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: SINGULAR SECOND-ORDER MULTIPOINT DYNAMIC BOUNDARY VALUE PROBLEMS WITH MIXED DERIVATIVES | SINGULAR SECOND-ORDER MULTIPOINT DYNAMIC BOUNDARY VALUE PROBLEMS WITH MIXED DERIVATIVES MARTIN BOHNER AND HUA LUO Received 12 September 2005 Accepted 26 October 2005 We study a certain singular second-order m-point boundary value problem on a time scale and establish the existence of a solution. The proof of our main result is based upon the Leray-Schauder continuation theorem. Copyright 2006 M. Bohner and H. Luo. 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 Singular nonlinear boundary value problems for differential equations and difference equations have been extensively studied in the literature see 1 4 11 12 16 18-22 and the references therein. However the research for singular boundary value problems on time scales is still in its beginning stages. In 8 the authors investigate the existence of a positive solution for the three-point dynamic boundary value problem yAA f x y 0 x e 0 1 y 0 0 y p y ơ2 1 where T is a time scale the interval 0 1 n T is abbreviated by 0 1 p e 0 1 is fixed and f x y is singular at y 0 and possibly at x 0 y TO. Throughout we denote by T a time scale that is a nonempty closed subset of the real numbers. In this paper we study the singular second-order m-point dynamic boundary value problem xA f t x xA e t t e a b Á . m 2 xA a 0 x ơ b y aix i i 1 where a e R ỉi e a ơ b i e 1 2 . m - 2 and f a ơ b X R2 R satisfies the Caratheodory conditions that is for each x y e R2 the function f x y is measurable Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 54989 Pages 1-15 DOI ADE 2006 54989 2 Singular multipoint dynamic boundary value problems on a Ơ b and for see Definition . t E a Ơ b the function f t is continuous on R2. Here we allow f and e to be singular at t Ơ b . In particular when the nonlinearity
