A Kannappan-sine subtraction law on semigroups

Let $S$ be a semigroup, $z_0$ a fixed element in $S$ and $\sigma :S⟶S$ an involutive automorphism. We determine the complex-valued solutions of Kannappan-sine subtraction law $f(x\sigma (y)z_0)=f(x)g(y)-f(y)g(x),x,y\in S$. As an application we solve the following variant of Kannappan-sine subtraction law viz. $f(x\sigma (y)z_0)=f(x)g(y)-f(y)g(x)+\lambda g(x\sigma (y)z_0),x,y\in S,$ where $\lambda \in {\mathbb{C}}^{\ast }$. The continuous solutions on topological semigroups are given and an example to illustrate the main results is also given.