| 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 |
| . |
