On a Generalized Dhombres Functional Equation Ii

We consider the functional equation f(xf(x)) = φ(f(x)) where φ : J → J is a given increasing homeomorphism of an open interval J ⊂ (0,∞) and f : (0,∞) → J is an unknown continuous function. In a previous paper we proved that no continuous solution can cross the line y = p where p is a fixed point of φ, with a possible exception for p = 1. The range of any non-constant continuous solution is an interval whose end-points are fixed by φ and which contains in its interior no fixed point except for 1. We also gave a characterization of the class of continuous monotone solutions and proved a sufficient condition for any continuous function to be monotone. In the present paper we give a characterization of the equations (or equivalently, of the functions φ) which have all continuous solutions monotone. In particular, all continuous solutions are monotone if either (i) 1 is an end-point of J and J contains no fixed point of φ, or (ii) 1 ∈ J and J contains no fixed points different from 1.