Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces

In 1940, Ulam 1 gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems, containing the stability problem of homomorphisms as follows Let G1 be a group and let G2 be a metric group with the metric d ·, · . Given ε > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d h xy , h x h y < δ for all x, y ∈ G1, then there is a homomorphism H : G1 → G2 with d h x ,H x < ε for all x ∈ G1? Hyers 2 proved the stability problem of additive mappings in Banach spaces. Rassias 3 provided a generalization of Hyers theorem which allows the Cauchy difference to be unbounded: let f : E → E be a mapping from a normed vector space E into a Banach space E subject to the inequality