The stability of additive (alpha, beta)-functional equations

Abstract

In this paper, we investigate the following \lt inline-formula \gt (α,β) \lt /inline-formula \gt -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)), ~~~(0.1) \\ \label{0.2} 2f(x)+2f(y)=f(x+y)+\alpha^{-1}f(\alpha(x+z)) +\beta^{-1}f(\beta(y-z)), ~~~(0.2) \end{eqnarray} where \lt inline-formula \gt α,β \lt /inline-formula \gt are fixed nonzero real numbers with \lt inline-formula \gt α−1+β−1≠3 \lt /inline-formula \gt . Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the \lt inline-formula \gt (α,β) \lt /inline-formula \gt -functional equations \lt inline-formula \gt (0.1) \lt /inline-formula \gt and \lt inline-formula \gt (0.2) \lt /inline-formula \gt in non-Archimedean Banach spaces.