Fixed-point theorem

fixed point theoremfixed point theoryfixed-point theoryfixed pointfixed point setfixed-point propertyList of fixed point theorems
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.wikipedia
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Banach fixed-point theorem

Banach fixed point theoremBanach contraction principleContraction mapping theorem
The Banach fixed-point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
In mathematics, the Banach–Caccioppoli 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 certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

Brouwer fixed-point theorem

Brouwer fixed point theoremBrouwer's fixed point theoremBrouwer's fixed-point theorem
By contrast, the Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer.

Fixed point (mathematics)

fixed pointfixed pointsfixpoint
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.

Denotational semantics

denotationalfully abstractHistory of denotational semantics
In denotational semantics of programming languages, a special case of the Knaster–Tarski theorem is used to establish the semantics of recursive definitions.
So by a fixed-point theorem (specifically Bourbaki–Witt theorem), there exists a fixed point for this iterative process.

Borel fixed-point theorem

Borel fixed point theorem
In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem.

Lefschetz fixed-point theorem

Lefschetz fixed point theoremLefschetz numberLefschetz trace formula
The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology is notable because it gives, in some sense, a way to count fixed points.

Kakutani fixed-point theorem

Kakutani fixed point theoremeponymous fixed-point theoremKakutani's fixed point theorem
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions.

Nielsen theory

Nielsen fixed-point theorem
The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology is notable because it gives, in some sense, a way to count fixed points.
Nielsen theory is a branch of mathematical research with its origins in topological fixed point theory.

Topological degree theory

topological degreedegree
It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory.

Injective metric space

injectiveinjective metric
Every injective space is a complete space, and every metric map (or, equivalently, nonexpansive mapping, or short map) on a bounded injective space has a fixed point.

Schauder fixed-point theorem

Schauder fixed point theoremLeray-Schauder fixed point theoremSchaefer's fixed point theorem

Mathematics

mathematicalmathmathematician
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.

Function (mathematics)

functionfunctionsmathematical function
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.

Iteration

iterativeiteratediteratively
The Banach fixed-point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.

Constructivism (philosophy of mathematics)

constructive mathematicsconstructivismconstructive
By contrast, the Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).

Continuous function

continuouscontinuitycontinuous map
By contrast, the Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).

Unit sphere

unit ballclosed unit ballsphere
By contrast, the Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).

Euclidean space

EuclideanspaceEuclidean vector space
By contrast, the Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).

Sperner's lemma

Sperner coloring
By contrast, the Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).

Trigonometric functions

cosinetrigonometric functiontangent
For example, the cosine function is continuous in [−1,1] and maps it into [−1, 1], and thus must have a fixed point.

Algebraic topology

algebraicalgebraic topologistalgebraic topological
The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology is notable because it gives, in some sense, a way to count fixed points.