Accelero-summation of the formal solutions of nonlinear difference equations

Geertrui Klara Immink[1]

  • [1] University of Groningen Faculty of Economics P.O. Box 800 9700 AV Groningen (The Netherlands)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 1, page 1-51
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Immink, Geertrui Klara. "Accelero-summation of the formal solutions of nonlinear difference equations." Annales de l’institut Fourier 61.1 (2011): 1-51. <http://eudml.org/doc/219670>.

@article{Immink2011,
abstract = {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.},
affiliation = {University of Groningen Faculty of Economics P.O. Box 800 9700 AV Groningen (The Netherlands)},
author = {Immink, Geertrui Klara},
journal = {Annales de l’institut Fourier},
keywords = {Nonlinear difference equation; formal solution; accelero-summation; quasi-function; nonlinear difference equation},
language = {eng},
number = {1},
pages = {1-51},
publisher = {Association des Annales de l’institut Fourier},
title = {Accelero-summation of the formal solutions of nonlinear difference equations},
url = {http://eudml.org/doc/219670},
volume = {61},
year = {2011},
}

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
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 1
SP - 1
EP - 51
AB - 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.
LA - eng
KW - Nonlinear difference equation; formal solution; accelero-summation; quasi-function; nonlinear difference equation
UR - http://eudml.org/doc/219670
ER -

References

top
  1. B.L.J. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier 42 (1992), 517-540 Zbl0759.34003MR1182640
  2. B.L.J. Braaksma, Borel transforms and multisums, Revista del Seminario Iberoamericano de Matemáticas V (1997), 27-44 
  3. B.L.J. Braaksma, B.F. Faber, Multisummability for some classes of difference equations, Ann. Inst. Fourier 46 (1996), 183-217 Zbl0837.39001MR1385515
  4. B.L.J. Braaksma, B.F. Faber, G.K. Immink, Summation of formal solutions of a class of linear difference equations, Pacific J. Math. 195 (2000), 35-65 Zbl1008.39003MR1781613
  5. J. Ecalle, The acceleration operators and their applications, Proc. Internat. Congr. Math., Kyoto (1990), Vol. 2 (1991), 1249-1258, Springer-Verlag Zbl0741.30030MR1159310
  6. J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, (1992), Actualités Math., Hermann, Paris MR1399559
  7. J. Ecalle, Cohesive functions and weak accelerations, J. Anal. Math. 60 (1993), 71-97 Zbl0808.30002MR1253230
  8. G.K. Immink, Asymptotics of analytic difference equations, 1085 (1984), Springer Verlag, Berlin Zbl0548.39001MR765699
  9. G.K. Immink, A particular type of summability of divergent power series, with an application to difference equations, Asymptotic Analysis 25 (2001), 123-148 Zbl0981.30002MR1818642
  10. G.K. Immink, Summability of formal solutions of a class of nonlinear difference equations, Journal of Difference Equations and Applications 7 (2001), 105-126 Zbl0973.39003MR1809599
  11. G.K. Immink, Existence theorem for nonlinear difference equations, Asymtotic Analysis 44 (2005), 173-220 Zbl1083.39001MR2176272
  12. G.K. Immink, Gevrey type solutions of nonlinear difference equations, Asymtotic Analysis 50 (2006), 205-237 Zbl1122.39018MR2294599
  13. G.K. Immink, On the Gevrey order of formal solutions of nonlinear difference equations, Journal of Difference Equations and Applications 12 (2006), 769-776 Zbl1106.39001MR2243835
  14. G.K. Immink, Exact asymptotics of nonlinear difference equations with levels 1 and 1 + , Ann. Fac. Sci. Toulouse 17 (2008), 309-356 Zbl1160.39003MR2487857
  15. B. Malgrange, Sommation des séries divergentes, Expo. Math. 13 (1995), 163-222 Zbl0836.40004MR1346201
  16. B. Malgrange, J.-P. Ramis, Fonctions multisommables, Ann. Inst. Fourier 41-3 (1991), 1-16 MR1162566
  17. C. Praagman, The formal classification of linear difference operators, Proceedings Kon. Nederl. Ac. van Wetensch., ser. A, 86 (2) (1983), 249-261 Zbl0519.39003MR705431
  18. J.P. Ramis, Séries divergentes et théories asymptotiques, Panoramas et synthèses 121 (1993), 651-684, ParisSoc. Math. FranceS. M. F., Paris MR1272100
  19. J.P. Ramis, Y. Sibuya, A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier 44 (1994), 811-848 Zbl0812.34005MR1303885

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.