A fixed point of a function $F:A\to A$ is a value $x\in A$ that the function leaves unchanged:

$$F(x)=x.$$

For example, if

$$F(x)=\cos x,$$

then any solution of

$$x=\cos x$$

is a fixed point of $F$.

Fixed points arise whenever an object is defined in terms of a transformation of itself. A recursive equation can ask for a value $x$ satisfying

$$x=F(x),$$

while an iterative algorithm may repeatedly apply $F$ in the hope of approaching such a value.

A function can have no fixed points, one fixed point or many. Existence and uniqueness therefore require additional assumptions; the equation $F(x)=x$ alone does not guarantee either.

The concept is broader than numerical root finding. Recursive program meanings, recursive definitions and iterative program analyses can all be formulated as fixed-point problems. What differs between applications is the set $A$, the structure available on it and the method used to obtain or characterize an appropriate fixed point.