Exact asymptotics of nonlinear difference equations with levels 1 and 1 +

G.K Immink[1]

  • [1] Faculty of Economics, University of Groningen, P.O. Box 800, 9700 AV Groningen

Annales de la faculté des sciences de Toulouse Mathématiques (2008)

  • Volume: 17, Issue: 2, page 309-356
  • ISSN: 0240-2963

Abstract

top
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.

How to cite

top

Immink, G.K. "Exact asymptotics of nonlinear difference equations with levels $1$ and $1^+$." Annales de la faculté des sciences de Toulouse Mathématiques 17.2 (2008): 309-356. <http://eudml.org/doc/10088>.

@article{Immink2008,
abstract = {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.},
affiliation = {Faculty of Economics, University of Groningen, P.O. Box 800, 9700 AV Groningen},
author = {Immink, G.K},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Gevrey asymptotics; formal power series solutions; nonlinear difference equations; analytic solutions},
language = {eng},
month = {6},
number = {2},
pages = {309-356},
publisher = {Université Paul Sabatier, Toulouse},
title = {Exact asymptotics of nonlinear difference equations with levels $1$ and $1^+$},
url = {http://eudml.org/doc/10088},
volume = {17},
year = {2008},
}

TY - JOUR
AU - Immink, G.K
TI - Exact asymptotics of nonlinear difference equations with levels $1$ and $1^+$
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 2
SP - 309
EP - 356
AB - 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.
LA - eng
KW - Gevrey asymptotics; formal power series solutions; nonlinear difference equations; analytic solutions
UR - http://eudml.org/doc/10088
ER -

References

top
  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). Zbl1008.39003MR1781613
  2. Écalle (J.).— Les Fonctions Résurgentes, tome III, Publ. Math. d’Orsay, Université de Paris-Sud, Paris (1985). Zbl0602.30029
  3. Écalle (J.).— The acceleration operators and their applications, In Proc. Internat. Congr. Math., Kyoto (1990), Vol. 2, pages 1249–1258, Springer-Verlag (1991). Zbl0741.30030MR1159310
  4. Écalle (J.).— Cohesive functions and weak accelerations. Journal d’An. Math., 60:71–97, 1993. Zbl0808.30002
  5. Immink (G.K.).— Asymptotics of analytic difference equations. In Lecture Notes in Mathematics 1085, Berlin, 1984. Springer Verlag. Zbl0548.39001MR765699
  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). Zbl0872.39002MR1433913
  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). Zbl0981.30002MR1818642
  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). Zbl0973.39003MR1809599
  9. Immink (G.K.).— Existence theorem for nonlinear difference equations, Asymptotic Analysis, 44, p.173–220 (2005). Zbl1083.39001MR2176272
  10. Immink (G.K.).— Gevrey type solutions of nonlinear difference equations, Asymptotic Analysis, 50, p. 205–237 (2006). Zbl1122.39018MR2294599
  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). Zbl0519.39003MR705431
  12. Ramis (J.P.).— Séries divergentes et théories asymptotiques, In Panoramas et synthèses, volume 121, pages 651–684. Soc. Math. France, Paris (1993). Zbl0830.34045MR1272100
  13. Wasow (W.).— Asymptotic Expansions for Ordinary Differential Equations, Interscience Publishers, New York (1965). Zbl0133.35301MR203188

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.