The converse problem for a generalized Dhombres functional equation

L. Reich; Jaroslav Smítal; M. Štefánková

Mathematica Bohemica (2005)

  • Volume: 130, Issue: 3, page 301-308
  • ISSN: 0862-7959

Abstract

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We consider the functional equation f ( x f ( x ) ) = ϕ ( f ( x ) ) where ϕ J J is a given homeomorphism of an open interval J ( 0 , ) and f ( 0 , ) J is an unknown continuous function. A characterization of the class 𝒮 ( J , ϕ ) 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 ϕ is increasing. In the present paper we solve the converse problem, for which continuous maps f ( 0 , ) J , where J is an interval, there is an increasing homeomorphism ϕ of J such that f 𝒮 ( J , ϕ ) . We also show why the similar problem for decreasing ϕ is difficult.

How to cite

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Reich, L., Smítal, Jaroslav, and Štefánková, M.. "The converse problem for a generalized Dhombres functional equation." Mathematica Bohemica 130.3 (2005): 301-308. <http://eudml.org/doc/249586>.

@article{Reich2005,
abstract = {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.},
author = {Reich, L., Smítal, Jaroslav, Štefánková, M.},
journal = {Mathematica Bohemica},
keywords = {iterative functional equation; equation of invariant curves; general continuous solution; converse problem; iterative functional equation; equation of invariant curves; general continuous solution},
language = {eng},
number = {3},
pages = {301-308},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The converse problem for a generalized Dhombres functional equation},
url = {http://eudml.org/doc/249586},
volume = {130},
year = {2005},
}

TY - JOUR
AU - Reich, L.
AU - Smítal, Jaroslav
AU - Štefánková, M.
TI - The converse problem for a generalized Dhombres functional equation
JO - Mathematica Bohemica
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 130
IS - 3
SP - 301
EP - 308
AB - 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.
LA - eng
KW - iterative functional equation; equation of invariant curves; general continuous solution; converse problem; iterative functional equation; equation of invariant curves; general continuous solution
UR - http://eudml.org/doc/249586
ER -

References

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  2. 10.1007/s000100050044, Aequationes Math. 56 (1998), 63–68. (1998) MR1628303DOI10.1007/s000100050044
  3. 10.1007/PL00000138, Aequationes Math. 62 (2001), 18–29. (2001) MR1849137DOI10.1007/PL00000138
  4. On a generalized Dhombres functional equation II, Math. Bohem. 127 (2002), 547–555. (2002) MR1942640
  5. The continuous solutions of a generalized Dhombres functional equation, Math. Bohem. 129 (2004), 399–410. (2004) MR2102613
  6. Functional Equations in a Single Variable, Polish Scientific Publishers, Warsawa, 1968. (1968) Zbl0196.16403MR0228862
  7. Iterative Functional Equations, Encyclopedia of mathematics and its applications, 32, Cambridge University Press, Cambridge, 1990. (1990) MR1067720

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