Stability of a Cauchy-Jensen Functional Equation in Quasi-Banach Spaces

In 1940, Ulam proposed the general Ulam stability problem see 1 . 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? In 1941, this problem was solved by Hyers 2 in the case of Banach space. Thereafter, we call that type the Hyers-Ulam stability. Throughout this paper, letX and Y be vector spaces. Amapping g : X → Y is called an additive mapping respectively, an affine mapping if g satisfies the Cauchy functional equation g x y g x g y respectively, the Jensen functional equation 2g x y /2 g x g y . Aoki 3 and Rassias 4, 5 extended the Hyers-Ulam stability by considering variables for Cauchy equation. Using the method introduced in 3 , Jung 6 obtained a result for Jensen equation. It also has been generalized to the function case by Găvruta 7 and Jung 8 for Cauchy equation, and by Lee and Jun 9 for Jensen equation.