An extension of Kannappan's functional equation on semigroups (Q6623513)

Consider a semigroup \(S\) with center \(Z(S):=\{s\in S\colon xs=sx\;\mbox{for all}\;x\in S\}\). The authors determine solutions of the complex-valued solutions of Kannappan-d'Alembert's functional equation \N\[\N\int\limits_Sf(xyt)d\nu(t) +\int\limits_S f(\sigma (y)xt)d\nu(t) =2f(y)f(x), \qquad x,y \in S,\N\]\Nand of the Kannappan-Wilson's functional equation \N\[\N\int\limits_Sf(xyt)d\nu(t) + \int\limits_Sf(\sigma (y)xt)d\nu(t)= 2f(y)g(x), \qquad x,y\in S.\N\]\NHere \(\mu\) 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 \(i\in I\) and \(\sigma \colon S\to S\) is an involutive automorphism or an involutive anti-automorphism for the first equation and an involutive automorphism for the second one. As\Na byproduct, the solutions of the Wilson-Kannappan's functional equation \N\[\Nf(xyz_0) + f(\sigma(y)xz_0)= 2f(y)g(x), \qquad x,y\in S,\N\]\Nand the Jensen-Kannappan's functional equation \N\[\Nf(xyz_0) + f (\sigma(y)xz_0)=2f(y),\qquad x,y\in S,\N\]\Nwith fixed \(z_0\in Z(S)\), are found.