The converse problem for a generalized Dhombres functional equation

Reich, L., Smítal, Jaroslav, and Štefánková, M.. "The converse problem for a generalized Dhombres functional equation." Mathematica Bohemica 130.3 (2005): 301-308. <http://eudml.org/doc/249586>.

@article{Reich2005,
abstract = {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.},
author = {Reich, L., Smítal, Jaroslav, Štefánková, M.},
journal = {Mathematica Bohemica},
keywords = {iterative functional equation; equation of invariant curves; general continuous solution; converse problem; iterative functional equation; equation of invariant curves; general continuous solution},
language = {eng},
number = {3},
pages = {301-308},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The converse problem for a generalized Dhombres functional equation},
url = {http://eudml.org/doc/249586},
volume = {130},
year = {2005},
}

TY - JOUR
AU - Reich, L.
AU - Smítal, Jaroslav
AU - Štefánková, M.
TI - The converse problem for a generalized Dhombres functional equation
JO - Mathematica Bohemica
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 130
IS - 3
SP - 301
EP - 308
AB - 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.
LA - eng
KW - iterative functional equation; equation of invariant curves; general continuous solution; converse problem; iterative functional equation; equation of invariant curves; general continuous solution
UR - http://eudml.org/doc/249586
ER -