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

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Abstract

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.

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  1. Hochschule für Technik Stuttgart, D-70174, Stuttgart, Germany

    Oldřich Kopeček

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  1. Oldřich Kopeček
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Correspondence to Oldřich Kopeček.

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Kopeček, O. Equation f(p(x)) = q(f(x)) for given real functions p, q . Czech Math J 62, 1011–1032 (2012). https://doi.org/10.1007/s10587-012-0061-2

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  • DOI: https://doi.org/10.1007/s10587-012-0061-2

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