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Báo cáo hóa học: " A functional equation related to inner product spaces in non-Archimedean normed spaces"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A functional equation related to inner product spaces in non-Archimedean normed spaces | Gordji et al. Advances in Difference Equations 2011 2011 37 http www.advancesindifferenceequations.eom content 2011 1 37 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access A functional equation related to inner product spaces in non-Archimedean normed spaces 1.2 1.2 1.2 3 4 Madjid Eshaghi Gordji 1 Razieh Khodabakhsh 1 Hamid Khodaei 1 Choonkil Park and Dong Yun shin Correspondence dyshin@uos.ac. kr 4Department of Mathematics University of Seoul Seoul 130-743 Republic of Korea Full list of author information is available at the end of the article Springer Abstract In this paper we prove the Hyers-Ulam stability of a functional equation related to inner product spaces in non-Archimedean normed spaces. 2010 Mathematics Subject Classification Primary 46S10 39B52 47S10 26E30 12J25. Keywords non-Archimedean spaces additive and quadratic functional equation Hyers-Ulam stability 1. Introduction and preliminaries One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows When is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation The first stability problem concerning group homomorphisms was raised by Ulam 1 in 1940 and affirmatively solved by Hyers 2 . The result of Hyers was generalized by Aoki 3 for approximate additive mappings and by Rassias 4 for approximate linear mappings by allowing the difference Cauchy equation 11 f x1 x2 - fx1 - fx2 11 to be controlled by e X1 p x2 p . Taking into consideration a lot of influence of Ulam Hyers and Rassias on the development of stability problems of functional equations the stability phenomenon that was proved by Rassias is called Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. In 1994 a generalization of the Rassias theorem was obtained by Găvruta 5 who replaced e X1 p x2 p by a general control .