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 On the Stability of Generalized Additive Functional Inequalities in Banach Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 210626 13 pages doi 2008 210626 Research Article On the Stability of Generalized Additive Functional Inequalities in Banach Spaces Jung Rye Lee 1 Choonkil Park 2 and Dong Yun Shin3 1 Department of Mathematics Daejin University Kyeonggi 487-711 South Korea 2 Department of Mathematics Hanyang University Seoul 133-791 South Korea 3 Department of Mathematics University of Seoul Seoul 130-743 South Korea Correspondence should be addressed to Choonkil Park baak@ Received 18 February 2008 Accepted 2 May 2008 Recommended by Ram Verma We study the following generalized additive functional inequality af x bf y cf z f ax fy yz II associated with linear mappings in Banach spaces. Moreover we prove the Hyers-Ulam-Rassias stability of the above generalized additive functional inequality associated with linear mappings in Banach spaces. Copyright 2008 Jung Rye Lee et al. 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 Rassias 4 for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gavruta 5 by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias approach. Rassias 6 during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p 1. Gajda 7 following the same approach as in Rassias 4 gave an affirmative solution
