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- Title
The kernel of the second order Cauchy difference on semigroups.
- Authors
Stetkær, Henrik
- 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.
- Subjects
KERNEL (Mathematics); CAUCHY problem; SEMIGROUPS (Algebra); MATHEMATICAL functions; FUNCTIONAL equations
- Publication
Aequationes Mathematicae, 2017, Vol 91, Issue 2, p279
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-016-0453-8