In Analysis
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 and maps it into, 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 intersects the line . Numerically, the fixed point is approximately (thus ).
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 spaces and further; these are applied in PDE theory. See fixed-point theorems in infinite-dimensional spaces.
The collage theorem in fractal compression proves that, for many images, there exists a relatively small description of a function that, when iteratively applied to any starting image, rapidly converges on the desired image.
Read more about this topic: Fixed-point Theorem
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