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

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Oldrich Kopecek

**Abstract:** 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.

**Keywords:** homomorphism of mono-unary algebras, functional equation, strictly increasing continuous real functions

**Classification (MSC 2010):** 08A60, 65Q20, 97I70

**Full text** available as PDF.

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