This is an old pre-lim question I've been working on
if $f(x)$ is continuously differentiable on $(-a,a), f(0)=0$ and $|f'(y)| \leq k < 1$ on $(-a,a)$, Then $\exists \epsilon > 0$ and a unique diff. $g$ on $(-\epsilon, \epsilon)$ s.t. $x=g(x)+f(g(x))$
the only thing I can think of is that x+f(x) should have an inverse by CH 5 exercise 3 of Rudin.