Averaging Theorems for Ordinary Differential Equations and Retarded Functional Differential Equations

Mustapha Lakrib (1) , Tewfik Sari (2)

1 UDL - جامعة جيلالي اليابس [سيدي بلعباس، الجزائر] = Université Djillali Liabès [Sidi Bel Abbès, Algérie] = Djillali Liabes University [Sidi Bel Abbès, Algeria]
2 LMIA - Laboratoire de Mathématiques Informatique et Applications [UHA]

We prove averaging theorems for ordinary differential equations and retarded functional differential equations. 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 interval. We extend this result to the case of vector fields which are continuous in the spatial variable uniformly with respect to time and without any assumption on the interval of existence of the solutions of the nonautonmous differential equation. Our results are formulated in classical mathematics. Their proofs use nonstandard analysis.

Domaines: Systèmes dynamiques [math.DS]

Mustapha Lakrib, Tewfik Sari. Averaging Theorems for Ordinary Differential Equations and Retarded Functional Differential Equations. 2006. ⟨hal-00115539⟩