tailieunhanh - báo cáo hóa học:" Review Article An Overview of the Lower and Upper Solutions Method with Nonlinear Boundary Value Conditions"

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: Review Article An Overview of the Lower and Upper Solutions Method with Nonlinear Boundary Value Conditions | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 893753 18 pages doi 2011 893753 Review Article An Overview of the Lower and Upper Solutions Method with Nonlinear Boundary Value Conditions Alberto Cabada Departamento de Anứlise Matemdtica Facultade de Matemdticas Universidade de Santiago de Compostela 15782 Santiago de Compostela Spain Correspondence should be addressed to Alberto Cabada Received 19 April 2010 Accepted 7 July 2010 Academic Editor Gennaro Infante Copyright 2011 Alberto Cabada. 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. The aim of this paper is to point out recent and classical results related with the existence of solutions of second-order problems coupled with nonlinear boundary value conditions. 1. Introduction The first steps in the theory of lower and upper solutions have been given by Picard in 1890 1 for Partial Differential Equations and three years after in 2 for Ordinary Differential Equations. In both cases the existence of a solution is guaranteed from a monotone iterative technique. Existence of solutions for Cauchy equations have been proved by Perron in 1915 3 . In 1927 Muller extended Perron s results to initial value systems in 4 Dragoni 5 introduces in 1931 the notion of the method of lower and upper solutions for ordinary differential equations with Dirichlet boundary value conditions. In particular by assuming stronger conditions than nowadays the author considers the second-order boundary value problem u t f t u t u t t e a b I u a A u b B for f I X R2 R a continuous function and A B e R. The most usual form to define a lower solution is to consider a function a e C2 I that satisfies the inequality a t f t a t a t 2 Boundary Value Problems together with a a A a b B. In the same way an upper .