On a composite functional equation fulfilled by modulus of an additive function

Abstract

We deal with the problem of determining general solutions f : R → R of the following composite functional equation introduced by Fechner: f(f(x) − f(y)) = f(x + y) + f(x − y) − f(x) − f(y). Our result gives a partial answer to this problem under some assumptions upon f(R). We are applying a theorem of Simon and Volkmann concerning a certain characterization of modulus of an additive function. A new proof of their result is also presented.