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arXiv · 2105.03385

Continuous solutions of an iterative equation with multiplication

Chaitanya Gopalakrishna·Murugan Veerapazham·Suyun Wang·Weinian Zhang
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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 $\mathbb{R}_-$.

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Chaitanya Gopalakrishna, Murugan Veerapazham, Suyun Wang, Weinian Zhang. 2021-05-07. Continuous solutions of an iterative equation with multiplication. https://arxiv.org/abs/2105.03385

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