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 Necessary and Sufﬁcient Conditions for the Existence of Positive Solution for Singular Boundary Value Problems on Time Scales | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 737461 14 pages doi 2009 737461 Research Article Necessary and Sufficient Conditions for the Existence of Positive Solution for Singular Boundary Value Problems on Time Scales Meiqiang Feng 1 Xuemei Zhang 2 3 Xianggui Li 1 and Weigao Ge3 1 School of Science Beijing Information Science Technology University Beijing 100192 China 2 Department of Mathematics and Physics North China Electric Power University Beijing 102206 China 3 Department of Applied Mathematics Beijing Institute of Technology Beijing 100081 China Correspondence should be addressed to Xuemei Zhang zxm74@ Received 27 March 2009 Revised 3 July 2009 Accepted 15 September 2009 Recommended by Alberto Cabada By constructing available upper and lower solutions and combining the Schauder s fixed point theorem with maximum principle this paper establishes sufficient and necessary conditions to guarantee the existence of Cid 0 1 t as well as C fO 1 t positive solutions for a class of singular boundary value problems on time scales. The results significantly extend and improve many known results for both the continuous case and more general time scales. We illustrate our results by one example. Copyright 2009 Meiqiang Feng 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 Recently there have been many papers working on the existence of positive solutions to boundary value problems for differential equations on time scales see for example fl-22 . This has been mainly due to its unification of the theory of differential and difference equations. An introduction to this unification is given in 10 14 23 24 . Now this study is still a new area of fairly theoretical exploration in mathematics. However it has led to several important .