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Báo cáo hóa học: " Research Article Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces"

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 Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 527462 9 pages doi 10.1155 2009 527462 Research Article Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces M. Eshaghi Gordji and M. B. Savadkouhi Department of Mathematics Semnan University P.O. Box 35195-363 Semnan Iran Correspondence should be addressed to M. Eshaghi Gordji madjid.eshaghi@gmail.com Received 22 June 2009 Accepted 5 August 2009 Recommended by Patricia J. Y. Wong We obtain the stability result for the following functional equation in random normed spaces in the sense of Sherstnev under arbitrary f-norms f x 2y f x - 2y 4 f x y f x - y -24f y - 6f x 3f 2y . Copyright 2009 M. Eshaghi Gordji and M. B. Savadkouhi. 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 The stability problem of functional equations originated from a question of Ulam 1 in 1940 concerning the stability of group homomorphisms. Let G1 be a group and let G2 d be a metric group with the metric d - . Given e 0 does there exist a Ỗ 0 such that if a mapping h G1 G2 satisfies the inequality d h x y h. x hf-yf Ỗ for all x y e G1 then there exists a homomorphism H G1 G2 with d h x H x e for all x e G1 In other words under what condition does there exist a homomorphism near an approximate homomorphism The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941 Hyers 2 gave a first affirmative answer to the question of Ulam for Banach spaces. Let f E E1 be a mapping between Banach spaces such that Ilf x y - f x - f y Ỗ 1.1 for all x y e E and for some Ỗ 0. Then there exists a unique additive mapping T E E such that Ilf x - T x Ô 1.2 2 Journal of Inequalities and Applications for all