On a generalized Dhombres functional equation

Abstract

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.