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 $x^*$.