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On the functional equation \(f(x+g(y))-f(y+g(y))=f(x)-f(y)\) on groups

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We solve the equation

$$f(x+g(y)) - f(y + g(y)) = f(x) - f(y)$$

in the class of pairs of functions (f, g), where \(f : G \rightarrow H\), \(g : Y \rightarrow G,\quad (G,+) \quad {\rm and} \quad (H,+)\) are abelian groups and Y is a non-empty subset of G. We make it by proving that the equation above and some conditional Cauchy equation are equivalent.

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Correspondence to Marcin Balcerowski.

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Manuscript received: September 24, 2007 and, in final form, December 15, 2008.

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Balcerowski, M. On the functional equation \(f(x+g(y))-f(y+g(y))=f(x)-f(y)\) on groups. Aequat. Math. 78, 247 (2009).

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Mathematics Subject Classification (2000).