Solutions and stability of generalized Kannappan's and Van Vleck's functional equations

We study the solutions of the integral Kannappan’s and Van Vleck’s functional equations $${\int }_{S}f(xyt)d\mu (t)+{\int }_{S}f(x\sigma (y)t)d\mu (t)=2f(x)f(y),x,y\in S;$$ $${\int }_{S}f(x\sigma (y)t)d\mu (t)-{\int }_{S}f(xyt)d\mu (t)=2f(x)f(y),x,y\in S,$$ where $S$ is a semigroup, $\sigma$ is an involutive automorphism of $S$ and $\mu$ is a linear combination of Dirac measures $({\delta }_{z_i}{)}_{i\in I}$, such that for all $i\in I$, $z_i$ is contained in the center of $S$. We show that the solutions of these equations are closely related to the solutions of the d’Alembert’s classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems that these functional equations are superstable in the general case, where $\sigma$ is an involutive morphism.