Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 13 2 2022 07 01 An additive $(u, \beta)$-functional equation and linear mappings in Banach modules 1747 1755 6651 10.22075/ijnaa.2021.22580.2386 EN Siriluk Paokanta Department of Mathematics, Hanyang University, Seoul 04763, Korea Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea Jung Rye Lee Department of Data Science, Daejin University, Kyunggi 11159, Korea Journal Article 2021 02 05 Let $A$ be a unital $C^*$-algebra. In this paper, we investigate  the additive $(u, \beta)$-functional equation \begin{eqnarray}\label{0.1}<br />f(x)+ u^* f(u y)+  f( z) =  \beta^{-1} f(\beta(x+y+z))<br />\end{eqnarray}<br />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. https://ijnaa.semnan.ac.ir/article_6651_437e94f4c94563c52125130526bb17f1.pdf