# 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.

### Diagonal lemma

diagonalizationdiagonal argumentdiagonalisation

### 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

### Ryll-Nardzewski fixed-point theorem

Ryll-Nardzewski fixed point theorem

### Kleene fixed-point theorem

Kleene fixpoint theorem
* Other fixed-point theorems

### 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.