Abstract
In this paper, we solve the following bi-additive $s$-functional inequalities: $$ \displaylines { \, \| f(x+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \hfill \cr \hfill \quad \le \Bigl \|s \Bigl (2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) + 2f\Bigl (\frac {x-y}{2}, z+w\Bigr ) - 2f(x,z )+ 2 f(y, w)\Bigr )\Bigr \| ,\! \cr \, \Bigl \|2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) +2 f\Bigl (\frac {x-y}{2}, z+w\Bigr ) -2 f(x,z )+2 f(y, w)\Bigr \| \hfill \cr \hfill \le \|s ( f(x+y, z-w) + f(x-y, z+w) -2f(x,z) +2 f(y, w) )\|, \cr \, } $$ where $s$ is a fixed nonzero complex number with $|s |\lt 1$. We also prove the Hyers-Ulam stability of quasi-multipliers on Banach algebras and unital $C^*$-algebras associated with the bi-additive $s$-functional inequalities above.
Funding Statement
The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology, grant No. NRF-2017R1D1A1B04032937. The second author was supported by the National Natural Science Foundation of China, grant No. 11761074.
Citation
Choonkil Park. Yuanfeng Jin. Xiaohong Zhang. "Bi-additive $s$-functional inequalities and quasi-multipliers on Banach algebras." Rocky Mountain J. Math. 49 (2) 593 - 607, 2019. https://doi.org/10.1216/RMJ-2019-49-2-593
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