tailieunhanh - Báo cáo hóa học: " Research Article First-Order Singular and Discontinuous Differential Equations"

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 First-Order Singular and Discontinuous Differential Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 507671 25 pages doi 2009 507671 Research Article First-Order Singular and Discontinuous Differential Equations Daniel C. Biles1 and Rodrigo Lopez Pouso2 1 Department of Mathematics Belmont University 1900 Belmont Blvd. Nashville TN 37212 USA 2 Department of Mathematical Analysis University of Santiago de Compostela 15782 Santiago de Compostela Spain Correspondence should be addressed to Rodrigo Lopez Pouso Received 10 March 2009 Accepted 4 May 2009 Recommended by Juan J. Nieto We use subfunctions and superfunctions to derive sufficient conditions for the existence of extremal solutions to initial value problems for ordinary differential equations with discontinuous and singular nonlinearities. Copyright 2009 D. C. Biles and R. Lopez Pouso. 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 Let t0 x0 e R and L 0 be fixed and let f t0 t0 L X R R be a given mapping. We are concerned with the existence of solutions of the initial value problem x f t x t e I t0 t0 L x t0 x0. It is well-known that Peano s theorem ensures the existence of local continuously differentiable solutions of in case f is continuous. Despite its fundamental importance it is probably true that Peano s proof of his theorem is even more important than the result itself which nowadays we know can be deduced quickly from standard fixed point theorems see 1 Theorem for a proof based on the Schauder s theorem . The reason for believing this is that Peano s original approach to the problem in 2 consisted in obtaining the greatest solution as the pointwise infimum of strict upper solutions. Subsequently this idea was improved by Perron in 3 who also adapted it to study the Laplace equation by means of what we call