An extension of Kannappan’s functional equation on semigroups

Let S be a semigroup, Z(S) the center of S. In this paper, we determine the complex-valued solutions of Kannappan–d’Alembert’s functional equation \(\begin{aligned}\displaystyle \int _{S} f(xyt)d\mu (t) +\displaystyle \int _{S} f(\sigma (y)xt)d\mu (t)= 2f(y)f(x),\ x,y\in S,\end{aligned}\) $$\begin{array}{c}\hfill {\displaystyle {\int }_{S}f(xyt)d\mu (t)+{\int }_{S}f(\sigma (y)xt)d\mu (t)=2f(y)f(x),x,y\in S,}\end{array}$$ and Kannappan–Wilson’s functional equation \(\begin{aligned}\displaystyle \int _{S} f(xyt)d\mu (t) +\displaystyle \int _{S} f(\sigma (y)xt)d\mu (t)= 2f(y)g(x),\ x,y\in S,\end{aligned}\) $$\begin{array}{c}\hfill {\displaystyle {\int }_{S}f(xyt)d\mu (t)+{\int }_{S}f(\sigma (y)xt)d\mu (t)=2f(y)g(x),x,y\in S,}\end{array}$$ where \(\mu \) $\mu$ is a measure that is a linear combination of Dirac measures \((\delta _{z_i})_{i\in I}\) ${({\delta }_{z_i})}_{i\in I}$ , such that \(z_i\in Z(S)\) $z_i\in Z(S)$ for all \(i\in I\) $i\in I$ , and \(\sigma :S\rightarrow S\) $\sigma :S\to S$ is an involutive automorphism or an involutive anti-automorphism for the first equation and an involutive automorphism for the second one. We also give some interesting applications.