ORTHOGONALLY ADDITIVE AND ORTHOGONALLY QUADRATIC FUNCTIONAL EQUATION

Abstract

Using the fixed point method, we prove the Ulam-Hyers stability of the orthogonally additive and orthogonally quadratic functional equation

$f(\frac{x}{2}+y)+f(\frac{x}{2}-y)+f(\frac{x}{2}+z)+f(\frac{x}{2}-z)=3f(x)-1f(-x)+f(y)+f(-y)+f(z)+f(-z)$

for all $x,y,z$ with $x\perp y$, in orthogonality Banach spaces and in non-Archimedean orthogonality Banach spaces.