MATHEMATICA BOHEMICA, Vol. 130, No. 3, pp. 301-308 (2005)

The converse problem for a generalized Dhombres functional equation

L. Reich, J. Smital, M. Stefankova

Ludwig Reich, Institut f. Mathematik, Karl-Franzens-Universität Graz, A-8010 Graz, Austria, e-mail:; Jaroslav Smital, Marta Stefankova, Mathematical Institute, Silesian University, CZ-746 01 Opava, Czech Republic, e-mail:,

Abstract: We consider the functional equation $f(xf(x))=\varphi(f(x))$ where $\varphi J\rightarrow J$ is a given homeomorphism of an open interval $J\subset(0,\infty)$ and $f (0,\infty) \rightarrow J$ is an unknown continuous function. A characterization of the class $\Cal S(J,\varphi)$ of continuous solutions $f$ is given in a series of papers by Kahlig and Smital 1998-2002, and in a recent paper by Reich et al. 2004, in the case when $\varphi$ is increasing. In the present paper we solve the converse problem, for which continuous maps $f (0,\infty)\rightarrow J$, where $J$ is an interval, there is an increasing homeomorphism $\varphi$ of $J$ such that $f\in\Cal S(J,\varphi)$. We also show why the similar problem for decreasing $\varphi$ is difficult.

Keywords: iterative functional equation, equation of invariant curves, general continuous solution, converse problem

Classification (MSC2000): 39B12, 39B22, 26A18

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