On a generalized Dhombres functional equation. II.

P. Kahlig; Jaroslav Smítal

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 4, page 547-555
  • 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 increasing homeomorphism of an open interval J ( 0 , ) and f ( 0 , ) 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 ϕ , with a possible exception for p = 1 . The range of any non-constant continuous solution is an interval whose end-points are fixed by ϕ 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 ϕ ) 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 ϕ , or (ii) 1 J and J contains no fixed points different from 1.

How to cite

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Kahlig, P., and Smítal, Jaroslav. "On a generalized Dhombres functional equation. II.." Mathematica Bohemica 127.4 (2002): 547-555. <http://eudml.org/doc/249022>.

@article{Kahlig2002,
abstract = {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.},
author = {Kahlig, P., Smítal, Jaroslav},
journal = {Mathematica Bohemica},
keywords = {iterative functional equation; invariant curves; monotone solutions; iterative functional equation; invariant curves; monotone solutions; generalized Dhombres functional equation; fixed point; continuous solutions},
language = {eng},
number = {4},
pages = {547-555},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a generalized Dhombres functional equation. II.},
url = {http://eudml.org/doc/249022},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Kahlig, P.
AU - Smítal, Jaroslav
TI - On a generalized Dhombres functional equation. II.
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 4
SP - 547
EP - 555
AB - 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.
LA - eng
KW - iterative functional equation; invariant curves; monotone solutions; iterative functional equation; invariant curves; monotone solutions; generalized Dhombres functional equation; fixed point; continuous solutions
UR - http://eudml.org/doc/249022
ER -

References

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  1. Applications associatives ou commutatives, C. R. Acad. Sci. Paris 281 (1975), 809–812. (1975) Zbl0344.39009MR0419662
  2. On the solutions of a functional equation of Dhombres, Results Math. 27 (1995), 362–367. (1995) MR1331109
  3. On a parametric functional equation of Dhombres type, Aequationes Math. 56 (1998), 63–68. (1998) MR1628303
  4. On a generalized Dhombres functional equation, Aequationes Math. 62 (2001), 18–29. (2001) MR1849137
  5. Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw, 1968. (1968) Zbl0196.16403MR0228862
  6. Iterative Functional Equations, Encyclopedia of Mathematics and its Applications Vol. 32, Cambridge University Press, Cambridge, 1990. (1990) MR1067720

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