P. Kahlig, Institute of Meteorology and Geophysics, University of Vienna, UZA2, A-1090 Vienna, Austria, e-mail: peter.kahlig@univie.ac.at; Jaroslav Smital, Mathematical Institute, Silesian University, CZ-746 01 Opava, Czech Republic, e-mail: jaroslav.smital@math.slu.cz
Abstract: We consider the functional equation $f(xf(x))=\varphi(f(x))$ where $\varphi J\rightarrow J$ is a given increasing homeomorphism of an open interval $J\subset(0,\infty)$ and $f (0,\infty)\rightarrow J$ is an unknown continuous function. In a previous paper we proved that no continuous solution can cross the line $y=p$ where $p$ is a fixed point of $\varphi$, with a possible exception for $p=1$. The range of any non-constant continuous solution is an interval whose end-points are fixed by $\varphi$ and which contains in its interior no fixed point except for $1$. We also gave a characterization of the class of continuous monotone solutions and proved a sufficient condition for any continuous function to be monotone. In the present paper we give a characterization of the equations (or equivalently, of the functions $\varphi$) which have all continuous solutions monotone. In particular, all continuous solutions are monotone if either (i) 1 is an end-point of $J$ and $J$ contains no fixed point of $\varphi$, or (ii) $1\in J$ and $J$ contains no fixed points different from 1.
Keywords: iterative functional equation, invariant curves, monotone solutions
Classification (MSC 2000): 39B12, 39B22, 26A18
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