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 An Approximation Approach to Eigenvalue Intervals for Singular Boundary Value Problems with Sign Changing and Superlinear | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 103867 34 pages doi 2009 103867 Research Article An Approximation Approach to Eigenvalue Intervals for Singular Boundary Value Problems with Sign Changing and Superlinear Nonlinearities Haishen Lu 1 Ravi P. Agarwal 2 3 and Donal O Regan4 1 Department of Applied Mathematics Hohai University Nanjing 210098 China 2 Department of Mathematical Sciences Florida Institute of Technology Melbourne FL 32901-6975 USA 3 KFUPM Chair Professor Mathematics and Statistics Department King Fahd University of Petroleum and Minerals Dhahran 31261 Saudi Arabia 4 Department of Mathematics National University of Ireland Galway Ireland Correspondence should be addressed to Haishen Lu haishen2001@ Received 25 June 2009 Accepted 5 October 2009 Recommended by Ivan T. Kiguradze This paper studies the eigenvalue interval for the singular boundary value problem -u g t u Xhft u t e 0 1 u 0 0 u 1 where g h may be singular at u 0 t 0 1 and may change sign and be superlinear at u x . The approach is based on an approximation method together with the theory of upper and lower solutions. Copyright 2009 Haishen Lu 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. 1. Introduction The singular boundary value problems of the form -u f t u t e 0 1 u 0 0 u 1 occurs in several problems in applied mathematics see 1-6 and their references. In many papers a critical condition is that f t r 0 for t r e 0 1 X 0 to 2 Boundary Value Problems or there exists a constant L 0 such that for any compact set K c 0 1 there is e eK 0 such that f f r L Vf e K r e 0 e lim f r 0 Vf e 0 1 . r TO r We refer the reader to 1-4 . In the case when f f r may change sign in a neighborhood of r 0 and limsupr TO f f r r TO for f e 0 1 very few existence results .