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Title: Equation $f(p(x)) = q(f(x))$ for given real functions $p$, $q$ (English)
Author: Kopeček, Oldřich
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 62
Issue: 4
Year: 2012
Pages: 1011-1032
Summary lang: English
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Category: math
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Summary: We investigate functional equations $f(p(x)) = q(f(x))$ where $p$ and $q$ are given real functions defined on the set ${\Bbb R}$ of all real numbers. For these investigations, we can use methods for constructions of homomorphisms of mono-unary algebras. Our considerations will be confined to functions $p, q$ which are strictly increasing and continuous on ${\Bbb R}$. In this case, there is a simple characterization for the existence of a solution of the above equation. First, we give such a characterization. Further, we present a construction of any solution of this equation if some exists. This construction is demonstrated in detail and discussed by means of an example. (English)
Keyword: homomorphism of mono-unary algebras
Keyword: functional equation
Keyword: strictly increasing continuous real functions
MSC: 08A60
MSC: 65Q20
MSC: 97I70
idZBL: Zbl 1274.08022
idMR: MR3010254
DOI: 10.1007/s10587-012-0061-2
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Date available: 2012-11-10T21:39:14Z
Last updated: 2020-07-03
Stable URL: http://hdl.handle.net/10338.dmlcz/143042
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