The converse problem for a generalized Dhombres functional equation
Result description
We consider the functional equation $f(xf(x))=varphi(f(x))$ where $varphi Jrightarrow J$ is a given homeomorphism of an open interval $Jsubset(0,infty)$ and $f (0,infty) rightarrow J$ is an unknown continuous function. A characterization of the class $Cal S(J,varphi)$ of continuous solutions $f$ is given in a series of papers by Kahlig and Smital 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 $finCal S(J,varphi)$. We also show why the similar problem for decreasing $varphi$ is difficult.
Keywords
iterative functional equationequation of invariant curvesgeneral continuous solutionconverse problem
The result's identifiers
Result code in IS VaVaI
Result on the web
—
DOI - Digital Object Identifier
—
Alternative languages
Result language
angličtina
Original language name
The converse problem for a generalized Dhombres functional equation
Original language description
We consider the functional equation $f(xf(x))=varphi(f(x))$ where $varphi Jrightarrow J$ is a given homeomorphism of an open interval $Jsubset(0,infty)$ and $f (0,infty) rightarrow J$ is an unknown continuous function. A characterization of the class $Cal S(J,varphi)$ of continuous solutions $f$ is given in a series of papers by Kahlig and Smital 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 $finCal S(J,varphi)$. We also show why the similar problem for decreasing $varphi$ is difficult.
Czech name
Obrácený problém pro zobecněnou Dhombresovu funkcionální rovnici
Czech description
Uvažujeme funkcionální rovnici $f(xf(x))=varphi(f(x))$ kde $varphi Jrightarrow J$ je daný homeomorfizmus otevřeného intervalu $Jsubset(0,infty)$ a $f (0,infty) rightarrow J$ je neznámá funkce. Charakterizac třídy $Cal S(J,varphi)$ spojitých řešení $f$ jlze najít v sérii prací Kahliga a Smítala 1988 - 2002 a v nedávné práci Reich et al. 2004 v případě, kdy $varphi$ je rostoucí. V této práci řešíme obrácený problém, která spojitá zobrazení $f (0,infty)rightarrow J$, kde $J$ je daný interval, existuje rostoucí homeomorfizmus $varphi$ z $J$ na $J$ tak, že $finCal S(J,varphi)$. Ukazujeme též, proč podobný problém s klesající funkcí $varphi$ je obtížný.
Classification
Type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
—
Result continuities
Project
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Z - Vyzkumny zamer (s odkazem do CEZ)
Others
Publication year
2005
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Mathematica Bohemica
ISSN
ISSN0862-7959
e-ISSN
—
Volume of the periodical
130
Issue of the periodical within the volume
3
Country of publishing house
CZ - CZECH REPUBLIC
Number of pages
8
Pages from-to
301-308
UT code for WoS article
—
EID of the result in the Scopus database
—
Basic information
Result type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP
BA - General mathematics
Year of implementation
2005