Equation f ( p ( x ) ) = q ( f ( x ) ) for given real functions p , q

Oldřich Kopeček

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 4, page 1011-1032
  • ISSN: 0011-4642

Abstract

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We investigate functional equations f ( p ( x ) ) = q ( f ( x ) ) where p and q are given real functions defined on the set 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 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.

How to cite

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Kopeček, Oldřich. "Equation $f(p(x)) = q(f(x))$ for given real functions $p$, $q$." Czechoslovak Mathematical Journal 62.4 (2012): 1011-1032. <http://eudml.org/doc/247138>.

@article{Kopeček2012,
abstract = {We investigate functional equations $f(p(x)) = q(f(x))$ where $p$ and $q$ are given real functions defined on the set $\{\mathbb \{R\}\}$ 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 $\{\mathbb \{R\}\}$. 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.},
author = {Kopeček, Oldřich},
journal = {Czechoslovak Mathematical Journal},
keywords = {homomorphism of mono-unary algebras; functional equation; strictly increasing continuous real functions; homomorphism of monounary algebras; functional equation; strictly increasing functions},
language = {eng},
number = {4},
pages = {1011-1032},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Equation $f(p(x)) = q(f(x))$ for given real functions $p$, $q$},
url = {http://eudml.org/doc/247138},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Kopeček, Oldřich
TI - Equation $f(p(x)) = q(f(x))$ for given real functions $p$, $q$
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 1011
EP - 1032
AB - We investigate functional equations $f(p(x)) = q(f(x))$ where $p$ and $q$ are given real functions defined on the set ${\mathbb {R}}$ 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 ${\mathbb {R}}$. 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.
LA - eng
KW - homomorphism of mono-unary algebras; functional equation; strictly increasing continuous real functions; homomorphism of monounary algebras; functional equation; strictly increasing functions
UR - http://eudml.org/doc/247138
ER -

References

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