Brouwer's fixed-point theorem is a
fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms.
In mathematica ...
in
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, named after
L. E. J. (Bertus) Brouwer. It states that for any
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
mapping a nonempty
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
convex set
In geometry, a set of points is convex if it contains every line segment between two points in the set.
For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
to itself, there is a point
such that
. The simplest forms of Brouwer's theorem are for continuous functions
from a closed interval
in the real numbers to itself or from a closed
disk to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset
of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
to itself.
Among hundreds of
fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms.
In mathematica ...
s, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the
Jordan curve theorem
In topology, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region Boundary (topology), bounded by the curve (not to be ...
, the
hairy ball theorem
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous function, continuous tangent vector field on even-dimensional n‑sphere, ''n''-spheres. For the ord ...
, the
invariance of dimension and the
Borsuk–Ulam theorem
In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they ar ...
. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about
differential equations and is covered in most introductory courses on
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
. It appears in unlikely fields such as
game theory
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...
. In economics, Brouwer's fixed-point theorem and its extension, the
Kakutani fixed-point theorem
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed poi ...
, play a central role in the
proof of existence of
general equilibrium
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...
in market economies as developed in the 1950s by economics Nobel prize winners
Kenneth Arrow
Kenneth Joseph Arrow (August 23, 1921 – February 21, 2017) was an American economist, mathematician and political theorist. He received the John Bates Clark Medal in 1957, and the Nobel Memorial Prize in Economic Sciences in 1972, along with ...
and
Gérard Debreu
Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize ...
.
The theorem was first studied in view of work on differential equations by the French mathematicians around
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
and
Charles Émile Picard
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was ...
. Proving results such as the
Poincaré–Bendixson theorem
In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.
Theorem
Given a differentiable real dynamical system defined on an op ...
requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The case of differentiable mappings of the -dimensional closed ball was first proved in 1910 by
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations.
Biography
The son of a tea ...
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations.
Biography
The son of a tea ...
:
Note sur quelques applications de l'indice de Kronecker
' in Jules Tannery
Jules Tannery (24 March 1848 – 11 December 1910) was a French mathematician, who notably studied under Charles Hermite and was the PhD advisor of Jacques Hadamard. Tannery's theorem on interchange of limits and series is named after him. H ...
: ''Introduction à la théorie des fonctions d'une variable'' (Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477 (French) and the general case for continuous mappings by Brouwer in 1911.
Statement
The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows:
:;In the plane: Every
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
from a
closed disk to itself has at least one fixed point.
This can be generalized to an arbitrary finite dimension:
:;In Euclidean space:Every continuous function from a
closed ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defin ...
of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
into itself has a fixed point.
A slightly more general version is as follows:
:;Convex compact set:Every continuous function from a nonempty
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
subset ''K'' of a Euclidean space to ''K'' itself has a fixed point.
An even more general form is better known under a different name:
:;
Schauder fixed point theorem:Every continuous function from a nonempty convex compact subset ''K'' of a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
to ''K'' itself has a fixed point.
Importance of the pre-conditions
The theorem holds only for functions that are ''endomorphisms'' (functions that have the same set as the domain and codomain) and for nonempty sets that are ''compact'' (thus, in particular, bounded and closed) and ''convex'' (or
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to convex). The following examples show why the pre-conditions are important.
The function ''f'' as an endomorphism
Consider the function
:
with domain
1,1 The range of the function is
,2 Thus, f is not an endomorphism.
Boundedness
Consider the function
:
which is a continuous function from
to itself. As it shifts every point to the right, it cannot have a fixed point. The space
is convex and closed, but not bounded.
Closedness
Consider the function
:
which is a continuous function from the open interval
to itself. Since the point
is not part of the interval, there is no point in the domain such that
. The set
is convex and bounded, but not closed. On the other hand, the function
does have a fixed point in the ''closed'' interval