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Báo cáo hóa học: " Research Article Multiplicity of Positive Periodic Solutions of Singular Semipositone Third-Order Boundary Value Problems"

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 of Positive Periodic Solutions of Singular Semipositone Third-Order Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2008 Article ID 574842 9 pages doi 10.1155 2008 574842 Research Article Multiplicity of Positive Periodic Solutions of Singular Semipositone Third-Order Boundary Value Problems Yigang Sun Department of Applied Mathematics Hohai University Nanjing 210098 China Correspondence should be addressed to Yigang Sun hongyimingsun@163.com Received 2 July 2007 Accepted 13 December 2007 Recommended by Colin Rogers We establish the existence of multiple positive solutions for a singular nonlinear third-order periodic boundary value problem. We are mainly interested in the semipositone case. The proof relies on a nonlinear alternative principle of Leray-Schauder together with a truncation technique. Copyright 2008 Yigang Sun. 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 In this paper we study the existence and multiplicity of positive periodic solutions of the following singular nonlinear third-order periodic boundary value problem u p3u f t Ù 0 t 2n u f 0 w 2n i 0 1 2. 1.1 Here p G 0 1 v3 is a positive constant and f t u is continuous in t u and 2n- is periodic in t. We are mainly interested in the case that f t u may be singular at u 0 and satisfies the following semipositone condition G1 There exists a constant L 0 such that F t u f t u L 0 for all t u G 0 2n x 0 ot . During the last two decades singular periodic problems have deserved the attention of many researchers 1-8 . Third-order boundary value problems have also been studied in 9-11 . For the problem 1.1 we recall the following results. In 12 by using Schauder fixed-point theorem together with perturbation technique it was established the existence of at least one positive solution under some suitable conditions of f t u . One hard restriction in 12 was the monotonicity on f t u