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 Existence of Positive Solutions of Nonlinear Second-Order Periodic Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 626054 18 pages doi 2010 626054 Research Article Existence of Positive Solutions of Nonlinear Second-Order Periodic Boundary Value Problems Ruyun Ma Chenghua Gao and Ruipeng Chen Department of Mathematics Northwest Normal University Lanzhou 730070 China Correspondence should be addressed to Ruyun Ma ruyumma@ Received 31 August 2010 Revised 30 October 2010 Accepted 8 November 2010 Academic Editor Irena Rachunkova Copyright 2010 Ruyun Ma et al. 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. This paper is devoted to study the existence of periodic solutions of the second-order equation x f t x where f is a Caratheodory function by combining a new expression of Green s function together with Dancer s global bifurcation theorem. Our main results are sharp and improve the main results by Torres 2003 . 1. Introduction Let us say that the following linear problem x a f x 0 t e 0 T x 0 x T x 0 x T is nonresonant when its unique solution is the trivial one. It is well known that is nonresonant then provided that h is a -function the Fredholm s alternative theorem implies that the inhomogeneous problem x a J x h f t e 0 T x 0 x T x 0 x T 2 Boundary Value Problems always has a unique solution which moreover can be written as x f J G t s hfs ds where G t s is the Green s function related to . In recent years the conditions H Problem is nonresonant and the corresponding Green s function G t s is positive on 0 T X 0 T H- Problem is nonresonant and the corresponding Green s function G t s is negative on 0 T X 0 T have become the assumptions in the searching for positive solutions of singular second-order equations and systems see for instance Chu and Torres 2 Chu et al. 3 Franco .
