Stability of equilibrium points

Question

Given the difference equation and the continuously differentiable function $g$: $$x(n+1)=x(n)+h\times g(x(n))$$ Determine conditions on $h$ for which an equilibrium point is asymptotically stable, respectively unstable.

Answer 1

Hint: From the comments, I infer that you are familiar with the requirements for stability of fixed points for the difference equation $x(n+1)=f(x(n))$. In the case you are asking about, you have $$ f(x)=x+hg(x) $$ for some function $g$ and a constant $h$. So you just need to insert this function $f$ into the criteria you already know, maybe pretty up the result a little, and you're done!