Approximate Quartic and Quadratic Mappings in Quasi-Banach Spaces

The stability problem of functional equations originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms. Let G1, · be a group, and let G2, ∗ be a metric group with the metric d ·, · . Given > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d h x · y , h x ∗ h y < δ for all x, y ∈ G1, then there exists a homomorphism H : G1 → G2 with d h x ,H x < for all x ∈ G1? In other words, under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers 2 gave a first affirmative answer to the question of Ulam for Banach spaces. Let f : X → X′ be a mapping between Banach spaces such that