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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

$\left\{\begin{array}{c}f\left(xy\right)+f\left(x{y}^{-1}\right)-2f\left(x\right)-f\left(y\right)-f\left({y}^{-1}\right)=0\hfill \\ f\left(yx\right)+f\left({y}^{-1}x\right)-2f\left(x\right)-f\left(y\right)-f\left({y}^{-1}\right)=0\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $x,y\in G$ is called stable if for any $f:G\to ℝ$ satisfying the system of inequalities

$\left\{\begin{array}{c}|f\left(xy\right)+f\left(x{y}^{-1}\right)-2f\left(x\right)-f\left(y\right)-f\left({y}^{-1}\right)|\le \delta \hfill \\ |f\left(yx\right)+f\left({y}^{-1}x\right)-2f\left(x\right)-f\left(y\right)-f\left({y}^{-1}\right)|\le \delta \hfill \end{array}\right\$

for some positive number $\delta$, there is a solution $\varphi$ of ($*$) and a positive number $\epsilon$ such that $|f\left(x\right)-\varphi \left(x\right)|\le \epsilon$ $\left(x\in G\right)$.

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

$UT\left(3,K\right)\left\{\left[\begin{array}{ccc}1& y& t\\ 0& 1& x\\ 0& 0& 1\end{array}\right]:x,y,t\in K\right\},$

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 V. A. Fauiziev and P. K. Sahoo [Stability of Drygas functional equation on $T\left(3,ℝ\right)$, Int. J. Appl. Math. Stat. 7, No. Fe07, 70–81 (2007)].

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges 39B72 Systems of functional equations and inequalities