Reference request: using integral equations to study asymptotics of ODEs 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.
 A: I'm not sure precisely what you're looking for, but the asymptotics of solution curves towards an equilibrium point (i.e. finding stable and unstable manifolds) can be studied using the (Cotton-)Lyapunov-Perron method, which is formulated as a fixed point problem for a Picard-like integral operator that is a contraction on an apropriate space of curves.
Basically you rewrite the Picard integral equation for an ODE with part of the coordinate components integrated over the infinite interval from +/- infinity (depending on whether you're looking for the stable or unstable manifold). Then for curves that are bounded in these components, this defines a contraction map that converges to a solution curve on the (un)stable manifold to the equilibrium point.
Some of the original articles are:
Cotton, Sur les solutions asymptotiques des équations différentielles 1911), Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen (1929), Perron, Die Stabilitätsfrage bei Differentialgleichungen (1930).
For a more modern, introductory reference, see for example Chicone's ODE book, section 4.1.
A: Fedoryuk, Mikhail V. Asymptotic analysis. Linear ordinary differential equations. Springer-Verlag, Berlin, 1993. viii+363 pp. ISBN: 3-540-54810-6 
Wasow, Wolfgang Asymptotic expansions for ordinary differential equations. Reprint of the 1976 edition. Dover Publications, Inc., New York, 1987. x+374 pp. ISBN: 0-486-65456-7 
