I have the following implicit equation
$$ x= f(x) $$
which I solve by starting with some value for $x$, then setting $x$ to the new value $f(x)$ and so forth until convergence.
How is that method called? What are its convergence properties, what are caveats one should be aware of, especially wrt the convexity of $f$? In my case $f$ is infinitely derivable.