Comptes Rendus
no. 2
Dynamical Systems/Ordinary Differential Equations
A Note on non-autonomous scalar functional differential equations with small delay
[Une Note sur les équations fonctionnelles scalaires non-autonomes à petit retard]

Présenté par : Étienne Ghys

Ana I. Alonso1 ; Rafael Obaya1 ; Ana M. Sanz2
1 Departamento de Matemática Aplicada, E.T.S. de Ingenieros Industriales, Paseo del Cauce s/n, 47011 Valladolid, Spain
2 Departamento de Análisis Matemático y Didáctica de la Matemática, Facultad de Ciencias, Prado de la Magdalena s/n, 47005 Valladolid, Spain
Comptes Rendus. Mathématique, Volume 340 (2005) no. 2, pp. 155-160.

Dans cette Note on montre que les ensembles minimaux pour les semiflots engendrés par les solutions des équations fonctionnelles non-autonomes à petit retard sont des extensions presque automorphes de la base. Ce résultat n'est plus vrai pour un retard arbitraire. C'est la condition sur le retard qui garantit l'existence de solutions dites solutions spéciales. Ces solutions-ci nous permettent de considérer notre problème au moyen d'un autre plus facile relatif aux équations différentielles ordinaires.

We prove that the minimal sets in the skew-product semiflows generated from a non-autonomous scalar functional differential equation with a small delay are all almost automorphic extensions of the base. This result is not true for arbitrary delay equations. The point is that, for a small delay, so-called special solutions exist and permit us to tackle the problem by means of some related scalar ODE's for which the study is much simpler.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.11.027
Ana I. Alonso 1 ; Rafael Obaya 1 ; Ana M. Sanz 2

1 Departamento de Matemática Aplicada, E.T.S. de Ingenieros Industriales, Paseo del Cauce s/n, 47011 Valladolid, Spain
2 Departamento de Análisis Matemático y Didáctica de la Matemática, Facultad de Ciencias, Prado de la Magdalena s/n, 47005 Valladolid, Spain 
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Ana I. Alonso; Rafael Obaya; Ana M. Sanz. A Note on non-autonomous scalar functional differential equations with small delay. Comptes Rendus. Mathématique, Volume 340 (2005) no. 2, pp. 155-160. doi : 10.1016/j.crma.2004.11.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.11.027/

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