Iterative functional equations related to a competition equation
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| 024 | 7 | 0 | |a 10.1007/s00010-013-0248-0 |2 doi |
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| 245 | 0 | 0 | |a Iterative functional equations related to a competition equation |h [Elektronische Daten] |c [Peter Kahlig, Janusz Matkowski] |
| 520 | 3 | |a The diagonalization of a two-variable functional equation (related to competition) leads to the iterative equation $$f\left( \frac{2x}{1-x^{2}}\right) =\frac{2f( x)}{1+f( x) ^{2}},\quad x\in {\mathbb{R}},\, x^{2}\neq 1.$$ f 2 x 1 - x 2 = 2 f ( x ) 1 + f ( x ) 2 , x ∈ R , x 2 ≠ 1 . It was shown in Kahlig (Appl Math 39:293-303, 2012) that if a function $${f:{\mathbb{R}}\rightarrow {\mathbb{R}}}$$ f : R → R , such that f(0) =0, satisfies this equation for all $${x\in (-1,1),}$$ x ∈ ( - 1 , 1 ) , and is twice differentiable at the point 0, then $${f=\tanh \circ (p\,\tan ^{-1}) }$$ f = tanh ∘ ( p tan - 1 ) for some real p. In this paper we prove the following stronger result. A function $${f:{\mathbb{R}} \rightarrow {\mathbb{R}},\;f(0) =0}$$ f : R → R , f ( 0 ) = 0 , differentiable at the point 0, satisfies this functional equation if, and only if, there is a real p such that $${f=\tanh \circ (p\,\tan ^{-1})}$$ f = tanh ∘ ( 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 $${(n=5, 6, 7, \ldots)}$$ ( n = 5 , 6 , 7 , ... ) . | |
| 540 | |a Springer Basel, 2014 | ||
| 690 | 7 | |a Functional equation |2 nationallicence | |
| 690 | 7 | |a competition equation |2 nationallicence | |
| 690 | 7 | |a iterative functional equation |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, Sect. 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 89/3(2015-06-01), 613-624 |x 0001-9054 |q 89:3<613 |1 2015 |2 89 |o 10 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00010-013-0248-0 |q text/html |z Onlinezugriff via DOI |
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| 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/s00010-013-0248-0 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Kahlig |D Peter |u Science Pool Vienna, Sect. 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 89/3(2015-06-01), 613-624 |x 0001-9054 |q 89:3<613 |1 2015 |2 89 |o 10 | ||