Equation $f(p(x)) = q(f(x))$ for given real functions $p$, $q$
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 1011-1032.

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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 ${\Bbb R}$ 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.
DOI : 10.1007/s10587-012-0061-2
Classification : 08A60, 65Q20, 97I70
Mots-clés : homomorphism of mono-unary algebras; functional equation; strictly increasing continuous real functions 
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     title = {Equation $f(p(x)) = q(f(x))$ for given real functions $p$, $q$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1011--1032},
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Kopeček, Oldřich. Equation $f(p(x)) = q(f(x))$ for given real functions $p$, $q$. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 1011-1032. doi : 10.1007/s10587-012-0061-2. https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-012-0061-2/

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