Functional equations on finite groups of substitutions
type: Item
instance of: scholarly article
title: Functional equations on finite groups of substitutions
zbMATH Open document ID: 1261.39025
author: Mihály Bessenyei
author: Gábor Horváth
author: Csaba G. Kézi
published in: Expositiones Mathematicae
publication date: 2012-11-27T00:00:00Z
review text: Motivated by Babbage, the authors study the functional equation \[ F(f\circ g_1(t), \dots, f\circ g_r(t), t) =0, \tag{1} \] where the functions \(g_1, \dots, g_r\) and \(F\) are given and \(f\) is to be determined. However, under such general circumstances, there is no hope to give the set of solutions. As Babbage illustrates it via several examples, it may occur that, having a particular solution, infinitely many functions \(f\) can be created satisfying the functional equation (1), although the family of such functions may not be the complete collection of all solutions, see [\textit{C. G. Small}, Functional equations and how to solve them. Problem Books in Mathematics. New York, NY: Springer (2007; Zbl 1109.39022)]. The problem becomes more tractable if the functions \(g_1, \dots, g_r\) form a group under composition on their domain. It turns out that the algebraic structure of a stabilizer determines the number of initial value conditions for the functional equation. The first systematic investigations in this direction are due to \textit{L. Presić} [Publ. Fac. Electrotech. Univ. Belgrade, Ser. Math. Phys. 115--121, 21--28 (1963; Zbl 0166.41202)], who characterized all solutions of (1) when \(F\) is linear and does not depend on its last variable. The authors give examples and the method given in the examples is applied in more general settings. Composing both sides of (1) by the elements of the group, a system of nonlinear equations is obtained where \(\Phi_i = f\circ g_i\) play the role of unknowns. Then, under additional regularity assumptions, the implicit function theorem offers a possible approach to provide a solution. Such an approach was presented by \textit{M. Bessenyei} [Am. Math. Mon. 117, No. 10, 921--927 (2010; Zbl 1208.39032)] under the assumptions that the group is an abelian one and its elements have a common fixed point in the interior of the domain. But the paper concerns only a special case of (1). The general form of (1) was investigated by …
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 6110674
zbMATH Keywords: functional equation of single variable
zbMATH Keywords: implicit function
zbMATH Keywords: Cauchy problem
zbMATH Keywords: finite group
zbMATH Keywords: coset of finite function groups
reviewed by: Choonkil Park
MaRDI profile type: Publication
full work available at URL: j.exmath.2012.03.004
OpenAlex ID: W2008202673
cites work: Q5514010
cites work: Q3708715
cites work: Functional Equations and Finite Groups of Substitutions
cites work: Functional equations and group substitutions
cites work: Ordinary Differential Equations with Applications
cites work: Q4882944
cites work: Q3887378
cites work: Representations of permutation groups. Part I
cites work: Q5439382
cites work: Q4040882
cites work: A Uniqueness Theorem for y � = f(x, y), y(x 0 ) = y 0
cites work: Q5552516
cites work: Q5552514
cites work: Q5526189
cites work: Functional equations and how to solve them
cites work: Uniqueness of Solutions of Ordinary Differential Equations
DOI: 10.1016/J.EXMATH.2012.03.004
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description: scientific article; zbMATH DE number 6110674