About: On generalized Dhombres functional equation

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
Description	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. 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 containsin 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 which have all continuous solutions monotone. 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. 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. 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 containsin 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 which have all continuous solutions monotone. 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. (en)
Title	On generalized Dhombres functional equation On generalized Dhombres functional equation (en)
skos:prefLabel	On generalized Dhombres functional equation On generalized Dhombres functional equation (en)
skos:notation	RIV/47813059:19610/01:00000054!RIV/2002/GA0/196102/N
http://linked.open.../vavai/riv/strany	%2212;29%22
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(GA201/97/0001), Z(MSM 192400002)
http://linked.open...iv/cisloPeriodika	1
http://linked.open...vai/riv/dodaniDat	2002
http://linked.open...aciTvurceVysledku	Smítal, Jaroslav
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Slezská univerzita v Opavě / Matematický ústav v Opavě
http://linked.open...dnocenehoVysledku	689880
http://linked.open...ai/riv/idVysledku	RIV/47813059:19610/01:00000054
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	%22iterative functional equation; invariant curves; monotone solutions%22 (en)
http://linked.open.../riv/klicoveSlovo	invariant curves %22iterative functional equation monotone solutions%22
http://linked.open...odStatuVydavatele	CH - Švýcarská konfederace
http://linked.open...ontrolniKodProRIV	[9D69C05DAD38]
http://linked.open...i/riv/nazevZdroje	Aequationes Mathematicae
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...ocetUcastnikuAkce	0 (xsd:int)
http://linked.open...nichUcastnikuAkce	0 (xsd:int)
http://linked.open...vavai/riv/projekt	Dynamical systems
http://linked.open...UplatneniVysledku	2001
http://linked.open...v/svazekPeriodika	62
http://linked.open...iv/tvurceVysledku	Smítal, Jaroslav
http://linked.open...n/vavai/riv/zamer	http://linked.opendata.cz/resource/domain/vavai/zamer/MSM%20192400002
issn	0001-9054
number of pages	18 (xsd:int)
http://localhost/t...ganizacniJednotka	19610

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