caa a22 4500 605508089 CHVBK 20210128100635.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s00010-014-0272-8 doi (NATIONALLICENCE)springer-10.1007/s00010-014-0272-8 On regular solutions of the generalized Dhombres equation [Elektronische Daten] [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 $${\mathbb{R}_{+} \rightarrow \mathbb{R}_{+}}$$ R + → R + . A solution f is singular if there are $${0 < a \leq b< \infty}$$ 0 < a ≤ b < ∞ such that $${f|_{(0,a)} > 1, f|_{[a,b]} \equiv 1}$$ f | ( 0 , a ) > 1 , f | [ a , b ] ≡ 1 , and $${f|_{(b,\infty)} < 1}$$ f | ( b , ∞ ) < 1 ; other solutions are regular. It is known that the range R f of a singular solution can contain periodic orbits of $${\varphi}$$ φ of all periods. In this paper we show that the range of a regular solution f contains no periodic point of $${\varphi}$$ φ of period different from $${2^n, n \in \mathbb{N}}$$ 2 n , n ∈ N so that $${\varphi|_{R_f}}$$ φ | R f has zero topological entropy. It follows, that the regular solutions are just the solutions f satisfying one of the conditions: (i) $${R_{f} \subseteq (0,1]}$$ R f ⊆ ( 0 , 1 ] , (ii) $${R_{f} \subseteq [1, \infty)}$$ R f ⊆ [ 1 , ∞ ) , (iii) there are $${0 <a \leq b< \infty}$$ 0 < a ≤ b < ∞ such that $${f|_{(0,a)} < 1}$$ f | ( 0 , a ) < 1 , $${f|_{[a,b]} \equiv 1}$$ f | [ a , b ] ≡ 1 , and $${f|_{(b,\infty)}>1}$$ f | ( b , ∞ ) > 1 . Springer Basel, 2014 Iterative functional equation nationallicence Invariant curves nationallicence Real solutions nationallicence Periodic orbits nationallicence Smítal J. Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic aut Štefánková M. Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic aut Aequationes mathematicae Springer Basel 89/1(2015-02-01), 57-61 0001-9054 89:1<57 2015 89 10 https://doi.org/10.1007/s00010-014-0272-8 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/s00010-014-0272-8 text/html Onlinezugriff via DOI 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- Aequationes mathematicae Springer Basel 89/1(2015-02-01), 57-61 0001-9054 89:1<57 2015 89 10