Electron. J. Diff. Equ., Vol. 2010(2010), No. 40, pp. 1-24.

Time averaging for ordinary differential equations and retarded functional differential equations

Mustapha Lakrib, Tewfik Sari

We prove averaging theorems for non-autonomous 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 non-autonomous 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@univ-sba.dz
  Tewfik Sari
Laboratoire de Mathématiques, Informatique et Applications
Universite de Haute Alsace, 4 rue des frères Lumière, 68093 Mulhouse
pl. Viala, 34060 Montpellier, France
email: Tewfik.Sari@uha.fr

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