Asymptotic behavior of $a_{n+1}=a_n + f(a_n)$

Question

Let $f$ be $C^1([0,1])$ function. I'm looking for a good reference for the asymptotic behavior of $$a_{n+1}=a_n + f(a_n)$$ and it's relation to dynamical systems.

Answer 1

As a first cut, I would write $a_{n+1}-a_n \approx a'(x)$ so my guess would be $a'(x) \approx f(a(x))$.

Answer 2

Clearly, if $a_n \to L$, then $f(L)=0$.

Convergence will be guaranteed if $g:x \mapsto x+f(x)$ is a contraction, as long as $g([0,1]) \subseteq [0,1]$:

• $|g'(x)|<1$ iff $-2 < f(x) < 0$.

• $g([0,1]) \subseteq [0,1]$ iff $-x \le f(x) \le 1-x$.