In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the

Brouwer fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simpl ...

. 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 function. The function is often thought of as an "unknown" to be solved for, similarly to ...

s. The first result in the field was the Schauder fixed-point theorem, proved in 1930 by

Juliusz Schauder Juliusz Paweł Schauder (; 21 September 1899, Lwów, Austria-Hungary – September 1943, Lwów, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equations and m ...

(a previous result in a different vein, the

Banach fixed-point theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...

for contraction mappings in complete

metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

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 Algebraic topology is a branch of mathematics that 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 classif ...

, 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 ...

es, to spaces of infinite dimension. For example, the research of

Jean Leray Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. Life and career He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...

who founded

sheaf theory In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...

came out of efforts to extend Schauder's work.

Schauder fixed-point theorem: Let ''C'' be a nonempty closed
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 ...
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 ve ...
''V''. If ''f'' : ''C'' → ''C'' is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
with a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
image, then ''f'' has a fixed point.

Tikhonov (Tychonoff) fixed-point theorem: Let ''V'' be a
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topologica ...
. 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 In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. Definition A uniformly convex space is a ...
. Then any non-expansive function ''f'' : ''K'' → ''K'' has a fixed point. (A function $f$ is called non-expansive if $\, f(x)-f(y)\, \leq \, x-y\,$ for each $x$ and $y$ .)

Other results include the Markov–Kakutani fixed-point theorem (1936-1938) and the

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 that is compact under the weak topology, then every group (or equivalentl ...

(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 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 poin ...
: 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.

References

* Vasile I. Istratescu, ''Fixed Point Theory, An Introduction'', D.Reidel, Holland (1981). . * Andrzej Granas and

James Dugundji James Dugundji (August 30, 1919 – January 8, 1985) was an American mathematician, a professor of mathematics at the University of Southern California.. See in particulap. 244for a brief biography of 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

See also

References

External links