Exact asymptotics of nonlinear difference equations with levels $1$ and $1^{+}$

Immink, G.K

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 2, pp. 309-356.

On étudie une classe d’équations aux différences finies, nonlinéaires, possédants une solution formelle en forme de série $1$-Gevrey qui, en général, n’est pas Borel-sommable. En utilisant des inverses à droite d’un opérateur aux différences associé, définies sur des espaces Banach de quasi-fonctions, on démontre qu’à la solution formelle peut être associée, de façon unique, une solution analytique sur un domaine approprié, qui est une accéléro-somme de la solution formelle.

We study a class of nonlinear difference equations admitting a $1$-Gevrey formal power series solution which, in general, is not $1$- (or Borel-) summable. Using right inverses of an associated difference operator on Banach spaces of so-called quasi-functions, we prove that this formal solution can be lifted to an analytic solution in a suitable domain of the complex plane and show that this analytic solution is an accelero-sum of the formal power series.

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