## 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:

  Aczél, J.,Lectures of functional equations and their applications. Academic Press, New York-London, 1966. · Zbl 0139.09301  Blanton, G. andBaker, J.A.,Iteration groups generated by C n functions. Arch. Math.18 (1982), 121–128. · Zbl 0518.26002  Boruvka, O.,Linear differential transformations of the second order. The English University Press, London, 1971. · Zbl 0222.34002  Boruvka, O.,Sur une classe des groupes continus à un paramètre formés des functions réeles d’une variable. Ann. Polon. Math.42 (1982), 27–37.  Choczewski, B.,On differentiable solutions of a functional equation. Ann. Polon. Math.13 (1963), 133–138. · Zbl 0118.32401  Kuczma. M.,Functional equations in a single variable. Monografie Mat., Tom 46, PWN, Warsaw, 1968. · Zbl 0196.16403  Neuman, F.,On solutions of the vector functional equation y({$$\xi$$}(x))=f(x){$$\cdot$$}A{$$\cdot$$}y(x).Aequationes Math. 16 (1977), 245–257. · Zbl 0375.34013  Neuman, F.,Criterion of global equivalence of linear differential equations. Proc. Roy. Soc. Edinburgh97 A (1984), 217–221. · Zbl 0552.34009  Neuman, F.,Stationary groups of linear differential equations. Czechoslovak Math. J.34(109) (1984) 645–663. · Zbl 0573.34028
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