Accelero-summation of the formal solutions of nonlinear difference equations

Immink, Geertrui Klara

Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 1-51.

En 1996, Braaksma et Faber ont établi la multi-sommabilité, sur des multi-intervalles convenables, des solutions formelles d’équations aux différences nonlinéaires, localement analytiques, sous la condition que le niveau $1^{+}$ ne se présente pas. En combinant leurs résultats avec d’autres récents pour le cas des deux niveaux $1$ et $1^{+}$, on démontre, pour une classe très générale d’équations, l’accéléro-sommabilité de la solution formelle. L’accéléro-somme est solution analytique de l’équation, admettant la solution formelle comme développement asymptotique à l’infini.

In 1996, Braaksma and Faber established the multi-summability, on suitable multi-intervals, of formal power series solutions of locally analytic, nonlinear difference equations, in the absence of “level $1^{+}$”. Combining their approach, which is based on the study of corresponding convolution equations, with recent results on the existence of flat (quasi-function) solutions in a particular type of domains, we prove that, under very general conditions, the formal solution is accelero-summable. Its sum is an analytic solution of the equation, represented asymptotically by the formal solution in a certain unbounded domain.

Classification : 39A10, 30E15, 40G10
Keywords: Nonlinear difference equation, formal solution, accelero-summation, quasi-function
Mot clés : équation aux différences nonlinéaire, solution formelle, accéléro-sommation, quasi-fonction

[1] Braaksma, B.L.J. Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, Volume 42 (1992), pp. 517-540 | DOI | Numdam | MR | Zbl

[2] Braaksma, B.L.J. Borel transforms and multisums, Revista del Seminario Iberoamericano de Matemáticas, Volume V (1997), pp. 27-44

[3] Braaksma, B.L.J.; Faber, B.F. Multisummability for some classes of difference equations, Ann. Inst. Fourier, Volume 46 (1996) no. 1, pp. 183-217 | DOI | Numdam | MR | Zbl

[4] Braaksma, B.L.J.; Faber, B.F.; Immink, G.K. Summation of formal solutions of a class of linear difference equations, Pacific J. Math., Volume 195 (2000) no. 1, pp. 35-65 | DOI | MR | Zbl

[5] Ecalle, J. The acceleration operators and their applications, Proc. Internat. Congr. Math., Kyoto (1990), Vol. 2, Springer-Verlag (1991), pp. 1249-1258 | MR | Zbl

[6] Ecalle, J. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Math., Hermann, Paris, 1992 | MR

[7] Ecalle, J. Cohesive functions and weak accelerations, J. Anal. Math., Volume 60 (1993), pp. 71-97 | MR | Zbl

[8] Immink, G.K. Asymptotics of analytic difference equations, 1085, Springer Verlag, Berlin, 1984 | MR | Zbl

[9] Immink, G.K. A particular type of summability of divergent power series, with an application to difference equations, Asymptotic Analysis, Volume 25 (2001), pp. 123 -148 | MR | Zbl

[10] Immink, G.K. Summability of formal solutions of a class of nonlinear difference equations, Journal of Difference Equations and Applications, Volume 7 (2001), pp. 105 -126 | DOI | MR | Zbl

[11] Immink, G.K. Existence theorem for nonlinear difference equations, Asymtotic Analysis, Volume 44 (2005), pp. 173 -220 | MR | Zbl

[12] Immink, G.K. Gevrey type solutions of nonlinear difference equations, Asymtotic Analysis, Volume 50 (2006), pp. 205 -237 | MR | Zbl

[13] Immink, G.K. On the Gevrey order of formal solutions of nonlinear difference equations, Journal of Difference Equations and Applications, Volume 12 (2006), pp. 769-776 | DOI | MR | Zbl

[14] Immink, G.K. Exact asymptotics of nonlinear difference equations with levels 1 and $1^{+}$, Ann. Fac. Sci. Toulouse, Volume 17 (2008), pp. 309-356 | DOI | Numdam | MR | Zbl

[15] Malgrange, B. Sommation des séries divergentes, Expo. Math., Volume 13 (1995), pp. 163-222 | MR | Zbl

[16] Malgrange, B.; Ramis, J.-P. Fonctions multisommables, Ann. Inst. Fourier, Volume 41-3 (1991), pp. 1 -16 | MR

[17] Praagman, C. The formal classification of linear difference operators, Proceedings Kon. Nederl. Ac. van Wetensch., ser. A, 86 (2) (1983), pp. 249-261 | MR | Zbl

[18] Ramis, J.P.; Paris, Soc. Math. France Séries divergentes et théories asymptotiques, Panoramas et synthèses, Volume 121, Paris (1993), pp. 651-684 | MR

[19] Ramis, J.P.; Sibuya, Y. A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, Volume 44 (1994) no. 3, pp. 811-848 | DOI | Numdam | MR | Zbl