TAILIEUCHUNG - Báo cáo hóa học: " Research Article On the Cauchy Functional Inequality in Banach Modules"

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 Cauchy Functional Inequality in Banach Modules | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 904852 8 pages doi 2008 904852 Research Article On the Cauchy Functional Inequality in Banach Modules Choonkil Park Department of Mathematics Hanyang University Seoul 133791 South Korea Correspondence should be addressed to Choonkil Park baak@ Received 17 January 2008 Revised 21 March 2008 Accepted 16 April 2008 Recommended by Ram Mohapatra We investigate the following functional inequality f x f y f z f x y z in Banach modules over a c -algebra and prove the generalized Hyers-Ulam stability of linear mappings in Banach modules over a c -algebra. 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. Let X and Y be 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. A generalization of the Th. M. Rassias theorem was obtained by Găvruta 5 by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias approach. The result of Gdvrula 5 is a special case of a more general theorem which was obtained by Forti 6 . Th. M. Rassias 7 during the 27th international symposium on functional equations asked the question whether such a theorem can also be proved for p 1. Gajda 8 following the same approach as in Th. M. Rassias 4 gave an affirmative solution to this question for p 1. It was shown by Gajda 8 as well as by Th. M. Rassias and Semrl 9 that one cannot prove a Th. .

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