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Wilson’s functional equation with an anti-endomorphism

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Abstract

Let M be a topological monoid, let \(\psi :M\rightarrow M\) be a continuous anti-homomorphism of M, and let \(\mu :M\rightarrow {\mathbb {C}}\) be a continuous multiplicative function such that \(\mu (x\psi (x))=1\) for all \( x\in M\). We describe, in terms of multiplicative functions, additive functions and characters of 2-dimensional representations of M, the solutions (w, g), where \(w,g:M\rightarrow {\mathbb {C}}\), such that w is central and g is continuous, of the new functional equation

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

We also treat the equation on compact groups and the special equation (when \( \mu =g=1\)): Jensen’s functional equation.

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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. We would also like to express our thanks to the referees for useful comments.

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  1. Department of Mathematics, E.N.S.A.M, Moulay Ismaïl University, B.P: 15290, Al Mansour, Meknes, Morocco

    M. Ayoubi, D. Zeglami & Y. Aissi

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  1. M. Ayoubi
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  2. D. Zeglami
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  3. Y. Aissi
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Correspondence to D. Zeglami.

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Ayoubi, M., Zeglami, D. & Aissi, Y. Wilson’s functional equation with an anti-endomorphism. Aequat. Math. 95, 535–549 (2021). https://doi.org/10.1007/s00010-020-00748-9

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  • DOI: https://doi.org/10.1007/s00010-020-00748-9

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