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.