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 Nonlinear Systems of Second-Order ODEs Patricio Cerda and Pedro Ubilla | Hindawi Publishing Corporation Boundary Value Problems Volume 2008 Article ID 236386 9 pages doi 2008 236386 Research Article Nonlinear Systems of Second-Order ODEs Patricio Cerda and Pedro Ubilla Departamento de Matematica y C. C. Universidad de Santiago de Chile Casilla 307 Correo 2 Santiago Chile Correspondence should be addressed to Pedro Ubilla pubilla@ Received 2 February 2007 Accepted 16 November 2007 Recommended by Jean Mawhin We study existence of positive solutions of the nonlinear system - p1 t u v u h1 t f1 t u v in 0 1 - p2 t u v v h2 t f2 t u v in 0 1 u 0 u 1 v 0 v 1 0 where p1 t u v 1 a1 t c1g1 u v and p2 t u v 1 a2 t c2g2 u v . Here it is assumed that g1 g2 are nonnegative continuous functions a1 t a2 t are positive continuous functions c1 c2 0 h1 h2 E L1 0 1 and that the nonlinearities f1 f2 satisfy superlinear hypotheses at zero and TO. The existence of solutions will be obtained using a combination among the method of truncation a priori bounded and Krasnosel skii well-known result on fixed point indices in cones. The main contribution here is that we provide a treatment to the above system considering differential operators with nonlinear coefficients. Observe that these coefficients may not necessarily be bounded from below by a positive bound which is independent of u and v. Copyright 2008 P. Cerda and P Ubilla. 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 We study existence of positive solutions for the following nonlinear system of second-order ordinary differential equations -f u s h1 t f1 t u v in 0 1 U1 t c1g1 u v - h2 t f2 t u v in 0 1 d2 t c2g2 u v J u 0 u 1 v 0 v 1 0 where c1 c2 are nonnegatives constants the functions a1 a2 0 1 0 ro are continuous the functions f1 f2 0 1 X 0 2 0 ro are continuous and h1 h2 E L1 0 1 . We will suppose the
