An additive $(u, \beta)$-functional equation and linear mappings in Banach modules

Document Type : Special issue editorial


1 Department of Mathematics, Hanyang University, Seoul 04763, Korea

2 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

3 Department of Data Science, Daejin University, Kyunggi 11159, Korea


Let $A$ be a unital $C^*$-algebra. In this paper, we investigate  the additive $(u, \beta)$-functional equation \begin{eqnarray}\label{0.1}
f(x)+ u^* f(u y)+  f( z) =  \beta^{-1} f(\beta(x+y+z))
for all  unitary elements $u$ in $A$  and for a fixed nonzero complex number $\beta$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability  of the additive $(u, \beta)$-functional equation (\ref{0.1})  in  Banach modules.


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Volume 13, Issue 2
July 2022
Pages 1747-1755
  • Receive Date: 05 February 2021
  • Revise Date: 14 February 2021
  • Accept Date: 13 March 2021