In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a fixed-point theorem is a result saying that a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 F will have at least one fixed point

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

 (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics.

Well-known fixed-point theorems include:

• Atiyah–Bott fixed-point theorem
  Atiyah–Bott fixed-point theorem
  In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M , which uses an elliptic complex on M...
  
  
• Banach fixed-point theorem, for contraction mapping
  Contraction mapping
  In mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...
  
  s
• 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. The result was proved by .-Statement of the theorem:...
  
  
• Brouwer fixed point theorem
  Brouwer fixed point theorem
  Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
  
  , for continuous
  Continuous function
  In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
  
   functions from the closed unit ball in n-dimensional Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   to itself
• Caristi fixed point theorem
  Caristi fixed point theorem
  In mathematics, the Caristi fixed-point theorem generalizes the Banach fixed point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem is a variation of the ε-variational principle of Ekeland...
  
  
• Fixed-point lemma for normal functions
  Fixed-point lemma for normal functions
  The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points...
  
  , for continuous strictly increasing functions from ordinals
  Ordinal number
  In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
  
   to ordinals
• Fixed point theorems in infinite-dimensional spaces
  Fixed point theorems in infinite-dimensional spaces
  In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations....
  
  
• Kakutani fixed-point theorem
• Kleene fixpoint theorem
  Kleene fixpoint theorem
  In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following:...
  
  
• Lefschetz fixed-point theorem
  Lefschetz fixed-point theorem
  In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...
  
  
• Nielsen fixed-point theorem
  Nielsen theory
  Nielsen theory is a branch of mathematical research with its origins in topological fixed point theory. Its central ideas were developed by Danish mathematician Jakob Nielsen, and bear his name....
  
  
• Knaster–Tarski theorem, which states that any order-preserving function on a complete lattice
  Complete lattice
  In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
  
   has a smallest fixed point
• Tychonoff fixed point theorem
• Woods Hole fixed-point theorem

See also

• Fixed-point property
• Fixed-point combinator
• Collage theorem
  Collage theorem
  In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions whose images, as a collage or union when mapping the given set, are arbitrarily close to the given...
  
  
• Diagonal lemma
  Diagonal lemma
  In mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of self-referential sentences in certain formal theories of the natural numbers -- specifically those theories that are strong enough to represent all computable functions...
  
  , also known as the fixed-point lemma, for producing self-referential sentences of first-order logic
  First-order logic
  First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
  
  
• Sperner's lemma
  Sperner's lemma
  In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it. Sperner's lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors...
  
  
• Bourbaki–Witt theorem
• Injective metric space
  Injective metric space
  In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces...
  
  
• Topological degree theory
  Topological degree theory
  In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can...
  
  

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a fixed-point theorem is a result saying that a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 F will have at least one fixed point

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

 (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics.

Well-known fixed-point theorems include:

• Atiyah–Bott fixed-point theorem
  Atiyah–Bott fixed-point theorem
  In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M , which uses an elliptic complex on M...
  
  
• Banach fixed-point theorem, for contraction mapping
  Contraction mapping
  In mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...
  
  s
• 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. The result was proved by .-Statement of the theorem:...
  
  
• Brouwer fixed point theorem
  Brouwer fixed point theorem
  Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
  
  , for continuous
  Continuous function
  In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
  
   functions from the closed unit ball in n-dimensional Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   to itself
• Caristi fixed point theorem
  Caristi fixed point theorem
  In mathematics, the Caristi fixed-point theorem generalizes the Banach fixed point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem is a variation of the ε-variational principle of Ekeland...
  
  
• Fixed-point lemma for normal functions
  Fixed-point lemma for normal functions
  The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points...
  
  , for continuous strictly increasing functions from ordinals
  Ordinal number
  In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
  
   to ordinals
• Fixed point theorems in infinite-dimensional spaces
  Fixed point theorems in infinite-dimensional spaces
  In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations....
  
  
• Kakutani fixed-point theorem
• Kleene fixpoint theorem
  Kleene fixpoint theorem
  In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following:...
  
  
• Lefschetz fixed-point theorem
  Lefschetz fixed-point theorem
  In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...
  
  
• Nielsen fixed-point theorem
  Nielsen theory
  Nielsen theory is a branch of mathematical research with its origins in topological fixed point theory. Its central ideas were developed by Danish mathematician Jakob Nielsen, and bear his name....
  
  
• Knaster–Tarski theorem, which states that any order-preserving function on a complete lattice
  Complete lattice
  In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
  
   has a smallest fixed point
• Tychonoff fixed point theorem
• Woods Hole fixed-point theorem

See also

• Fixed-point property
• Fixed-point combinator
• Collage theorem
  Collage theorem
  In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions whose images, as a collage or union when mapping the given set, are arbitrarily close to the given...
  
  
• Diagonal lemma
  Diagonal lemma
  In mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of self-referential sentences in certain formal theories of the natural numbers -- specifically those theories that are strong enough to represent all computable functions...
  
  , also known as the fixed-point lemma, for producing self-referential sentences of first-order logic
  First-order logic
  First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
  
  
• Sperner's lemma
  Sperner's lemma
  In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it. Sperner's lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors...
  
  
• Bourbaki–Witt theorem
• Injective metric space
  Injective metric space
  In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces...
  
  
• Topological degree theory
  Topological degree theory
  In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can...
  
  

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a fixed-point theorem is a result saying that a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 F will have at least one fixed point

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

 (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics.

Well-known fixed-point theorems include:

• Atiyah–Bott fixed-point theorem
  Atiyah–Bott fixed-point theorem
  In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M , which uses an elliptic complex on M...
  
  
• Banach fixed-point theorem, for contraction mapping
  Contraction mapping
  In mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...
  
  s
• 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. The result was proved by .-Statement of the theorem:...
  
  
• Brouwer fixed point theorem
  Brouwer fixed point theorem
  Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
  
  , for continuous
  Continuous function
  In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
  
   functions from the closed unit ball in n-dimensional Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   to itself
• Caristi fixed point theorem
  Caristi fixed point theorem
  In mathematics, the Caristi fixed-point theorem generalizes the Banach fixed point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem is a variation of the ε-variational principle of Ekeland...
  
  
• Fixed-point lemma for normal functions
  Fixed-point lemma for normal functions
  The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points...
  
  , for continuous strictly increasing functions from ordinals
  Ordinal number
  In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
  
   to ordinals
• Fixed point theorems in infinite-dimensional spaces
  Fixed point theorems in infinite-dimensional spaces
  In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations....
  
  
• Kakutani fixed-point theorem
• Kleene fixpoint theorem
  Kleene fixpoint theorem
  In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following:...
  
  
• Lefschetz fixed-point theorem
  Lefschetz fixed-point theorem
  In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...
  
  
• Nielsen fixed-point theorem
  Nielsen theory
  Nielsen theory is a branch of mathematical research with its origins in topological fixed point theory. Its central ideas were developed by Danish mathematician Jakob Nielsen, and bear his name....
  
  
• Knaster–Tarski theorem, which states that any order-preserving function on a complete lattice
  Complete lattice
  In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
  
   has a smallest fixed point
• Tychonoff fixed point theorem
• Woods Hole fixed-point theorem

See also

• Fixed-point property
• Fixed-point combinator
• Collage theorem
  Collage theorem
  In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions whose images, as a collage or union when mapping the given set, are arbitrarily close to the given...
  
  
• Diagonal lemma
  Diagonal lemma
  In mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of self-referential sentences in certain formal theories of the natural numbers -- specifically those theories that are strong enough to represent all computable functions...
  
  , also known as the fixed-point lemma, for producing self-referential sentences of first-order logic
  First-order logic
  First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
  
  
• Sperner's lemma
  Sperner's lemma
  In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it. Sperner's lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors...
  
  
• Bourbaki–Witt theorem
• Injective metric space
  Injective metric space
  In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces...
  
  
• Topological degree theory
  Topological degree theory
  In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can...