Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias Stabilities of an Additive Functional Equation in Several Variables

In 1940, Ulam [13] proposed the Ulam stability problem of additive mappings. In the next year, Hyers [5] considered the case of approximately additive mappings f : E→ E′, where E and E′ are Banach spaces and f satisfies inequality ‖ f (x+ y)− f (x)− f (y)‖ ≤ ε for all x, y ∈ E. It was shown that the limit L(x) = limn→∞ 2−n f (2nx) exists for all x ∈ E and that L is the unique additive mapping satisfying ‖ f (x)−L(x)‖ ≤ ε. In 1978, Rassias [14] generalized the result to an approximation involving a sum of powers of norms. In 1982–1989, Rassias [8–11] treated the Ulam-Gavruta-Rassias stability on linear and nonlinear mappings and generalized Hyers result to the following theorem.