tailieunhanh - báo cáo hóa học:" Research Article Approximation of Analytic Functions by Kummer Functions"

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 Approximation of Analytic Functions by Kummer Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 898274 11 pages doi 2010 898274 Research Article Approximation of Analytic Functions by Kummer Functions Soon-Mo Jung Mathematics Section College of Science and Technology Hongik University Jochiwon 339-701 Republic of Korea Correspondence should be addressed to Soon-Mo Jung smjung@ Received 3 February 2010 Revised 27 March 2010 Accepted 31 March 2010 Academic Editor Alberto Cabada Copyright 2010 Soon-Mo Jung. 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. We solve the inhomogeneous Kummer differential equation of the form xy fl - x y - ay Sm 0 amxm and apply this result to the proof of a local Hyers-Ulam stability of the Kummer differential equation in a special class of analytic functions. 1. Introduction Assume that X and Y are a topological vector space and a normed space respectively and that I is an open subset of X. If for any function f I Y satisfying the differential inequality a x y n x an-1 x y n-1 x ai x y x a0 x y x h x II for all x e I and for some 0 there exists a solution f 0 I Y of the differential equation an x y n x an-i x y -1 x a1 x y x a0 x y x h x 0 such that f x -f0 x II K s for any x e I where K s depends on only then we say that the above differential equation satisfies the Hyers-Ulam stability or the local Hyers-Ulam stability if the domain I is not the whole space X . We may apply this terminology for other differential equations. For more detailed definition of the Hyers-Ulam stability refer to 1-6 . Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations see 7 8 . Here we will introduce a result of Alsina and Ger see 9 . If a differentiable function f I R is a solution of the differential .