I'm having problems with my difference equations problem set. The current topic is equilibrium points of non-autonomous equations. Let's see:
Consider $y_{k+1}=f(y_{k},k)$, where $f(y_{k},k)$ is a continuous and differentiable function on $y$.
1) Show that if an equilibrium exists, then it should satisfy $\lim_{k\rightarrow\infty}f(y,k)=y$
2) Show that if $y^*$ is a solution and $\lim_{k\rightarrow\infty}|f_{y}(y^{*},k)|>1$ so the sequence $y_{n}$ does not converge to $y^*$
This seems totally intuitive for me, but I can't provide a formal proof. Any help is absolutely welcome! Thanks in advance!