On a generalized Dhombres functional equation. II.

Kahlig, P., and Smítal, Jaroslav. "On a generalized Dhombres functional equation. II.." Mathematica Bohemica 127.4 (2002): 547-555. <http://eudml.org/doc/249022>.

@article{Kahlig2002,
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.},
author = {Kahlig, P., Smítal, Jaroslav},
journal = {Mathematica Bohemica},
keywords = {iterative functional equation; invariant curves; monotone solutions; iterative functional equation; invariant curves; monotone solutions; generalized Dhombres functional equation; fixed point; continuous solutions},
language = {eng},
number = {4},
pages = {547-555},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a generalized Dhombres functional equation. II.},
url = {http://eudml.org/doc/249022},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Kahlig, P.
AU - Smítal, Jaroslav
TI - On a generalized Dhombres functional equation. II.
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 4
SP - 547
EP - 555
AB - 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.
LA - eng
KW - iterative functional equation; invariant curves; monotone solutions; iterative functional equation; invariant curves; monotone solutions; generalized Dhombres functional equation; fixed point; continuous solutions
UR - http://eudml.org/doc/249022
ER -