On the Generalized Ulam-Gavruta-Rassias Stability of Mixed-Type Linear and Euler-Lagrange-Rassias Functional Equations

In 1940, Ulam [1] proposed the famous Ulam stability problem of linear mappings. In 1941, Hyers [2] considered the case of approximately additive mappings f : E→ E′, where E and E′ are Banach spaces and f satisfies Hyers 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 : E→ E′ is the unique additive mapping satisfying ‖ f (x)−L(x)‖ ≤ ε. In 1982– 1998, Rassias [3–9] generalized the result to include the following theorem.