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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:

[1] Aczél, J.,Lectures of functional equations and their applications. Academic Press, New York-London, 1966. · Zbl 0139.09301
[2] Blanton, G. andBaker, J.A.,Iteration groups generated by C n functions. Arch. Math.18 (1982), 121–128. · Zbl 0518.26002
[3] Boruvka, O.,Linear differential transformations of the second order. The English University Press, London, 1971. · Zbl 0222.34002
[4] 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.
[5] Choczewski, B.,On differentiable solutions of a functional equation. Ann. Polon. Math.13 (1963), 133–138. · Zbl 0118.32401
[6] Kuczma. M.,Functional equations in a single variable. Monografie Mat., Tom 46, PWN, Warsaw, 1968. · Zbl 0196.16403
[7] 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
[8] Neuman, F.,Criterion of global equivalence of linear differential equations. Proc. Roy. Soc. Edinburgh97 A (1984), 217–221. · Zbl 0552.34009
[9] Neuman, F.,Stationary groups of linear differential equations. Czechoslovak Math. J.34(109) (1984) 645–663. · Zbl 0573.34028
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