Asymptotic Solutions of nonlinear difference equations

I. P. van den Berg[1]

  • [1] Departamento de Matemática, Universidade de Évora. Colégio Luis António Verney, 7000-671 Évora, Portugal.

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

  • Volume: 17, Issue: 3, page 635-660
  • ISSN: 0240-2963

Abstract

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We study the asymptotics of first-order nonlinear difference equations. In particular we present an asymptotic functional equation for potential asymptotic behaviour, and a theorem stating sufficient conditions for the existence of an actual solution with such asymptotic behaviour.

How to cite

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van den Berg, I. P.. "Asymptotic Solutions of nonlinear difference equations." Annales de la faculté des sciences de Toulouse Mathématiques 17.3 (2008): 635-660. <http://eudml.org/doc/10099>.

@article{vandenBerg2008,
abstract = {We study the asymptotics of first-order nonlinear difference equations. In particular we present an asymptotic functional equation for potential asymptotic behaviour, and a theorem stating sufficient conditions for the existence of an actual solution with such asymptotic behaviour.},
affiliation = {Departamento de Matemática, Universidade de Évora. Colégio Luis António Verney, 7000-671 Évora, Portugal.},
author = {van den Berg, I. P.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {asymptotics; stability; rivers; nonstandard analysis; change of scale; nonlinear difference equations},
language = {eng},
month = {6},
number = {3},
pages = {635-660},
publisher = {Université Paul Sabatier, Toulouse},
title = {Asymptotic Solutions of nonlinear difference equations},
url = {http://eudml.org/doc/10099},
volume = {17},
year = {2008},
}

TY - JOUR
AU - van den Berg, I. P.
TI - Asymptotic Solutions of nonlinear difference equations
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 3
SP - 635
EP - 660
AB - We study the asymptotics of first-order nonlinear difference equations. In particular we present an asymptotic functional equation for potential asymptotic behaviour, and a theorem stating sufficient conditions for the existence of an actual solution with such asymptotic behaviour.
LA - eng
KW - asymptotics; stability; rivers; nonstandard analysis; change of scale; nonlinear difference equations
UR - http://eudml.org/doc/10099
ER -

References

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  1. Artigue (M.), Gautheron (V.), Isambert (E.).— Une notion non standard d’attracteurs : les fleuves. Mathématiques finitaires Analyse non standard (Diener, M. and Wallet, G. eds.), Tome 2, Publ. Math. Univ. Paris VII, p. 191-208 (1989). Zbl0684.58021MR1028798
  2. Benoît (E.), Callot (J.-L.), Diener (F.), Diener (M.) .— Chasse au canard, Collect. Math. 32, no. 1-4, p. 37-119 (1981). Zbl0529.34046MR643399
  3. Benzaid (Z.), Lutz (D. A.) .— Asymptotic representation of solutions of perturbed systems of linear difference equations. Stud. Appl. Math. 77, no. 3, p. 195-221 (1987). Zbl0628.39002MR1002291
  4. Van den Berg (I. P.) .— On solutions of polynomial growth of ordinary differential equations J. Differential Equations, 81 (2), p. 368-402 (1989). Zbl0699.34059MR1016088
  5. Van den Berg (I. P.) .— Macroscopic rivers. In : Dynamic bifurcations (Benoît, E. ed.), Lect. N. Math. 1493, p. 190-209, Springer (1991). Zbl0875.34016MR1167005
  6. Van den Berg (I. P.) .— Extended use of IST, Ann. Pure Appl. Logic 58 (1992). Zbl0777.03019MR1169787
  7. Van den Berg (I. P.) .— Asymptotics of families of solutions of nonlinear difference equations, Logic and Analysis 2, p. 153-185 (2008). Zbl1170.39006MR2403502
  8. Diener (F.), Diener (M.) (eds.) .— Nonstandard analysis in practice. Universitext, Springer (1995). Zbl0848.26015MR1396794
  9. Diener (M.) .— Détermination et existence des fleuves en dimension deux, C. R. Acad. Sci. Paris Sér. I Math. 301, no. 20, p. 899-902 (1985). Zbl0582.34066MR829057
  10. Diener (M.), Reeb (G.) .— Champs polynômiaux : nouvelles trajectoires remarquables, Bull. Soc. Math. Belg. Sér. A 38, p. 131-150 (1987). Zbl0681.34034MR885526
  11. Elaydi (S.) .— An introduction to difference equations. 3 rd ed., Springer (2005). Zbl1071.39001MR2128146
  12. Fruchard (A.), Schäfke (R.) .— Analytic solutions of difference equations with small step size. In memory of W. A. Harris, J. Differ. Equations Appl. 7, no. 5, p. 651-684 (2001). Zbl1006.39003MR1871573
  13. Harris (W. A. Jr.), Sibuya (Y.) .— Asymptotic solutions of systems of nonlinear difference equations. Arch. Rational Mech. Anal. 15, p. 377-395 (1964). Zbl0122.09704MR159148
  14. Harris (W. A. Jr.), Sibuya (Y.) .— On asymptotic solutions of systems of nonlinear difference equations. J. Reine Angew. Math. 222, p. 120-135 (1966). Zbl0142.05703MR188653
  15. Immink (G. K.).— Asymptotics of analytic difference equations. Springer Lect. N. Math. 1085 (1984). Zbl0548.39001MR765699
  16. Lakshmikantham (V.), Trigiante (D.) .— Theory of difference equations : numerical methods and applications. Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 251. Marcel Dekker, Inc., New York (2002). Zbl1014.39001MR1988945
  17. Nelson (E.) .— Internal Set Theory, Bull. Amer. Math. Soc. 83, no. 6, p. 1165-1198 (1977). Zbl0373.02040MR469763
  18. Nörlund (N. E.) .— Sur les équations linéaires aux différences finies à coefficients rationnels. Acta Math. 40, no. 1, p. 191-249 (1916). Zbl46.0703.06MR1555137
  19. Nörlund (N. E.) .— Mémoire sur le calcul aux différences finies. Acta Math. 44, no. 1, p. 71-212 (1923). Zbl49.0322.02MR1555184
  20. Poincaré (R.) .— Sur les équations linéaires aux différences ordinaires et aux différences finies, American Journal of Mathematics 7, 203-258 (1885). Zbl17.0290.01
  21. Robinson (A.) .— Non-standard analysis. 3 rd ed., Princ. Univ. Press (1996). Zbl0843.26012MR1373196

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