Matematický ústav

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Preprint MA 22/2000

On a generalized dhombres functional equation II

By P. Kahlig, J. Smital

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 condition sufficient 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.

Mathematics Subject Classification. Primary 39B12, 39B22. Secondary 26A18

Poslední aktualizace stránky dne 6. dubna 2001, 11:18.

Tato stránka byla od 10. dubna 2002 navštívena 1502 krát.