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.