There is an exercise I came across that I struggle with. Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be lipschitz continuous in the second argument. How to choose the step size $h$ in the $m$-step method $$ \sum_{k=0}^{m} \alpha_k u_{j+k} = h \sum_{k=0}^{m} \beta_k f(t_{j+k}, u_{j+k}) \quad j= 0, ..., n-m \qquad (1)$$ under the assumption $\beta_m \neq 0$ for the solution of the IVP $$ u'(t) = f(t,y), t > t_0$$ $$ u(t_0) = u_0$$ such that the system of equations in (1) can be uniquely solved for $u_{j+m}$?
I know that implicit methods can only be solved iteratively. Also, I was told to consider Banach's fixed-point theorem. Nevertheless, I do not seem to get a grip on the general idea behind the exercise.