Fixed-point theorem

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. Results of this kind are amongst the most generally useful in mathematics.

Well-known fixed-point theorems include:

• Atiyah–Bott fixed-point theorem
• Banach fixed-point theorem, for contraction mappings
• Borel fixed-point theorem
• Brouwer fixed point theorem, for continuous functions from the closed unit ball in n-dimensional Euclidean space to itself
• Caristi fixed point theorem
• Fixed-point lemma for normal functions, for continuous strictly increasing functions from ordinals to ordinals
• Fixed point theorems in infinite-dimensional spaces
• Kakutani fixed-point theorem
• Kleene fixpoint theorem
• Lefschetz fixed-point theorem
• Nielsen fixed-point theorem
• Knaster–Tarski theorem, which states that any order-preserving function on a complete lattice has a smallest fixed point
• Tychonoff fixed point theorem
• Woods Hole fixed-point theorem

See also

• Fixed-point property
• Fixed-point combinator
• Collage theorem
• Diagonal lemma, also known as the fixed-point lemma, for producing self-referential sentences of first-order logic
• Sperner's lemma
• Bourbaki–Witt theorem
• Injective metric space
• Topological degree theory