Fixed point theorems in infinite-dimensional spaces
Encyclopedia
In 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
s for partial differential equation
s.
The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder
. 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
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.
The Schauder fixed-point theorem states, in one version, that if C is a nonempty closed
convex
subset of a Banach space
V and f is a continuous map
from C to C whose image is compact, then f has a fixed point.
The Tikhonov (Tychonoff) fixed point theorem is applied to any locally convex topological vector space
V. It states that for any non-empty compact convex set X in V, any continuous function
has a fixed point.
Other results are the Kakutani
and Markov fixed point theorems, as well as the Ryll-Nardzewski fixed point theorem
(1967).
Kakutani's fixed-point theorem
states that:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a number of fixed-point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorem
Existence theorem
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist ..', or more generally 'for all x, y, ... there exist ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not...
s for partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s.
The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder
Juliusz Schauder
Juliusz Paweł Schauder was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equation and mathematical physics.Born on September 21, 1899 in Lemberg, he had to fight in World War I right after his graduation from school...
. 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
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, first proved for finite simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es, to spaces of infinite dimension. For example, the research of Jean Leray
Jean Leray
Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....
who founded sheaf theory came out of efforts to extend Schauder's work.
The Schauder fixed-point theorem states, in one version, that if C is a nonempty closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
subset of a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
V and f is a continuous map
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
from C to C whose image is compact, then f has a fixed point.
The Tikhonov (Tychonoff) fixed point theorem is applied to any locally convex topological vector space
Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces which generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of ...
V. It states that for any non-empty compact convex set X in V, any continuous function
- ƒ:X → X,
has a fixed point.
Other results are the Kakutani
Shizuo Kakutani
was a Japanese-born American mathematician, best known for his eponymous fixed-point theorem.Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. Early in his career he spent two years at the Institute for Advanced Study in Princeton at the invitation of the...
and Markov fixed point theorems, as well as the Ryll-Nardzewski fixed point theorem
Ryll-Nardzewski fixed point theorem
In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if E is a normed vector space and K is a nonempty convex subset of E which is compact under the weak topology, then every group of affine isometries of K has at least one fixed point...
(1967).
Kakutani's fixed-point theorem
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 point, i.e. a point which is mapped to a set containing...
states that:
- 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.