Integral Kannappan-cosine addition law on semigroups

Abstract

Let  $S$ be a semigroup,  $\sigma:S \longrightarrow S$ be an involutive automorphism, $\mu$ be a complex measure that is a  linear combination of Dirac measures and $\alpha \in \mathbb{C}$. We determine the  complex-valued solutions of  the following integral Kannappan-cosine addition law  with an additional term $$\int_{S}g(x\sigma(y)t) d\mu(t)=g(x)g(y)-f(x)f(y)+\alpha \int_{S}f(x\sigma(y)t) d\mu(t)  ,\; x,y \in S.$$   As application we solve two functional equations that have not been studied until now. The continuous solutions on topological semigroups are found.