Generalized U - H Stability of New n - type of Additive Quartic Functional Equation in Non - Archimedean Orthogonally Space

Abstract

Using direct method, we prove the stability of the orthogonally additive-quartic functional equation of the form $f(nx+n^{2}y+n^{3}z)+f(nx-n^{2}y+n^{3}z)$$+f(nx+n^{2}y-n^{3}z)+f(-nx+n^{2}y+n^{3}z) $$= 2[f(nx+n^{2}y)+f(nx-n^{2}y)] + 2[f(n^{2}y+n^{3}z) $$+ f(n^{2}y-n^{3}z)]+ 2[f(nx+n^{3}z)+f(nx-n^{3}z)]$$-3n[f(x)-f(-x)]-3n^{2}[f(y)-f(-y)] -3n^{3}[f(z)-f(-z)]-2n^{4}[f(x)$$+f(-x)]-2n^{8}[f(y) +f(-y)]-2n^{12}[f(z)+f(-z)]$ for all $x,y,z$ in $X$ with $x\bot y$, $y\bot z$ and $z\bot x$ in Banach non-Archimedean Orthogonally spaces. Here $\bot$ is the orthogonally in the sense of Ratz.