Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 187827, 16 pages
doi:10.1155/2010/187827

Symmetric positive solutions for nonlinear singular fourth-order eigenvalue problems with nonlocal boundary condition

Fuyi Xu1 and Jian Liu2

1School of Science, Shandong University of Technology, Zibo, Shandong 255049, China
 2School of Statistics and Mathematics Science, Shandong Economics University, Jinan, 250014 Shandong, China

Abstract

We investigate nonlinear singular fourth-order eigenvalue problems with nonlocal boundary condition u(4)(t)-λh(t)f(t,u,u′′)=0, 0<t<1, u(0)=u(1)=∫01a(s)u(s)ds, u′′(0)=u′′(1)=∫01b(s)u′′(s)ds, where a,b∈L1[0,1], λ>0, h may be singular at t=0 and/or 1. Moreover f(t,x,y) may also have singularity at x=0 and/or y=0. By using fixed point theory in cones, an explicit interval for λ is derived such that for any λ in this interval, the existence of at least one symmetric positive solution to the boundary value problem is guaranteed. Our results extend and improve many known results including singular and nonsingular cases. The associated Green's function for the above problem is also given.