Biderivations and bihomomorphisms in Banach algebras

Abstract

In this paper, we solve the following  bi-additive s-functional inequalities
\begin{eqnarray*} && \| f(x+y, z+w) + f(x+y, z-w) + f(x-y, z+w) + f(x-y, z-w) -4f(x,z)\| \\ &&  \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{eqnarray*}
and
\begin{eqnarray*} && \left\|2f\left(x+y, z-w\right) +2 f\left(x-y, z+w\right) -4 f(x,z )+4 f(y, w)\right\| \\ &&  \le \left \|s \left(  f(x+y, z+w) + f(x+y, z-w) + f(x-y, z+w) + f(x-y, z-w) -4f(x,z)  \right)\right\| ,
\end{eqnarray*}
where s is a fixed nonzero complex number with $|s |< 1$.

Moreover, we prove the Hyers-Ulam stability of  biderivations and bihomomorphismsions in  Banach algebras and unital $C^*$-algebras, associated with  the above bi-additive s-functional inequalities.