Journal of Applied Mathematics
Volume 2012 (2012), Article ID 160891, 23 pages
Research Article

Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term

1College of International Economics and Trade, Jilin University of Finance and Economics, Jilin, Changchun 130117, China
2School of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, China
3School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China
4Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 13 July 2011; Accepted 3 November 2011

Academic Editor: Andrew Pickering

Copyright © 2012 Yuefeng Han 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.


By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation Lu=p(t)f(t,u(t),u(t))-g(t,u(t),u(t)),0<t<1,α1u(0)-β1u'(0)=0,γ1u(1)+δ1u'(1)=0,α2u(0)-β2u(0)=0,γ2u(1)+δ2u(1)=0, with αi,βi,γi,δi0 and βiγi+αiγi+αiδi>0,    i=1,2, where L denotes the linear operator Lu:=(ru)'-qu,rC1([0,1],(0,+)), and qC([0,1],[0,+)). This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed term g:(0,1)×[0,+)×(-,+)(-,+) only satisfies the global Carathéodory conditions, which implies that the perturbed effect of g on f is quite large so that the nonlinearity can tend to negative infinity at some singular points.