Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 826130, 17 pages doi:10.1155/2009/826130 Research Article Solution and Stability of a Mixed Type Additive, Quadratic, and Cubic Functional Equation M. Eshaghi Gordji, 1 S. Kaboli Gharetapeh, 2 J. M. Rassias, 3 and S. Zolfaghari 1 1 Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran 2 Department of Mathematics, Payame Noor University of Mashhad, Mashhad, Iran 3 Section of Mathematics and Informatics, Pedagogical Department, National and Capodistrian University of Athens, 4 Agamemnonos St., Aghia Paraskevi, Athens 15342, Greece Correspondence should be addressed to M. Eshaghi Gordji, madjid.eshaghi@gmail.com Received 24 January 2009; Revised 13 April 2009; Accepted 26 June 2009 Recommended by Patricia J. Y. Wong We obtain the general solution and the generalized Hyers-Ulam-Rassias stability of the mixed type additive, quadratic, and cubic functional equation f x  2y - f x - 2y 2f x  y - f x - y  2f 3y - 6f 2y 6f y. Copyright q 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 G 1 , · be a group, and let G 2 , ∗ be a metric group with the metric d·, ·. Given ǫ> 0, does there exist a δ> 0, such that if a mapping h : G 1 → G 2 satisfies the inequality dhx · y,hx ∗ hy <δ for all x, y ∈ G 1 , then there exists a homomorphism H : G 1 → G 2 with dhx,Hx <ǫ for all x ∈ G 1 ? In other words, under what condition does there exist a homomorphism near an approximate homomorphism? In 1941, Hyers 2 gave a first affirmative answer to the question of Ulam for Banach spaces. Let f : E → E ′ be a mapping between Banach spaces such that   f ( x  y ) - f x - f ( y )  ≤ δ, 1.1 for all x, y ∈ E and for some δ> 0. Then there exists a unique additive mapping T : E → E ′ such that   f x - T x   ≤ δ, 1.2