Integral Kannappan-Sine subtraction and addition law on semigroups
Let $S$ be a semigroup, $μ$ a discrete measure on $S$ and $σ:S \longrightarrow S$ is an involutive automorphism. We determine the complex-valued solutions of the integral Kannappan-Sine subtraction law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)-f(y)g(x),\; x,y \in S,$$ and the integral Kannappan-Sine addition law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)+f(y)g(x),\; x,y \in S.$$ We express the solutions by means of exponentials on S, the solutions of the special sine addition law $f(xy)=f(x)χ(y)+f(y)χ(x),$ $x,y\in S$ and the solutions of of the special case of the integral Kannappan-Sine addition law $\int_{S}f(xσ(y)t)dμ(t)=[f(x)χ(y)+f(y)χ(x)]\int_{S}χ(t)dμ(t), $ $x,y\in S$, and where $χ$: $S\longrightarrow \mathbb{C}$ is an exponential. The continuous solutions on topological semigroups are also given.