Non-Archimedean stability of an AQQ functional equation (Q2917662)

The article deals with the stability properties of the following functional equation NEWLINE\[NEWLINE\begin{multlined} f(x + 2y) + f(x - 2y) = 2f(x + y) + 2f(-x - y) + 2f(x - y) + 2f(y - x) \\ -4f(-x) - 2f(x) + f(2y) + f(-2y) - 4f(y) - 4f(-y);\end{multlined}\tag{1}NEWLINE\]NEWLINE here \(f:\;G \to X\), \(G\) is an additive semigroup, \(X\) a complete non-Archimedean space. The authors consider functions \(f:\;G \to X\) satisfying the inequality NEWLINE\[NEWLINE\|\chi_f(x,y)\| \leq \zeta(x,y), \qquad x, y \in G,NEWLINE\]NEWLINE where \(\chi_f(x,y)\) is the difference between the left and right hand sides of (1), and \(\zeta:\;G \times G \to [0,+\infty)\) is a given function. Under some natural conditions for \(\zeta\), the authors construct a function \(\alpha:\;G \to X\) satisfying the conditions NEWLINE\[NEWLINE\|f(x) - \alpha(x)\| \leq |3|^{-1} \, \Delta(x), \qquad x \in G,\tag{2}NEWLINE\]NEWLINE where \(\Delta:\;G \to [0,\infty)\) is defined in the explicit form; moreover, they present additional properties of \(\zeta\) which guarantee the uniqueness of \(\alpha\) (Theorems 2.1 and 2.2). Similar results are proved if (2) is changed into the inequality NEWLINE\[NEWLINE\|f(2x) - 4f(x) - \beta(x)\| \leq |16|^{-1} \Delta^2(x), \qquad x \in G,NEWLINE\]NEWLINE (Theorem 3.1 and 3.2), NEWLINE\[NEWLINE\|f(2x) - 16f(x) - \beta(x)\| \leq |4|^{-1} \Delta^3(x), \qquad x \in G,NEWLINE\]NEWLINE (Theorem 3.3 and 3.4) and NEWLINE\[NEWLINE\bigg\|f(x) - \frac1{12} \alpha(x) - \frac1{12} \beta(x)\bigg\| \leq \max \;\bigg\{\frac1{|192|} \Delta^1(x),\frac1{|48|} \Delta^3(x)\bigg\}, \qquad x \in G,NEWLINE\]NEWLINE (Theorems 4.1 and 4.2).