Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Approximately cubic functional equations and cubic multipliers | Bodaghi et al. Journal of Inequalities and Applications 2011 2011 53 http content 2011 1 53 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Approximately cubic functional equations and cubic multipliers Abasalt Bodaghi1 Idham Arif Alias2 and Mohammad Hossein Ghahramani1 Correspondence abasalt. bodaghi@ department of Mathematics Garmsar Branch Islamic Azad University Garmsar Iran Full list of author information is available at the end of the article Springer Abstract In this paper we prove the Hyers-Ulam stability and the superstability for cubic functional equation by using the fixed point alternative theorem. As a consequence we show that the cubic multipliers are superstable under some conditions. 2000 Mathematics Subject Classification 39B82 39B52. Keywords cubic functional equation multiplier Hyers-Ulam stability Superstability 1. Introduction The stability problem for functional equations is related to the following question originated by Ulam 1 in 1940 concerning the stability of group homomorphisms Let G1 . be a group and let G2 be a metric group with the metric d . . . Given 0 does there exist Ỏ 0 such that if a mapping h G1 G2 satisfies the inequality d h h x h y Ỗ for all x y e G1 then there exists a homomorphism H G1 G2 with d h x H x for all x e G1 In 1941 Hyers 2 gave a first affirmative answer to the question of Ulam for Banach spaces. Later Rassias in 3 provided a remarkable generalization of the Hyers result by allowing the Cauchy difference to be bounded for the first time in the subject of functional equations and inequalities. Găvruta then generalized the Rassias result in 4 for the unbounded Cauchy difference. The functional equation f x y f x - y 2f x 2f y is called quadratic functional equation. Also every solution for example f x ax2 of functional Equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic .
