Comptes Rendus
no. 7-8
Ordinary Differential Equations
Existence of periodic solutions of a ordinary differential equation perturbed by a small parameter: An averaging approach
[Sur l'existence de solutions périodiques pour une équation différentielle perturbée par un petit paramètre : une approche par moyennisation]

Présenté par : Haïm Brezis

Anastasia Gudovich1, 2 ; Mikhail Kamenskii3 ; Marc Quincampoix2
1 Department of Applied Mathematics and Mechanics, Voronezh State University, 1 Universitetskaya pl., 394006 Voronezh, Russia
2 Laboratoire de mathématiques, UMR CNRS 6205, Université de Bretagne Occidentale, 6, avenue Victor-Le-Gorgeu, 29200 Brest, France
3 Department of Mathematics, Voronezh State University, 1 Universitetskaya pl., 394006 Voronezh, Russia
Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 369-374.

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.

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.

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Accepté le :
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DOI : 10.1016/j.crma.2009.02.012
Anastasia Gudovich 1, 2 ; Mikhail Kamenskii 3 ; Marc Quincampoix 2

1 Department of Applied Mathematics and Mechanics, Voronezh State University, 1 Universitetskaya pl., 394006 Voronezh, Russia
2 Laboratoire de mathématiques, UMR CNRS 6205, Université de Bretagne Occidentale, 6, avenue Victor-Le-Gorgeu, 29200 Brest, France
3 Department of Mathematics, Voronezh State University, 1 Universitetskaya pl., 394006 Voronezh, Russia 
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Anastasia Gudovich; Mikhail Kamenskii; Marc Quincampoix. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.012/

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