Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa-00127, 10 Pages doi:10.5899/2012/jnaa-00127 Research Article On the Paper by A. Najati and S.-M. Jung: The Hyers-Ulam Stability of Approximately Quadratic Mapping on Restricted Domains Elqorachi Elhoucien 1* , Manar Youssef 1 (1) Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco. Copyright 2012 c ⃝ Elqorachi Elhoucien and Manar Youssef. 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. Abstract In [32], A. Najati and S.-M. Jung obtained the Hyers-Ulam stability of the generalized quadratic functional equation f (rx + sy)+ rsf (x − y)= rf (x)+ sf (y) with r + s = 1, r ̸= 0, s ̸= 0, provided that f is an even mapping. The purpose of this paper is to remove this restriction. Keywords : Hyers-Ulam stability; Quadratic functional equation; Quadratic mapping; Approximate quadratic mapping; Asymptotic behavior. 1 Introduction In 1940, S. M. Ulam [50] raised a question concerning the stability of group homomor- phisms: Let G 1 be a group and let G 2 be a metric group with the metric d(·, ·). Given ϵ> 0, does there exist a δ> 0 such that if a function h : G 1 → G 2 satisfies the inequality d(h(xy),h(x)h(y)) <δ for all x, y ∈ G 1 , then there exists a homomorphism a : G 1 → G 2 with d(h(x),a(x)) <ϵ for all x ∈ G 1 . The problem for the case of approximately additive mappings was solved by D. H. Hyers [13] on Banach spaces. In 1950 T. Aoki [1] provided a generalization of the Hyers’s theorem for additive mappings and in 1978 Th. M. Rassias [40] generalized the Hyers’ Theorem for linear mappings by allowing the Cauchy difference to be unbounded. The result of Rassias’ Theorem has been generalized by P. Gˇavruta [10] who permitted the Cauchy difference * Corresponding author. Email address: elqorachi@hotmail.com 1