Comptes Rendus
no. 2
Dynamical Systems/Ordinary Differential Equations
A Note on non-autonomous scalar functional differential equations with small delay

Presented by: 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.

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.

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.

Received:
Accepted:
Published online:
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 
@article{CRMATH_2005__340_2_155_0,
     author = {Ana I. Alonso and Rafael Obaya and Ana M. Sanz},
     title = {A {Note} on non-autonomous scalar functional differential equations with small delay},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {155--160},
     publisher = {Elsevier},
     volume = {340},
     number = {2},
     year = {2005},
     doi = {10.1016/j.crma.2004.11.027},
     language = {en},
}
TY  - JOUR
AU  - Ana I. Alonso
AU  - Rafael Obaya
AU  - Ana M. Sanz
TI  - A Note on non-autonomous scalar functional differential equations with small delay
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 155
EP  - 160
VL  - 340
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2004.11.027
LA  - en
ID  - CRMATH_2005__340_2_155_0
ER  - 
%0 Journal Article
%A Ana I. Alonso
%A Rafael Obaya
%A Ana M. Sanz
%T A Note on non-autonomous scalar functional differential equations with small delay
%J Comptes Rendus. Mathématique
%D 2005
%P 155-160
%V 340
%N 2
%I Elsevier
%R 10.1016/j.crma.2004.11.027
%G en
%F CRMATH_2005__340_2_155_0
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/

[1] A.I. Alonso; R. Obaya The structure of the bounded trajectories set of a scalar convex differential equation, Proc. Roy. Soc. Edinburgh Sect. A, Volume 133 (2003), pp. 237-263

[2] R.D. Driver Linear differential systems with small delays, J. Differential Equations, Volume 21 (1976), pp. 149-167

[3] H. Furstenberg; B. Weiss On almost 1–1 extensions, Israel J. Math., Volume 65 (1989) no. 3, pp. 311-322

[4] R. Johnson A linear, almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc., Volume 82 (1981) no. 2, pp. 199-205

[5] R. Johnson On almost-periodic linear differential systems of Millionščikov and Vinograd, J. Math. Anal. Appl., Volume 85 (1982), pp. 452-460

[6] T. Krisztin; H.-O. Walther; J. Wu Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Positive Feedback, Fields Institute Monograph Series, vol. 11, Amer. Math. Soc., Providence, RI, 1999

[7] S. Novo; R. Obaya; A.M. Sanz Attractor minimal sets for cooperative and strongly convex delay differential systems, J. Differential Equations, Volume 208 (2005), pp. 86-123

[8] S. Novo, R. Obaya, A.M. Sanz, Almost periodic and almost automorphic dynamics for scalar convex differential equations, Israel J. Math., in press

[9] M. Pituk Convergence to equilibria in scalar nonquasimonotone functional differential equations, J. Differential Equations, Volume 193 (2003), pp. 95-130

[10] W. Shen; Y. Yi Almost Automorphic and Almost Periodic Dynamics in Skew-Products Semiflows, Mem. Amer. Math. Soc., vol. 647, Amer. Math. Soc., Providence, RI, 1998

[11] V.V. Zhikov; B.M. Levitan Favard theory, Russian Math. Surveys, Volume 32 (1977), pp. 129-180

Cited by Sources:

Comments - Policy