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 Existence of Positive Solution to Second-Order Three-Point BVPs on Time Scales | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 685040 6 pages doi 2009 685040 Research Article Existence of Positive Solution to Second-Order Three-Point BVPs on Time Scales Jian-Ping Sun Department of Applied Mathematics Lanzhou University of Technology Lanzhou Gansu 730050 China Correspondence should be addressed to Jian-Ping Sun jpsun@ Received 19 April 2009 Accepted 14 September 2009 Recommended by Kanishka Perera We are concerned with the following nonlinear second-order three-point boundary value problem on time scales -xAA t f t x t t e a b T x a 0 x ơ2 b 6x n where a b e T with a b n e a b T and 0 6 ơ2 b - a n - a . A new representation of Green s function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution for the above nonlinear boundary value problem are established by using the iterative method. Copyright 2009 Jian-Ping Sun. 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 Let T be a time scale that is T is an arbitrary nonempty closed subset of R. For each interval I of R we define IT I T. For more details on time scales one can refer to 1-5 . Recently three-point boundary value problems BVPs for short for second-order dynamic equations on time scales have received much attention. For example in 2002 Anderson 6 studied the following second-order three-point BVP on time scales MAV t a f f u ff 0 t e 0 T T u 0 0 u T au fi where 0 T e T n e 0 p T T and 0 a T n. Some existence results of at least one positive solution and of at least three positive solutions were established by using the well-known Krasnoselskii and Leggett-Williams fixed point theorems. In 2003 Kaufmann 7 applied the Krasnoselskii fixed point theorem to obtain the existence of multiple .
