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 simples ...

. 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 equations. The first result in the field was the

Schauder fixed-point theorem The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if K is a nonempty convex closed subset of a Hausdorff topological vector space V ...

, 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 settin ...

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 classify ...

, first proved for finite simplicial complexes, 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 The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if K is a nonempty convex closed subset of a Hausdorff topological vector space V ...
: Let ''C'' be a
nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
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 ''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 ...
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 topological ...
. 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 $f$ 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 In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space ha ...

(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 In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into itself to have a fixed point. The result was ...

(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 set, convex, compact set, compact subset of a Euclidean sp ...
: 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 Brailey Sims (born 26 October 1947) is an Australian mathematician born and educated in Newcastle, New South Wales. He received his BSc from the University of New South Wales in 1969 and, under the supervision of J. R. Giles, a PhD from the s ...

, ''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