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 integral(s) f(sigma(y)xt)d mu(t) - integral(s) f(xyt)d mu(t) = 2f (x) f (y), x, y is an element of 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(zi))(i is an element of I), such that for all i is an element of 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 integral(s) f(xty)d nu(t) - integral(s) f(sigma(y)tx)d nu(t) = 2f (x) f (y), x, y is an element of S, where S is a topological semigroup, sigma is a continuous involutive automorphism of S, and v is a complex measure with compact support and which is sigma-invariant. (3) We prove the superstability theorems of the first functional equation.