# On a generalized Dhombres functional equation. II.

Mathematica Bohemica (2002)

• Volume: 127, Issue: 4, page 547-555
• ISSN: 0862-7959

top Access to full text Full (PDF)

## Abstract

top
We consider the functional equation $f\left(xf\left(x\right)\right)=\varphi \left(f\left(x\right)\right)$ where $\varphi \phantom{\rule{0.222222em}{0ex}}J\to J$ is a given increasing homeomorphism of an open interval $J\subset \left(0,\infty \right)$ and $f\phantom{\rule{0.222222em}{0ex}}\left(0,\infty \right)\to J$ 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.

## Citations in EuDML Documents

top

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.