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Báo cáo hóa học: " Research Article Stability Problem of Ulam for Euler-Lagrange Quadratic Mappings"

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 Problem of Ulam for Euler-Lagrange Quadratic Mappings | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 10725 15 pages doi 10.1155 2007 10725 Research Article Stability Problem of Ulam for Euler-Lagrange Quadratic Mappings Hark-Mahn Kim John Michael Rassias and Young-Sun Cho Received 26 May 2007 Revised 9 August 2007 Accepted 9 November 2007 Recommended by Ondrej Dosly We solve the generalized Hyers-Ulam stability problem for multidimensional Euler-Lagrange quadratic mappings which extend the original Euler-Lagrange quadratic mappings. Copyright 2007 Hark-Mahn Kim 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 In 1940 Ulam 1 proposed at the University of Wisconsin the following problem give conditions in order for a linear mapping near an approximately linear mapping to exist. In 1968 Ulam proposed the general Ulam stability problem when is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is how do the solutions of the inequality differ from those of the given functional equation If the answer is affirmative we would say that the equation is stable. In 1978 Gruber 2 remarked that Ulam problem is of particular interest in probability theory and in the case of functional equations of different types. We wish to note that stability properties of different functional equations can have applications to unrelated fields. For instance Zhou 3 used a stability property of the functional equation f x - y f x y 2 f x to prove a conjecture of Z. Ditzian about the relationship between the .