caa a22 4500 605508259 CHVBK 20210128100635.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00010-013-0248-0 doi (NATIONALLICENCE)springer-10.1007/s00010-013-0248-0 Iterative functional equations related to a competition equation [Elektronische Daten] [Peter Kahlig, Janusz Matkowski] 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 , ... ) . Springer Basel, 2014 Functional equation nationallicence competition equation nationallicence iterative functional equation nationallicence differentiable solution nationallicence solution depending on an arbitrary function nationallicence Kahlig Peter Science Pool Vienna, Sect. 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 89/3(2015-06-01), 613-624 0001-9054 89:3<613 2015 89 10 https://doi.org/10.1007/s00010-013-0248-0 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/s00010-013-0248-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kahlig Peter Science Pool Vienna, Sect. 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 89/3(2015-06-01), 613-624 0001-9054 89:3<613 2015 89 10