Stability of Maximum Functional Equation and Some Properties of Groups

Abstract

In this research paper, we deal with the problem of determining the function χ:G→R, which is the solution to the maximum functional equation (MFE) max{χ(xy),χ(xy−1)}=χ(x)χ(y), when the domain is a discretely normed abelian group or any arbitrary group G. We also analyse the stability of the maximum functional equation max{χ(xy),χ(xy−1)}=χ(x)+χ(y) and its solutions for the function χ:G→R, where G be any group and also investigate the connection of the stability with commutators and free abelian group K that can be embedded into a group G.