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Title: On a generalized Dhombres functional equation. II. (English)
Author: Kahlig, P.
Author: Smítal, J.
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 127
Issue: 4
Year: 2002
Pages: 547-555
Summary lang: English
Category: math
Summary: 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. (English)
Keyword: iterative functional equation
Keyword: invariant curves
Keyword: monotone solutions
MSC: 26A18
MSC: 39B12
MSC: 39B22
idZBL: Zbl 1007.39016
idMR: MR1942640
DOI: 10.21136/MB.2002.133958
Date available: 2009-09-24T22:04:57Z
Last updated: 2020-07-29
Stable URL:
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Reference: [2] P. Kahlig, J. Smítal: On the solutions of a functional equation of Dhombres.Results Math. 27 (1995), 362–367. MR 1331109, 10.1007/BF03322840
Reference: [3] P. Kahlig, J. Smítal: On a parametric functional equation of Dhombres type.Aequationes Math. 56 (1998), 63–68. MR 1628303, 10.1007/s000100050044
Reference: [4] P. Kahlig, J. Smítal: On a generalized Dhombres functional equation.Aequationes Math. 62 (2001), 18–29. MR 1849137, 10.1007/PL00000138
Reference: [5] M. Kuczma: Functional Equations in a Single Variable.Polish Scientific Publishers, Warsaw, 1968. Zbl 0196.16403, MR 0228862
Reference: [6] M. Kuczma, B. Choczewski, R. Ger: Iterative Functional Equations.Encyclopedia of Mathematics and its Applications Vol. 32, Cambridge University Press, Cambridge, 1990. MR 1067720


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