Editing Fixed point (mathematics) (section)

==Fixed-point theorems==
{{Main|Fixed-point theorems}}

A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition.<ref>{{cite book
 | editor     = Brown, R. F.
 | title      = Fixed Point Theory and Its Applications
 | year       = 1988
 | publisher  = American Mathematical Society
 | isbn       = 0-8218-5080-6
}}
</ref>

For example, the [[Banach fixed-point theorem]] (1922) gives a general criterion guaranteeing that, if it is satisfied, [[fixed-point iteration]] will always converge to a fixed point.

The [[Brouwer fixed-point theorem]] (1911) 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.

The [[Lefschetz fixed-point theorem]] (and the [[Nielsen theory|Nielsen fixed-point theorem]]) from [[algebraic topology]] give a way to count fixed points.