# 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**

### Euclidean space

**EuclideanspaceEuclidean vector space**

### Sperner's lemma

**Sperner coloring**

### 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.