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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 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 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=0.7390910582853 (thus x=\cos(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 Space s and further; these are applied in PDE 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