On the Alienation of Multiplicative and Additive Functions

Abstract Given S a semigroup. We study two Pexider-type functional equations f x y + g x y = 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 ∫ S f x y t d μ t + ∫ S g x y t d μ 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 z i ) 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 x y = f x + f y   , g x y = g x g y ,         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 ∫ S f x y t d μ t = f x + f y , ∫ S g x y t d μ 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),} } \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.