Approximately quintic and sextic mappings on the probabilistic normed spaces (Q2880473)

The article deals with two systems of functional equations NEWLINE\[NEWLINE\begin{cases} f(ax_1 + bx_2,y) + f(ax_1 - bx_2,y) = 2a^2f(x_1,y) + 2b^2f(x_2,y), \\ f(x,ay_1 + by_2) + f(x,ay_1 - by_2)\\ \qquad = ab^2f(x,y_1 + y_2) + f(x,y_1 - y_2) + 2a(a^2 - b^2)f(x,y_1) ,\end{cases}\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{cases} f(ax_1 + bx_2,y,z) + f(ax_1 - bx_2,y,z) = 2af(x_1,y,z), \\ f(x,ay_1 + by_2,z) + f(x,ay_1 - by_2,z) = 2a^2f(x,y_1,z) + 2b^2f(x,y_2,z), \\ f(x,y,az_1 + bz_2) + f(x,yaz_1 - bz_2)\\ \qquad = ab^2(f(x,y,z_1 + z_2) + f(x,y,z_1 - z_2) + 2a(a^2 - b^2)f(x,y,z_1) \end{cases}\tag{2}NEWLINE\]NEWLINE with \(a,b \in {\mathbb Z} \setminus \{0\}\), \(a \neq \pm 1, \pm b\) (the function \(f(x,y) = cx^2y^3\) is a solution to (1), the function \(f(x,y,z) = cxy^2z^3\) is a solution to (2)). For both systems, the following stability problem is investigated: if there exists an approximate solution \(\widetilde f\) satisfying the systems (1) (correspondingly (2)) with a sufficiently small residual, then there exists the exact solution \(f\) of the system (1) ((2)) sufficiently close to \(\widetilde f\). The smallness of the residuals and the proximity between \(f\) and \(\widetilde f\) are estimated in the probabilistic (due to Šerstnev) norm.