Nice results about quadratic type functional equations on semigroups

Let ( S , + ) be an abelian semigroup, let σ be an involution of S , let X be a linear space over the field K ∈ { R , C } and let μ , ν be linear combinations of Dirac measures. In the present paper, we find the general solution f : S → X of the following functional equation ∫ S f ( x + y + t ) d μ ( t ) + ∫ S f ( x + σ ( y ) + t ) d ν ( t ) = f ( x ) + f ( y ) , x , y ∈ S , in terms of additive and bi-additive maps. Many consequences of this result are presented.