AIMS Mathematics, 2020, 5(3): 2196-2210. doi: 10.3934/math.2020145

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Bihomomorphisms and biderivations in Lie Banach algebras

1 Mathematics Branch, Seoul Science High School, Seoul 03066, Korea
2 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

In this paper, we solve the following bi-additive $s$-functional inequality
$\begin{array}{*{20}{c}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left\| {f(x - y,y + z) + f\left( {y + z,z - x} \right) + f\left( {z + x,x - z} \right) - f\left( {x - y,x + y} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {0.1} \right)} \right.}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{ \le \left\| {s\left( {f\left( {y - z,z + x} \right) + f\left( {z + x,x - y} \right) + f\left( {x + y,y - x} \right) - f\left( {y - z,y + z} \right)} \right)} \right\|,}\end{array}$
where $s$ is a fixed nonzero complex number satisfying $|s|<1$. Furthermore, we prove the Hyers-Ulam stability of bihomomorphisms and biderivations in Lie Banach algebras associated with the bi-additive $s$-functional inequality (0.1).
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