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Iterated Lie brackets in limit processes in ordinary differential equations. (English) Zbl 0663.34043

A rather general form of the averaging principle [which was announced by the authors in Z. Angew. Math. Phys. 38, 241-256 (1987; Zbl 0616.34004)] is proved. Let m be an integer, \(m>1\), \(\phi_ i: R\to R\), \(i=1,2,...,r\) locally integrable. Denote by \(\int \phi_ i\) a primitive of \(\phi_ i\), by\(\int \phi_ j\int \phi_ i\) a primitive of \(\phi_ j\int \phi_ i\), by \(\int \phi_ k\int \phi_ j\int \phi_ i\) a primitive of \(\phi_ k\int \phi_ j\int \phi_ i\) etc. Let \(f_ j: R^ n\to R^ n\) be of class \(C^{(m)}\), \([f_ j,f_ k]\) denoting the Lie brackets. Theorem: Let \(\phi_ j\) fulfill the following conditions: \((1)\quad \int^{s+1}_{s}| \phi_ i| dt\leq C\) for \(s\in R\), \(i=1,2,...,r\), (2) if \(j<m\), \(i_ 1,i_ 2,...,i_ j\in \{1,2,...,r\}\), then the mean value of \(\phi_{i_ 1}\int \phi_{i_ 2}...\int \phi_{i_ j}\) is zero (i.e. the corresponding primitive is bounded), (3) if \(i_ 1,...,i_ m\in \{1,2,...,r\}\), then the mean value of \(\phi_{i_ 1}\int \phi_{i_ 2}...\int \phi_{i_ m}\) is \(\lambda_{i_ 1i_ 2...i_ m}\in R\). Then the solutions of \[ (4)\quad \dot x=f_ 0(x)+\epsilon^{-\alpha}\sum^{r}_{i=1}f_ i(x)\phi_ i(\frac{t}{\epsilon}),\quad x(s)=y \] tend for \(\epsilon\) \(\to 0\) to the solution of (5) \(\dot x=f_ 0(x)\), \(x(s)=y\) in case that \(\alpha <(m-1)/m\) and to the solution of \[ (6)\quad \dot x=f_ 0(x)+\frac{(-1)^{m-1}}{m}\sum^{r}_{i_ 1,...,i_ m=1}[...[f_{i_ 1},f_{i_ 2}]...f_{i_ m}]\lambda_{i_ 1i_ 2...i_ m}, \] x(s)\(=y\) in case that \(\alpha =(m-1)/m\).
Reviewer: J.Kurzweil

MSC:

34C29 Averaging method for ordinary differential equations

Keywords:

Lie brackets

Citations:

Zbl 0616.34004
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References:

[1] J. Kurzweil and J. Jarník: A convergence effect in ordinary differential equations. In: Asymptotic Methods of the Mathematical Physics. Dedicated to the 70th birthday of Acad. Ju. A. Mitropolskij. Kijev, to appear
[2] J. Kurzweil and J. Jarník: Limit processes in ordinary differential equations. Journal of Applied Math, and Physics (ZAMP), 38 0987), 241–256. · Zbl 0616.34004
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