## 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

Lie brackets

Zbl 0616.34004
Full Text:

### 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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