Aequationes mathematicae

, Volume 89, Issue 3, pp 613–624 | Cite as

Iterative functional equations related to a competition equation

  • Peter KahligEmail author
  • Janusz Matkowski


The diagonalization of a two-variable functional equation (related to competition) leads to the iterative equation
$$f\left( \frac{2x}{1-x^{2}}\right) =\frac{2f( x)}{1+f( x) ^{2}},\quad x\in {\mathbb{R}},\, x^{2}\neq 1.$$
It was shown in Kahlig (Appl Math 39:293–303, 2012) that if a function \({f:{\mathbb{R}}\rightarrow {\mathbb{R}}}\), such that f(0) = 0, satisfies this equation for all \({x\in (-1,1),}\) and is twice differentiable at the point 0, then \({f=\tanh \circ (p\,\tan ^{-1}) }\) for some real p. In this paper we prove the following stronger result. A function \({f:{\mathbb{R}} \rightarrow {\mathbb{R}},\;f(0) =0}\), differentiable at the point 0, satisfies this functional equation if, and only if, there is a real p such that \({f=\tanh \circ (p\,\tan ^{-1})}\). We also show that the assumption of the differentiability of f at 0 cannot be replaced by the continuity of f. The corresponding result for the iterative equation coming from a three- respectively four-variable competition equation is also proved. Our conjecture is that analogous results hold true for the diagonalization of any n-variable competition equation \({(n=5, 6, 7, \ldots)}\).

Mathematics Subject Classification (1991)

Primary 39B12 39B22 


Functional equation competition equation iterative functional equation differentiable solution solution depending on an arbitrary function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kahlig P.: A model of competition. Appl. Math. 39, 293–303 (2012)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Kahlig, P., Matkowski, J.: On a functional equation related to competition, Aequ. Math. (accepted 2013)Google Scholar
  3. 3.
    Kuczma, M.: Functional equations in a single variable. Monografie Matematyczne, vol. 46, PWN Warszawa (1968)Google Scholar
  4. 4.
    Kuczma M., Choczewski B., Ger R.: Iterative Functional Equations, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  5. 5.
    Matkowski J.: On the uniqueness of differentiable solutions of a functional equation. Bull. Acad. Polon. Sci. 18, 253–255 (1970)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Matkowski J.: On the existence of differentiable solutions of a functional equation. Bull. Acad. Polon. Sci. 19, 19–22 (1971)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Matkowski J.: The uniqueness of solutions of a system of functional equations in some classes of functions. Aequ. Math. 8, 233–237 (1972)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Science Pool Vienna, Sect. HydrometeorologyViennaAustria
  2. 2.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GoraPoland

Personalised recommendations