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 Multiplicity Results Using Bifurcation Techniques for a Class of Fourth-Order m-Point Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 970135 20 pages doi 2009 970135 Research Article Multiplicity Results Using Bifurcation Techniques for a Class of Fourth-Order m-Point Boundary Value Problems Yansheng Liu1 and Donal O Regan2 1 Department of Mathematics Shandong Normal University Jinan 250014 China 2 Department of Mathematics National University of Ireland Galway Ireland Correspondence should be addressed to Yansheng Liu yanshliu@ Received 13 March 2009 Accepted 12 April 2009 Recommended by Juan J. Nieto By using bifurcation techniques this paper investigates the existence of nodal solutions for a class of fourth-order m-point boundary value problems. Our results improve those in the literature. Copyright 2009 Y. Liu and D. O Regan. 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 Consider the following fourth order m-point boundary value problem BVP for short u 4 f f u if u tf t e 0 1 m-2 u 0 0 u 1 y uírì i 1 u 0 0 u 1 u rì i 1 where f R X R R is a given sign-changing continuous function m 3 n e 0 1 and ai 0 for i 1 . . m - 2 with m-2 ai 1. i 1 2 Boundary Value Problems Multi-point boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics. The existence of solutions of the second order multi-point boundary value problems has been studied by many authors and the methods used are the nonlinear alternative of Leray-Schauder coincidence degree theory fixed point theorems in cones and global bifurcation techniques see 1-9 and the references therein . In 5 Ma investigated the existence and multiplicity of nodal solutions for u f f u fy 0 t e 0 1 m-2 u 0 0 u 1 y uim i 1 when n e Q i 1 2 . m - 2 with 0 n1 n2 nm-2 1 and ai 0 for i 1 . . m - 2 satisfying . He
