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 ʃs f(σ (y)xt)dμ(t)- ʃs (xyt)dμ(t) = 2f(x)f(y); x; y ϵ s,where s is a semigroup, σ is an involutive morphism of s, and μ is a complex measure that is linear combinations of dirac measures (δ zi) iϵi, such that for all i ϵ i, zi 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 ʃs f(xty)dμ(t) + ʃs f(σ(y)tx)dμ(t)= 2f(x)f(y); x; y ϵ s, where s is a topological semigroup, σ is a continuous involutive automorphism of s, and  is a complex measure with compact support and which is σ-invariant. (3) we prove the superstability theorems of the first functional equation.