Generalized Dhombres functional equation

@inbook{29851,
   author = {Smítal, Jaroslav and Štefánková, Marta},
   address = {Cham (Switzerland)},
   booktitle = {Developments in Functional Equations and Related Topics},
   doi = {http://dx.doi.org/10.1007/978-3-319-61732-9_13},
   editor = {Janusz Brzdek, Krzysztof Ciepliński, Themistocles M. Rassias},
   keywords = {Invariant curves; Iterative functional equations; Periodic orbits; Real solutions; Topological entropy},
   howpublished = {tištěná verze "print"},
   language = {eng},
   location = {Cham (Switzerland)},
   isbn = {978-3-319-61731-2},
   pages = {297-303},
   publisher = {Springer International Publishing},
   title = {Generalized Dhombres functional equation},
   url = {https://link.springer.com/chapter/10.1007/978-3-319-61732-9_13},
   year = {2017}
}

TY  - CHAP
ID  - 29851
AU  - Smítal, Jaroslav - Štefánková, Marta
PY  - 2017
TI  - Generalized Dhombres functional equation
VL  - Springer Optimization and Its Applications
PB  - Springer International Publishing
CY  - Cham (Switzerland)
SN  - 9783319617312
KW  - Invariant curves
KW  - Iterative functional equations
KW  - Periodic orbits
KW  - Real solutions
KW  - Topological entropy
UR  - https://link.springer.com/chapter/10.1007/978-3-319-61732-9_13
L2  - https://link.springer.com/chapter/10.1007/978-3-319-61732-9_13
N2  - We consider the equation f(xf(x)) = phi(f(x)), x > 0, where phi is given, and f is an unknown continuous function (0,infinity)->(0,infinity). This equation was for the first time studied in 1975 by Dhombres (with phi(y) = y^2), later it was considered for other particular choices of phi, and since 2001 for arbitrary continuous function phi. The main problem, a classification of possible solutions and a description of the structure of periodic points contained in the range of the solutions (which appeared to be important way of the classification of solutions), was basically solved. This process involved not only methods from one-dimensional dynamics but also some new methods which could be useful in other problems. In this paper we provide a brief survey.
ER  - 

SMÍTAL, Jaroslav a Marta ŠTEFÁNKOVÁ. Generalized Dhombres functional equation. In Janusz Brzdek, Krzysztof Ciepli\'nski, Themistocles M. Rassias. \textit{Developments in Functional Equations and Related Topics}. Cham (Switzerland): Springer International Publishing, 2017, s.~297-303. Springer Optimization and Its Applications. ISBN~978-3-319-61731-2. Dostupné z: https://dx.doi.org/10.1007/978-3-319-61732-9\_{}13.