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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 Solutions to a Three-Point Boundary Value Problem | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 894135 20 pages doi 10.1155 2011 894135 Research Article Solutions to a Three-Point Boundary Value Problem Jin Liang1 and Zhi-Wei Lv2 3 1 Department of Mathematics Shanghai Jiao Tong University Shanghai 200240 China 2 Department of Mathematics and Physics Anyang Institute of Technology Anyang Henan 455000 China 3 Department of Mathematics University of Science and Technology of China Hefei Anhui 230026 China Correspondence should be addressed to Jin Liang jinliang@sjtu.edu.cn Received 25 November 2010 Accepted 19 January 2011 Academic Editor Toka Diagana Copyright 2011 J. Liang and Z.-W. Lv. 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. By using the fixed-point index theory and Leggett-Williams fixed-point theorem we study the existence of multiple solutions to the three-point boundary value problem u t a t f t u t u t 0 0 t 1 u 0 u 0 0 u 1 - au ỳ X- where n 0 1 2 a e 1 2y 1 y are constants d e 0 to is a parameter and a f are given functions. New existence theorems are obtained which extend and complement some existing results. Examples are also given to illustrate our results. 1. Introduction It is known that when differential equations are required to satisfy boundary conditions at more than one value of the independent variable the resulting problem is called a multipoint boundary value problem and a typical distinction between initial value problems and multipoint boundary value problems is that in the former case one is able to obtain the solutions depend only on the initial values while in the latter case the boundary conditions at the starting point do not determine a unique solution to start with and some random choices among the solutions that satisfy these starting boundary conditions are normally not to .