Electron. J. Diff. Equ.,
Vol. 2010(2010), No. 40, pp. 124.
Time averaging for ordinary differential equations and retarded
functional differential equations
Mustapha Lakrib, Tewfik Sari
Abstract:
We prove averaging theorems for nonautonomous ordinary
differential equations and retarded functional differential equations
in the case where the vector fields are continuous in the spatial
variable uniformly with respect to the time and the solution of
the averaged system exists on some given interval.
Our assumptions are weaker than those required in the results of
the existing literature. Usually, we require that the nonautonomous
differential equation and the autonomous averaged equation
are locally Lipschitz and that the solutions of both equations
exist on some given interval. Our results are formulated in classical
mathematics. Their proofs use the stroboscopic method which is
a tool of the nonstandard asymptotic theory of differential equations.
Submitted September 15, 2009. Published March 19, 2010.
Math Subject Classifications: 34C29, 34C15, 34K25, 34E10, 34E18.
Key Words: Averaging; ordinary differential equations; stroboscopic method;
retarded functional differential equations; nonstandard analysis.
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Mustapha Lakrib
Laboratoire de Mathématiques, Université Djillali Liabès
B.P. 89, 22000 Sidi Bel Abbès, Algérie
email: m.lakrib@univsba.dz 

Tewfik Sari
Laboratoire de Mathématiques, Informatique et Applications
Universite de Haute Alsace, 4 rue des frères Lumière, 68093
Mulhouse
and EPI MERE (INRIAINRA), UMR MISTEA, INRA 2
pl. Viala, 34060 Montpellier, France
email: Tewfik.Sari@uha.fr 
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