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: Research Article Fixed Points in Functional Inequalities Choonkil Park | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 298050 8 pages doi 2008 298050 Research Article Fixed Points in Functional Inequalities Choonkil Park Department of Mathematics Hanyang University Seoul 133791 South Korea Correspondence should be addressed to Choonkil Park baak@ Received 29 September 2008 Accepted 16 December 2008 Recommended by Shusen Ding Using fixed point methods we prove the generalized Hyers-Ulam stability of the following functional inequalities f x f y f z f x y z and f x f y 2f z 2f x y 2 z in the spirit of Th. M. Rassias stability approach. Copyright 2008 Choonkil Park. 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 and preliminaries The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers Theorem was generalized by Aoki 3 for additive mappings and by Th. M. Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias 4 has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Gavruta 5 by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias approach. In 1982 J. M. Rassias 6 followed the innovative approach of the Th. M. Rassias theorem 4 in which he replaced the factor x p y p by x p llyh for p q e R with p q 1. The functional equation f x y f x - y 2f x 2f y C1-1 is called a quadratic functional equation. In particular every solution of the quadratic functional equation is said to be a quadratic mapping. A
