On solutions and stability of a generalized quadratic equation on non-Archimedean normed spaces (Q2919916)

Suppose that \(X\) and \(Y\) are non-Archimedean Banach spaces and \(f:X \to Y\) is a function. The authors investigate the Hyers-Ulam-Rassias stability of the following functional equation NEWLINE\[NEWLINE\begin{multlined} f(x-\sum_{i=1}^kx_i)+(k-1)f(x)+(k-1)\sum_{i=1}^kf(x_i)=\\ f(x-x_1)+\sum_{i=2}^kf(x_i-x)+\sum_{i=1}^k\sum_{j=1,~~j>i}^kf(x_i+x_j),\end{multlined}NEWLINE\]NEWLINE where \(k\geq 2\). They also prove that the above functional equation is equivalent with the quadratic functional equation NEWLINE\[NEWLINEf(x+y)+f(x-y)=2f(x)+2f(y).NEWLINE\]