Abstract
We investigate the general solution of the quadratic functional equation , in the class of all functions between quasi- -normed spaces, and then we prove the generalized Hyers-Ulam stability of the equation by using direct method and fixed point method.
Highlights
In 1940, Ulam 1 gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms
Given > 0, does there exist a δ > 0 such that if f : G1 → G2 satisfies ρ f xy, f x f y < δ for all x, y ∈ G1, a homomorphism h : G1 → G2 exists with ρ f x, h x < for all x ∈ G1?
In 1941, the first result concerning the stability of functional equations was presented by Hyers 2
Summary
Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, South Korea Copyright q 2010 Hark-Mahn Kim et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the general solution of the quadratic functional equation f 2x y 3f 2x − y 4f x − y 12f x , in the class of all functions between quasi-β-normed spaces, and then we prove the generalized Hyers-Ulam stability of the equation by using direct method and fixed point method.
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