Alienation of the quadratic, exponential and d’Alembert functional equations

Abstract

Let $$(S,+,0)$$ ( S , + , 0 ) be a commutative monoid, $$\sigma :S\rightarrow S$$ σ : S → S be an endomorphism with $$\sigma ^2=id$$ σ 2 = i d and let K be a field of characteristic different from 2. We study the solutions $$f,g,h:S\rightarrow K$$ f , g , h : S → K of the Pexider type functional equation $$\begin{aligned} f(x+y)+f(x+\sigma y)+g(x+y)=2f(x)+2f(y)+g(x)g(y) \end{aligned}$$ f ( x + y ) + f ( x + σ y ) + g ( x + y ) = 2 f ( x ) + 2 f ( y ) + g ( x ) g ( y ) resulting from summing up the well known quadratic and exponential functional equations side by side. We show that under some additional assumptions the above equation forces f and g to split back into the system of two equations $$\begin{aligned} \left\{ \begin{array}{ll}f(x+y)+f(x+\sigma y)=2f(x)+2f(y)\\ g(x+y)=g(x)g(y)\end{array}\right. \end{aligned}$$ f ( x + y ) + f ( x + σ y ) = 2 f ( x ) + 2 f ( y ) g ( x + y ) = g ( x ) g ( y ) for all $$x,y\in S$$ x , y ∈ S (alienation phenomenon). We also consider an analogous problem for the quadratic and d’Alembert functional equations as well as for the quadratic, exponential and d’Alembert functional equations.