Tham khảo luận văn - đề án 'báo cáo hóa học: "research article a multidimensional functional equation having quadratic forms as solutions"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 24716 8 pages doi 2007 24716 Research Article A Multidimensional Functional Equation Having Quadratic Forms as Solutions Won-Gil Park and Jae-Hyeong Bae Received 7 July 2007 Accepted 3 September 2007 Recommended by Vijay Gupta We obtain the general solution and the stability of the m-variable quadratic functional equation f X1 y1 . Xm ym f X1 - y1 . Xm - ym 2 f X1 . Xm 2 f y1 . ym . The quadratic form f x1 . Xm y 1si j maijXiXj is a solution of the given functional equation. Copyright 2007 . Park and . Bae. 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 In this paper let X and Y be real vector spaces. A mapping f is called a quadratic form if there exist aij e R 1 i j m such that f x1 . Xm aijXiXj for all x1 . Xm e X. For a mapping f Xm Y consider the m-variable quadratic functional equation f X1 y1 . Xm ym f X1 -y1 . Xm - ym 2f X1 . Xm 2f y1 . ym . When X Y R the quadratic form f Rm R given by f X1 . Xm y aijXiXj is a solution of . 2 Journal of Inequalities and Applications For a mapping g X- Y consider the quadratic functional equation g x y g x - y 2g x 2g y . In 1989 Aczel 1 proposed the solution of . Later many different quadratic functional equations were solved by numerous authors 2-6 . In this paper we investigate the relation between and . And we find out the general solution and the generalized Hyers-Ulam stability of . 2. Results The m-variable quadratic functional equation induces the quadratic functional equation as follows. Theorem . Let f Xm Y be a mapping satisfying and let g X - Y be the mapping given by g x f x . x for all x e X then g satisfies . Proof. By and g x y g x - y f x y . x y f x - y .