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Keywords:
homomorphism of mono-unary algebras; functional equation; strictly increasing continuous real functions
homomorphism of mono-unary algebras; functional equation; strictly increasing continuous real functions
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.
References:
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