Stability of Quadratic Functional Equations via the Fixed Point and Direct Method
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Research Article Stability of Quadratic Functional Equations via the Fixed Point and Direct Method Eunyoung Son, Juri Lee, and Hark-Mahn Kim Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea Correspondence should be addressed to Hark-Mahn Kim, [email protected] Received 13 October 2009; Accepted 19 January 2010 Academic Editor: Yeol Je Cho Copyright q 2010 Eunyoung Son 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. C˘adariu and Radu applied the fixed point theorem to prove the stability theorem of Cauchy and Jensen functional equations. In this paper, we prove the generalized Hyers-Ulam stability via the fixed point method and investigate new theorems via direct method concerning the stability of a general quadratic functional equation.
1. Introduction 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. Let G be a group and let G be a metric group with metric ρ·, ·. Given > 0, does there exist a δ > 0 such that if f : G → G satisfies ρfxy, fxfy < δ for all x, y ∈ G, then a homomorphism h : G → G exists with ρfx, hx < for all x ∈ G? The concept of stability for functional equations arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus we say that a functional equation E1 f E2 f is stable if any mapping g approximately satisfying the equation dE1 g, E2 g ≤ ϕx is near to a true solution f such that E1 f E2 f and dfx, gx ≤ Φx for some function Φ depending on the given function ϕ. In 1941, the first result concerning the stability of functional equations for the case where G1 and G2 are Banach spaces was presented by Hyers 2. In fact, he proved that each solution f of the inequality fx y − fx − fy ≤ for all x, y ∈ G1 can be approximated by a unique additive function L : G1 → G2 defined by Lx limn → ∞ f2n x/2n such that fx − Lx ≤ for every x ∈ G1 . Moreover, if ftx is continuous in t ∈ R for each fixed x ∈ G1 , then the function L is linear. And then Aoki 3, Bourgin 4, and Forti 5 have investigated the
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Journal of Inequalities and Applications
stability theorems of functional equations which generalize the Hyers’ result. In 1978, Rassias 6 attempted to weaken the condition for the bound of Cauchy difference controlled by a sum of unbounded function εxp yp , 0 < p < 1, and provided a generalization of Hyers’ theorem. In 1991, Gajda 7 gave an affirmative solution to this question for p > 1 by following the same approach as in 6. Rassias 8 established a similar stability theorem for the unbounded Cauchy difference controlled by a product of unbounded function εxp · 1. G˘avru
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