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On the Pexider difference. (English) Zbl 0715.39012

The authors consider the Pexider difference \(f(x+y)-g(x)-h(y),\) where f,g,h: \(G\to H\), G is a groupoid with identity and H is a group. They prove that if the Pexider difference assume values in a subgroup K of H, then there are functions k,\(\ell: G\to K\), \(\phi: G\to H\) and constants a and b in H such that \(\phi (x+y)-\phi (x)-\phi (y)\in K\) and \(f(x)=k(x)+\phi (x)+a,\) \(g(x)=b-a+\phi (x)+a,\) \(h(x)=\ell (x)+\phi (x)+a- b.\) Afterwards they obtain the representation of the Cauchy difference \(\phi (x+y)-\phi (x)-\phi (y)\in K\) under various topological and measure-theoretic conditions.
Reviewer: G.L.Forti

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
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