Stability of functional equations of several variables which are additive or quadratic in each variables in Fréchet spaces (Q445788)

In the present paper, the authors investigate the generalized Hyers-Ulam-Rassias stability problem of the following system of functional equations: \[ \begin{multlined} f(x_1,x_2,\dots{},x_{i_r-1},a+b,x_{i_r+1},\dots{},x_n)\\ =f(x_1,x_2,\dots{},x_{i_r-1},a,x_{i_r+1},\dots{},x_n)+f(x_1,x_2,\dots{},x_{i_r-1},b,\dots{},x_n),\end{multlined} \] \[ \begin{multlined} f(x_1,x_2,\dots{},x_{j_s-1},a+b,x_{j_s+1},\dots{},x_n)+f(x_1,x_2,\dots{},x_{j_s-1},a-b,..,x_{j_s+1},\dots{},x_n)\\ =2f(x_1,x_2,\dots{},x_{i_r-1},a,x_{i_r+1},\dots{},x_n)+2 f(x_1,x_2,\dots{},x_{i_r-1},b,\dots{},x_n),\end{multlined} \] where \(n>1\), \(1\leq i_1<i_2<\dots{}<i_k<n\), \(1<j_1<j_2<\dots{}<j_{n-k}\), and \(\{1,2,\dots{},n\}=\{i_1,i_2,\dots{},i_k\}\bigcup\{j_1,j_2,\dots{},j_{n-k}\}\).. It is assumed that \(f:G^n\to Y\), where \(G\) is an additive group and \(Y\) is a real Fréchet space.