On a generalized Dhombres functional equation

Summary.

We consider the functional equation $f(xf(x))=\varphi (f(x))$ where $\varphi: J\rightarrow J$ is a given increasing homeomorphism of an open interval $J \subset (0,\infty)$, and $f:(0,\infty )\rightarrow J$ is an unknown continuous function. If 1 is a fixed point of $\varphi$ then the solutions are pointwise complete in the class of continuous monotone functions, i.e., for any point $(x_0,y_0) \in (0,\infty) \times J$ there is a continuous monotone solution passing through it. On the other hand, if 1 is not fixed by $\varphi$ then there exist exceptional values y ₀ if and only if there is a minimal open $\varphi$-invariant interval $K \subset J$ (i.e., $\varphi (K)=K$) containing 1; the exceptional values of y ₀ are then just the elements of K. We also show that the continuous solutions cannot cross the line y = p where p is a fixed point of $\varphi$, unless p = 1.¶We give a characterization of the class of monotone continuous solutions; it depends on a family of monotone functions defined on a compact “initial” interval. We provide a sufficient condition for any continuous solution to be monotone. It implies, among others, that 0 or $\infty$; is an end-point of J. We conjecture that this condition is also necessary.