Kurzweil, Jaroslav; Jarník, Jiří Iterated Lie brackets in limit processes in ordinary differential equations. (English) Zbl 0663.34043 Result. Math. 14, No. 1-2, 125-137 (1988). 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 Cited in 1 ReviewCited in 11 Documents MSC: 34C29 Averaging method for ordinary differential equations Keywords:Lie brackets Citations:Zbl 0616.34004 PDF BibTeX XML Cite \textit{J. Kurzweil} and \textit{J. Jarník}, Result. Math. 14, No. 1--2, 125--137 (1988; Zbl 0663.34043) Full Text: DOI OpenURL 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.