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Aequationes mathematicae

, Volume 89, Issue 1, pp 107–117 | Cite as

Second order iterative functional equations related to a competition equation

  • Peter Kahlig
  • Janusz MatkowskiEmail author
Open Access
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Abstract

The functional equation related to competition ([2])
$$f\left( \frac{x+y}{1-xy}\right) =\frac{f\left( x\right) +f\left(y\right)} {1+f\left( x\right) f\left( y\right)},\qquad x,y\in\mathbb{R}, xy\neq 1,$$
for y = cx with a fixed c > 0, leads to the equation
$$f\left( \frac{\left( 1+c\right) x}{1-cx^{2}}\right) =\frac{f\left(x\right) +f\left( cx\right)} {1+f\left( x\right) f\left( cx\right)},\qquad x\in \mathbb{R}, \left\vert x \right\vert <\frac{1}{\sqrt{c}}.$$
The case c = 1 (a first order iterative functional equation) was treated in [3]. In this paper we consider the case c ≠ 1 (when the equation is of the second order). We show that a function \({f:\mathbb{R} \rightarrow \mathbb{R},\,f\left( 0\right) =0}\), differentiable at the point 0 satisfies this functional equation iff there is a real p such that \({f=\tanh \circ \left( p\tan ^{-1} \right) }\) which extends the main result of [3].

Mathematics Subject Classification

Primary 39B12 39B22 

Keywords

Functional equation competition equation iterative functional equation of the second order differentiable solution solution depending on an arbitrary function 

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Copyright information

© The Author(s) 2014

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.Science Pool ViennaSection of HydrometeorologyViennaAustria
  2. 2.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GoraPoland

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