Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of biderivations and bihomomorphisms in Banach algebras and unital \(C^*\)-algebras, associated with the bi-additive functional inequality:$$\begin{aligned}&\Vert f(x+y, z+w) + f(x+y, z-w) + f(x-y, z+w) \nonumber \\&\quad + f(x-y, z-w) -4f(x,z)\Vert \nonumber \\&\quad \le \left\| s \left( 2f\left( x+y, z-w\right) + 2f\left( x-y, z+w\right) - 4f(x,z )+ 4 f(y, w)\right) \right\| , \end{aligned}$$(1)where s is a fixed nonzero complex number with \(|s |< 1\).

中文翻译：

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Banach代数中双同态和除态的Ulam稳定性

使用固定点法和直接法中，我们证明在Banach代数和酉biderivations和bihomomorphisms的Hyers-乌拉姆稳定性\（C ^ * \） -代数，与双官能添加剂相关的不等式：$$ \ {开始对齐}＆\ Vert f（x + y，z + w）+ f（x + y，zw）+ f（xy，z + w）\ nonumber \\＆\ quad + f（xy，zw）-4f（ x，z）\ Vert \ nonumber \\＆\ quad \ le \ left \ | s \ left（2f \ left（x + y，zw \ right）+ 2f \ left（xy，z + w \ right）-4f（x，z）+ 4 f（y，w）\ right）\ right \ | ，\ end {aligned} $$（1），其中s是带有\（| s | <1 \）的固定非零复数。