Abstract
Let \(S\) be a semigroup, let \(H\) be a \(2\)-torsion free abelian group, and let \(\varphi,\psi\colon S\to S\) be two endomorphisms that need not be involutive. We express the solutions \(f\colon S\to H\) of generalized variant of Jensen's functional equation \begin{equation*} f(x\varphi(y))+f(\psi(y)x)=2f(x), \quad x,y\in S, \end{equation*} in terms of additive maps. In all results it is supposed that at least one of the endomorphisms \(\varphi\) and \(\psi\) is surjective. Many consequences of this result are presented.
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Acknowledgement
I would like to thank the referee for a number of constructive comments which have led to an essential improvement of the paper.
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Akkaoui, A. Jensen’s functional equation on semigroups. Acta Math. Hungar. 170, 261–268 (2023). https://doi.org/10.1007/s10474-023-01341-7
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DOI: https://doi.org/10.1007/s10474-023-01341-7