The kernel of the second order Cauchy difference on semigroups

Henrik Stetkær*

*Corresponding author af dette arbejde

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Abstract

Let S be a semigroup, H a 2-torsion free, abelian group and C 2f the second order Cauchy difference of a function f: S→ H. Assuming that H is uniquely 2-divisible or S is generated by its squares we prove that the solutions f of C 2f= 0 are the functions of the form f(x) = j(x) + B(x, x) , where j is a solution of the symmetrized additive Cauchy equation and B is bi-additive. Under certain conditions we prove that the terms j and B are continuous, if f is. We relate the solutions f of C 2f= 0 to Fréchet’s functional equation and to polynomials of degree less than or equal to 2.

OriginalsprogEngelsk
TidsskriftAequationes Mathematicae
Vol/bind91
Nummer2
Sider (fra-til)279–288
Antal sider10
ISSN0001-9054
DOI
StatusUdgivet - 2017

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