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

Abstract

We deal with the problem of determining general solutions ${f\colon\mathbb{R}\to\mathbb{R}}$ 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 ${f(\mathbb{R})}$. 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.