Fixed-point theorem

In mathematics, a fixed-point theorem is a result saying that a function <math>F<math> will have at least one fixed point, under some conditions on <math>F<math> that can be stated in general term. Results of this kind are amongst the most generally useful in mathematics.

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.

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 does not describe how to find the fixed point (see also Sperner's lemma).

For example, the cosine function is continuous in [-1,1] and maps it into [-1, 1], and thus should have a fixed point.

The Lefschetz fixed-point theorem from algebraic topology is notable because it gives, in some sense, a way to count fixed points.

There are a number of generalisations to Banach spaces and further; these are applied in partial differential equation theory. See fixed point theorems in infinite-dimensional spaces.

The Knaster-Tarski theorem is somewhat removed from analysis and does not deal with continuous functions. It states that any order-preserving function on a complete lattice has a fixed point, and indeed a smallest fixed point. See also Bourbaki-Witt theorem.

A common theme in lambda calculus is to find fixed points of given lambda expressions. Every lambda expression has a fixed point, and a fixed point combinator is a "function" which takes as input a lambda expression and produces as output a fixed point of that expression. An important fixed point combinator is the Y combinator used to give recursive definitions.

The above technique of iterating a function to find a fixed point can also be used in set theory; the fixed-point lemma for normal functions states that any continuous strictly increasing function from ordinals to ordinals has one (and indeed many) fixed points.

Every closure operator on a poset has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place.

See also: Woods hole fixed-point theorem.

References

• Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, Holland (1981). ISBN 90–277–1224–7 provides an undergraduate level introduction.
• Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0–387–00173–5.
• William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, London ISBN 0–7923–7073–2.