On the stability of Drygas functional equation on groups

Abstract

In this paper, we study the stability of the system of functional equations $f(xy)+f(xy^{-1})=2f(x)+f(y)+f(y^{-1})$ and $f(yx)+f(y^{-1}x)=2f(x)+f(y)+f(y^{-1})$ on groups. Here $f$ is a real-valued function that takes values on a group. Among others we proved the following results: 1) the system, in general, is not stable on an arbitrary group; 2) the system is stable on Heisenberg group $UT(3, K)$, where $K$ is a commutative field with characteristic different from two; 3) the system is stable on certain class of $n$-Abelian groups; 4) any group can be embedded into a group where this system is stable.