In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a number of
fixed-point theorems in infinite-dimensional spaces generalise the
Brouwer fixed-point theorem. They have applications, for example, to the proof of
existence theorem
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ...
s for
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s.
The first result in the field was the
Schauder fixed-point theorem, proved in 1930 by
Juliusz Schauder (a previous result in a different vein, the
Banach fixed-point theorem for
contraction mappings in complete
metric spaces was proved in 1922). Quite a number of further results followed. One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of
algebraic topology, first proved for finite
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
es, to spaces of infinite dimension. For example, the research of
Jean Leray who founded
sheaf theory came out of efforts to extend Schauder's work.
Schauder fixed-point theorem: Let ''C'' be a nonempty closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
convex subset of a Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
''V''. If ''f'' : ''C'' → ''C'' is continuous with a compact image, then ''f'' has a fixed point.
Tikhonov (Tychonoff) fixed-point theorem: Let ''V'' be a locally convex topological vector space. For any nonempty compact convex set ''X'' in ''V'', any continuous function ''f'' : ''X'' → ''X'' has a fixed point.
Browder fixed-point theorem: Let ''K'' be a nonempty closed bounded convex set in a uniformly convex Banach space. Then any non-expansive function ''f'' : ''K'' → ''K'' has a fixed point. (A function is called non-expansive if for each and .)
Other results include the
Markov–Kakutani fixed-point theorem (1936-1938) and the
Ryll-Nardzewski fixed-point theorem (1967) for continuous affine self-mappings of compact convex sets, as well as the
Earle–Hamilton fixed-point theorem (1968) for holomorphic self-mappings of open domains.
Kakutani fixed-point theorem: Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.
See also
*
Topological degree theory
References
* Vasile I. Istratescu, ''Fixed Point Theory, An Introduction'', D.Reidel, Holland (1981). .
* Andrzej Granas and
James Dugundji, ''Fixed Point Theory'' (2003) Springer-Verlag, New York, .
* William A. Kirk and
Brailey Sims, ''Handbook of Metric Fixed Point Theory'' (2001), Kluwer Academic, London {{isbn, 0-7923-7073-2.
External links
PlanetMath article on the Tychonoff Fixed Point Theorem
Fixed-point theorems
fr:Théorème du point fixe de Schauder