Geodesic
The continuous solutions of a generalized Dhombres functional equation
Mathematica Bohemica, Tome 129 (2004) no. 4, pp. 399-410

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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 $I\subset (0,\infty )$.
DOI : 10.21136/MB.2004.134048
Classification : 26A18, 39B12, 39B22
Keywords: iterative functional equation; equation of invariant curves; general continuous solution 
@article{10_21136_MB_2004_134048,
     author = {Reich, L. and Sm{\'\i}tal, J. and \v{S}tef\'ankov\'a, M.},
     title = {The continuous solutions of a generalized {Dhombres} functional equation},
     journal = {Mathematica Bohemica},
     pages = {399--410},
     publisher = {mathdoc},
     volume = {129},
     number = {4},
     year = {2004},
     doi = {10.21136/MB.2004.134048},
     mrnumber = {2102613},
     zbl = {1080.39505},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2004.134048/}
}
TY  - JOUR
AU  - Reich, L.
AU  - Smítal, J.
AU  - Štefánková, M.
TI  - The continuous solutions of a generalized Dhombres functional equation
JO  - Mathematica Bohemica
PY  - 2004
SP  - 399
EP  - 410
VL  - 129
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2004.134048/
DO  - 10.21136/MB.2004.134048
LA  - en
ID  - 10_21136_MB_2004_134048
ER  - 
%0 Journal Article
%A Reich, L.
%A Smítal, J.
%A Štefánková, M.
%T The continuous solutions of a generalized Dhombres functional equation
%J Mathematica Bohemica
%D 2004
%P 399-410
%V 129
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2004.134048/
%R 10.21136/MB.2004.134048
%G en
%F 10_21136_MB_2004_134048
Reich, L.; Smítal, J.; Štefánková, M. The continuous solutions of a generalized Dhombres functional equation. Mathematica Bohemica, Tome 129 (2004) no. 4, pp. 399-410. doi: 10.21136/MB.2004.134048

Cité par Sources :