An alternative Cauchy functional equation on a semigroup

Abstract

More than 33 years ago M. Kuczma and R. Ger posed the problem of solving the alternative Cauchy functional equation \({f(xy) - f(x) - f(y) \in \{ 0, 1\}}\) where \({f : S \to \mathbb{R}, S}\) is a group or a semigroup. In the case when the Cauchy functional equation is stable on S, a method for the construction of the solutions is known (see Forti in Abh Math Sem Univ Hamburg 57:215–226, 1987). It is well known that the Cauchy functional equation is not stable on the free semigroup generated by two elements. At the 44th ISFE in Louisville, USA, Professor G. L. Forti and R. Ger asked to solve this functional equation on a semigroup where the Cauchy functional equation is not stable. In this paper, we present the first result in this direction providing an answer to the problem of G. L. Forti and R. Ger. In particular, we determine the solutions \({f : H \to \mathbb{R}}\) of the alternative functional equation on a semigroup \({H = \langle a, b| a^2 = a, b^2 = b \rangle }\) where the Cauchy equation is not stable.