mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compactness, compact convex set to itself, the ...

. 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 involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

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

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 Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, first proved for finite

simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

es, to spaces of infinite dimension. For example, the research of Jean Leray who founded

sheaf theory In mathematics, a sheaf (: sheaves) 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 d ...

came out of efforts to extend Schauder's work.

Schauder fixed-point theorem: Let ''C'' be a
nonempty In mathematics, the empty set or void 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, whi ...
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 (, ) 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 ...
''V''. If ''f'' : ''C'' → ''C'' is 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
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 vec ...
. 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 (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. Also, Aniki & Rauf (2019) presented some interesting results on the stability of partially ordered metric spaces for coupled fixed point iteration procedures for mixed monotone mappings.

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 ...
: 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 . * Samuel A. Aniki and Kamilu Rauf, ''Some stability results in partially ordered metric spaces for coupled fixed point iteration of procedures for mixed monotone mappings'' (2019), Islamic University Multidisciplinary Journal, 6(3), 175-186 https://www.iuiu.ac.ug/journaladmin/iumj/ArticleFiles/49305.pdf

External links

PlanetMath article on the Tychonoff Fixed Point Theorem
{{Webarchive, url=https://web.archive.org/web/20100620174722/http://planetmath.org/encyclopedia/TychonoffFixedPointTheorem.html , date=2010-06-20 Fixed-point theorems fr:Théorème du point fixe de Schauder

See also

References

External links