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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 Eigenvalue Problem and Unbounded Connected Branch of Positive Solutions to a Class of Singular Elastic Beam Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 594128 21 pages doi 10.1155 2011 594128 Research Article Eigenvalue Problem and Unbounded Connected Branch of Positive Solutions to a Class of Singular Elastic Beam Equations Huiqin Lu School of Mathematical Sciences Shandong Normal University Jinan 250014 Shandong China Correspondence should be addressed to Huiqin Lu lhy@sdu.edu.cn Received 16 October 2010 Revised 22 December 2010 Accepted 27 January 2011 Academic Editor Kanishka Perera Copyright 2011 Huiqin Lu. 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. This paper investigates the eigenvalue problem for a class of singular elastic beam equations where one end is simply supported and the other end is clamped by sliding clamps. Firstly we establish a necessary and sufficient condition for the existence of positive solutions then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from 0 0 . Our nonlinearity f t u v w may be singular at u v t 0 and or t 1. 1. Introduction Singular differential equations arise in the fields of gas dynamics Newtonian fluid mechanics the theory of boundary layer and so on. Therefore singular boundary value problems have been investigated extensively in recent years see 1-4 and references therein . This paper investigates the following fourth-order nonlinear singular eigenvalue problem u 4 t Xf t u f u f u tỴ t e 0 1 u 0 u 1 u 0 u 1 0 1.1 where X e 0 TO is a parameter and f satisfies the following hypothesis H f e C 0 1 X 0 TO X 0 TO X -TO 0 0 to and there exist constants ai pi Ni i 1 2 3 -TO a1 0 p1 TO -TO a2 0 p2 TO 0 a3 p3 1 2 Boundary Value Problems S3 1 pi 1 0 Ni 1 i 1 2 3 such that for any t e 0 1 u v e 0 to w e -TO 0 f satisfies cP1 f t u v w f t cu v w caỵ f t u v w V0 c N1 cp2 f t