Consider a function $f(x)=g(x,h(x))$, which we know has a unique root $x^*$. The functions $f$, $g$ and $h$ are all continuous in $x$ and behave nicely. Iteratively solving $g(x_{i+1}, h(x_i))=0$ with some starting guess $x_0$, we are able to find $x^*$.

My goal is to prove that this iteration indeed works. Can one give simple sufficient conditions on $f$, $g$ and $h$, that would indeed guarantee convergence? I guess this is a pretty high level question, but perhaps people can point me in the right direction.

Example. $f(x)=x^2+\log[x]$ and take $h(x)=x^2$, then $x_{i+1}=\exp[-x_{i}^2]$ will converge to $x^*=0.652919$ for any $x_0\in \mathbb{R}$.