Abstract In this paper, we prove the general solution and generalized Hyers–Ulam stability of n-dimensional functional equations of the form ∑ i = 1 i ≠ j ≠ k n f ⁢ ( - x i - x j - x k + ∑ l = 1 l ≠ i ≠ j ≠ k n x l ) = ( n 3 - 9 ⁢ n 2 + 20 ⁢ n - 12 6 ) ⁢ ∑ i = 1 n f ⁢ ( x i ) , \sum_{\begin{subarray}{c}i=1\\ i\neq j\neq k\end{subarray}}^{n}f\biggl{(}-x_{i}-x_{j}-x_{k}+\sum_{% \begin{subarray}{c}l=1\\ l\neq i\neq j\neq k\end{subarray}}^{n}x_{l}\biggr{)}=\biggl{(}\frac{n^{3}-9n^{% 2}+20n-12}{6}\biggr{)}\sum_{i=1}^{n}f(x_{i}), where n is a fixed positive integer with ℕ - { 0 , 1 , 2 , 3 , 4 } \mathbb{N}-\{0,1,2,3,4\} , in a Banach space via direct and fixed point methods.

中文翻译：

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一个n维函数方程的解与稳定性

摘要 本文证明了∑ i = 1 i ≠ j ≠ knf ⁢ ( - xi - xj - xk + ∑ l = 1 l ≠ i ≠ j ≠ knxl ) = ( n 3 - 9 ⁢ n 2 + 20 ⁢ n - 12 6 ) ⁢ ∑ i = 1 nf ⁢ ( xi ) , \sum_{\begin{subarray}{c}i=1\\ i \neq j\neq k\end{subarray}}^{n}f\biggl{(}-x_{i}-x_{j}-x_{k}+\sum_{% \begin{subarray}{c} l=1\\ l\neq i\neq j\neq k\end{subarray}}^{n}x_{l}\biggr{)}=\biggl{(}\frac{n^{3}-9n ^{% 2}+20n-12}{6}\biggr{)}\sum_{i=1}^{n}f(x_{i})，其中 n 是固定的正整数，其中 ℕ - { 0 ， 1 , 2 , 3 , 4 } \mathbb{N}-\{0,1,2,3,4\} ，在 Banach 空间中通过直接和不动点方法。