On the Alienation of Multiplicative and Additive Functions

Given S a semigroup. We study two Pexider-type functional equations fxy+gxy=fx+fy+gxgy, 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 ∫Sfxytdμt+∫Sgxytdμt=fx+fy+gxgy, 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 fxy=fx+fy ,gxy=gxgy, x, y∈S, \left\{ {\matrix{ {f\left( {xy} \right) = f\left( x \right) + f\left( y \right)\;,} \hfill \cr {g\left( {xy} \right) = g\left( x \right)g\left( y \right), \;x,\;y \in S,} \hfill \cr } } \right. and ∫Sfxytdμt=fx+fy, ∫Sgxytdμt=gxgy, x, y∈S, \left\{ {\matrix{ {\int_S {f\left( {xyt} \right)d\mu \left( t \right) = f\left( x \right) + f\left( y \right),} } \hfill \cr {\int_S {g\left( {xyt} \right)d\mu \left( t \right) = g\left( x \right)g\left( y \right), \;x,\;y \in S,} } \hfill \cr } } \right. respectively.