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On the stability of Drygas functional equation on groups. (English) Zbl 1130.39023
Suppose that $G$ is a group. The system of functional equations $$\cases f(xy)+f(xy^{-1})-2f(x)-f(y)-f(y^{-1})=0\\ f(yx)+f(y^{-1}x)-2f(x)-f(y)-f(y^{-1})=0\endcases \tag $*$ $$ where $x, y\in G $ is called stable if for any $f:G\to \Bbb{R}$ satisfying the system of inequalities $$\cases \vert f(xy)+f(xy^{-1})-2f(x)-f(y)-f(y^{-1})\vert \leq \delta\\ \vert f(yx)+f(y^{-1}x)-2f(x)-f(y)-f(y^{-1})\vert \leq \delta\endcases$$ for some positive number $\delta$, there is a solution $\varphi$ of ($*$) and a positive number $\varepsilon$ such that $\vert f(x)-\varphi(x)\vert \leq \varepsilon$ $(x\in G)$. In this paper the authors prove that the system ($*$) is not stable on an arbitrary group, in general; the system is stable on Heisenberg group $$ UT(3, K)\left\{ \left[\matrix 1 & y & t\\ 0&1 &x\\0&0&1\\ \endmatrix \right]: x, y, t \in K \right \}, $$ where $K$ is a (commutative) field with characteristic different from two; the system is stable on certain class of $n$-Abelian groups; and finally that any group can be embedded into a group, where the system ($*$) is stable. See also {\it V. A. Fauiziev} and {\it P. K. Sahoo} [Stability of Drygas functional equation on $T(3,\Bbb R)$, Int. J. Appl. Math. Stat. 7, No. Fe07, 70--81 (2007)].

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
39B72Systems of functional equations and inequalities
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