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 ”. 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.
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 ne se présente pas. En combinant leurs résultats avec d’autres récents pour le cas des deux niveaux et , 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.
Keywords: Nonlinear difference equation, formal solution, accelero-summation, quasi-function
Mot clés : équation aux différences nonlinéaire, solution formelle, accéléro-sommation, quasi-fonction
@article{AIF_2011__61_1_1_0, author = {Immink, Geertrui Klara}, title = {Accelero-summation of the formal solutions of nonlinear difference equations}, journal = {Annales de l'Institut Fourier}, pages = {1--51}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {1}, year = {2011}, doi = {10.5802/aif.2596}, zbl = {1225.39005}, mrnumber = {2828125}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2596/} }
TY - JOUR AU - Immink, Geertrui Klara TI - Accelero-summation of the formal solutions of nonlinear difference equations JO - Annales de l'Institut Fourier PY - 2011 SP - 1 EP - 51 VL - 61 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2596/ DO - 10.5802/aif.2596 LA - en ID - AIF_2011__61_1_1_0 ER -
%0 Journal Article %A Immink, Geertrui Klara %T Accelero-summation of the formal solutions of nonlinear difference equations %J Annales de l'Institut Fourier %D 2011 %P 1-51 %V 61 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2596/ %R 10.5802/aif.2596 %G en %F AIF_2011__61_1_1_0
Immink, Geertrui Klara. Accelero-summation of the formal solutions of nonlinear difference equations. Annales de l'Institut Fourier, Volume 61 (2011) no. 1, pp. 1-51. doi : 10.5802/aif.2596. http://www.numdam.org/articles/10.5802/aif.2596/
[1] 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] Borel transforms and multisums, Revista del Seminario Iberoamericano de Matemáticas, Volume V (1997), pp. 27-44
[3] Multisummability for some classes of difference equations, Ann. Inst. Fourier, Volume 46 (1996) no. 1, pp. 183-217 | DOI | Numdam | MR | Zbl
[4] 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] The acceleration operators and their applications, Proc. Internat. Congr. Math., Kyoto (1990), Vol. 2, Springer-Verlag (1991), pp. 1249-1258 | MR | Zbl
[6] Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Math., Hermann, Paris, 1992 | MR
[7] Cohesive functions and weak accelerations, J. Anal. Math., Volume 60 (1993), pp. 71-97 | MR | Zbl
[8] Asymptotics of analytic difference equations, 1085, Springer Verlag, Berlin, 1984 | MR | Zbl
[9] 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] 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] Existence theorem for nonlinear difference equations, Asymtotic Analysis, Volume 44 (2005), pp. 173 -220 | MR | Zbl
[12] Gevrey type solutions of nonlinear difference equations, Asymtotic Analysis, Volume 50 (2006), pp. 205 -237 | MR | Zbl
[13] 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] Exact asymptotics of nonlinear difference equations with levels 1 and , Ann. Fac. Sci. Toulouse, Volume 17 (2008), pp. 309-356 | DOI | Numdam | MR | Zbl
[15] Sommation des séries divergentes, Expo. Math., Volume 13 (1995), pp. 163-222 | MR | Zbl
[16] Fonctions multisommables, Ann. Inst. Fourier, Volume 41-3 (1991), pp. 1 -16 | MR
[17] The formal classification of linear difference operators, Proceedings Kon. Nederl. Ac. van Wetensch., ser. A, 86 (2) (1983), pp. 249-261 | MR | Zbl
[18] Séries divergentes et théories asymptotiques, Panoramas et synthèses, Volume 121, Paris (1993), pp. 651-684 | MR
[19] 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
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