Differentiable solutions of an equation with product of iterates

TL;DR

This paper investigates differentiable solutions to a specific iterative equation involving products of iterates, extending previous work on continuous solutions by establishing existence and uniqueness results on positive and negative real axes.

Contribution

It introduces a method to analyze differentiable solutions on non-compact intervals by reducing the problem to polynomial-like iterative equations and proves their existence and uniqueness on  and .

Findings

Existence of differentiable solutions on  and .

Uniqueness of these solutions on  and .

Extension of previous continuous solution results to differentiable solutions on  and .

Abstract

In the previous work [2] (i.e., arXiv:2105.03385), we considered continuous solutions of an iterative equation involving the multiplication of iterates. In this paper, we continue to investigate this equation for differentiable solutions. Similar to continuous solutions until [2], there is no obtained result on differentiable solutions of such an equation on non-compact intervals of $\mathbb{R}$. Although our strategy here is to use conjugation to reduce the equation to the well-known polynomial-like iterative equation as in [2], all known results on differentiable solutions of the latter are given on compact intervals. We re-explore polynomial-like iterative equation on the whole of R and prove the existence and uniqueness of differentiable solutions of our equation on $\mathbb{R}_+$ and $\mathbb{R}_-$.