# Article

 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: http://hdl.handle.net/10338.dmlcz/133958 . Reference: [1] J. Dhombres: Applications associatives ou commutatives.C. R. Acad. Sci. Paris 281 (1975), 809–812. Zbl 0344.39009, MR 0419662 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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