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báo cáo hóa học:" Research Article Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales"

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 Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 937064 19 pages doi 10.1155 2009 937064 Research Article Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales Liang-Gen Hu 1 Ti-Jun Xiao 2 and Jin Liang3 1 Department of Mathematics University of Science and Technology of China Hefei 230026 China 2 School of Mathematical Sciences Fudan University Shanghai 200433 China 3 Department of Mathematics Shanghai Jiao Tong University Shanghai 200240 China Correspondence should be addressed to Jin Liang jinliang@sjtu.edu.cn Received 21 March 2009 Accepted 1 July 2009 Recommended by Juan Jose Nieto We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result. Copyright 2009 Liang-Gen Hu 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 In this paper we are concerned with the following singular three-point boundary value problem BVP for short for delay higher-order dynamic equations on time scales -1 nuA2n t w t f t u t - c t e a b u t y t t e a - c a . 1.1 uA a - pi 1 lA a ai 1uA w Ỵi 1uầZi w uA2i b 0 i n - 1 where c e 0 b - a 2 w e a b pi 0 1 fi b - a pi w - a pi 0 ai b - ỴịW yi - 1 a - pi b - w i 1 2 . n and y e C a - c a . The functional w a b 0 x is continuous and f a b X 0 x 0 x is continuous. Our 2 Boundary Value Problems nonlinearity w may have singularity at t a and or t b and f may have singularity at u 0. To understand the notations used in 1.1 we recall the following definitions which can be found in 1 2 . a A time scale T is a nonempty closed subset of the real numbers R.