Abstract
Let \(S\) be a semigroup, \(Z(S)\) the center of \(S\) and \(\sigma \colon S \rightarrow S\) is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation
\(\int_{S} f(xyt)\, d\mu(t) + \int_{S} f(\sigma(y)xt)\, d\mu(t)= 2f(x)g(y),\ \ x,y\in S,\)
and the Van Vleck-Wilson functional equation
\(\int_{S} f(xyt)\, d\mu(t) - \int_{S} f(\sigma(y)xt)\, d\mu(t)= 2f(x)g(y),\ \ x,y\in S,\)
where \(\mu\) is a measure that is a linear combination of Dirac measures \((\delta_{z_i})_{i\in I}\), such that \(z_i\in Z(S)\) for all \(i\in I\). Interesting consequences of these results are presented.
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Acknowledgements
Our sincere regards and gratitude go to Professor Henrik Stetkær for many valuable comments on our papers. We would also like to express our thanks to the referees for useful comments.
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Aserrar, Y., Elqorachi, E. Kannappan–Wilson and Van Vleck–Wilson functional equations on semigroups. Acta Math. Hungar. (2024). https://doi.org/10.1007/s10474-024-01433-y
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DOI: https://doi.org/10.1007/s10474-024-01433-y