The converse problem for a generalized Dhombres functional equation

summary: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 $\mathcal S(J,\varphi )$ of continuous solutions $f$ is given in a series of papers by Kahlig and Smítal 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 $f\in \mathcal S(J,\varphi )$. We also show why the similar problem for decreasing $\varphi $ is difficult