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The kernel of the second order Cauchy difference on semigroups

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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 \(f:S \rightarrow H\). 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.

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  1. Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000, Aarhus C, Denmark

    Henrik Stetkær

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  1. Henrik Stetkær
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Correspondence to Henrik Stetkær.

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Stetkær, H. The kernel of the second order Cauchy difference on semigroups. Aequat. Math. 91, 279–288 (2017). https://doi.org/10.1007/s00010-016-0453-8

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  • DOI: https://doi.org/10.1007/s00010-016-0453-8

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