On regular solutions of the generalized Dhombres equation
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| 001 | 605508089 | ||
| 003 | CHVBK | ||
| 005 | 20210128100635.0 | ||
| 007 | cr unu---uuuuu | ||
| 008 | 210128e20150201xx s 000 0 eng | ||
| 024 | 7 | 0 | |a 10.1007/s00010-014-0272-8 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s00010-014-0272-8 | ||
| 245 | 0 | 0 | |a On regular solutions of the generalized Dhombres equation |h [Elektronische Daten] |c [J. Smítal, M. Štefánková] |
| 520 | 3 | |a 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 $${0 1}$$ f | ( b , ∞ ) > 1 . | |
| 540 | |a Springer Basel, 2014 | ||
| 690 | 7 | |a Iterative functional equation |2 nationallicence | |
| 690 | 7 | |a Invariant curves |2 nationallicence | |
| 690 | 7 | |a Real solutions |2 nationallicence | |
| 690 | 7 | |a Periodic orbits |2 nationallicence | |
| 700 | 1 | |a Smítal |D J. |u Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic |4 aut | |
| 700 | 1 | |a Štefánková |D M. |u Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic |4 aut | |
| 773 | 0 | |t Aequationes mathematicae |d Springer Basel |g 89/1(2015-02-01), 57-61 |x 0001-9054 |q 89:1<57 |1 2015 |2 89 |o 10 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00010-014-0272-8 |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/s00010-014-0272-8 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Smítal |D J. |u Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Štefánková |D M. |u Mathematical Institute, Silesian University, 746 01, Opava, Czech Republic |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Aequationes mathematicae |d Springer Basel |g 89/1(2015-02-01), 57-61 |x 0001-9054 |q 89:1<57 |1 2015 |2 89 |o 10 | ||