Boundary Value Problems
Volume 2010 (2010), Article ID 368169, 15 pages
doi:10.1155/2010/368169
Research Article

Positive Solutions of Singular Complementary Lidstone Boundary Value Problems

Ravi P. Agarwal,1 Donal O'Regan,2 and Svatoslav Staněk3

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA
2Department of Mathematics, National University of Ireland, Galway, Ireland
3Department of Mathematical Analysis, Faculty of Science, Palacký University, Tř. 17. listopadu 12, 771 46 Olomouc, Czech Republic

Received 7 October 2010; Accepted 21 November 2010

Academic Editor: Irena Rachůnková

Copyright © 2010 Ravi P. Agarwal 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.

Abstract

We investigate the existence of positive solutions of singular problem ( − 1 ) 𝑚 𝑥 ( 2 𝑚 + 1 ) = 𝑓 ( 𝑡 , 𝑥 , … , 𝑥 ( 2 𝑚 ) ) , 𝑥 ( 0 ) = 0 , 𝑥 ( 2 𝑖 − 1 ) ( 0 ) = 𝑥 ( 2 𝑖 − 1 ) ( 𝑇 ) = 0 , 1 ≤ 𝑖 ≤ 𝑚 . Here, 𝑚 ≥ 1 and the Carathéodory function 𝑓 ( 𝑡 , 𝑥 0 , … , 𝑥 2 𝑚 ) may be singular in all its space variables 𝑥 0 , … , 𝑥 2 𝑚 . The results are proved by regularization and sequential techniques. In limit processes, the Vitali convergence theorem is used.