I was told by my supervisor that one way to study the asymptotic behaviour of solutions to ODEs is "to reformulate them as integral equations, and use fixed-point kind theorems on the resulting integral equations".
Since my supervisor is currently away, I would like to ask for references about the topic above, namely: studying the asymptotic nature of solutions to (possibly nonlinear) differential equations using the integral equation formulation.
Thank you for any reference in that matter!
EDIT:
I'm interested in ODEs of low order ($n \leq 2$). I'm aware that Taylor's theorem gives
$$y(x)=y(x_0)+y'(x_0)(x-x_0)+\int_{x_0}^x (x-t)y''(t) \mathrm{d}t $$
and I'm interested in different formulations (possibly, involving integrating factors).
Lastly, I'm the most interested in the limit $x \to \infty$.
Sorry for not being specific in the first place.