I found the following interesting equation on some web page I cannot remember:

$f(f(x))=\cos(x)$

Out of curiosity I tried to solve it, but realized that I do not have a clue how to approach such an iterative equation except for trial and error. I also realized that the solution might not be unique, from the solution of a simpler problem

$f(f(x)) = x$

which has for example the solutions $f(x) = x$ or $f(x) = \frac{x+1}{x-1}$.

Is there a general solution strategy to equations of this kind? Can you perhaps point me to some literature about these kind of equations? And what is the solution for $f(f(x))=\cos(x)$ ?