Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
iterative functional equation; equation of invariant curves; general continuous solution; converse problem
iterative functional equation; equation of invariant curves; general continuous solution; converse problem
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.
References:
[1] J. Dhombres: Applications associatives ou commutatives. C. R. Acad. Sci. Paris, Sér. A 281 (1975), 809–812. MR 0419662 | Zbl 0344.39009
[2] P. Kahlig, J. Smítal: On a parametric functional equation of Dhombres type. Aequationes Math. 56 (1998), 63–68. DOI 10.1007/s000100050044 | MR 1628303
[3] P. Kahlig, J. Smítal: On a generalized Dhombres functional equation. Aequationes Math. 62 (2001), 18–29. DOI 10.1007/PL00000138 | MR 1849137
[4] P. Kahlig, J. Smítal: On a generalized Dhombres functional equation II. Math. Bohem. 127 (2002), 547–555. MR 1942640
[5] L. Reich, J. Smítal, M. Štefánková: The continuous solutions of a generalized Dhombres functional equation. Math. Bohem. 129 (2004), 399–410. MR 2102613
[6] M. Kuczma: Functional Equations in a Single Variable. Polish Scientific Publishers, Warsawa, 1968. MR 0228862 | Zbl 0196.16403
[7] M. Kuczma, B. Choczewski, R. Ger: Iterative Functional Equations. Encyclopedia of mathematics and its applications, 32, Cambridge University Press, Cambridge, 1990. MR 1067720
Search
Partner of
