On a \(\rho \) -Orthogonally Additive Mappings

We show that a real normed linear space endowed with the $$\rho $$ -orthogonality relation, in general need not be an orthogonality space in the sense of Rätz. However, we prove that $$\rho $$ -orthogonally additive mappings defined on some classical Banach spaces have to be additive. Moreover, additivity (and approximate additivity) under the condition of an approximate orthogonality is considered.