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 Recent Existence Results for Second-Order Singular Periodic Differential Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 540863 20 pages doi 2009 540863 Research Article Recent Existence Results for Second-Order Singular Periodic Differential Equations Jifeng Chu1 2 and Juan J. Nieto3 1 Department of Mathematics College of Science Hohai University Nanjing 210098 China 2 Department of Mathematics Pusan National University Busan 609-735 South Korea 3 Departamento de Analisis Matemdtico Facultad de Matemdticas Universidad de Santiago de Compostela 15782 Santiago de Compostela Spain Correspondence should be addressed to Jifeng Chu jifengchu@ Received 12 February 2009 Accepted 29 April 2009 Recommended by Donal O Regan We present some recent existence results for second-order singular periodic differential equations. A nonlinear alternative principle of Leray-Schauder type a well-known fixed point theorem in cones and Schauder s fixed point theorem are used in the proof. The results shed some light on the differences between a strong singularity and a weak singularity. Copyright 2009 J. Chu and J. J. Nieto. 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 The main aim of this paper is to present some recent existence results for the positive T-periodic solutions of second order differential equation x a Jfx f t x e Jf where a f e t are continuous and T-periodic functions. The nonlinearity f t x is continuous in t x and T-periodic in t. We are mainly interested in the case that f t x has a repulsive singularity at x 0 lim f t x x uniformly in t. x 0 y It is well known that second order singular differential equations describe many problems in the applied sciences such as the Brillouin focusing system 1 and nonlinear elasticity 2 . Therefore during the last two decades singular equations have attracted many .