We prove the Hyers–Ulam stability of the functional equation

$$\begin{aligned}&f(a_1x_1+a_2x_2,b_1y_1+b_2y_2)=C_{1}f(x_1,y_1)\nonumber \\ \nonumber \\&\quad +C_{2}f(x_1,y_2)+C_{3}f(x_2,y_1)+C_{4}f(x_2,y_2) \end{aligned}$$(*)

in the class of functions from a real or complex linear space into a Banach space over the same field. We also study, using the fixed point method, the generalized stability of \((*)\) in the same class of functions. Our results generalize some known outcomes.

中文翻译：

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关于一般双线性函数方程的稳定性

我们证明了函数方程的Hyers-Ulam稳定性

$$\begin{对齐}&f(a_1x_1+a_2x_2,b_1y_1+b_2y_2)=C_{1}f(x_1,y_1)\nonumber \\ \nonumber \\&\quad +C_{2}f(x_1,y_2) +C_{3}f(x_2,y_1)+C_{4}f(x_2,y_2) \end{aligned}$$ (*)

在同一域上从实数或复数线性空间到 Banach 空间的函数类中。我们还使用不动点方法研究了\((*)\)在同一类函数中的广义稳定性。我们的结果概括了一些已知的结果。