A Kannappan-sine subtraction law on semigroups

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