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D’Alembert’s Functional Equations on Monoids with an Anti-endomorphism

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Abstract

Let M be a topological monoid. Our main goal is to introduce the functional equation

$$\begin{aligned} g(xy)+\mu (y)g(x\psi (y))=2g(x)g(y),\ \ x,y\in M, \end{aligned}$$

where \(\psi :M\rightarrow M\) is a continuous anti-endomorphism that need not be involutive and \(\mu :M\rightarrow \mathbb {C}\) is a continuous multiplicative function such that \(\mu (x\psi (x))=1\) for all \(x\in M\). We exploit results on the pre-d’Alembert functional equation by Davison (Publ Math Debrecen 75(1–2):41–66, 2009) and Stetkær (Functional equations on groups. World Scientific Publishing Company, Singapore (2013)) to prove that the continuous solutions \( g:M\rightarrow \mathbb {C}\) of this equation can be expressed in terms of multiplicative functions and characters of 2-dimensional representations of M. Interesting consequences of this result are presented.

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Acknowledgements

Our sincere regards and gratitude go to Professor Henrik Stetkær for fruitful discussions and for many valuable comments which have led to an essential improvement of the paper.

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Correspondence to Driss Zeglami.

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Ayoubi, M., Zeglami, D. D’Alembert’s Functional Equations on Monoids with an Anti-endomorphism. Results Math 75, 74 (2020). https://doi.org/10.1007/s00025-020-01199-z

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