Second order iterative functional equations related to a competition equation
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| 008 | 210128e20150201xx s 000 0 eng | ||
| 024 | 7 | 0 | |a 10.1007/s00010-014-0307-1 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s00010-014-0307-1 | ||
| 245 | 0 | 0 | |a Second order iterative functional equations related to a competition equation |h [Elektronische Daten] |c [Peter Kahlig, Janusz Matkowski] |
| 520 | 3 | |a 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]. | |
| 540 | |a The Author(s), 2014 | ||
| 690 | 7 | |a Functional equation |2 nationallicence | |
| 690 | 7 | |a competition equation |2 nationallicence | |
| 690 | 7 | |a iterative functional equation of the second order |2 nationallicence | |
| 690 | 7 | |a differentiable solution |2 nationallicence | |
| 690 | 7 | |a solution depending on an arbitrary function |2 nationallicence | |
| 700 | 1 | |a Kahlig |D Peter |u Science Pool Vienna, Section of Hydrometeorology, Altmannsdorfer Str. 21/5/2, 1120, Vienna, Austria |4 aut | |
| 700 | 1 | |a Matkowski |D Janusz |u Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4A, 65-516, Zielona Gora, Poland |4 aut | |
| 773 | 0 | |t Aequationes mathematicae |d Springer Basel |g 89/1(2015-02-01), 107-117 |x 0001-9054 |q 89:1<107 |1 2015 |2 89 |o 10 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00010-014-0307-1 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s00010-014-0307-1 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Kahlig |D Peter |u Science Pool Vienna, Section of Hydrometeorology, Altmannsdorfer Str. 21/5/2, 1120, Vienna, Austria |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Matkowski |D Janusz |u Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4A, 65-516, Zielona Gora, Poland |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Aequationes mathematicae |d Springer Basel |g 89/1(2015-02-01), 107-117 |x 0001-9054 |q 89:1<107 |1 2015 |2 89 |o 10 | ||