In this Note, we obtain two new results for existence of periodic solutions for differential equations perturbed by a small parameter. The first one is based on a new fixed point theorem previously obtained by the authors. The second one is based on study of suitable linearized equations. Our approach deals with degree theory and nonsmooth analysis.
Dans cette Note, nous donnons deux résultats nouveaux sur l'existence de solutions périodiques pour des équations différentielles perturbées par un petit paramètre. Le premier résultat est lié à un nouveau théorème de point fixe précédemment obtenu par les auteurs ; le second est déduit de l'étude d'une équation différentielle linéarisée. Cette approche relève de la théorie du degré topologique et de l'analyse non régulière.
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@article{CRMATH_2009__347_7-8_369_0, author = {Gudovich, Anastasia and Kamenskii, Mikhail and Quincampoix, Marc}, title = {Existence of periodic solutions of a ordinary differential equation perturbed by a small parameter: {An} averaging approach}, journal = {Comptes Rendus. Math\'ematique}, pages = {369--374}, publisher = {Elsevier}, volume = {347}, number = {7-8}, year = {2009}, doi = {10.1016/j.crma.2009.02.012}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.02.012/} }
TY - JOUR AU - Gudovich, Anastasia AU - Kamenskii, Mikhail AU - Quincampoix, Marc TI - Existence of periodic solutions of a ordinary differential equation perturbed by a small parameter: An averaging approach JO - Comptes Rendus. Mathématique PY - 2009 SP - 369 EP - 374 VL - 347 IS - 7-8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.02.012/ DO - 10.1016/j.crma.2009.02.012 LA - en ID - CRMATH_2009__347_7-8_369_0 ER -
%0 Journal Article %A Gudovich, Anastasia %A Kamenskii, Mikhail %A Quincampoix, Marc %T Existence of periodic solutions of a ordinary differential equation perturbed by a small parameter: An averaging approach %J Comptes Rendus. Mathématique %D 2009 %P 369-374 %V 347 %N 7-8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.02.012/ %R 10.1016/j.crma.2009.02.012 %G en %F CRMATH_2009__347_7-8_369_0
Gudovich, Anastasia; Kamenskii, Mikhail; Quincampoix, Marc. Existence of periodic solutions of a ordinary differential equation perturbed by a small parameter: An averaging approach. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 369-374. doi : 10.1016/j.crma.2009.02.012. http://www.numdam.org/articles/10.1016/j.crma.2009.02.012/
[1] On the second theorem of N.N. Bogoljubov in the averaging principle for functional-differential equations of neutral type, Differ. Uravn., Volume 10 (1974), pp. 537-540 (in Russian)
[2] Asymptotic Methods in the Theory of Non-Linear Oscillations, Gosudarstv. Izdat. Fiz-Mat. Lit., Moscow, 1963 (in Russian)
[3] Necessary and sufficient conditions for the existence of (generalized) equilibria on a compact epilipschitzian domain, Comm. Appl. Nonlinear Anal., Volume 7 (2000), pp. 21-53
[4] Euler characteristic and fixed-point theorems, Positivity, Volume 6 (2002) no. 3, pp. 243-260
[5] On existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite-dimensional space, J. Differential Equations, Volume 185 (2002) no. 2, pp. 483-512
[6] De l'existence de solutions d'équations différentielles assujetties à rester dans un ensemble fermé, C. R. Acad. Sci. Paris, Ser. I., Volume 333 (2001), pp. 913-918
[7] A new approach to the theory of ordinary differential equations with small parameter, Dokl. Akad. Nauk, Volume 388 (2003) no. 4, pp. 439-442
[8] Small parameter perturbations of nonlinear periodic systems, Nonlinearity, Volume 17 (2004), pp. 193-205
[9] Existence of fixed points on compact epilipschitz sets without invariance conditions, Fixed Point Theory Appl., Volume 3 (2005), pp. 267-279
[10] J. Mawhin, Periodic solutions of weakly nonlinear differential systems, in: Proceedings of the Fifth International Conference on Nonlinear Vibrations, Kiev, 1969
[11] Équations différentielles ordinaires, tome 2: stabilité et solutions périodiques, Masson et Cie, Paris, 1973 (266 p)
[12] Periodic and stationary trajectories of flows and ordinary differential equations, Univ. Iagel., Volume 27 (1988), pp. 29-37
[13] Periodic solutions of differential inclusions on compact subsets of , J. Math. Anal. Appl., Volume 148 (1990) no. 1, pp. 202-212
[14] Clarke's tangent cones and the boundaries of closed sets in , Nonlinear Anal., Volume 3 (1979), pp. 145-154
[15] Vibrational control of singularly perturbed systems, Lectures Notes in Control and Information Science, vol. 259, 2001, pp. 397-408
[16] Algebraic Topology, McGraw–Hill, 1966
[17] A theorem concerning the existence of periodic solutions of systems of differential equations with delayed arguments, Mat. Zametki, Volume 8 (1970) no. 2, pp. 229-234 (in Russian)
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