Abstract
We establish the general solution of the functional equation for fixed integers with and investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.
Highlights
The stability problem of functional equations originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms
Given ε > 0, does there exist a δ > 0, such that if a mapping h : G1 → G2 satisfies the inequality d h x · y, h x ∗ h y < δ for all x, y ∈ G1, there exists a homomorphism H : G1 → G2 with d h x, H x < ε for all x ∈ G1? In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation
The generalized Hyers-Ulam stability problem for the quadratic functional equation 1.3 was proved by Skof for mappings f : A → B, where A is a normed space and B is a Banach space see 8
Summary
The stability problem of functional equations originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms. Let f : E → E be a mapping between Banach spaces such that f x y −f x −f y ≤δ
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