caa a22 4500 606200517 CHVBK 20210128100944.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00025-015-0437-3 doi (NATIONALLICENCE)springer-10.1007/s00025-015-0437-3 On Regular Solutions of the Generalized Dhombres Equation II [Elektronische Daten] [L. Reich, J. Smítal, M. Štefánková] We consider continuous solutions $${f : \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}=(0,\infty )}$$ f : R + → R + = ( 0 , ∞ ) of the functional equation $${f(xf(x))=\varphi (f(x))}$$ f ( x f ( x ) ) = φ ( f ( x ) ) where $${\varphi}$$ φ is a given continuous map from $${\mathbb{R}_{+}}$$ R + to $${\mathbb{R}_{+}}$$ R + . A solution f is singular if there are a and b such that $${0<a\le b<\infty}$$ 0 < a ≤ b < ∞ , $${f|_{(0,a)} > 1}$$ f | ( 0 , a ) > 1 , $${f|_{[a,b]} \equiv 1}$$ f | [ a , b ] ≡ 1 , and f|(b,∞)<1; all other solutions are regular. It is known that the range R f of a singular solution can contain, for every positive integer n, a periodic point of $${\varphi}$$ φ of period n. In this paper we show that the range of a regular solution f contains no periodic point of $${\varphi}$$ φ of period different from 1 and 2. Our proof is essentially based on a recent result that, for regular solutions, $${\varphi |_{R_f}}$$ φ | R f has zero topological entropy. Since there are regular solutions containing periodic points of period 2 in the range, we get the best possible result. Springer Basel, 2015 Iterative functional equation nationallicence invariant curves nationallicence real solutions nationallicence regular solutions nationallicence topological entropy nationallicence periodic orbits nationallicence Reich L. Institut für Mathematik, Karl-Franzens Universität Graz, 8010, Graz, Austria aut Smítal J. Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic aut Štefánková M. Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic aut Results in Mathematics Springer Basel 67/3-4(2015-06-01), 521-528 1422-6383 67:3-4<521 2015 67 25 https://doi.org/10.1007/s00025-015-0437-3 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/s00025-015-0437-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Reich L. Institut für Mathematik, Karl-Franzens Universität Graz, 8010, Graz, Austria aut NATIONALLICENCE 700 1- Smítal J. Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic aut NATIONALLICENCE 700 1- Štefánková M. Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic aut NATIONALLICENCE 773 0- Results in Mathematics Springer Basel 67/3-4(2015-06-01), 521-528 1422-6383 67:3-4<521 2015 67 25