Averaging results for functional differential equations

Siberian Mathematical Journal

Volumn 45, Issue 2, 2004, Pages 311-320

  Lakrib, M ^a   Sari, T ^a  

^a NONE

Author keywords

Averaging method; Exponential stability; Functional differential equation; Nonstandard analysis

Indexed keywords

EID: 3843124230     PISSN: 00374466     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:SIMJ.0000021286.49651.c9     Document Type: Article

References (33)

1. Bogoliubov N. N. and Mitropolsky Y. A., Asymptotic Methods in the Theory of Nonlinear Oscillations, New York, Gordon and Breach Science Publ. (1961). (International Monographs on Advanced Mathematics and Physics).
2. Guckenheimer J. and Holmes P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York (1983). (Applied Mathematical Sciences; 42).
3. Guckenheimer J. and Holmes P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York (1983). (Applied Mathematical Sciences; 42).
4. Hale J. K., Ordinary Differential Equations, Wiley-Interscience, New York (1969). (Pure and Applied Mathematics; 21).
5. Hale J. K., Ordinary Differential Equations, Wiley-Interscience, New York (1969). (Pure and Applied Mathematics; 21).
6. Lakrib M., "On the validity of the averaging method for all time," Maghreb Math. Rev., 8, No. 1-2, 119-128 (1999).
7. Lochak P. and Meunier C., Multiphase Averaging for Classical Systems, Springer-Verlag, New York (1988). (Applied Mathematical Sciences; 72).
8. Lochak P. and Meunier C., Multiphase Averaging for Classical Systems, Springer-Verlag, New York (1988). (Applied Mathematical Sciences; 72).
9. Sanders J. A. and Verhulst F., Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York (1985). (Applied Mathematical Sciences; 59).
10. Sanders J. A. and Verhulst F., Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York (1985). (Applied Mathematical Sciences; 59).
11. Sari T., "Stroboscopy and averaging," Colloque Trajectorien à la Mémoire de Georges Reeb et Jean-Louis Callot (Strasbourg-Obernai, 1995) (A. Fruchard and A. Troesch, eds.), Prépublication de l'Institut de Recherche Mathématique Avancée, Université Louis Pasteur, Strasbourg, 1995, pp. 95-124.
12. Halanay A., "The method of averaging in equations with retardation," Rev. Math. Pur. Appl. Acad. R.P.R., 4, 467-483 (1959).
13. Halanay A., "On the method of averaging for differential equations with retarded argument," J. Math. Anal. Appl., 14, 70-76 (1966).
14. Hale J. K., "Averaging methods for differential equations with retarded arguments and a small parameter," J. Differential Equations, 2, 57-73 (1966).
15. Hale J. K. and Verduyn Lunel S.M., "Averaging in infinite dimensions," J. Integral Equations Appl., 2, No. 4, 463-494 (1990).
16. Medvedev G. N., "Asymptotic solutions of some systems of differential equations with deviating argument," Soviet Math. Dokl., 9, 85-87 (1968).
17. Volosov V. M., Medvedev G. N., and Morgunov B. I., "On the applications of the averaging method for certain systems of differential equations with delay," Vestnik Moskov. Univ. Ser. III Fiz. Astronom., 8, 251-294 (1968).
18. Lakrib M., "The method of averaging and functional differential equations with delay," Internat. J. Math. Math. Sci., 26, No. 8, 497-511 (2001).
19. Robinson A., Nonstandard Analysis, American Elsevier, New York (1974).
20. Nelson E., "Internal set theory: a new approach to nonstandard analysis," Bull. Amer. Math. Soc., 83, No. 6, 1165-1198 (1977).
21. Diener F., Diener M. (eds.) Nonstandard Analysis in Practice, Springer-Verlag, Berlin (1995). (Universitext).
22. Diener F. and Reeb G., Analyse non Standard, Hermann, Paris (1989). (Collection Enseignement des Sciences; 40).
23. Diener F. and Reeb G., Analyse non Standard, Hermann, Paris (1989). (Collection Enseignement des Sciences; 40).
24. Diener M., Lobry C. (eds.) Analyse non Standard et Représentation du Réel, Actes de l'école d'été, Office des Publications Universitaires Alger (1985).
25. Diener M., Wallet G. (eds.) Mathématiques Finitaires & Analyse non Standard. T. 1 and 2, Publication Mathématiques de l'Université Paris VII, Vol. 31, Paris (1989).
26. Lutz R. and Goze M., Nonstandard Analysis: A Practical Guide with Applications, Springer-Verlag, New York (1981). (Lecture Notes in Mathematics; 881).
27. Lutz R. and Goze M., Nonstandard Analysis: A Practical Guide with Applications, Springer-Verlag, New York (1981). (Lecture Notes in Mathematics; 881).
28. Lutz R. and Sari T. "Applications of nonstandard analysis to boundary value problems in singular perturbation theory," in: Theory and Applications of Singularly Perturbations (Oberwolfach, 1981), Springer-Verlag, Berlin, 1982, pp. 113-135. (Lecture Notes in Mathematics; 942).
29. Lutz R. and Sari T. "Applications of nonstandard analysis to boundary value problems in singular perturbation theory," in: Theory and Applications of Singularly Perturbations (Oberwolfach, 1981), Springer-Verlag, Berlin, 1982, pp. 113-135. (Lecture Notes in Mathematics; 942).
30. Sari T., "Nonstandard perturbation theory of differential equations," Invited Talk at the International Research Symposium on Nonstandard Analysis and Its Applications, ICMS, Edinburgh, 11-17 August 1996, http://www.math.uha.fr/~geometry/sari/papers.html.
31. van den Berg I., Nonstandard Asymptotic Analysis, Springer, Berlin (1987). (Lecture Notes in Mathematics; 1249).
32. van den Berg I., Nonstandard Asymptotic Analysis, Springer, Berlin (1987). (Lecture Notes in Mathematics; 1249).
33. Lakrib M., "Time averaging for functional differential equations," J. Appl. Math., 1, 1-16 (2003).