Abstract

In this paper, we solve the following quadratic <TEX>${\rho}-functional$</TEX> inequalities <TEX>${\parallel}f({\frac{x+y+z}{2}})+f({\frac{x-y-z}{2}})+f({\frac{y-x-z}{2}})+f({\frac{z-x-y}{2}})-f(x)-f(y)f(z){\parallel}$</TEX> (0.1) <TEX>${\leq}{\parallel}{\rho}(f(x+y+z)+f(x-y-z)+f(y-x-z)+f(z-x-y)-4f(x)-4f(y)-4f(z)){\parallel}$</TEX>, where <TEX>${\rho}$</TEX> is a fixed non-Archimedean number with <TEX>${\mid}{\rho}{\mid}$</TEX> < <TEX>${\frac{1}{{\mid}4{\mid}}}$</TEX>, and <TEX>${\parallel}f(x+y+z)+f(x-y-z)+f(y-x-z)+f(z-x-y)-4f(x)-4f(y)-4f(z){\parallel}$</TEX> (0.2) <TEX>${\leq}{\parallel}{\rho}(f({\frac{x+y+z}{2}})+f({\frac{x-y-z}{2}})+f({\frac{y-x-z}{2}})+f({\frac{z-x-y}{2}})-f(x)-f(y)f(z)){\parallel}$</TEX>, where <TEX>${\rho}$</TEX> is a fixed non-Archimedean number with <TEX>${\mid}{\rho}{\mid}$</TEX> < <TEX>${\mid}8{\mid}$</TEX>. Using the direct method, we prove the Hyers-Ulam stability of the quadratic <TEX>${\rho}-functional$</TEX> inequalities (0.1) and (0.2) in non-Archimedean Banach spaces and prove the Hyers-Ulam stability of quadratic <TEX>${\rho}-functional$</TEX> equations associated with the quadratic <TEX>${\rho}-functional$</TEX> inequalities (0.1) and (0.2) in non-Archimedean Banach spaces.