Integral Kannappan-Sine subtraction and addition law on semigroups

Abstract

Let $S$ be a semigroup, $\mu$ a discrete measure on $S$ and $\sigma :S⟶S$ is an involutive automorphism. We determine the complex-valued solutions of the integral Kannappan-Sine subtraction law $${\int }_{S}f(x\sigma (y)t)d\mu (t)=f(x)g(y)-f(y)g(x),x,y\in S,$$ and the integral Kannappan-Sine addition law $${\int }_{S}f(x\sigma (y)t)d\mu (t)=f(x)g(y)+f(y)g(x),x,y\in S.$$ We express the solutions by means of exponentials on S, the solutions of the special sine addition law $f(xy)=f(x)\chi (y)+f(y)\chi (x),$ $x,y\in S$ and the solutions of of the special case of the integral Kannappan-Sine addition law ${\int }_{S}f(x\sigma (y)t)d\mu (t)=[f(x)\chi (y)+f(y)\chi (x)]{\int }_S\chi (t)d\mu (t),$ $x,y\in S$, and where $\chi$: $S⟶\mathbb{C}$ is an exponential. The continuous solutions on topological semigroups are also given.