Main Question
I am looking for a reference, where a connection is made between solutions of recursive formulae and Ordinary Differential Equations (ODEs) which look similar. In my case the recursive formula is $x_{k+1}=a Q(x_k)$ and the ODE is $f'(x)=aQ(f(x))-f(x)$.
Elaboration
Let $a < 1$ be some constant and $Q:(0,1]\rightarrow (0,1]$ be some smooth function satisfying $Q(x) < x$, in particular we use the example $Q(x)=x^2$. Let $(x_k)_k$ be a sequence defined by $x_0=1$ and the recursive formula : $$ x_{k+1} = a \cdot Q(x_k), $$ Furthermore we define the function: $$ f'(x) = a Q(f(x)) - f(x), \qquad f(0)=a, $$ I have found there to be quite a strong relation between the solution $(x_k)_k$ of the recursive formula and the solution of the ODE $f(x)$. In particular, both have the same fix-points. When one converges to zero, the other also converges to zero. In case of $Q(x)=x^2$, we have both the sequence and the function converging to zero and one can show that $\int_0^\infty f(x) \, dx \leq \sum_k x_k$, furthermore one can show that $\lim_{a\rightarrow 1} \frac{\int_0^\infty f(x)}{\sum_k x_k} = \frac{\log(2)}{2-1}$.