Continuous solutions of an iterative equation with multiplication

TL;DR

This paper investigates continuous solutions to a specific iterative equation involving multiplication, using exponential functions to transform the problem and establishing existence, uniqueness, and stability of solutions on positive and negative real numbers.

Contribution

It introduces a novel approach to solving an iterative equation with multiplication by reducing it to a polynomial-like form and extends solutions across the entire real line.

Findings

Existence and uniqueness of solutions on +

Stability of solutions established

Solutions extended to -

Abstract

Iterative equation is an equality with an unknown function and its iterates. There were not found a result on iterative equations with multiplication of iterates of the unknown function on $\mathbb{R}$. In this paper we use an exponential function to reduce the equation in conjugation to the well-known form of polynomial-like iterative equation, but we encountered two difficulties: the reduction restricts our discussion of the equation to $\mathbb{R}_+$; the reduced polynomial-like iterative equation is defined on the whole $\mathbb{R}$ but known results were given on comapct intervals. We revisit the polynomial-like iterative equation on the whole $\mathbb{R}$ and give existence, uniqueness, stability and construction of continuous solutions of our equation on $\mathbb{R}_+$. Then we technically extend our solutions from $\mathbb{R}_+$ to $\mathbb{R}_-$.