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.

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 doesn't 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 must have a fixed point. This is clear when examining a sketched graph of the cosine function; the fixed point occurs where the cosine curve y=cos(x) intersects the line y=x. Numerically, the fixed point is approximately x=0.73908513321516 (thus x=cos(x) for this value of x).

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.

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

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