| Title:
|
The converse problem for a generalized Dhombres functional equation (English) |
| Author:
|
Reich, L. |
| Author:
|
Smítal, J. |
| Author:
|
Štefánková, M. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
130 |
| Issue:
|
3 |
| Year:
|
2005 |
| Pages:
|
301-308 |
| 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 homeomorphism of an open interval $J\subset (0,\infty )$ and $f\: (0,\infty ) \rightarrow J$ is an unknown continuous function. A characterization of the class $\mathcal S(J,\varphi )$ of continuous solutions $f$ is given in a series of papers by Kahlig and Smítal 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 $f\in \mathcal S(J,\varphi )$. We also show why the similar problem for decreasing $\varphi $ is difficult. (English) |
| Keyword:
|
iterative functional equation |
| Keyword:
|
equation of invariant curves |
| Keyword:
|
general continuous solution |
| Keyword:
|
converse problem |
| MSC:
|
26A18 |
| MSC:
|
39B12 |
| MSC:
|
39B22 |
| idZBL:
|
Zbl 1110.39014 |
| idMR:
|
MR2164659 |
| DOI:
|
10.21136/MB.2005.134093 |
| . |
| Date available:
|
2009-09-24T22:21:27Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134093 |
| . |
| Reference:
|
[1] J. Dhombres: Applications associatives ou commutatives.C. R. Acad. Sci. Paris, Sér. A 281 (1975), 809–812. Zbl 0344.39009, MR 0419662 |
| Reference:
|
[2] P. Kahlig, J. Smítal: On a parametric functional equation of Dhombres type.Aequationes Math. 56 (1998), 63–68. MR 1628303, 10.1007/s000100050044 |
| Reference:
|
[3] P. Kahlig, J. Smítal: On a generalized Dhombres functional equation.Aequationes Math. 62 (2001), 18–29. MR 1849137, 10.1007/PL00000138 |
| Reference:
|
[4] P. Kahlig, J. Smítal: On a generalized Dhombres functional equation II.Math. Bohem. 127 (2002), 547–555. MR 1942640 |
| Reference:
|
[5] L. Reich, J. Smítal, M. Štefánková: The continuous solutions of a generalized Dhombres functional equation.Math. Bohem. 129 (2004), 399–410. MR 2102613 |
| Reference:
|
[6] M. Kuczma: Functional Equations in a Single Variable.Polish Scientific Publishers, Warsawa, 1968. Zbl 0196.16403, MR 0228862 |
| Reference:
|
[7] M. Kuczma, B. Choczewski, R. Ger: Iterative Functional Equations.Encyclopedia of mathematics and its applications, 32, Cambridge University Press, Cambridge, 1990. MR 1067720 |
| . |
