ABSTRACT
Let A be a set of positive reals, I be a real interval and be a set of functions of I into itself. We determine necessary and sufficient conditions on A, and I for the system of Poincaré functional equation to have a continuous increasing solution which is non-constant. It turns out that such a solution exists whenever A is closed under multiplication and is a multiplicative iteration semigroup of increasing continuous functions satisfying a specific density condition and I is a half-open interval. Finally, we show that such a solution is unique up to an internal multiplicative constant.
Disclosure statement
No potential conflict of interest was reported by the author(s).