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Aequationes mathematicae

, Volume 87, Issue 3, pp 301–308 | Cite as

On a functional equation related to competition

  • Peter Kahlig
  • Janusz Matkowski
Open Access
Article

Abstract

The functional equation
$$f \left(\frac{x + y}{1 - xy}\right) = \frac{f\left(x\right) + f\left(y\right)} {1 + f\left(x\right) f\left(y\right)}, \quad xy < 1,$$
(introduced by the first author in a competition model) is considered. The main result says that a function \({f : \mathbb{R} \rightarrow \mathbb{R}}\) satisfies this equation if, and only if, \({f = {\rm tanh} \circ \, \alpha \circ {\rm tan}^{-1}}\) , where \({\alpha : \mathbb{R} \rightarrow \mathbb{R}}\) is an additive function.

Mathematics Subject Classification (1991)

Primary 39B12 39B22 

Keywords

Functional equation additive function general solution competition model 

References

  1. 1.
    Kahlig P.: A model of competition. Appl. Math. 39, 293–303 (2012)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Kahlig P.: Note to the paper “A model of competition”. Appl. Math. 40, 127 (2013)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Matkowski, J.: The uniqueness of solutions of a system of functional equations in some classes of functions. Aequ. Math. 8, 233–237 (1972)Google Scholar
  4. 4.
    Kuczma, M.: An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality. Uniwersytet Śla̧ski - PWN, Warszawa - Krakow - Katowice, 1985 (2nd edn., edited with a preface of Attila Gilányi, Birkhäuser verlag, Basel, 2009)Google Scholar

Copyright information

© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.ViennaAustria
  2. 2.Faculty of Mathematics Computer Science and EconometricsUniversity of Zielona GóraZielonaGóraPoland

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