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@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.
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