Multisummability for some classes of difference equations

Braaksma, Boele L. J.; Faber, Bernard F.

Braaksma, Boele L. J. ; Faber, Bernard F.

Annales de l'Institut Fourier, Tome 46 (1996), p. 183-217 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

On considère des équations aux différences finies $y (x + 1) = G (x, y)$ où $G$ prend ses valeurs dans $C^{n}$ et $G$ est méromorphe en $x$ au voisinage de $\infty$ dans $C$ et holomorphe en $y$ au voisinage de 0 dans $C^{n}$ . On montre que sous certaines conditions sur la partie linéaire de $G$ , des solutions formelles dans $C^{n} [[x^{- 1 / p}]], p \in N,$ sont multisommables. De plus on montre que ces solutions formelles se relèvent toujours en des solutions holomorphes dans des demi-plans supérieurs et inférieurs, mais généralement ces solutions ne sont pas déterminées uniquement par les solutions formelles.

This paper concerns difference equations $y (x + 1) = G (x, y)$ where $G$ takes values in $C^{n}$ and $G$ is meromorphic in $x$ in a neighborhood of $\infty$ in $C$ and holomorphic in a neighborhood of 0 in $C^{n}$ . It is shown that under certain conditions on the linear part of $G$ , formal power series solutions in $x^{- 1 / p}, p \in N,$ are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions.

Publié le : 1996-01-01

MR 97e:39002 | Zbl 0837.39001 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1511

@article{AIF_1996__46_1_183_0,
     author = {Braaksma, Boele L. J. and Faber, Bernard F.},
     title = {Multisummability for some classes of difference equations},
     journal = {Annales de l'Institut Fourier},
     volume = {46},
     year = {1996},
     pages = {183-217},
     doi = {10.5802/aif.1511},
     mrnumber = {97e:39002},
     zbl = {0837.39001},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1996__46_1_183_0}
}

Braaksma, Boele L. J.; Faber, Bernard F. Multisummability for some classes of difference equations. Annales de l'Institut Fourier, Tome 46 (1996) pp. 183-217. doi : 10.5802/aif.1511. http://gdmltest.u-ga.fr/item/AIF_1996__46_1_183_0/

Bibliographie

[Bal94] W. Balser, From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics 1582. Springer Verlag, Heidelberg, 1994. | MR 96d:34071 | Zbl 0810.34046

[BIS] B.L.J. Braaksma, G.K. Immink, and Y. Sibuya, Cauchy-Heine and Borel transforms and Stokes phenomena, in preparation.

[Bra91] B.L.J. Braaksma, Multisummability and Stokes multipliers of linear meromorphic differential equations, J. Diff. Eq., 92 (1991), 45-75. | MR 93c:34010 | Zbl 0729.34005

[Bra92] B.L.J. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, 42-3 (1992), 517-540. | Numdam | MR 93j:34006 | Zbl 0759.34003

[BT93] W. Balser and A. Tovbis, Multisummability of iterated integrals, Asympt. Anal., 7 (1993), 121-127. | MR 94g:30002 | Zbl 0787.30026

[Duv83] A. Duval, Lemmes d'Hensel et factorisation formelle pour les opérateurs aux différences, Funkcial. Ekvac., 26 (1983), 349-368. | MR 86h:12011 | Zbl 0543.12018

[Eca87] J. Ecalle, L'accélération des fonctions résurgentes, manuscrit, 1987.

[Eca93] J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Hermann, Paris, 1993.

[GLS] R. Gérard, D.A. Lutz, and R. Schäfke, Analytic reduction of some nonlinear difference equations, To appear in Funkcial. Ekvac. | Zbl 0847.39003

[Imm] G.K. Immink, On the summability of the formal solutions of a class of inhomogeneous linear difference equations, submitted to Funkcial. Ekvac. | Zbl 0872.39002

[Imm84] G.K. Immink, Asymptotics of Analytic Difference Equations, Lecture Notes in Mathematics 1085, Springer Verlag, Heidelberg, 1984. | MR 86c:39002 | Zbl 0548.39001

[Imm91] G.K. Immink, Reduction to canonical forms and the Stokes phenomenon in the theory of linear difference equations, SIAM J. Math. Anal., 22 (1991), 238-259. | MR 92c:39005 | Zbl 0733.39004

[Mal] B. Malgrange, Sommation des séries divergentes, Prépublication de l'Inst. Fourier, no. 268, 1994.

[Mal91] B. Malgrange, Equations différentielles à coefficients polynomiaux, Progress in Math. 96, Birkhäuser, Basel, 1991. | MR 92k:32020 | Zbl 0764.32001

[MalR91] B. Malgrange and J.-P. Ramis, Fonctions multisommables, Ann. Inst. Fourier, 42-1/2 (1992), 353-368. | Numdam | MR 93e:40007 | Zbl 0759.34007

[MarR91] J. Martinet and J.-P. Ramis, Elementary acceleration and multisummability, Ann. Inst. H. Poincaré, Phys. Théor., 54 (1991), 331-401. | Numdam | MR 93a:32036 | Zbl 0748.12005

[Pra83] C. Praagman, The formal classification of linear difference operators, Proc. Kon. Ned. Ac. Wet., Ser A, 86 (1983), 249-261. | MR 85c:12006 | Zbl 0519.39003

[Ram93] J.-P. Ramis, Séries divergentes et théories asymptotiques, Panoramas et synthèses 121, Soc. Math. France, 1993. | MR 95h:34074 | Zbl 0830.34045

[vd PS] M. Van Der Put and M.F. Singer, Galois theory of difference equations, Book to appear, 1996. | Zbl 0930.12006

[Tur60] H.L. Turrittin, The formal theory of systems of irregular homogeneous linear difference and differential equations, Bol. Soc. Mat. Mexicana, (1960), 225-264. | MR 25 #349 | Zbl 0100.08201