Abstract
we establish the general solution for a mixed type functional equation of aquartic and a quadratic mapping in linear spaces. In addition, we investigate the generalized Hyers‐Ulam stability in p‐Banach spaces.
Highlights
Introduction and PreliminariesThe stability problem of functional equations originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms
Given > 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 < for all x ∈ G1? In other words, under what condition does there exists 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
Let f : X → X be a mapping between Banach spaces such that f x y − f x − f y ≤ δ, 1.1
Summary
The stability problem of functional equations originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms. A Hyers-Ulam stability problem for the quadratic functional equation 1.3 was proved by Skof for functions f : X → Y , where X is normed space and Y is Banach space see 11. F 2x y f 2x − y 4f x y 4f x − y 24f x − 6f y They proved that a function f between two real vector spaces X and Y is a solution of 1.5 if and only if there exists a unique symmetric biquadratic function B2 : X × X → Y such that f x B2 x, x for all x ∈ X. We investigate the general solution of functional equation 1.7 when f is a function between vector spaces, and we prove the generalized Hyers-Ulam stability of 1.7 in the spirit of Hyers, Ulam, and Rassias using the direct method. I1 for a positive real number p with p ≤ 1
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