The continuous solutions of a generalized Dhombres functional equation

Reich, L., Smítal, Jaroslav, and Štefánková, M.. "The continuous solutions of a generalized Dhombres functional equation." Mathematica Bohemica 129.4 (2004): 399-410. <http://eudml.org/doc/249393>.

@article{Reich2004,
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 series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under $\varphi $ and which contains in its interior no fixed point except for $1$. They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution to be monotone. In the present paper we give a characterization of the class of continuous solutions of this equation: We describe a method of constructing solutions as pointwise limits of solutions which are piecewise monotone on every compact subinterval. And we show that any solution can be obtained in this way. In particular, we show that if there exists a solution which is not monotone then there is a continuous solution which is monotone on no subinterval of a compact interval $I\subset (0,\infty )$.},
author = {Reich, L., Smítal, Jaroslav, Štefánková, M.},
journal = {Mathematica Bohemica},
keywords = {iterative functional equation; equation of invariant curves; general continuous solution; iterative functional equation; equation of invariant curves; general continuous solution},
language = {eng},
number = {4},
pages = {399-410},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The continuous solutions of a generalized Dhombres functional equation},
url = {http://eudml.org/doc/249393},
volume = {129},
year = {2004},
}

TY - JOUR
AU - Reich, L.
AU - Smítal, Jaroslav
AU - Štefánková, M.
TI - The continuous solutions of a generalized Dhombres functional equation
JO - Mathematica Bohemica
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 129
IS - 4
SP - 399
EP - 410
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 series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under $\varphi $ and which contains in its interior no fixed point except for $1$. They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution to be monotone. In the present paper we give a characterization of the class of continuous solutions of this equation: We describe a method of constructing solutions as pointwise limits of solutions which are piecewise monotone on every compact subinterval. And we show that any solution can be obtained in this way. In particular, we show that if there exists a solution which is not monotone then there is a continuous solution which is monotone on no subinterval of a compact interval $I\subset (0,\infty )$.
LA - eng
KW - iterative functional equation; equation of invariant curves; general continuous solution; iterative functional equation; equation of invariant curves; general continuous solution
UR - http://eudml.org/doc/249393
ER -