Solutions and stability of a variant of Van Vleck's and d'Alembert's functional equations

In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck’s functional equation $${\int }_{S}f(\sigma (y)xt)d\mu (t)-{\int }_{S}f(xyt)d\mu (t)=2f(x)f(y),x,y\in S,$$ where $S$ is a semigroup, $\sigma$ is an involutive morphism of $S$, and $\mu$ is a complex measure that is linear combinations of Dirac measures $({\delta }_{z_i}{)}_{i\in I}$, such that for all $i\in I$, $z_i$ is contained in the center of $S$. (2) We determine the complex-valued continuous solutions of the following variant of d’Alembert’s functional equation $${\int }_{S}f(xty)d\upsilon (t)+{\int }_{S}f(\sigma (y)tx)d\upsilon (t)=2f(x)f(y),x,y\in S,$$ where $S$ is a topological semigroup, $\sigma$ is a continuous involutive automorphism of $S$, and $\upsilon$ is a complex measure with compact support and which is $\sigma$-invariant. (3) We prove the superstability theorems of the first functional equation.