Stability of a 2-Dimensional Functional Equation in a Class of Vector Variable Functions

In 1940, Ulam proposed the 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. The authors investigated various functional equations and their Hyers-Ulam stability 3–8 . This Hyers-Ulam stability is a classical type of stability, but there is another kind of stability introduced by Risteski 9 for functional equations spanned over an n-dimensional complex vector space too. Let X and Y be real or complex vector spaces. For a mapping g : X → Y , consider the quadratic functional equation