caa a22 4500 605508038 CHVBK 20210128100634.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s00010-014-0307-1 doi (NATIONALLICENCE)springer-10.1007/s00010-014-0307-1 Second order iterative functional equations related to a competition equation [Elektronische Daten] [Peter Kahlig, Janusz Matkowski] The functional equation related to competition ([2]) $$f\left( \frac{x+y}{1-xy}\right) =\frac{f\left( x\right) +f\left(y\right)} {1+f\left( x\right) f\left( y\right)},\qquad x,y\in\mathbb{R}, xy\neq 1,$$ f x + y 1 - x y = f x + f y 1 + f x f y , x , y ∈ R , x y ≠ 1 , for y=cx with a fixed c>0, leads to the equation $$f\left( \frac{\left( 1+c\right) x}{1-cx^{2}}\right) =\frac{f\left(x\right) +f\left( cx\right)} {1+f\left( x\right) f\left( cx\right)},\qquad x\in \mathbb{R}, \left\vert x \right\vert <\frac{1}{\sqrt{c}}.$$ f 1 + c x 1 - c x 2 = f x + f c x 1 + f x f c x , x ∈ R , x < 1 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 $${f:\mathbb{R} \rightarrow \mathbb{R},\,f\left( 0\right) =0}$$ f : R → R , f 0 = 0 , differentiable at the point 0 satisfies this functional equation iff there is a real p such that $${f=\tanh \circ \left( p\tan ^{-1} \right) }$$ f = tanh ∘ p tan - 1 which extends the main result of [3]. The Author(s), 2014 Functional equation nationallicence competition equation nationallicence iterative functional equation of the second order nationallicence differentiable solution nationallicence solution depending on an arbitrary function nationallicence Kahlig Peter Science Pool Vienna, Section of Hydrometeorology, Altmannsdorfer Str. 21/5/2, 1120, Vienna, Austria aut Matkowski Janusz Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4A, 65-516, Zielona Gora, Poland aut Aequationes mathematicae Springer Basel 89/1(2015-02-01), 107-117 0001-9054 89:1<107 2015 89 10 https://doi.org/10.1007/s00010-014-0307-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0307-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kahlig Peter Science Pool Vienna, Section of Hydrometeorology, Altmannsdorfer Str. 21/5/2, 1120, Vienna, Austria aut NATIONALLICENCE 700 1- Matkowski Janusz Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4A, 65-516, Zielona Gora, Poland aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/1(2015-02-01), 107-117 0001-9054 89:1<107 2015 89 10