Abstract

In this paper, we solve the additive <TEX>${\rho}$</TEX>-functional equations <TEX>$$(0.1)\;f(x+y)+f(x-y)-2f(x)={\rho}\left(2f\left({\frac{x+y}{2}}\right)+f(x-y)-2f(x)\right)$$</TEX>, where <TEX>${\rho}$</TEX> is a fixed non-Archimedean number with <TEX>${\mid}{\rho}{\mid}$</TEX> < 1, and <TEX>$$(0.2)\;2f\left({\frac{x+y}{2}}\right)+f(x-y)-2f(x)={\rho}(f(x+y)+f(x-y)-2f(x))$$</TEX>, where <TEX>${\rho}$</TEX> is a fixed non-Archimedean number with <TEX>${\mid}{\rho}{\mid}$</TEX> < |2|. Furthermore, we prove the Hyers-Ulam stability of the additive <TEX>${\rho}$</TEX>-functional equations (0.1) and (0.2) in non-Archimedean Banach spaces.