Stability of an n-Dimensional Mixed-Type Additive and Quadratic Functional Equation in Random Normed Spaces

In 1940, Ulam 1 gave a wide-ranging talk before a mathematical colloquium at the University of Wisconsin, in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms. Let G1 be a group, and let G2 be a metric group with a metric d ·, · . Given ε > 0, does there exist a δ > 0 such that if a function 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? If the answer is affirmative, we say that the functional equation for homomorphisms is stable. Hyers 2 was the first mathematician to present the result concerning the stability of functional equations. He answered the question of Ulam for the case where G1 and G2 are assumed to be Banach spaces. This result of Hyers is stated as follows. Let f : E1 → E2 be a function between Banach spaces such that