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 of Positive Solutions of Fourth-Order Problems with Integral Boundary Conditions Ruyun Ma and Tianlan Chen | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 297578 17 pages doi 2011 297578 Research Article Existence of Positive Solutions of Fourth-Order Problems with Integral Boundary Conditions Ruyun Ma and Tianlan Chen Department of Mathematics Northwest Normal University Lanzhou 730070 China Correspondence should be addressed to Ruyun Ma ruyunma@ Received 5 May 2010 Accepted 7 July 2010 Academic Editor Daniel Franco Copyright 2011 R. Ma and T. Chen. 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. We study the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions u 4 t f t u t u t t e 0 1 m 0 J1 g s u s ds u 1 0 u 0 J1 h s w s ds u 1 0 where f 0 1 X 0 TO X -TO 0 0 TO is continuous g h e L1 0 1 are nonnegative. The proof of our main result is based upon the Krein-Rutman theorem and the global bifurcation techniques. 1. Introduction The deformations of an elastic beam in an equilibrium state whose both ends are simple supported can be described by the fourth-order boundary value problem u 4 t f t u t u t t e 0 1 u 0 u 1 u 0 u 1 0 where f 0 1 X R X R R is continuous see Gupta 1 2 . In the past twenty more years the existence of solutions and positive solutions of these kinds of problems and the Lidstone problem has been extensively studied see 3-9 and the references therein. In 3 Ma was concerned with the existence of positive solutions of and under the assumptions H1 f 0 1 X 0 to X -TO 0 0 to is continuous and there exist constants a b c d e 0 to with a b 0 c d 0 such that f ft up au - bp o u p as u p 0 2 Boundary Value Problems uniformly for t e 0 1 and f t u p cu - dp o u p as u p TO uniformly for t e 0 1 where u p ựu2 p2 H2 f t u p 0 for t e 0 1 and u p e 0 TO X -TO 0 0 0 H3 .
