Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A fixed-point approach to the stability of a functional equation on quadratic forms | Bae and Park Journal of Inequalities and Applications 2011 2011 82 http content 2011 1 82 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access A fixed-point approach to the stability of a functional equation on quadratic forms Jae-Hyeong Bae 1 and Won-Gil Park2 3 Correspondence wgpark@ department of Mathematics Education College of Education Mokwon University Daejeon 302729 Korea Full list of author information is available at the end of the article Springer Abstract Using the fixed-point method we prove the generalized Hyers-Ulam stability of the functional equation f x y z w f x y z w 2f x z 2f y w . The quadratic form f R X R R given by f x y ax2 bxy cy2 is a solution of the above functional equation. Keywords alternative of fixed point functional equation quadratic form stability 1. Introduction In 1940 S. M. Ulam 1 gave a wide-ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question 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 a Ỗ 0 such that if a function h G1 G2 satisfies the inequality d h xy h x h y Ỏ for all x y e Gt then there is a homomorphism H G1 G2 with d h x H x for all x e Gt The case of approximately additive mappings was solved by D. H. Hyers 2 under the assumption that G1 and G2 are Banach spaces. Thereafter many authors investigated solutions or stability of various functional equations see 3-11 . Let X be a set. A function d X X X 0 is called a generalized metric on X if d satisfies 1 d x y 0 if and only if x y 2 d x y d y x for all x y e X 3 d x z d x y d y z for all x y z e X. Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity. Throughout this paper let X and Y .