Increasing solutions of simultaneous Poincaré equations

ABSTRACT

Let A be a set of positive reals, I be a real interval and $\mathcal{A}\{f_{\alpha }:\alpha \in A\}$ be a set of functions of I into itself. We determine necessary and sufficient conditions on A, $\mathcal{A}$ and I for the system of Poincaré functional equation $\psi \left(\alpha x\right)=f_{\alpha }\left(\psi \right(x\left)\right)(\alpha \in A,x\in {\mathbb{R}}^{+})$to have a continuous increasing solution $\psi :{\mathbb{R}}^{+}\to I$ which is non-constant. It turns out that such a solution exists whenever A is closed under multiplication and $\mathcal{A}$ 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.

Keywords:

• Poincaré functional equation
• iteration semigroup
• increasing function

MSC (2020) Classifications:

• 39B12
• 39B22
• 26A18

Disclosure statement

No potential conflict of interest was reported by the author(s).