Generalized Dhombres Functional Equation

Jaroslav Smítal () and Marta Štefánková ()
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Jaroslav Smítal: Silesian University
Marta Štefánková: Silesian University

Chapter Chapter 13 in Developments in Functional Equations and Related Topics, 2017, pp 297-303 from Springer

Abstract: 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.

Keywords: Iterative functional equations; Invariant curves; Real solutions; Topological entropy; Periodic orbits; Primary 39B12; Secondary 26A18 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-319-61732-9_13

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DOI: 10.1007/978-3-319-61732-9_13

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