mapping...
.
Topological degree theory has applications in complementarity problems, differential equations, differential inclusions and dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s.
Further reading
- Topological fixed point theory of multivalued mappings, Lech Górniewicz, Springer, 1999, ISBN 9780792360018
- Topological degree theory and applications, Donal O'Regan, Yeol Je Cho, Yu Qing Chen, CRC Press, 2006, ISBN 9781584886488
- Mapping Degree Theory, Enrique Outerelo, Jesus M. Ruiz, AMS Bookstore, 2009, ISBN 9780821849156