Stability of a Cauchy-Jensen Functional Equation in Quasi-Banach Spaces
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Research Article Stability of a Cauchy-Jensen Functional Equation in Quasi-Banach Spaces Jae-Hyeong Bae1 and Won-Gil Park2 1 2
College of Liberal Arts, Kyung Hee University, Yongin 446-701, South Korea Division of Computational Sciences in Mathematics, National Institute for Mathematical Sciences, 385-16 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea
Correspondence should be addressed to Won-Gil Park, [email protected] Received 16 October 2009; Accepted 30 January 2010 Academic Editor: Yeol Je Cho Copyright q 2010 J.-H. Bae and W.-G. Park. 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 obtain the generalized Hyers-Ulam stability of the Cauchy-Jensen functional equation 2fx y, z w/2 fx, z fx, w fy, z fy, w.
1. Introduction In 1940, Ulam proposed the general Ulam stability problem see 1. Let G1 be a group and let G2 be a metric group with the metric d·, ·. Given ε > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality dhxy, hxhy < δ for all x, y ∈ G1 then there is a homomorphism H : G1 → G2 with dhx, Hx < ε for all x ∈ G1 ? In 1941, this problem was solved by Hyers 2 in the case of Banach space. Thereafter, we call that type the Hyers-Ulam stability. Throughout this paper, let X and Y be vector spaces. A mapping g : X → Y is called an additive mapping respectively, an affine mapping if g satisfies the Cauchy functional equation gxy gxgy respectively, the Jensen functional equation 2gxy/2 gxgy. Aoki 3 and Rassias 4, 5 extended the Hyers-Ulam stability by considering variables for Cauchy equation. Using the method introduced in 3, Jung 6 obtained a result for Jensen equation. It also has been generalized to the function case by G˘avruta 7 and Jung 8 for Cauchy equation, and by Lee and Jun 9 for Jensen equation. Definition 1.1. A mapping f : X × X → Y is called a Cauchy-Jensen mapping if f satisfies the system of equations
2
Journal of Inequalities and Applications f x y, z fx, z f y, z , 1.1 yz f x, y fx, z. 2f x, 2
When X Y R, the function f : R × R → R given by fx, y : axy bx is a solution of 1.1. In particular, letting x y, we get a function g : R → R given by gx : fx, x ax2 bx. For a mapping f : X × X → Y , consider the functional equation z w 2f x y, fx, z fx, w f y, z f y, w . 2
1.2
Definition 1.2 see 10, 11. Let X be a real linear space. A quasi-norm is real-valued function on X satisfying the following. i x ≥ 0 for all x ∈ X and x 0 if and only if x 0. ii λx |λ|x for all λ ∈ R and all x ∈ X. iii There is a constant K ≥ 1 such that x y ≤ Kx y for all x, y ∈ X. The pair X, · is called a quasi-normed space if · is a quasi-norm on X. The smallest possible K is called the modulus of concavit
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