logo AFST
Exact asymptotics of nonlinear difference equations with levels 1 and 1 +
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.

Reçu le : 2006-06-20
Accepté le : 2007-09-23
Publié le : 2008-12-11
DOI : https://doi.org/10.5802/afst.1185
@article{AFST_2008_6_17_2_309_0,
     author = {G.K Immink},
     title = {Exact asymptotics of nonlinear difference equations with levels $1$ and $1^+$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {309--356},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 17},
     number = {2},
     year = {2008},
     doi = {10.5802/afst.1185},
     zbl = {1160.39003},
     mrnumber = {2487857},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2008_6_17_2_309_0/}
}
G.K Immink. Exact asymptotics of nonlinear difference equations with levels $1$ and $1^+$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 2, pp. 309-356. doi : 10.5802/afst.1185. https://afst.centre-mersenne.org/item/AFST_2008_6_17_2_309_0/

[1] Braaksma (B.L.J.), Faber ( B.F.), and Immink (G.K.).— Summation of formal solutions of a class of linear difference equations, Pacific Journal of Mathematics, 195, 1, p. 35–65 (2000). | MR 1781613 | Zbl 1008.39003

[2] Écalle (J.).— Les Fonctions Résurgentes, tome III, Publ. Math. d’Orsay, Université de Paris-Sud, Paris (1985). | Zbl 0602.30029

[3] Écalle (J.).— The acceleration operators and their applications, In Proc. Internat. Congr. Math., Kyoto (1990), Vol. 2, pages 1249–1258, Springer-Verlag (1991). | MR 1159310 | Zbl 0741.30030

[4] Écalle (J.).— Cohesive functions and weak accelerations. Journal d’An. Math., 60:71–97, 1993. | Zbl 0808.30002

[5] Immink (G.K.).— Asymptotics of analytic difference equations. In Lecture Notes in Mathematics 1085, Berlin, 1984. Springer Verlag. | MR 765699 | Zbl 0548.39001

[6] Immink (G.K.).— On the summability of the formal solutions of a class of inhomogeneous linear difference equations, Funk. Ekv., 39-3, p. 469–490 (1996). | MR 1433913 | Zbl 0872.39002

[7] Immink (G.K.).— A particular type of summability of divergent power series, with an application to difference equations, Asymptotic Analysis, 25, p. 123–148 (2001). | MR 1818642 | Zbl 0981.30002

[8] Immink (G.K.).— Summability of formal solutions of a class of nonlinear difference equations, Journal of Difference Equations and Applications, 7, p. 105–126 (2001). | MR 1809599 | Zbl 0973.39003

[9] Immink (G.K.).— Existence theorem for nonlinear difference equations, Asymptotic Analysis, 44, p.173–220 (2005). | MR 2176272 | Zbl 1083.39001

[10] Immink (G.K.).— Gevrey type solutions of nonlinear difference equations, Asymptotic Analysis, 50, p. 205–237 (2006). | MR 2294599 | Zbl 1122.39018

[11] Praagman (C.).— The formal classification of linear difference operators, In Proceedings Kon. Nederl. Ac. van Wetensch., ser. A, 86 (2), pages 249–261 (1983). | MR 705431 | Zbl 0519.39003

[12] Ramis (J.P.).— Séries divergentes et théories asymptotiques, In Panoramas et synthèses, volume 121, pages 651–684. Soc. Math. France, Paris (1993). | MR 1272100 | Zbl 0830.34045

[13] Wasow (W.).— Asymptotic Expansions for Ordinary Differential Equations, Interscience Publishers, New York (1965). | MR 203188 | Zbl 0133.35301