Differentiable solutions of an equation with product of iterates

Article Type

Research Article

Publication Title

Aequationes Mathematicae

Abstract

The author, together with M. Veerapazham, S. Wang and W. Zhang, considered the existence and uniqueness of continuous solutions to an iterative equation involving the multiplication of iterates in Gopalakrishna et al. (Sci. China Math., to appear). In this paper we continue to investigate this equation for differentiable solutions. Similarly to continuous solutions until Gopalakrishna et al. (Sci. China Math., to appear), no result on differentiable solutions of this equation on non-compact intervals of R has been obtained until now. Although our strategy here is to use the logarithmic conjugation to reduce the equation to the well-known polynomial-like iterative equation, all known results on differentiable solutions of the latter are given on compact intervals. We revisit polynomial-like iterative equations on the whole of R and use the Banach contraction principle to prove the existence and uniqueness of differentiable solutions to our equation on R+. We then technically extend our results on R+ to R- and show that none of the pairs of solutions obtained on R+ and R- can be combined at the origin to obtain even a continuous solution of the equation on the whole R.

First Page

853

Last Page

870

DOI

https://10.1007/s00010-023-00952-3

Publication Date

8-1-2023

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