Abstract : We deal with the Hyers-Ulam stability problem of linear mappings from a vector space into a Banach one with respect to the following functional equation: $$ f\left(\frac{-x + y}{3}\right) + f\left(\frac{x - 3z}{3}\right) + f\left(\frac{3x - y + 3z}{3}\right) = f(x). $$ We then combine this equation with other ones and establish the Hyers-Ulam stability of several kinds of linear mappings, among which the algebra ($*$-) homomorphisms, the derivations, the multipliers and others. We thus repair and improve some previous assertions in the literature.

Keywords : Hyers-Ulam stability of functional equations, Hyers-Ulam stability of additive mappings, ring homomorphisms, ring derivations