The method of averaging and functional differential equations with delay

Mustapha Lakrib

doi:10.1155/s0161171201010407

Abstract

We present a natural extension of the method of averaging to fast oscillating functional differential equations with delay. Unlike the usual approach where the analysis is kept in an infinite‐dimensional Banach space, our analysis is achieved in ℝn. Our results are formulated in classical mathematics. They are proved within Internal Set Theory which is an axiomatic description of nonstandard analysis.

Highlights

An important tool in the rigorous study of differential equations with a small parameter is the method of averaging, which is well known for ordinary differential equations [1, 9, 13, 14] and for functional differential equations with small delay [6, 7, 17]
We develop an improved theory of averaging for functional differential equations with delay under smoothness hypotheses that are less restrictive than those of [8]
By the uniform attractiveness of ye (see Lemma 3.10), through the transformation t −t, the solution of (2.2) with the initial function z1(−t; r , x, t2) for t ∈ [−r − t1,2, −t1,2] which coincides with z1(·; r , x, t2) (by uniqueness; hypothesis (H.3)) is defined for all t > −t1,2 and satisfies z1(−t; r , x, t2) ye for t + t1,2 +∞

Summary

Introduction

An important tool in the rigorous study of differential equations with a small parameter is the method of averaging, which is well known for ordinary differential equations [1, 9, 13, 14] and for functional differential equations with small delay [6, 7, 17]. Let E and F be standard metric spaces, and g be an internal function defined on Ᏸ(g) ⊂ E and with values in F .

Results

Conclusion