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The stability of additive (α, β) -functional equations
Ziying lu,Gang Lu,Yuanfeng Jin,Choonkil Park
Keywords:Hyers-Ulam stability, additive $(\alpha,\beta)$-functional equation, fixed point method, direct method,non-Archimedean Banach space
Abstract:
      In this paper, we investigate the following $(\alpha,\beta)$-functional equations \begin{eqnarray}\label{0.1} 2f(x)+2f(z)=f(x-y)+\alpha^{-1}f(\alpha (x+z))+\beta^{-1}f(\beta(y+z)), \end{eqnarray} \begin{eqnarray}\label{0.2} 2f(x)+2f(y)=f(x+y)+\alpha^{-1}f(\alpha(x+z)) +\beta^{-1}f(\beta(y-z)), \end{eqnarray} where $\alpha,\beta$ are fixed nonzero real numbers with $\alpha^{-1}+\beta^{-1}\neq 3$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the $(\alpha,\beta)$-functional equations $(\ref{0.1})$ and $(\ref{0.2})$ in non-Archimedean Banach spaces.