On the Alienation of Multiplicative and Additive Functions

Given S a semigroup. We study two Pexider-type functional equations f(xy)+g(xy)=f(x)+f(y)+g(x)g(y), x,y ∈ S, f\left( {xy} \right) + g\left( {xy} \right) = f\left( x \right) + f\left( y \right) + g\left( x \right)g\left( y \right),\,\,\,\,\,\,\,x,y\,\, \in \,\,S, and ∫Sf(xyt)dμ(t)+∫sg(xyt)dμ(t)=f(x)+f(y)+g(x)g(y), x,y∈S, \int_S {f\left( {xyt} \right)d\mu \left( t \right) + \int_s {g\left( {xyt} \right)d\mu \left( t \right) = f\left( x \right) + f\left( y \right) + g\left( x \right)g\left( y \right),\,\,\,\,\,\,\,x,y \in S,} } for unknown functions f and g mapping S into ℂ, where µ is a linear combination of Dirac measures (δ_zi )_i∈I for some fixed elements (z_i)_i∈I contained in S such that ∫_S dµ(t) = 1.

The main goal of this paper is to solve the above two functional equations and examine whether or not they are equivalent to the systems of equations { f(xy)=f(x)+f(y),g(xy)=g(x)g(y), x,y∈S, {\left\{ {\matrix{{f\left( {xy} \right) = f\left( x \right) + f\left( y \right),} \cr {g\left( {xy} \right) = g\left( x \right)g\left( y \right),\,\,\,\,\,x,y \in S,} \cr } } \right.} and { ∫Sf(xyt)dμ(t)=f(x)+f(y),∫Sg(xyt)dμ(t)=g(x)g(y), x,y∈S, \left\{ {\matrix{{\int_S {f\left( {xyt} \right)d\mu \left( t \right)} = f\left( x \right) + f\left( y \right),} \cr {\int_S {g\left( {xyt} \right)d\mu \left( t \right) = g\left( x \right)g\left( y \right),\,\,\,\,\,x,y \in S,} } \cr } } \right. respectively.