Alienation of Cauchy’s and the quadratic functional equations on semigroups

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.

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Alienation of Cauchy’s and the quadratic functional equations on semigroups

Cited by 4 publications

References 13 publications

Alienation of Drygas’, Cauchy’s, Jensen’s and the quadratic equations on semigroups with an involutive automorphism

Alienation of Drygas’, Cauchy’s, Jensen’s and the quadratic equations on semigroups with an involutive automorphism

Computer assisted investigation of alienness of linear functional equations

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

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