Abstract
We consider the equation f(xf(x)) = φ(f(x)), x > 0, where φ is given, and f is an unknown continuous function (0, ∞) → (0, ∞). This equation was for the first time studied in 1975 by Dhombres (with φ(y) = y 2), later it was considered for other particular choices of φ, and since 2001 for arbitrary continuous function φ. 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.
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Smítal, J., Štefánková, M. (2017). Generalized Dhombres Functional Equation. In: Brzdęk, J., Ciepliński, K., Rassias, T. (eds) Developments in Functional Equations and Related Topics . Springer Optimization and Its Applications, vol 124. Springer, Cham. https://doi.org/10.1007/978-3-319-61732-9_13
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DOI: https://doi.org/10.1007/978-3-319-61732-9_13
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