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Research Article Quadratic-Quartic Functional Equations in RN-Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 868423 14 pages doi 2009 868423 Research Article Quadratic-Quartic Functional Equations in RN-Spaces M. Eshaghi Gordji 1 M. Bavand Savadkouhi 1 and Choonkil Park2 1 Department of Mathematics Semnan University . Box 35195-363 Semnan Iran 2 Department of Mathematics Hanyang University Seoul 133-791 South Korea Correspondence should be addressed to Choonkil Park baak@ Received 20 July 2009 Accepted 3 November 2009 Recommended by Andrea Laforgia We obtain the general solution and the stability result for the following functional equation in random normed spaces in the sense of Sherstnev under arbitrary f-norms f 2x y f 2x - y 4lf x y f x - y i 2 f 2x - 4f x - 6f y . Copyright 2009 M. Eshaghi Gordji 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. 1. Introduction The stability problem of functional equations originated from a question of Ulam 1 in 1940 concerning the stability of group homomorphisms. Let G1 be a group and let G2 d be a metric group with the metric d - . Given e 0 does there exist a Ỗ 0 such that if a mapping h G1 G2 satisfies the inequality d h x y h. x hf-yf Ỗ for all x y e G1 then there exists a homomorphism H G1 G2 with d h x H x e for all x e 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. Hyers 2 gave a first affirmative answer to the question of Ulam for Banach spaces. Let f E E1 be a mapping between Banach spaces such that llf x y - f x - f y ll ỗ k1 for all x y e E and some Ỗ 0. Then there exists a unique additive mapping T E E such that Ilf x - T x Ô