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 Results for Higher-Order Boundary Value Problems on Time Scales | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 209707 18 pages doi 2009 209707 Research Article Existence Results for Higher-Order Boundary Value Problems on Time Scales Jian Liu1 and Yanbin Sang2 1 School of Mathematics and Statistics Shandong Economics University Jinan Shandong 250014 China 2 Department of Mathematics North University of China Taiyuan Shanxi 030051 China Correspondence should be addressed to Jian Liu kkword@ Received 22 March 2009 Revised 5 June 2009 Accepted 16 June 2009 Recommended by Victoria Otero-Espinar By using the fixed-point index theorem we consider the existence of positive solutions for the following nonlinear higher-order four-point singular boundary value problem on time scales ma t g t f u t uA t . uA 2 t 0 0 t T uAi 0 0 0 i n-3 auA 2 0 -puA 1 f 0 n 3 yuA -2 T 5uA -1 n 0 n 3 where a 0 p 0 Y 0 6 0 f n 0 T f n and g 0 T 0 x is rd-continuous. Copyright 2009 J. Liu and Y. Sang. 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 Time scales and time-scale notation are introduced well in the fundamental texts by Bohner and Peterson 1 2 respectively as important corollaries. In the recent years many authors have paid much attention to the study of boundary value problems on time scales see . 3-17 . In particular we would like to mention some results of Anderson et al. 3 5 6 14 16 DaCunha et al. 4 and Agarwal and O Regan 7 which motivate us to consider our problem. In 3 Anderson and Karaca discussed the dynamic equation on time scales -1 nyA2n t f i ya 0 0 t a bj ai--ỵA n pi 1yA2 1 a yA a Ỵi 1 yA n yA byr and the eigenvalue problem -1 nyA2n t Xf i yat 0 t a b 2 Advances in Difference Equations with the same boundary conditions where A is a positive parameter. They obtained some results for the .