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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 \[ \begin{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\end{cases} \tag \(*\) \] where \(x, y\in G \) is called stable if for any \(f:G\to \mathbb{R}\) satisfying the system of inequalities
\[ \begin{cases} | f(xy)+f(xy^{-1})-2f(x)-f(y)-f(y^{-1})| \leq \delta\\ | f(yx)+f(y^{-1}x)-2f(x)-f(y)-f(y^{-1})| \leq \delta\end{cases} \] for some positive number \(\delta\), there is a solution \(\varphi\) of (\(*\)) and a positive number \(\varepsilon\) such that \(| f(x)-\varphi(x)| \leq \varepsilon\) \((x\in G)\).
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 \[ UT(3, K)\left\{ \left[\begin{matrix} 1 & y & t\\ 0&1 &x\\0&0&1\\ \end{matrix} \right]: x, y, t \in K \right \}, \] 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 \(T(3,\mathbb R)\), Int. J. Appl. Math. Stat. 7, No. Fe07, 70–81 (2007)].

MSC:

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