Slightly generalizing, we may state   conditions in terms of $G(x,y):=g(x,h(y))$. For instance:   

> Let $G$ be a continuous function on $\mathbb{R}^2$. Assume that  the
> equation $G(x,x)=0$ has a unique solution $x^*$ and that for some $M$
> any solution of $G(x,y)=0$  verifies $y\le x\le M$.  Then any sequence
> $(x_i)$ such that $G(x_{i+1}, x_i)=0$ does converge to $x^*$.

Indeed by assumption the sequence $x_i$ must be increasing and bounded, thus convergent, and by continuity of $G$, its limit is a solution of $G(x,x)=0$, hence it's $x^*$.