On the Alienation of Multiplicative and Additive Functions

Abstract

Abstract Given S a semigroup. We study two Pexider-type functional equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mi>f</m:mi> <m:mfenced> <m:mrow> <m:mi>x</m:mi> <m:mi>y</m:mi> </m:mrow> </m:mfenced> <m:mo>+</m:mo> <m:mi>g</m:mi> <m:mfenced> <m:mrow> <m:mi>x</m:mi> <m:mi>y</m:mi> </m:mrow> </m:mfenced> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mfenced> <m:mi>x</m:mi> </m:mfenced> <m:mo>+</m:mo> <m:mi>f</m:mi> <m:mfenced> <m:mi>y</m:mi> </m:mfenced> <m:mo>+</m:mo> <m:mi>g</m:mi> <m:mfenced> <m:mi>x</m:mi> </m:mfenced> <m:mi>g</m:mi> <m:mfenced> <m:mi>y</m:mi> </m:mfenced> <m:mo>,</m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mo> </m:mo> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mstyle displaystyle="true"> <m:mrow> <m:msub> <m:mo>∫</m:mo> <m:mi>S</m:mi> </m:msub> <m:mrow> <m:mi>f</m:mi> <m:mfenced> <m:mrow> <m:mi>x</m:mi> <m:mi>y</m:mi> <m:mi>t</m:mi> </m:mrow> </m:mfenced> <m:mi>d</m:mi> <m:mo>μ</m:mo> <m:mfenced> <m:mi>t</m:mi> </m:mfenced> <m:mo>+</m:mo> <m:mstyle displaystyle="true"> <m:mrow> <m:msub> <m:mo>∫</m:mo> <m:mi>S</m:mi> </m:msub> <m:mrow> <m:mi>g</m:mi> <m:mfenced> <m:mrow> <m:mi>x</m:mi> <m:mi>y</m:mi> <m:mi>t</m:mi> </m:mrow> </m:mfenced> <m:mi>d</m:mi> <m:mo>μ</m:mo> <m:mfenced> <m:mi>t</m:mi> </m:mfenced> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mfenced> <m:mi>x</m:mi> </m:mfenced> <m:mo>+</m:mo> <m:mi>f</m:mi> <m:mfenced> <m:mi>y</m:mi> </m:mfenced> <m:mo>+</m:mo> <m:mi>g</m:mi> <m:mfenced> <m:mi>x</m:mi> </m:mfenced> <m:mi>g</m:mi> <m:mfenced> <m:mi>y</m:mi> </m:mfenced> <m:mo>,</m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> </m:mrow> </m:mrow> </m:mstyle> </m:mrow> </m:mrow> </m:mstyle> </m:mrow> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mfenced close="" open="{"> <m:mrow> <m:mtable columnalign="left" equalrows="true" equalcolumns="true"> <m:mtr columnalign="left"> <m:mtd columnalign=