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Báo cáo hóa học: "Research Article Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular m-Point Boundary Value Problems"

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 and Uniqueness of Smooth Positive Solutions to a Class of Singular m-Point Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 191627 13 pages doi 10.1155 2009 191627 Research Article Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular m-Point Boundary Value Problems Xinsheng Du and Zengqin Zhao School of Mathematics Sciences Qufu Normal University Qufu Shandong 273165 China Correspondence should be addressed to Xinsheng Du duxinsheng@mail.qfnu.edu.cn Received 2 April 2009 Revised 15 September 2009 Accepted 23 November 2009 Recommended by Donal O Regan This paper investigates the existence and uniqueness of smooth positive solutions to a class of singular m-point boundary value problems of second-order ordinary differential equations. A necessary and sufficient condition for the existence and uniqueness of smooth positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Our nonlinearity f t u v may be singular at v t 0 and or t 1. Copyright 2009 X. Du and Z. Zhao. 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 and the Main Results In this paper we will consider the existence and uniqueness of positive solutions to a class of second-order singular m-point boundary value problems of the following differential equation -u t f t u t u t t e 0 1 1.1 with m-2 u 0 ỵ ữiu n u 1 0 1.2 i 1 where 0 ai 1 i 1 2 . m - 2 0 n n2 nm-2 1 are constants m12ai 1 m 3 and f satisfies the following hypothesis 2 Boundary Value Problems H f t u v 0 1 X 0 to X 0 to 0 to is continuous nondecreasing on u and nonincreasing on V for each fixed t 0 1 there exists a real number b R such that for any r 0 1 f t u rv rhf t u v N t u V 0 1 X 0 to X 0 to 1.3 there exists a function g 1 to 0 to g l l and g l l2 is integrable on 1 to such that f t lu v g T f t u v V t u v 0 1 X 0 to X 0 to l 1 to . 1.4 Remark .