Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation in Non-Archimedean Spaces

The stability problem of functional equations was originated from a question of Ulam 1 concerning the stability of group homomorphisms. Let G1 be a group and let G2 be a metric group with the metric d ·, · . Given > 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 exists a homomorphism H : G1 → G2 with d h x ,H x < for all x ∈ G1? In other words, we are looking for situations when the homomorphisms are stable, that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. If we turn our attention to the case of functional equations, we can ask the following question. When the solutions of an equation differing slightly from a given one must be close to the true solution of the given equation. For Banach spaces, the Ulam problem was first solved by Hyers 2 in 1941, which states that, if δ > 0 and f : X → Y is a mapping, where X,Y are Banach spaces, such that