Singular solutions of the generalized Dhombres functional equation (Q480898)

The authors consider the generalized Dhombres functional equation \[ f(xf(x))=\phi(f(x)), \] where \(\phi\) is given and \(f\) is an unknown continuous map from \(\mathbb R_+\) into \(\mathbb R_+\); the set of the continuous solutions is denoted by \(S(\phi)\). A solution \(f\) is regular if the sets \(R_f \cap (0,1]\) and \(R_f \cap [1,\infty)\), where \(R_f\) is the range of \(f\), are \(\phi\)-invariant; otherwise \(f\) is singular. To each \(f\in S(\phi)\) is naturally associated the dynamical system \((R_f,\phi_{|R_f})\). The paper investigates the properties of the singular solution of the functional equation. The main results of the paper can be summarized as follows: (i) Every singular \(f\in S(\phi)\) is non-increasing on the set of points which are mapped by \(f\) into the set \(P(\phi)\) of periodic points of \(\phi\); (ii) There are a \(\phi\) and a singular \(f\in S(\phi)\) such that \(\phi_{|R_f}\) contains exactly one fixed point, exactly one periodic orbit of period \(2\), and no other periodic points; (iii) There is a singular \(f\in S(\phi)\) such that \(\phi_{|R_f}\) possesses a periodic point of period \(3\) hence, for every \(n\in \mathbb N\), a periodic orbit of period \(n\).