Stability of Approximate Quadratic Mappings

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. Among these was the following question concerning the stability of homomorphisms. Let G1 be a group and let G2 be a metric group with metric ρ ·, · . Given > 0, does there exist a δ > 0 such that if f : G1 → G2 satisfies ρ f xy , f x f y < δ for all x, y ∈ G1, then a homomorphism h : G1 → G2 exists with ρ f x , h x < for all x ∈ G1? In 1941, the first result concerning the stability of functional equations was presented by Hyers 2 . And then Aoki 3 and Bourgin 4 have investigated the stability theorems of functional equations with unbounded Cauchy differences. In 1978, Th. M. Rassias 5 provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. It was shown by Gajda 6 as well as by Th. M. Rassias and Šemrl 7 that one cannot prove the Rassias’ type theorem when p 1. Găvruta 8 obtained generalized result of Th. M. Rassias’ Theorem which allow the Cauchy difference to be controlled by a general unbounded function. J. M. Rassias 9, 10 established a similar stability theorem linear and nonlinear mappings with the unbounded Cauchy difference. Let E1 and E2 be real vector spaces. A function f : E1 → E2 is called a quadratic function if and only if f is a solution function of the quadratic functional equation: