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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.

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  • $\begingroup$ What order are your ODEs? What do you already know about recasting ODE problems in integral form? Are you looking at asymptotics near zero or infinity? $\endgroup$ Jul 18, 2015 at 15:48
  • $\begingroup$ @JoonasIlmavirta edited. $\endgroup$
    – user1337
    Jul 18, 2015 at 16:01

2 Answers 2

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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.

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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

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