Additive $ \rho $-functional inequalities in non-Archimedean 2-normed spaces

In this paper, we solve the additive $ \rho $-functional inequalities: $ \begin{align*} \|f(x+y)-f(x)-f(y)\| \leq \|\rho(2f(\frac{x+y}{2})-f(x)-f(y))\|, \\ \|2f(\frac{x+y}{2})-f(x)-f(y)\| \leq \|\rho(f(x+y)-f(x)-f(y))\|, \end{align*} $ where $ \rho $ is a fixed non-Archimedean number with $ |\rho| < 1 $. More precisely, we investigate the solutions of these inequalities in non-Archimedean $ 2 $-normed spaces, and prove the Hyers-Ulam stability of these inequalities in non-Archimedean $ 2 $-normed spaces. Furthermore, we also prove the Hyers-Ulam stability of additive $ \rho $-functional equations associated with these inequalities in non-Archimedean $ 2 $-normed spaces.