A vector functional equation and linear differential equations.(English)Zbl 0593.39006

Given an open real interval I, an integer $$n\geq 2$$ and a function $$y:I\to {\mathbb{R}}^ n$$ of the class $$C^{2n-1}$$ with Wroński determinant different from zero on I, consider the set $$G_ 0(y)$$ of all diffeomorphisms h of the class $$C^ n$$ mapping I onto itself and such that there exists a regular n by n matrix A and a nonvanishing function f:I$$\to {\mathbb{R}}$$ of the class $$C^ n$$ such that $$y(x)=Af(x)y[h(x)]$$ for all $$x\in I$$. This set $$G_ 0(y)$$ with the composition rule forms a group which is $$C^ n$$-conjugate to a (topologically) closed subgroup of the fundamental group F. (Here by a subgroup of F the author means a subset of F or a subset of restrictions of functions from F to an interval, that is a group with respect to the composition.) The maximal case where $$G_ 0(y)$$ is $$C^ n$$-conjugate to F occurs exactly when $$y(x)=D\beta (x)([\sin \gamma (x)]^{n-1},[\sin \gamma (x)]^{n-2}\cdot \cos \gamma (x),...,[\cos \gamma (x)]^{n-1})$$ for every $$x\in I$$, where D is a regular n by n matrix, $$\beta$$ :I$$\to {\mathbb{R}}$$ is a nonvanishing function of the class $$C^ n$$, and $$\gamma$$ is a diffeomorphism of the class $$C^ n$$ mapping I onto $${\mathbb{R}}$$.
Reviewer: K.Baron

MSC:

 39B62 Functional inequalities, including subadditivity, convexity, etc. 34A30 Linear ordinary differential equations and systems 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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References:

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