Stability of Functional Equations in Non-archimedean Spaces

A classical question in the theory of functional equations is the following: “When is it true that a function which approximately satisfies a functional equation E must be close to an exact solution of E?” If the problem accepts a solution, we say that the equation E is stable. The first stability problem concerning group homomorphisms was raised by Ulam [30] in 1940. We are given a group G and a metric group G with metric ρ(·, ·). Given > 0, does there exist a δ > 0 such that if f : G→ G satisfies ρ ( f(xy), f(x)f(y) ) < δ for all x, y ∈ G, then a homomorphism h : G → G exists with ρ ( f(x), h(x) ) < for all x ∈ G? Ulam’s problem was partially solved by Hyers [12] in 1941. Let E1 be a normed space, E2 a Banach space and suppose that the mapping f : E1 → E2 satisfies the inequality